Higher-Order Scheme-Independent Series Expansions of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ in Conformal Field Theories

TL;DR

This paper develops higher-order, scheme-independent series expansions for key quantities in conformal gauge theories, providing more precise tools to analyze their infrared behavior and compare with lattice results.

Contribution

It introduces the highest-order scheme-independent series expansions for the anomalous dimension and beta function derivative at the IR fixed point in gauge theories.

Findings

The anomalous dimension increases monotonically as the number of fermions decreases.

The series expansions are consistent with previous loop calculations and lattice data.

Results are extended to large N_c and N_f limits for fundamental representations.

Abstract

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling $\alpha=\alpha_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$ and (ii) the derivative $\beta' = d\beta/d\alpha$, to $O(\Delta_f^5)$, both evaluated at $\alpha_{IR}$, where $\Delta_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G={\rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, $\gamma_{\bar\psi\psi,IR}$, calculated to the order $\Delta_f^p$, with $1 \le p \le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed. We also present expansions for $\alpha_{IR}$ calculated to $O(\Delta_f^4)$.

Tables

Table 1. Table 1: Values of the anomalous dimension γ ψ ¯ ​ ψ , I ​ R , F subscript 𝛾 ¯ 𝜓 𝜓 𝐼 𝑅 𝐹 \gamma_{\bar{\psi}\psi,IR,F} calculated to O ​ ( Δ f p ) 𝑂 superscript subscript Δ 𝑓 𝑝 O(\Delta_{f}^{p}) , i.e., γ ψ ¯ ​ ψ , I ​ R , F , Δ f p subscript 𝛾 ¯ 𝜓 𝜓 𝐼 𝑅 𝐹 superscript subscript Δ 𝑓 𝑝 \gamma_{\bar{\psi}\psi,IR,F,\Delta_{f}^{p}} , with 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , for G = SU ​ ( N c ) 𝐺 SU subscript 𝑁 𝑐 G={\rm SU}(N_{c}) , as a function of N c subscript 𝑁 𝑐 N_{c} and N f subscript 𝑁 𝑓 N_{f} for 2 ≤ N c ≤ 4 2 subscript 𝑁 𝑐 4 2\leq N_{c}\leq 4 and N f subscript 𝑁 𝑓 N_{f} in the respective intervals I I ​ R ​ Z subscript 𝐼 𝐼 𝑅 𝑍 I_{IRZ} for each N c subscript 𝑁 𝑐 N_{c} . For comparison, we also include the n 𝑛 n -loop values γ ψ ¯ ​ ψ , I ​ R , F , n ​ ℓ subscript 𝛾 ¯ 𝜓 𝜓 𝐼 𝑅 𝐹 𝑛 ℓ \gamma_{\bar{\psi}\psi,IR,F,n\ell} with 2 ≤ n ≤ 4 2 𝑛 4 2\leq n\leq 4 from Table VI of bvh . Values that exceed the bound γ ψ ¯ ​ ψ , I ​ R ≤ 2 subscript 𝛾 ¯ 𝜓 𝜓 𝐼 𝑅 2 \gamma_{\bar{\psi}\psi,IR}\leq 2 in Eq. ( 18 ) are marked as unphysical (u). For notational brevity in this and successive tables, we omit the subscript ψ ¯ ​ ψ ¯ 𝜓 𝜓 \bar{\psi}\psi . See text for further details.

$N_c$	$N_f$	${\gamma }_{IR,F,2\ell }$	${\gamma }_{IR,F,3\ell }$	${\gamma }_{IR,F,4\ell }$	${\gamma }_{IR,F,{\Delta }_f}$	${\gamma }_{IR,F,{\Delta }_f^2}$	${\gamma }_{IR,F,{\Delta }_f^3}$	${\gamma }_{IR,F,{\Delta }_f^4}$
2	6	u	u	u	0.337	0.520	0.596	0.698
2	7	u	u	u	0.270	0.387	0.426	0.467
2	8	0.752	0.272	0.204	0.202	0.268	0.285	0.298
2	9	0.275	0.161	0.157	0.135	0.164	0.169	0.172
2	10	0.0910	0.0738	0.0748	0.0674	0.07475	0.07535	0.0755
3	9	u	u	u	0.374	0.587	0.687	0.804
3	10	u	u	u	0.324	0.484	0.549	0.615
3	11	1.61	0.439	0.250	0.274	0.389	0.428	0.462
3	12	0.773	0.312	0.253	0.224	0.301	0.323	0.338
3	13	0.404	0.220	0.210	0.174	0.221	0.231	0.237
3	14	0.212	0.146	0.147	0.125	0.148	0.152	0.153
3	15	0.0997	0.0826	0.0836	0.0748	0.0833	0.0841	0.0843
3	16	0.0272	0.0258	0.0259	0.0249	0.0259	0.0259	0.0259
4	11	u	u	u	0.424	0.694	0.844	1.029
4	12	u	u	u	0.386	0.609	0.721	0.8475
4	13	u	u	u	0.347	0.528	0.610	0.693
4	14	u	u	u	0.308	0.451	0.509	0.561
4	15	1.32	0.420	0.281	0.270	0.379	0.418	0.448
4	16	0.778	0.325	0.269	0.231	0.312	0.336	0.352
4	17	0.481	0.251	0.234	0.193	0.249	0.263	0.2705
4	18	0.301	0.189	0.187	0.154	0.190	0.197	0.200
4	19	0.183	0.134	0.136	0.116	0.136	0.139	0.140
4	20	0.102	0.0854	0.0865	0.0771	0.0860	0.0869	0.0871
4	21	0.0440	0.0407	0.0409	0.0386	0.0408	0.0409	0.0409

