Renormalization-Group Flows in Gauge Theories

Robert Shrock
YITP, Stony Brook University

SCGT14Mini Workshop, KMI, Nagoya University, 3/6/2014

Outline• Beta functions, renormalization-group flows in gauge theories• Higher-loop calculations of UV to IR evolution in asymptotically free gauge theories• Structural properties of β and results in the limit Nc → ∞, Nf → ∞ with Nf /Nc
fixed• Study of scheme dependence in higher-loop calculations• UV to IR evolution in asymptotically free chiral gauge theories• Study of RG flows in U(1) and non-Abelian gauge theories with many fermions (nonasymptotically free)• Conclusions

This talk contains material from the following papers since SCGT12 conference:• R. Shrock, “Higher-Loop Structural Properties of the β Function in AsymptoticallyFree Vectorial Gauge Theories”, Phys. Rev. D 87, 105005 (2013) [arXiv:1301.3209].
• R. Shrock, “Higher-Loop Calculations of the Ultraviolet to Infrared Evolution of a
Vectorial Gauge Theory in the Limit Nc → ∞, Nf → ∞ with Nf /Nc Fixed”,
Phys. Rev. D 87, 116007 (2013) [arXiv:1302.5434].
• R. Shrock, “Study of Scheme Transformations to Remove Higher-Loop Terms in theβ Function of a Gauge Theory”, Phys. Rev. D 88, 036003 (2013) [arXiv:1305.6524].
• T. Appelquist and R. Shrock, “On the Ultraviolet to Infrared Evolution of Chiral Gauge
Theories” Phys. Rev. D 88, 105012 (2013) [arXiv:1310.6076].
• R. Shrock, “Study of Possible Ultraviolet Zero of the Beta Function in Gauge Theories
with Many Fermions”, Phys. Rev. D 89, 045019 (2014) [arXiv:1311.5268].
together with some earlier work with T. Ryttov.

Beta functions, RG flows in gauge theories
Consider a gauge theory with gauge group G and a set of massless fermions in some
representation(s) R. If β < 0 so this theory is asymptotically free (AF), then it isweakly coupled, properties are perturbatively calculable for large Euclidean momenta µin the deep ultraviolet (UV). If β > 0, theory is infrared (IR)-free.
The renormalization-group (RG) flows in these theories as functions of µ are offundamental field-theoretic interest. For an AF theory, study how it flows from large µin the UV to small µ in the IR. For some fermion contents, theory may have an exact
or approximate IR fixed point (zero of β).
Denote running gauge coupling at scale µ as g = g(µ), and let
α(µ) = g(µ)2/(4π) and a(µ) = g(µ)2 /(16π 2) = α(µ)/(4π).

The dependence of α(µ) on µ is described by the renormalization group β function.
For the AF case,
∞
∞
X
X
dα
= −2α
bℓ aℓ = −2αb̄ℓ αℓ ,
βα ≡
dt
ℓ=1
ℓ=1

where dt = d ln µ, ℓ = loop order of the coeff. bℓ, and b̄ℓ = bℓ/(4π)ℓ.
Coefficients b1 and b2 in β are independent of regularization/renormalization scheme,
while bℓ for ℓ ≥ 3 are scheme-dependent.
Asymptotic freedom means b1 > 0, so β < 0 for small α(µ), in neighborhood of UV
fixed point (UVFP) at α = 0.
Consider vectorial AF case. As the scale µ decreases from large values, α(µ)
increases. Denote αcr as minimum value for formation of bilinear fermion condensates
and resultant spontaneous chiral symmetry breaking (SχSB).

Two generic possibilities for β and resultant UV to IR flow:• β has no IR zero, so as µ decreases, α(µ) increases, eventually beyond theperturbatively calculable region. This is the case for QCD.
• β has a IR zero, αIR, so as µ decreases, α → αIR . In this class of theories, there
are two further generic possibilities: αIR < αcr or αIR > αcr .
If αIR < αcr , the zero of β at αIR is an exact IR fixed point (IRFP) of the RG; asµ → 0 and α → αIR , β → β(αIR) = 0, and the theory becomes exactly
scale-invariant with nontrivial anomalous dimensions.
If β has no IR zero, or an IR zero at αIR > αcr , then as µ decreases through a scaleΛ, α(µ) exceeds αcr and SχSB occurs, so fermions gain dynamical masses ∼ Λ.
If SχSB occurs, then in low-energy effective field theory applicable for µ < Λ, one
integrates these fermions out, and β fn. becomes that of a pure gauge theory, with noIR zero. Hence, if β has a zero at αIR > αcr , this is only an approx. IRFP of RG.