Table 2. Table 2: Values of the scheme-independent γ I ​ R , F , Δ r p subscript 𝛾 𝐼 𝑅 𝐹 superscript subscript Δ 𝑟 𝑝 \gamma_{IR,F,\Delta_{r}^{p}} in the LNN limit ( 90 ) for 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , together with γ I ​ R , F , n ​ ℓ subscript 𝛾 𝐼 𝑅 𝐹 𝑛 ℓ \gamma_{IR,F,n\ell} with n = 2 , 3 , 4 𝑛 2 3 4 n=2,\ 3,\ 4 from Table V of lnn for comparison, as a function of r 𝑟 r for r ∈ I I ​ R ​ Z , r 𝑟 subscript 𝐼 𝐼 𝑅 𝑍 𝑟 r\in I_{IRZ,r} . Values that exceed the bound γ I ​ R ≤ 2 subscript 𝛾 𝐼 𝑅 2 \gamma_{IR}\leq 2 are marked as unphysical (u) or placed in parentheses. We also list the extrapolated estimate γ I ​ R , F , e ​ x ​ 234 subscript 𝛾 𝐼 𝑅 𝐹 𝑒 𝑥 234 \gamma_{IR,F,ex234} of γ I ​ R , F , Δ r ∞ subscript 𝛾 𝐼 𝑅 𝐹 superscript subscript Δ 𝑟 \gamma_{IR,F,\Delta_{r}^{\infty}} and, in the last column, the ratio γ I ​ R , F , e ​ x ​ 234 / γ I ​ R , F , Δ r 4 subscript 𝛾 𝐼 𝑅 𝐹 𝑒 𝑥 234 subscript 𝛾 𝐼 𝑅 𝐹 superscript subscript Δ 𝑟 4 \gamma_{IR,F,ex234}/\gamma_{IR,F,\Delta_{r}^{4}} .

$r$	${\gamma }_{{}_{IR,F,2\ell }}$	${\gamma }_{{}_{IR,F,3\ell }}$	${\gamma }_{{}_{IR,F,4\ell }}$	${\gamma }_{IR,F,{\Delta }_r}$	${\gamma }_{IR,F,{\Delta }_r^2}$	${\gamma }_{IR,F,{\Delta }_r^3}$	${\gamma }_{IR,F,{\Delta }_r^4}$	${\gamma }_{IR,F,ex234}$	$\frac{{\gamma }_{IR,F,ex234}}{{\gamma }_{IR,F,{\Delta }_r^4}}$
2.8	u	1.708	0.190	0.432	0.706	0.870	1.064	(2.09)	1.96
3.0	u	1.165	0.225	0.400	0.635	0.765	0.908	1.645	1.82
3.2	u	0.854	0.264	0.368	0.567	0.668	0.770	1.28	1.66
3.4	u	0.656	0.293	0.336	0.502	0.579	0.650	0.993	1.53
3.6	1.853	0.520	0.308	0.304	0.440	0.497	0.5445	0.763	1.40
3.8	1.178	0.420	0.306	0.272	0.381	0.422	0.452	0.584	1.29
4.0	0.785	0.341	0.288	0.240	0.325	0.353	0.371	0.444	1.20
4.2	0.537	0.277	0.257	0.208	0.272	0.290	0.300	0.337	1.12
4.4	0.371	0.222	0.217	0.176	0.2215	0.233	0.238	0.253	1.06
4.6	0.254	0.1735	0.1745	0.144	0.1745	0.1805	0.183	0.188	1.03
4.8	0.170	0.129	0.131	0.112	0.130	0.133	0.134	0.135	1.01
5.0	0.106	0.0889	0.0900	0.0800	0.0894	0.09045	0.0907	0.0905	1.00
5.2	0.0562	0.0512	0.0516	0.0480	0.0514	0.0516	0.0516	0.0516	1.00
5.4	0.0168	0.0164	0.0164	0.0160	0.0164	0.0164	0.0164	0.0164	1.00

Table 3. Table 3: Values of the anomalous dimension γ I ​ R , a ​ d ​ j , Δ f p subscript 𝛾 𝐼 𝑅 𝑎 𝑑 𝑗 superscript subscript Δ 𝑓 𝑝 \gamma_{IR,adj,\Delta_{f}^{p}} with 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , for N f = 2 subscript 𝑁 𝑓 2 N_{f}=2 and G = SU ​ ( N c ) 𝐺 SU subscript 𝑁 𝑐 G={\rm SU}(N_{c}) with N c = 2 , 3 subscript 𝑁 𝑐 2 3 N_{c}=2,\ 3 . For comparison, we also list our n 𝑛 n -loop values, γ I ​ R , a ​ d ​ j , n ​ ℓ subscript 𝛾 𝐼 𝑅 𝑎 𝑑 𝑗 𝑛 ℓ \gamma_{IR,adj,n\ell} for this theory from Table VIII of Ref. bvh .

$N_c$	${\gamma }_{IR,adj,2\ell }$	${\gamma }_{IR,adj,3\ell }$	${\gamma }_{IR,adj,4\ell }$	${\gamma }_{IR,adj,{\Delta }_f}$	${\gamma }_{IR,adj,{\Delta }_f^2}$	${\gamma }_{IR,adj,{\Delta }_f^3}$	${\gamma }_{IR,adj,{\Delta }_f^4}$
2	0.820	0.543	0.500	0.333	0.465	0.511	0.556
3	0.820	0.543	0.523	0.333	0.465	0.516	0.553

Table 4. Table 4: Values of the anomalous dimension γ I ​ R , S 2 , Δ f p subscript 𝛾 𝐼 𝑅 subscript 𝑆 2 superscript subscript Δ 𝑓 𝑝 \gamma_{IR,S_{2},\Delta_{f}^{p}} with 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , for G = SU ​ ( N c ) 𝐺 SU subscript 𝑁 𝑐 G={\rm SU}(N_{c}) with N c = 3 , 4 subscript 𝑁 𝑐 3 4 N_{c}=3,\ 4 and N f = 2 , 3 subscript 𝑁 𝑓 2 3 N_{f}=2,\ 3 (so N f ∈ I I ​ R ​ Z subscript 𝑁 𝑓 subscript 𝐼 𝐼 𝑅 𝑍 N_{f}\in I_{IRZ} ). For comparison, we also include values of γ I ​ R , S 2 , n ​ ℓ subscript 𝛾 𝐼 𝑅 subscript 𝑆 2 𝑛 ℓ \gamma_{IR,S_{2},n\ell} with 2 ≤ n ≤ 4 2 𝑛 4 2\leq n\leq 4 for this theory from Table XI in our Ref. bvh . Values that exceed the upper bound γ I ​ R < 2 subscript 𝛾 𝐼 𝑅 2 \gamma_{IR}<2 are marked as unphysical (u).

$N_c$	$N_f$	${\gamma }_{IR,S_2,2\ell }$	${\gamma }_{IR,S_2,3\ell }$	${\gamma }_{IR,S_2,4\ell }$	${\gamma }_{IR,S_2,{\Delta }_f}$	${\gamma }_{IR,S_2,{\Delta }_f^2}$	${\gamma }_{IR,S_2,{\Delta }_f^3}$	${\gamma }_{IR,S_2,{\Delta }_f^4}$
3	2	u	1.28	1.12	0.501	0.789	0.960	1.132
3	3	0.144	0.133	0.133	0.116	0.131	0.133	0.1335
4	2	u	u	1.79	0.581	0.966	1.242	1.536
4	3	0.381	0.313	0.315	0.232	0.294	0.312	0.319

Table 5. Table 5: Values of the anomalous dimension γ I ​ R , A 2 , Δ f p subscript 𝛾 𝐼 𝑅 subscript 𝐴 2 superscript subscript Δ 𝑓 𝑝 \gamma_{IR,A_{2},\Delta_{f}^{p}} calculated to order 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , for G = SU ​ ( 4 ) 𝐺 SU 4 G={\rm SU}(4) and N f ∈ I I ​ R ​ Z subscript 𝑁 𝑓 subscript 𝐼 𝐼 𝑅 𝑍 N_{f}\in I_{IRZ} . For comparison, we also include values of γ I ​ R , A 2 , n ​ ℓ subscript 𝛾 𝐼 𝑅 subscript 𝐴 2 𝑛 ℓ \gamma_{IR,A_{2},n\ell} with 2 ≤ n ≤ 4 2 𝑛 4 2\leq n\leq 4 for this theory from Table XII in bvh . Values that exceed the upper bound γ I ​ R < 2 subscript 𝛾 𝐼 𝑅 2 \gamma_{IR}<2 are marked as unphysical (u).