If αIR is only slightly greater than αcr , then, as α(µ) approaches αIR, since
β = dα/dt → 0, α(µ) varies very slowly as a function of the scale µ, i.e., there isapproximately scale-invariant “walking” behavior (Yamawaki et al; Holdom, Appelquist,
Wijewardhana...).
SχSB at Λ breaks the approximate dilatation symmetry and can lead to a resultant
approx. Nambu-Goldstone boson (NGB), the dilaton.
Consider AF chiral gauge theory (χGT). If this has IRFP at sufficiently weak coupling,
then it evolves from UV to a non-Abelian Coulomb phase in the IR. If the IRFP occurs
at stronger coupling, then there are several possibilities for the UV to IR evolution.
If theory satisfies ’t Hooft anomaly-matching conditions, then possible confinement
without SχSB, producing massless fermions; alternatively, fermion condensates may
form that break the chiral gauge symmetry and global flavor symmetries.
For a U(1) theory, or a non-Abelian theory with many fermions, Nf >> 1, β > 0;
calculate flow from weak coupling in the IR toward the UV. Study possibility of UVFP
and relate higher-loop perturbative calculations to exact results in Nf → ∞ limit.

First, discuss flows for AF gauge theories. AF requires
Nf < Nf,b1z = 11CA/(4Tf ), where CA ≡ C2(G) is quadratic Casimir invariant,
Tf ≡ T (R) is trace invariant. Focus here on G = SU(Nc); then, e.g., for R =fundamental rep. (fund. rep.), Nf < (11/2)Nc.
Denote the n-loop β fn. as βnℓ and the IR zero of βnℓ as αIR,nℓ.
At the n = 2 loop level, if b2 < 0, then two-loop beta function, β2ℓ, has IR zero at
αIR,2ℓ = −4πb1/b2. Interval I where this occurs is
I : Nf,b2z < Nf < Nf,b1z
2
where Nf,b2z = 34CA/[4Tf (5CA + 3Cf )]. For SU(2), I: 5.55 < Nf < 11;
for SU(3), I: 8.05 < Nf < 16.5;
as Nc → ∞, I: (34/13)Nc < Nf < (11/2)Nc, i.e.,
2.62Nc < Nf < 5.5Nc.
(expressions evaluated for Nf ∈ R, but it is understood that physical values of Nf arenonnegative integers.)
As Nf decreases from the upper to lower end of interval I, αIR increases. Denote
Nf,cr as value of Nf where IR behavior changes from non-Abelian Coulomb phasewith no Sχ SB to phase with SχSB.

Higher-Loop Corrections to UV → IR Evolution of
Asymptotically Free Gauge Theories
Just as higher-loop calculations have been important in calculating properties of QCD,
near to the UVFP, so also they are important for understanding properties of an IRFP
in asymptotically free gauge theories, especially for the case where αIR occurs at
moderately strong coupling (Gardi, Grunberg, Karliner..).
With T. Ryttov, we carried out these calculations of the IR zero in β and the resultant
value of the anomalous dimension γm ≡ γ for the fermion bilinear, evaluated at thispoint, for arbitrary G and fermions in general representations R, with numerical results
for G = SU(N ) and fermion reps. including fundamental, adjoint, and rank-2
symmetric and antisymmetric tensor, in T. A. Ryttov and R. Shrock, Phys. Rev. D 83,
056011 (2011) [arXiv:1011.4542]. Related work in Pica and Sannino, Phys. Rev. D 83,
035013 (2011) [arXiv:1011.5917]; results agree.
Our calculations were done up to the highest loop order (4 loops) for which the β andγ function coefficients are known (from M S calculations by Vermaseren, Larin, andvan Ritbergen).
Using the fact that b3 < 0 for Nf ∈ I, we showed that the location of the zero in β
(IRFP) decreases when one goes from 2-loop to 3-loop order.

We have extended this to an arbitrary scheme in (RS, Phys. Rev. D 87, 105005 (2013)[arxiv:1301.3209]).
At the 3-loop level, the expression for αIR,3ℓ is a solution ofpthe quadratic equation
b1 + b2a + b3a2 = 0 which thus involves a square root, b22 − 4b1b3. If a scheme
had b3 > 0 in I, then, since b2 → 0 at lower end of I, b22 − 4b1b3 would go
negative, so this scheme would yield a complex, unphysical αIR,3ℓ in this region.
Since the existence of the IR zero in β at 2-loop level is scheme-independent, one may
require that a scheme should maintain this property to higher-loop order, and hence
that b3 < 0 for Nf ∈ I. Given that b3 < 0 for Nf ∈ I, we then proved that
αIR,3ℓ < αIR,2ℓ holds in all such schemes, not just in MS.
At 4-loop level, αIR,4ℓ is determined as the physical root of the cubic equation
b1 + b2a + b3a2 + b4a3 = 0. We found that going from 3-loop to 4-loop level, there
is only a small change, so αIR,4ℓ < αIR,2ℓ.
Our result of smaller fractional change in value of IR zero of β at higher-loop orderagrees with expectation that calc. to higher loop order should give more stable result.

General result on the shift of an IR zero of β when calculated at next higher order (RS,PR D 87, 105005 (2013) [arxiv:1301.3209]). Assume fermion content is such thatb2 < 0, so theory has a 2-loop IR zero .
Consider a scheme in which the bℓ with ℓ = 3, ..., n + 1 have values that preserve theexistence of the scheme-independent 2-loop IR zero of β at higher-loop level.
Use fact that theory is asymptotically free, so β < 0 for 0 < α < αIR, and hencedβnℓ/dα > 0 for α ≃ αIR,nℓ.
Expand βnℓ in Taylor series around α = αIR,nℓ :′βnℓ = βIR,nℓ(α − αIR,nℓ) + O2(α − αIR,nℓ)

Now calculate β to the next-higher-loop order, i.e., β(n+1)ℓ, and solve for αIR,(n+1)ℓ .
To determine whether αIR,(n+1)ℓ is larger or smaller than αIR,nℓ , considerβ(n+1)ℓ − βnℓ = −2b̄n+1αn+2

In a scheme where bn+1 > 0, this difference, evaluated at α = αIR,nℓ, is negative,so, given that dβnℓ/dα|αIR,nℓ > 0, to compensate for this, the zero shifts to theright, whereas if bn+1 < 0, the difference is positive, so the zero shifts to the left.
If bn+1 > 0 ,then αIR,(n+1)ℓ > αIR,nℓ

If bn+1 < 0 ,then αIR,(n+1)ℓ < αIR,nℓThis general result is evident in our MS calculations.

b3 < 0,=⇒αIR,3ℓ < αIR,2ℓb4 > 0,=⇒αIR,4ℓ > αIR,3ℓ

We have calculated the anomalous dimension γm ≡ γ for the fermion bilinear at the
IRFP: γIR,nℓ ≡ γnℓ(α = αIR,nℓ). Recall results for illustrative cases Nc = 2, 3,
fermions in fund. rep.
Nc Nf γIR,2ℓ γIR,3ℓ γIR,4ℓ
2 7 (2.67) 0.457 0.0325
2 8 0.752 0.272 0.204
2 9 0.275 0.161 0.157
2 10 0.0910 0.0738 0.0748
3 10 (4.19) 0.647 0.156
3 11 1.61 0.439 0.250
3 12 0.773 0.312 0.253
3 13 0.404 0.220 0.210
3 14 0.212 0.146 0.147
3 15 0.0997 0.0826 0.0836
3 16 0.0272 0.0258 0.0259
As with αIR,nℓ , the fact that for a given Nc, Nf , the 3-loop and 4-loop results arecloser to each other than the 2-loop and 3-loop results shows the value of including
higher-loop terms to increase the accuracy of the calculation.

Thus, our higher-loop calcs. of αIR and γ allow an analysis of the theory to smaller
values of Nf and thus stronger couplings. Obviously, perturbative calculations are notapplicable when α is too large. It is still possible that γm ∼ 1 for Nf ≃ Nf,cr .
We have also performed these higher-loop calculations for larger fermion reps. R. In
general, we find that, for a given Nc, R, and Nf , the values of γIR,nℓ calculated to
3-loop and 4-loop order are smaller than the 2-loop value.
Extensive lattice gauge theory simulations have been performed to study IR propertiesof AF gauge theories with various fermion contents. It has been interesting to compare
our perturbative calculations of γIR,nℓ with lattice measurements.
For example, for SU(3), Nf = 12, we calculate γIR,2ℓ = 0.77, γIR,3ℓ = 0.31,
γIR,4ℓ = 0.25. Lattice results have been reported by LSD Collab. (Appelquist et al.);
LHC Collab. (Kuti et al.); LatKMI Collab. (Y. Aoki et al.); Hasenfratz et al; Degrand;
Itou...; values range from ∼ 0.2 to ∼ 0.5. Our 2-loop result is larger than latticemeasurements, and our higher-loop calculations yield results closer to thesemeasurements. This shows the value of these higher-loop calculations, since one thencan achieve both analytic understanding, from continuum QFT, and a numerical
understanding from the lattice, of γIR. More progress expected in future.

In addition to αIR,nℓ and γIR,nℓ, it is also of interest to investigate various structuralproperties of the n-loop beta function βnℓ, including
′
nℓ
≡ dβ
• the derivative βIR,nℓevaluated at αIR,nℓ .
dα• the magnitude and location of the minimum in βnℓ

We have calculated these structural properties analytically and numerically in RS, Phys.
Rev. D87, 105005 (2013) [arXiv:1301.3209].
In quasi-scale-invariant case where αIR >∼ αcr , dilaton mass relevant in dynamical
EWSB models depends on how small β is for α near to αIR and hence, at n-loop′, via the series expansion of βnℓ around αIR,nℓ,order, on βIR,nℓ′βnℓ(α) = βIR,nℓ(α − αIR,nℓ) + O2(α − αIR,nℓ)

Derivative of 2-loop β function at αIR,2ℓ:
2(11CA − 4Tf Nf )2
2b21
2b21
′
=
=
βIR,2ℓ = −
2
b2
|b2|
3[4(5CA + 3Cf )Tf Nf − 34CA]
At 3-loop level:′
βIR,3ℓ
=