$N_c$	$N_f$	${\gamma }_{IR,A_2,2\ell }$	${\gamma }_{IR,A_2,3\ell }$	${\gamma }_{IR,A_2,4\ell }$	${\gamma }_{IR,A_2,{\Delta }_f}$	${\gamma }_{IR,A_2,{\Delta }_f^2}$	${\gamma }_{IR,A_2,{\Delta }_f^3}$	${\gamma }_{IR,A_2,{\Delta }_f^4}$
4	5	u	u	u	0.5405	0.941	1.287	1.671
4	6	u	1.38	0.293	0.450	0.728	0.928	1.114
4	7	u	0.695	0.435	0.360	0.538	0.641	0.717
4	8	0.802	0.402	0.368	0.270	0.370	0.4135	0.438
4	9	0.331	0.228	0.232	0.180	0.225	0.237	0.242
4	10	0.117	0.101	0.103	0.0901	0.101	0.103	0.103

Table 6. Table 6: Values of the anomalous dimension γ I ​ R , T 2 , Δ f p subscript 𝛾 𝐼 𝑅 subscript 𝑇 2 superscript subscript Δ 𝑓 𝑝 \gamma_{IR,T_{2},\Delta_{f}^{p}} for T 2 = S 2 subscript 𝑇 2 subscript 𝑆 2 T_{2}=S_{2} or T 2 = A 2 subscript 𝑇 2 subscript 𝐴 2 T_{2}=A_{2} , calculated to order 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 , in the limit N c → ∞ → subscript 𝑁 𝑐 N_{c}\to\infty with N f ∈ I I ​ R ​ Z subscript 𝑁 𝑓 subscript 𝐼 𝐼 𝑅 𝑍 N_{f}\in I_{IRZ} for this limit, namely 3 ≤ N f ≤ 5 3 subscript 𝑁 𝑓 5 3\leq N_{f}\leq 5 .

$N_f$	${\gamma }_{IR,T_2,{\Delta }_f}$	${\gamma }_{IR,T_2,{\Delta }_f^2}$	${\gamma }_{IR,T_2,{\Delta }_f^3}$	${\gamma }_{IR,T_2,{\Delta }_f^4}$
3	0.5555	0.921	1.177	1.408
4	0.333	0.465	0.520	0.550
5	0.111	0.126	0.128	0.128

Table 7. Table 7: Signs of the d j , R subscript 𝑑 𝑗 𝑅 d_{j,R} coefficients for 2 ≤ j ≤ 5 2 𝑗 5 2\leq j\leq 5 for gauge group G = SU ​ ( N c ) 𝐺 SU subscript 𝑁 𝑐 G={\rm SU}(N_{c}) and fermion representations R 𝑅 R equal to F 𝐹 F (fundamental), a ​ d ​ j 𝑎 𝑑 𝑗 adj (adjoint), S 2 subscript 𝑆 2 S_{2} , and A 2 subscript 𝐴 2 A_{2} (symmetric and antisymmetric rank-2 tensor). Note that d 1 = 0 subscript 𝑑 1 0 d_{1}=0 for all G 𝐺 G and R 𝑅 R . In the case R = A 2 𝑅 subscript 𝐴 2 R=A_{2} , we restrict to N c ≥ 3 subscript 𝑁 𝑐 3 N_{c}\geq 3 .

$j$	$d_{j,F}$	$d_{j,adj}$	$d_{j,S_2}$	$d_{j,A_2}$
2	$+$	$+$	$+$	$+$
3	$+$	$+$	$+$	$+$
4	$-$	$+$	$+$	$-$ for $N_c=3,4,5$
				$+$ for $N_c\ge 6$
5	$-$	$-$	$-$	$-$

Table 8. Table 8: Scheme-independent values of β I ​ R , F , Δ f p ′ subscript superscript 𝛽 ′ 𝐼 𝑅 𝐹 superscript subscript Δ 𝑓 𝑝 \beta^{\prime}_{IR,F,\Delta_{f}^{p}} with 2 ≤ p ≤ 4 2 𝑝 4 2\leq p\leq 4 for G = SU ​ ( 2 ) 𝐺 SU 2 G={\rm SU}(2) , SU(3), and SU(4), as functions of N f subscript 𝑁 𝑓 N_{f} in the respective intervals I I ​ R ​ Z subscript 𝐼 𝐼 𝑅 𝑍 I_{IRZ} . For comparison, we list the n 𝑛 n -loop values of β I ​ R , F , n ​ ℓ ′ subscript superscript 𝛽 ′ 𝐼 𝑅 𝐹 𝑛 ℓ \beta^{\prime}_{IR,F,n\ell} with 2 ≤ n ≤ 5 2 𝑛 5 2\leq n\leq 5 , where β I ​ R , F , n ​ ℓ ′ subscript superscript 𝛽 ′ 𝐼 𝑅 𝐹 𝑛 ℓ \beta^{\prime}_{IR,F,n\ell} with n = 3 , 4 , 5 𝑛 3 4 5 n=3,\ 4,\ 5 are computed in the MS ¯ ¯ MS \overline{\rm MS} scheme. The notation a 𝑎 a e- n 𝑛 n means a × 10 − n 𝑎 superscript 10 𝑛 a\times 10^{-n} .