1 h

|b3|2

q
i
− 4|b2|(b22 + b1|b3|) + (b22 + 2b1|b3|) b22 + 4b1|b3|

We prove a general inequality: for a given gauge group G, fermion rep. R, andNf ∈ I (in a scheme with b3 < 0, which thus preserves the existence of the 2-loop IR
zero in β at 3-loop level),′′
βIR,3ℓ< βIR,2ℓ

We carry out a similar analysis of the derivative of the 4-loop β function evaluated at′, and find a similar decrease from 3-loop to 4-loop order. Some
αIR,4ℓ, denoted βIR,4ℓ
numerical values:′′′
Nc Nf βIR,2ℓ
βIR,3ℓβIR,4ℓ
2 7
1.20
0.728
0.677
2 8
0.400
0.318
0.300
2 9
0.126
0.115
0.110
2 10 0.0245 0.0239 0.0235
3 10
1.52
0.872
0.853
3 11 0.720
0.517
0.498
3 12 0.360 0.2955 0.282
3 13 0.174
0.156
0.149
3 14 0.0737 0.0699 0.0678
3 15 0.0227 0.0223 0.0220
3 16 0.00221 0.00220 0.00220

Illustrative figures for SU(2) with Nf = 8 fermions and SU(3) with Nf = 12
fermions:

0.15

0.1

β
0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α
–0.05

–0.1

Figure 1:βnℓ for SU(2), Nf = 8, at n = 2, 3, 4 loops. From bottom to top, curves are β2ℓ, β4ℓ, β3ℓ.

0.1
0.08

β

0.06
0.04
0.02
0
–0.02

0.2

0.4

0.6

0.8

1

α

–0.04
–0.06
–0.08
–0.1

Figure 2:βnℓ for SU(3), Nf = 12, at n = 2, 3, 4 loops. From bottom to top, curves are β2ℓ, β4ℓ, β3ℓ.

Interesting property: for R = fund. rep., αIR,nℓ Nc, γIR,nℓ, and other structuralproperties of βnℓ are similar in theories with different values of Nc and Nf if they haveequal or similar values of r = Nf /Nc.
This motivates a study of the UV to IR evolution of an SU(Nc) gauge theory with Nf
fermions in the fundamental rep. in the ’t Hooft-Veneziano limit Nc → ∞,Nf → ∞ with r ≡ Nf /Nc fixed and α(µ)Nc ≡ ξ(µ) independent of Nc.
Denote this as the LNN (large Nc, large Nf ) limit.
We have carried out this study in RS, Phys. Rev. D87, 116007 (2013)[arXiv:1302.5434]. Our results provide a unified quantitative understanding of thesimilarities in UV to IR evolution of SU(Nc) theories with different Nc and Nf but
similar r.
With ξ = αNc and x = aNc = ξ/(4π), define a rescaled beta function that is
finite in the LNN limit:βξ ≡with the expansiondξdt= lim βαNc
LN N

βξ ≡

dξ
dt

= −8πx∞Xℓ=1b̂ℓxℓ = −2ξ∞Xb̃ℓξ ℓ ,

ℓ=1

where
b̂ℓ = limbℓ

LN N N ℓc,b̃ℓ = limb̄ℓ

LN N N ℓcso b̃ℓ =

b̂ℓ
(4π)ℓ

1-loop and 2-loop coefficients in βξ :1b̂1 = (11 − 2r)
3
and
b̂2 =

1
3

(34 − 13r)

Asymptotic freedom requires r < 11/2. The interval where βξ,2ℓ has an IR zero is
Ir : 34/13 < r < 11/2 , i.e., 2.615 < r < 5.500.
2-loop IR zero of βξ,2ℓ is at
4π(11 − 2r)
ξIR,2ℓ =13r − 34

3-loop and 4-loop coefficients in βξ (in MS scheme):
b̂3 =

1
54

(2857 − 1709r + 112r 2) = 52.9074 − 31.6481r + 2.07407r 2

150473

 485513  8654 

3

4

√

C3ℓ ]

r + (11 − 5r + 21r 2)ζ(3)
486
1944
243
243
9
= 315.492 − 252.421 r + 46.832 r 2 + 0.534979 r 3
P∞(where ζ(s) = n=1 n−s is Riemann zeta fn.). The 3-loop β function βξ,3ℓ has an
IR zero at
b̂4 =

−

r+

ξIR,3ℓ =2

 130 r +12π[−3(13r − 34) +
D3ℓ

,

where
C3ℓ = −52450 + 41070r − 7779r 2 + 448r 3
D3ℓ = −2857 + 1709r − 112r 2

By same type of proof as given before, we showξIR,3ℓ ≤ ξIR,2ℓ
Further, since b̂4 reverses sign from neg. to pos. as r increases through r = 3.119,
ξIR,4ℓ < ξIR,3ℓ if 2.615 < r < 3.119, (where b̂4 < 0),ξIR,4ℓ > ξIR,3ℓ if 3.119 < r < 5.500, (where b̂4 > 0)
Numerical values given in next table. The magnitude of the fractional difference|ξIR,4ℓ − ξIR,3ℓ|
ξIR,4ℓis reasonably small.

r ξIR,2ℓ ξIR,3ℓ ξIR,4ℓ
2.8 28.274 3.573
3.323
3.0 12.566 2.938
2.868
3.2 7.606
2.458
2.494
3.4 5.174
2.076
2.168
3.6 3.731
1.759
1.873
3.8 2.774
1.489
1.601
4.0 2.095
1.252
1.349
4.2 1.586
1.041
1.115
4.4 1.192 0.8490 0.9003
4.6 0.8767 0.6725 0.7038
4.8 0.6195 0.5083 0.5244
5.0 0.4054 0.3538 0.3603
5.2 0.2244 0.2074 0.2089
5.4 0.06943 0.06769 0.06775

Anomalous dimension γm ≡ γ:γ=∞Xĉℓ xℓ =ℓ=1∞Xc̃ℓ ξ ℓℓ=1

where ĉℓ = limLN N (cℓ /Ncℓ) and c̃ℓ = ĉℓ/(4π)ℓ. The coefficients ĉℓ areĉ1 = 3 ,

ĉ3 =

11413
108

−



ĉ2 =

203

5

− r = 16.917 − 1.667r
12
3


1177
35 2
+ 12ζ(3) r − r = 105.676 − 36.221r − 1.296r 2
5427
460151 23816
899 2 83 31157 889
16 3
2
ĉ4 =
−
r+
r − r +−
r + 20r + r ζ(3)
576
81
162
81
9
3
9




+r 66 − 12r ζ(4) + − 220 + 160r ζ(5)

= 725.280 − 412.892r + 16.603r 2 + 1.1123r 3

Value of n-loop γ evaluated at n-loop ξIR,nℓ: γIR,nℓ ≡ γnℓ

ξ=ξIR,nℓ(11 − 2r)(1009 − 158r + 40r 2)
γIR,2ℓ =12(13r − 34)2
and so forth for higher-loop order. Numerical values:r γIR,2ℓ γIR,3ℓ γIR,4ℓ
3.6 1.853 0.5201 0.3083
3.8 1.178 0.4197 0.3061
4.0 0.7847 0.3414 0.2877
4.2 0.5366 0.2771 0.2664
4.4 0.3707 0.2221 0.2173
4.6 0.2543 0.1735 0.1745
4.8 0.1696 0.1294 0.1313
5.0 0.1057 0.08886 0.08999
5.2 0.05620 0.05123 0.05156
5.4 0.01682 0.01637 0.01638
General inequality as before: γIR,3ℓ < γIR,2ℓ .

;We have studied the approach to the LNN limit and find that this is quite rapid, withleading correction terms suppressed by 1/Nc2. For example,
αIR,2ℓNc =4π(11 − 2r)
13r − 34

γIR,2ℓ =++12πr(11 − 2r)(34 − 13r)2Nc2

+O

 1 
Nc4

(11 − 2r)(1009 − 158r + 40r 2)

12(13r − 34)2
(11 − 2r)(18836 − 5331r + 648r 2 − 140r 3)(13r − 34)3Nc2

+O

 1 
Nc4

(We have also done an analogous study for a supersymmetric gauge theory.)
These results provide an understanding of the approximate universality that is exhibitedin calculations of these quantities for different (finite) values of Nc and Nf with similar
or identical values of r.

Study of Scheme Dependence in Higher-Loop Calculations
Since coeffs. bn with n ≥ 3 in βnℓ are scheme-dependent, it is important to assess
the effects of this scheme dependence in higher-loop calculations. We have given new
results in RS, Phys. Rev D 88, 036003 (2013) [arXiv:1305.6524], extending the earlier
studies in Ryttov and RS, PRD 86, 065032 (2012) [arXiv:1206.2366] and PRD 86,
085005 (2012) [arXiv:1206.6895].
A physically acceptable ST must satisfy several conditions, which are easily met in thevicinity of a UVFP of an AF theory at α = 0 or the IRFP of an IR-free theory at
α = 0, but are significant constraints if the fixed point occurs at α ∼ O(1).
We have constructed several STs that are acceptable at an IRFP and have studied
scheme dependence of the IR zero of βnℓ using these. An exampleispa = (1/r) sinh(ra′), with inverse a′ = (1/r) ln[ra + 1 + (ra)2 ]. We findreasonably small scheme-dependence for moderate αIR. Our studies give a quantitative
assessment of the scheme dependence of an IR zero of β in such a theory.

Since the bℓ with ℓ ≥ 3 are scheme-dependent, one might expect that it would bepossible, at least in the vicinity of α = α′ = 0, to construct scheme transformations
that would set b′ℓ = 0 for some range of ℓ ≥ 3, and, indeed a ST that would do thisfor all ℓ ≥ 3, so that βα′ would consist only of the 1-loop and 2-loop terms (’t Hooft
scheme).
We have constructed scheme transformations denoted SR,m with m ≥ 2 that remove
the terms in the beta function at loop order ℓ = 3 to ℓ = m + 1, inclusive, and, inthe limit, m → ∞, a ST that can transform to the ’t Hooft scheme in the vicinity of
α = α′ = 0.
We have quantitatively determined the range of applicability of these scheme
transformations in an AF SU(Nc) gauge theory with Nf fermions.