$N_c$	$N_f$	${\beta }_{IR,F,2\ell }^{\prime }$	${\beta }_{IR,F,3\ell ,\overline{\mathrm{MS}}}^{\prime }$	${\beta }_{IR,F,4\ell ,\overline{\mathrm{MS}}}^{\prime }$	${\beta }_{IR,F,{\Delta }_f^2}^{\prime }$	${\beta }_{IR,F,{\Delta }_f^3}^{\prime }$	${\beta }_{IR,F,{\Delta }_f^4}^{\prime }$	${\beta }_{IR,F,{\Delta }_f^5}^{\prime }$
2	6	6.061	1.620	0.975	0.499	0.957	0.734	0.6515
2	7	1.202	0.728	0.677	0.320	0.554	0.463	0.436
2	8	0.400	0.318	0.300	0.180	0.279	0.250	0.243
2	9	0.126	0.115	0.110	0.0799	0.109	0.1035	0.103
2	10	0.0245	0.0239	0.0235	0.0200	0.0236	0.0233	0.0233
3	9	4.167	1.475	1.464	0.467	0.882	0.7355	0.602
3	10	1.523	0.872	0.853	0.351	0.621	0.538	0.473
3	11	0.720	0.517	0.498	0.251	0.415	0.3725	0.344
3	12	0.360	0.2955	0.282	0.168	0.258	0.239	0.228
3	13	0.174	0.1556	0.149	0.102	0.144	0.137	0.134
3	14	0.0737	0.0699	0.678	0.0519	0.0673	0.0655	0.0649
3	15	0.0227	0.0223	0.0220	0.0187	0.0220	0.0218	0.0217
3	16	2.21e-3	2.20e-3	2.20e-3	2.08e-3	2.20e-3	2.20e-3	2.20e-3
4	11	16.338	2.189	2.189	0.553	1.087	0.898	0.648
4	12	3.756	1.430	1.429	0.457	0.858	0.729	0.574
4	13	1.767	0.965	0.955	0.370	0.663	0.578	0.486
4	14	0.984	0.655	0.639	0.292	0.498	0.445	0.394
4	15	0.581	0.440	0.424	0.224	0.362	0.331	0.3045
4	16	0.348	0.288	0.276	0.1645	0.251	0.234	0.222
4	17	0.204	0.180	0.1725	0.114	0.164	0.156	0.1515
4	18	0.113	0.105	0.101	0.0731	0.0988	0.0955	0.0939
4	19	0.0558	0.0536	0.0522	0.0411	0.0520	0.0509	0.0505
4	20	0.0222	0.0218	0.0215	0.0183	0.0215	0.0213	0.0212
4	21	5.01e-3	4.99e-3	4.96e-3	4.57e-3	4.97e-3	4.96e-3	4.96e-3

Table 9. Table 9: Scheme-independent values of β I ​ R , Δ r p ′ subscript superscript 𝛽 ′ 𝐼 𝑅 superscript subscript Δ 𝑟 𝑝 \beta^{\prime}_{IR,\Delta_{r}^{p}} for 2 ≤ p ≤ 5 2 𝑝 5 2\leq p\leq 5 in the LNN limit ( 90 ) as functions of r = 5.5 − Δ r 𝑟 5.5 subscript Δ 𝑟 r=5.5-\Delta_{r} . For comparison, we also list the n 𝑛 n -loop values β I ​ R , n ​ ℓ ′ subscript superscript 𝛽 ′ 𝐼 𝑅 𝑛 ℓ \beta^{\prime}_{IR,n\ell} with 2 ≤ n ≤ 5 2 𝑛 5 2\leq n\leq 5 , where β I ​ R , n ​ ℓ ′ subscript superscript 𝛽 ′ 𝐼 𝑅 𝑛 ℓ \beta^{\prime}_{IR,n\ell} with n = 3 , 4 , 5 𝑛 3 4 5 n=3,\ 4,\ 5 are computed in the MS ¯ ¯ MS \overline{\rm MS} scheme. The notation a 𝑎 a e- n 𝑛 n means a × 10 − n 𝑎 superscript 10 𝑛 a\times 10^{-n} .