UV to IR Evolution in Asymptotically Free Chiral Gauge
Theories
With T. Appelquist, we have obtained new results on the UV to IR evolution of
asymptotically free chiral gauge theories (χGTs) in T. Appelquist and RS, Phys. Rev. D
88, 105012 (2013) [arXiv:1310.6076]. (These theories are free of any triangle anomalies
in gauged currents.) There are no fermion masses in the Lagrangians for these theories.
Some of these χGTs have IRFPs, and if these occur at sufficiently weak coupling, then
the theories flow from the UV to a non-Abelian Coulomb phase in the IR, with noconfinement or SχSB.
For theories with an IR zero in β at strong coupling or no IR zero in β, the flow fromthe UV leads to a strong coupling among the elementary fermions and gluons in the IR.
Several possible types of behavior can occur, including• confinement without any SχSB, leading to massless gauge-singlet composite
fermions, if anomalies in global flavor symmetries are matched between the fermionsin the Lagrangian and the composite fermions (’t Hooft anomaly-matchingconditions)• formation of bilinear fermion condensates, which break global flavor symmetries andcan break the chiral gauge symmetries in sequential stages
N.B.: sequential self-breaking of a strongly coupled chiral gauge theory was used heavily
in earlier constructions of moderately UV-complete ETC models, e.g., Appelquist andRS, Phys. Lett. B 548, 204 (2002); Phys. Rev. Lett. 90, 201801 (2003); Appelquist,
Piai, and RS, Phys. Rev. D 69, 015002 (2004). The sequential nature of thissymmetry breaking was the key to producing a hierarchy of different ETC scales andhence a generational hierarchy in the SM fermion masses.
If several possible bilinear fermion condensation channels are possible, a priori, anatural criterion for deciding which occurs is the most attractive channel (MAC)
criterion. For chiral fermions in reps. R1 and R2, consider a condensation channel (c)R1 × R2 → Rc. The MAC criterion is that the channel in which condensation
actually occurs is the one that maximizes∆C2 ≡ C2(R1) + C2(R2) − C2(Rc)
The MAC is motivated by its application to QCD, which explains why only 3 × 3̄ → 1
occurs (it is the MAC), and not 3 × 3̄ → 8, 3 × 3 → 3̄a, or 3 × 3 → 6s.
Of course since the condensation is nonperturbative, while the MAC issemiperturbative, it is only a rough guide.

Second-order phase transitions in statistical mechanics and condensed matter physics
motivate the notion that the UV to IR flow involves a Wilsonian thinning of degrees of
freedom (d.o.f.) in the fields. In an approach using finite temperature T as the RG
scale µ, and in a weakly interacting regime, one can enumerate these field degrees of
freedom via the coefficient f in the free energy density
π2 4
F (T ) = f T
90
where

7
f = 2NV + NF + NS ,4
where NV is the number of massless gauge fields, NF is the number of massless chiralcomponents of fermion fields, and NS is the number of massless real scalar fields.
Definelim f ≡ fU V ,µ→∞lim f ≡ fIR ,µ→0∆f ≡ fU V − fIR

A conjectured d.o.f. inequality (DFI) (Appelquist, Cohen, Schmaltz, Shrock, 1999)
motivated by the idea of thinning of d.o.f. in a UV to IR flow is∆f ≥ 0Lattice gauge simulations can be used to test the DFI conjecture for a vectorial gauge
theory (VGT), but to do this for a chiral gauge theory requires putting a χGT on thelattice, which is challenging.
A further conjecture is that if various different condensation channels are possible, apriori, the one that occurs is the one that maximizes ∆f (DFM conjecture).
Consider, e.g., a theory with G = SU(N ) and massless chiral fermions including a(ab)symmetric, rank-2 tensor (S) ψLab ≡ ψL , together with N + 4 copies (flavors) of
the conjugate fundamental (F̄ ) representation, χa,i,L, i = 1, ..., N + 4, wherea, b, .. are SU(Nc) gauge indices; i, j, ... are copy indices. We use N + 4 F̄
fermions to cancel gauge anomalies, since Anom(S) + (N + 4)Anom(F̄ ) = 0.
To this irreducibly chiral set of fermions we add a vectorlike set consisting of p copies ofa, j = 1, ..., p.
(F + F̄ ), χa,i,L, i = N + 4 + 1, ...N + 4 + p and ωj,LThis theory satisfies the conditions that would allow the formation of a set of masslessgauge-singlet composite fermions (Bars and Yankielowicz).
The AF property β < 0 requires that p < pb1z = (9/2)N − 3, and we assume this.

For a range of p < pb1z , this theory has a (scheme-independent) two-loop zero of β, at
8πN (9N − 6 − 2p)
αIR,2ℓ =−39N 3 + 90N 2 − 3N − 36 + p(26N 2 − 6)
If αIR,2ℓ is small, then the UV flow leads to (i) a non-Abelian Coulomb phase in theIR, and the DFI is satisfied. As p decreases, αIR,2ℓ increases.
For sufficiently small p and large IR coupling, there are several IR possibilities, including(ii) a phase with confinement and a massless sector consisting of gauge-singlet
composite fermions, with no gauge or chiral symmetry breaking.
(iii) sequential condensation in the respective S × F̄ → F (MAC) channel, breaking
the SU(N ) gauge symmetry completely, leaving a set of massless elementary fermionsand massless composite NGBs in the IR.
The fact that the MAC is S × F̄ → F follows since ∆C2 is greater for this channel
than for F × F̄ → 1:
N −1
>0
∆C2(S × F̄ → F ) − ∆C2(F × F̄ → 1) =N
The S × F̄ → F condensation breaks the SU(N ) gauge symmetry to SU(N − 1).
Denote the scale where this happens as ΛN . The fermion field components involved inthe condensate gain dynamical masses of order ΛN and are integrated out in the
low-energy effective field theory operative for µ < ΛN . There is then sequential
condensation in the S × F̄ → F channel in this SU(N − 1) theory, and so on tolower scales.
Two other possible types of UV to IR evolution are(iv) condensation of the p vectorlike fermions in the channel F × F̄ → 1 followed byconfinement with massless composite fermion formation, no further chiral symmetry
breaking, and no gauge-symmetry breaking, so that the IR effective FT consists of themassless composite fermions together with massless NGBs(v) condensation of the p vectorlike fermions in the channel F × F̄ → 1, followed bycondensation in the S × F̄ → F channel, again breaking the SU(N ) gauge symmetrycompletely, so that the IR particle content consists of massless NGBs and masslesselementary fermionsWe find that the MAC prediction for the most likely type of UV to IR evolution differsfrom the prediction from the ∆f maximization (DFM) conjecture. The MAC predicts
that (iii) will occur, but this would actually violate the ∆f ≥ 0 (DFI) conjecture for arange of larger values of p. For example, in the LNN limit N → ∞, p → ∞ holdingr = p/N fixed (with r < 11/2 to maintain AF), (iii) would violate DFI for
15/8 < r < 11/2, i.e., for 1.875 < r < 5.5.
Further, (iii) does not minimize fIR , i.e., maximize ∆f ; instead, (iv), namely,
F × F̄ → 1 condensation followed by confinement with massless gauge-singlet
fermions, maximizes ∆f and hence (iv) is favored by the DFM conjecture.
More details in our paper. These results show that the methods that one currently has
for analyzing the UV to IR evolution in asymptotically free chiral gauge theories do not,in general, agree in their predictions. There is a great opportunity here for further
progress in understanding this evolution better.

Study of RG Flows in Gauge Theories with Many Fermions
If the β function of a theory is positive near zero coupling, then this theory is IR-free; asµ increases from the IR to the UV, the coupling grows. It is of interest to investigate
whether a non-AF theory of this type might have a UV fixed point (UV zero of β).
In addition to performing perturbative calculations of β to search for such a UVFP in
an IR-free theory, one can use large-N methods. An explicit example is the O(N )nonlinear σ model in d = 2 + ǫ spacetime dimensions. From an exact solution of thismodel in the limit N → ∞, one finds that (for small ǫ)λ,β(λ) = ǫλ 1 −λcwhere λ is the effective coupling and λc = 2πǫ/N (W. Bardeen, B. W. Lee, and R.
Shrock, Phys. Rev. D 14, 985 (1976); E. Brézin and J. Zinn-Justin, Phys. Rev. B 14,3110 (1976)). Thus this theory has a UVFP at λc, so that if initial value of λ < λc,then λ ր λc as µ → ∞.
There has long been interest in RG properties of d = 4 QED and, more generally, U(1)gauge theory (Gell-Mann and Low; Johnson, Baker, and Willey; Adler; Miransky;
Yamawaki,...).

Consider a vectorial U(1) theory with Nf massless Dirac fermions of charge q. With noloss of generality, set q = 1. Write β function asβα = 2α∞Xb ℓ aℓℓ=1

The 1-loop and 2-loop coefficients are
4Nf,
b2 = 4Nf
3
These coefficients have the same sign, so the two-loop beta function, βα,2ℓ, does nothave a UV zero, and this is the maximal scheme-independent information about it. The
coefficients have been calculated up to five loops in the M S scheme.
b1 =The 3-loop coefficient (deRafael and Rosner, 1974) is negative:
22Nf 
b3 = −2Nf 1 +9Hence, βα,3ℓ has a UV zero, namely,
αU V,3ℓ = 4πaU V,3ℓ =p
4π[9 + 3(45 + 44Nf ) ]9 + 22Nf