$r$	${\beta }_{IR,2\ell }^{\prime }$	${\beta }_{IR,3\ell }^{\prime }$	${\beta }_{IR,4\ell }^{\prime }$	${\beta }_{IR,{\Delta }_r^2}^{\prime }$	${\beta }_{IR,{\Delta }_r^3}^{\prime }$	${\beta }_{IR,{\Delta }_r^4}^{\prime }$	${\beta }_{IR,{\Delta }_r^5}^{\prime }$
2.8	8.100	1.918	1.913	0.518	1.004	0.851	0.583
3.0	3.333	1.376	1.379	0.444	0.830	0.717	0.535
3.2	1.856	1.006	1.003	0.376	0.676	0.596	0.4755
3.4	1.153	0.7395	0.729	0.314	0.542	0.486	0.410
3.6	0.752	0.542	0.527	0.257	0.426	0.388	0.342
3.8	0.500	0.393	0.378	0.2055	0.327	0.303	0.276
4.0	0.333	0.279	0.267	0.160	0.243	0.229	0.214
4.2	0.219	0.193	0.184	0.120	0.174	0.166	0.159
4.4	0.139	0.128	0.122	0.0860	0.119	0.115	0.112
4.6	0.0837	0.0792	0.0766	0.0576	0.0756	0.0737	0.0726
4.8	0.0460	0.0445	0.0435	0.0348	0.0433	0.0426	0.0423
5.0	0.0215	0.0212	0.0208	0.0178	0.0209	0.0207	0.0206
5.2	0.714e-2	0.710e-2	0.706e-2	0.640e-2	0.707e-2	0.704e-2	0.704e-3
5.4	0.737e-3	0.736e-3	0.7356e-3	0.7111e-3	0.7358e-3	0.7355e-3	0.7355e-3

Table 10. Table 10: Values of the IR zero ξ I ​ R , n ​ ℓ subscript 𝜉 𝐼 𝑅 𝑛 ℓ \xi_{IR,n\ell} of the β ξ , n ​ ℓ subscript 𝛽 𝜉 𝑛 ℓ \beta_{\xi,n\ell} function in the LNN limit for 2 ≤ n ≤ 5 2 𝑛 5 2\leq n\leq 5 and r ∈ I r 𝑟 subscript 𝐼 𝑟 r\in I_{r} . Notation u (unphysical) means that there is no physical IR zero ξ I ​ R , 5 ​ ℓ subscript 𝜉 𝐼 𝑅 5 ℓ \xi_{IR,5\ell} of the 5-loop beta function.

$r$	${\xi }_{IR,2\ell }$	${\xi }_{IR,3\ell }$	${\xi }_{IR,4\ell }$	${\xi }_{IR,5\ell }$
2.8	28.274	3.573	3.323	u
3.0	12.566	2.938	2.868	u
3.2	7.606	2.458	2.494	u
3.4	5.174	2.076	2.168	u
3.6	3.731	1.759	1.873	u
3.8	2.774	1.489	1.601	u
4.0	2.095	1.252	1.349	u
4.2	1.586	1.041	1.115	u
4.4	1.192	0.8490	0.9003	1.0353
4.6	0.8767	0.6725	0.7038	0.7439
4.8	0.6195	0.5083	0.5244	0.5364
5.0	0.4054	0.3538	0.3603	0.3630
5.2	0.2244	0.2074	0.2089	0.2093
5.4	0.06943	0.06769	0.06775	0.06776

Table 11. Table 11: Values of α I ​ R , Δ f p subscript 𝛼 𝐼 𝑅 superscript subscript Δ 𝑓 𝑝 \alpha_{IR,\Delta_{f}^{p}} with 1 ≤ p ≤ 4 1 𝑝 4 1\leq p\leq 4 for N c = 2 , 3 , 4 subscript 𝑁 𝑐 2 3 4 N_{c}=2,\ 3,\ 4 and R = F 𝑅 𝐹 R=F , as functions of N f ∈ I I ​ R ​ Z subscript 𝑁 𝑓 subscript 𝐼 𝐼 𝑅 𝑍 N_{f}\in I_{IRZ} , together with α I ​ R , 2 ​ ℓ subscript 𝛼 𝐼 𝑅 2 ℓ \alpha_{IR,2\ell} and MS ¯ ¯ MS \overline{\rm MS} values of n 𝑛 n -loop α I ​ R , n ​ ℓ subscript 𝛼 𝐼 𝑅 𝑛 ℓ \alpha_{IR,n\ell} with 3 ≤ n ≤ 4 3 𝑛 4 3\leq n\leq 4 from bvh , for comparison.

$N_c$	$N_f$	${\alpha }_{IR,2\ell }$	${\alpha }_{IR,3\ell }$	${\alpha }_{IR,4\ell }$	${\alpha }_{IR,{\Delta }_f}$	${\alpha }_{IR,{\Delta }_f^2}$	${\alpha }_{IR,{\Delta }_f^3}$	${\alpha }_{IR,{\Delta }_f^4}$
2	6	11.42	1.645	2.395	0.941	1.098	1.979	1.951
2	7	2.83	1.05	1.21	0.753	0.853	1.305	1.293
2	8	1.26	0.688	0.760	0.565	0.621	0.8115	0.808
2	9	0.595	0.418	0.444	0.377	0.402	0.458	0.457
2	10	0.231	0.196	0.200	0.188	0.1945	0.202	0.2015
3	9	5.24	1.028	1.072	0.587	0.712	1.19	1.26
3	10	2.21	0.764	0.815	0.509	0.603	0.913	0.952
3	11	1.23	0.578	0.626	0.431	0.498	0.686	0.706
3	12	0.754	0.435	0.470	0.352	0.397	0.500	0.509
3	13	0.468	0.317	0.337	0.274	0.301	0.350	0.353
3	14	0.278	0.215	0.224	0.196	0.210	0.227	0.228
3	15	0.143	0.123	0.126	0.117	0.122	0.126	0.126
3	16	0.0416	0.0397	0.0398	0.0391	0.0397	0.0398	0.0398
4	11	14.00	0.972	0.943	0.474	0.592	1.042	1.1475
4	12	3.54	0.754	0.759	0.431	0.528	0.867	0.939
4	13	1.85	0.6035	0.628	0.388	0.467	0.713	0.7605
4	14	1.16	0.489	0.521	0.345	0.407	0.580	0.610
4	15	0.783	0.397	0.428	0.3015	0.349	0.465	0.483
4	16	0.546	0.320	0.345	0.258	0.294	0.367	0.376
4	17	0.384	0.254	0.271	0.215	0.240	0.282	0.2865
4	18	0.266	0.194	0.205	0.172	0.188	0.210	0.211
4	19	0.175	0.140	0.145	0.129	0.138	0.147	0.148
4	20	0.105	0.091	0.092	0.0861	0.09005	0.0928	0.0929
4	21	0.0472	0.044	0.044	0.0431	0.04405	0.0444	0.0444

Equations

$${\Delta }_f=N_u-N_f=\frac{11C_A}{4T_f}-N_f,$$

$${\gamma }_{\bar{\psi }\psi ,IR}=\sum _{j=1}^{\infty }{\kappa }_j{\Delta }_f^j$$

$${\beta }_{IR}^{\prime }=\sum _{j=1}^{\infty }{d}_j{\Delta }_f^j,$$

$$\beta=-2\alpha\sum_{\ell=1}^{\infty}b_{\ell}\,\Big{(}\frac{\alpha}{4\pi}\Big{)}^{\ell}\ .$$

$$D_{\mathcal{O}}=D_{\mathcal{O},free}-{\gamma }_{\mathcal{O}}.$$

$$\gamma_{\bar{\psi}\psi}=\sum_{\ell=1}^{\infty}c_{\ell}\Big{(}\frac{\alpha}{4\pi}\Big{)}^{\ell}\ ,$$

$$N_u=\frac{11C_A}{4T_f},$$

$${\alpha }_{IR,2\ell }=-\frac{4\pi b_1}{b_2}.$$

$$N_{\ell }=\frac{17C_A^2}{2T_f(5C_A+3C_f)}.$$

$$I_{IRZ}:N_{\ell }<N_f<N_u.$$

$$G_{fl}=\mathrm{SU}(N_f{)}_L\otimes \mathrm{SU}(N_f{)}_R\otimes \mathrm{U}(1{)}_V.$$

$${\alpha }_{IR}\equiv 4\pi a_{IR}=4\pi \sum _{j=1}^{\infty }{a}_j{\Delta }_f^j.$$