The 4-loop coefficient is (Gorishny, Kataev, Larin, Surguladze, 1991)h 760
832ζ(3) 
1232 2ib4 = Nf − 46 +
−
Nf −
Nf ,
27
9
243
Numerically,
b4 = −Nf (46 + 82.97533Nf + 5.06996Nf2]
This is negative for all Nf > 0.
Recently, b5 has been calculated (Kataev, Larin, 2012; Baikov, Chetyrkin, Kühn,
Rittinger, Sturm, 2012, 2013). Numerically,b5 = Nf (846.6966 + 798.8919Nf − 148.7919Nf2 + 9.22127Nf3
which is positive for all Nf > 0.
Using these results we have investigated whether βα,nℓ has a UV zero for n up to 5
loops for a large range of Nf . Our results are given in the table. The notation −means that for the given loop order and value of Nf , βnℓ has no UV zero.

Nf αU V,2ℓ αU V,3ℓ
αU V,4ℓ αU V,5ℓ
1−
10.2720 3.0400−
2−6.87002.4239−3−
5.36892.0776−4−4.50171.8463−5−
3.9279 1.67685 2.5570
6−
3.51561.5455 1.8469
7−
3.20271.4397 1.6243
8−
2.95551.3519 1.4851
9−2.75451.2776 1.3863
10−
2.5871
1.2135 1.3120
20−
1.72620.8483−100 −
0.7081 0.33265−
500 −
0.30380.1203−103 −0.2127 0.07678−
104 − 0.016614 0.016965 −A necessary condition for the perturbatively calculated β function to yield evidence fora stable UV zero is that it should remain present when one increases the loop order andthe fractional change in the value should decrease going from n to n + 1 loops.

As is evident from the table, we do not find that the UV zeros that we have calculated
at ℓ = 3, 4, 5 loop order for a large range of Nf values satisfy this necessary
condition. Hence, these calculations do not give evidence for a UVFP in this theory.
We have also carried out an analysis in the limit
Nf α(µ)Nf → ∞ with finite y(µ) ≡ Nf a(µ) =4π
We denote this as the LNF (large-Nf ) limit; analogous to N → ∞ limit in nonlinear
σ model.
We set b1 = b1,1Nf with b1,1 = 4/3. Further,
ℓ−1
X
bℓ =
bℓ,k Nfk for ℓ ≥ 2 ,k=1

where the bℓ,k are independent of Nf .
Hence,bℓ ∝ Nfℓ−1 for ℓ ≥ 2 as Nf → ∞We thus define the finite quantities
b̌ℓ ≡

bℓ
Nfℓ−1

for ℓ ≥ 2

solim b̌ℓ = bℓ,ℓ−1 for ℓ ≥ 2

Nf →∞We define a rescaled β function that is finite in the LNF limit as βy ≡ βαNf . Then


∞
X
1
βy = 8πb1,1 y 2 1 +bℓ y ℓ−1 .
b1,1Nf ℓ=2
The condition that the n-loop βy , βy,nℓ, has a zero at y 6= 0 is the equationn
1 X
bℓ y ℓ−1 = 0 .
1+
b1,1Nf ℓ=2
In the LNF limit, of the n − 1 roots of this equation, the relevant one has theapproximate form
1
 n−1

b1,1NfyU V,nℓ ∼ −bn,n−1Hence, βy,nℓ has a zero for y 6= 0 in the LNF limit if and only if bn,n−1 < 0.
However, even if this condition were to be met, it follows that, for fixed finite loop ordern, in the LNF limit, limNf →∞ yU V,nℓ = ∞.

One can reexpress βy as a series in powers of ν ≡ 1/Nf :∞hiXβy = 8πb1,1 y 2 1 +Fs(y)ν s
s=1

An exact integral representation of F1(y) is known (cf. Holdom, 2010). We have used
this representation to determine the signs of bn,n−1 up to n = 24 loops. We find thatthese signs are scattered, and show no indication of an onset of negative signs.
Thus, we do not find evidence of a UVFP in a U(1) gauge theory with Nf masslesscharged fermions for large Nf . Further nonperturbative results, such as calculations ofFs(y) for s ≥ 2, would give more information on this question.
We have also studied an SU(N ) non-Abelian gauge theory with Nf massless fermionsin a given representation for Nf . This theory is IR-free, and we again we do not findevidence of a UVFP.

Conclusions• We have reported further results on the UV to IR evolution of an asymptotically freegauge theory and the nature of the IR behavior, in particular, on effects of including
higher-loop terms in calculations and structural properties of the β function.
• Results on the limit Nc → ∞, Nf → ∞ with Nf /Nc fixed provide
understanding of similarities in UV to IR flows in theories with different Nc and Nf
but similar r.
• Effects of scheme-dependence of IR zero in higher-loop calculations have been
quantitatively studied.
• The varieties of UV to IR evolution in asymptotically free chiral gauge theories havebeen studied comparatively using different methods, including the MAC, aconjectured ∆f inequality, and a ∆f maximization conjecture.
• We have studied RG flows in U(1) and non-Abelian gauge theories with Nf fermions
for large Nf .