$$({\Delta }_f{)}_{max}=N_u-N_{\ell }$$

$$=$$

$$f(z)=\sum _{j=1}^{\infty }{f}_j{z}^j,$$

$$z_c=\underset{j\to \infty }{\lim }\frac{|f_{j-1}|}{|f_j|}.$$

$${\gamma }_{\bar{\psi }\psi ,IR}\le 2.$$

$$N_{u,SGT}=\frac{3C_A}{2T_f},$$

$$N_{\ell ,SGT}=\frac{3C_A}{4T_f}=\frac{N_u}{2},$$

$${\gamma }_{\bar{\psi }\psi ,IR,SGT}$$

$$=$$

$$D=7C_A+11C_f,$$

$${\kappa }_1=\frac{8C_f{T}_f}{C_{A}D},$$

$${\kappa }_2=\frac{4C_f{T}_f^2(5C_A+88C_f)(7C_A+4C_f)}{3C_A^2{D}^3},$$

$${\kappa }_3$$

$$-$$

$$+$$

$$\displaystyle\kappa_{4}=\frac{T_{f}^{2}}{3^{5}C_{A}^{5}D^{7}}\Bigg{[}C_{A}C_{f}T_{f}^{2}\bigg{(}19515671C_{A}^{6}-131455044C_{A}^{5}C_{f}+1289299872C_{A}^{4}C_{f}^{2}+2660221312C_{A}^{3}C_{f}^{3}$$

$$+$$

$$-$$

$$+$$

$$-$$

$$+$$

Full text

I Introduction

An important advance in the understanding of quantum field theory was the realization that the properties of a theory depend on the Euclidean energy/momentum scale $\mu$ at which they are measured. This is of particular interest in an asymptotically free non-Abelian gauge theory, in which the running gauge coupling $g(\mu)$ and the associated quantity $\alpha(\mu)=g(\mu)^{2}/(4\pi)$ approach zero at large $\mu$ in the deep ultraviolet (UV). We shall consider a theory of this type, with gauge group $G$ and $N_{f}$ massless fermions $\psi_{j}$, $j=1,...,N_{f}$, in a representation $R$ of $G$. The dependence of $\alpha(\mu)$ on $\mu$ is described by the renormalization-group (RG) rg beta function, $\beta=d\alpha(\mu)/dt$, where $dt=d\ln\mu$. The condition that the theory be asymptotically free implies that $N_{f}$ be less than a certain value, $N_{u}$, given below in Eq. (7). Since $\alpha(\mu)$ is small at large $\mu$, one can self-consistently calculate $\beta$ as a power series in $\alpha(\mu)$. As $\mu$ decreases from large values in the UV to small values in the infrared (IR), $\alpha(\mu)$ increases. A situation of special interest occurs if the beta function has a zero at some value away from the origin. For a given $G$ and $R$, this can happen for sufficiently large $N_{f}$, while still in the asymptotically free regime. In this case, as $\mu$ decreases from large values in the UV toward $\mu=0$ in the IR, the coupling increases, but approaches the value of $\alpha$ at this zero in the beta function, which is thus denoted $\alpha_{IR}$. Since $\beta=0$ at $\alpha=\alpha_{IR}$, the resultant theory in this IR limit is scale-invariant, and generically also conformally invariant scalecon ; fm . A fundamental question concerns the properties of the interacting theory at such an IR fixed point (IRFP) of the renormalization group. There is convincing evidence that if $\alpha_{IR}$ is small enough, then the IR theory is in a (deconfined) non-Abelian Coulomb phase (NACP), also called the conformal window nacp_comment . In terms of $N_{f}$, this phase occurs if $N_{f}$ is in the interval $N_{f,cr}<N_{f}<N_{u}$, where $N_{u}$ and $N_{f,cr}$ depend on $G$ and $R$. Here, $N_{f,cr}$ denotes the value of $N_{f}$ below which the running $\alpha(\mu)$ becomes large enough to cause spontaneous chiral symmetry breaking and dynamical fermion mass generation.

Physical quantities in the IR-limit theory at $\alpha_{IR}$ cannot depend on the scheme used for the regularization and subtraction procedure in renormalization. In conventional computations of these quantities, first, one expresses them as series expansions in powers of $\alpha$, calculated to $n$-loop order; second, one computes the IR zero of the beta function at the $n$-loop ($n\ell$) level, denoted $\alpha_{IR,n\ell}$; and third, one sets $\alpha=\alpha_{IR,n\ell}$ in the series expansion for the given quantity to obtain its value at the IR zero of the beta function to this $n$-loop order. However, these conventional series expansions in powers of $\alpha$, calculated to a finite order, are scheme-dependent beyond the leading one or two terms. Specifically, the terms in the beta function are scheme-dependent at loop order $\ell\geq 3$ and the terms in an anomalous dimension are scheme-dependent at loop order $\ell\geq 2$ gross75 . Indeed, as is well-known, the presence of scheme-dependence in higher-order perturbative calculations is a general property in quantum field theory.

It is therefore of great value to use a complementary approach in which one expresses these physical quantities at $\alpha_{IR}$ as an expansion in powers of a variable such that, at every order in this expansion, the result is scheme-independent. A very important property is that one can recast the expressions for physical quantities in a manner that is scheme-independent. A crucial point here is that, for a given gauge group $G$ and fermion representation $R$, as $N_{f}$ (formally generalized from non-negative integers to the real numbers) approaches the upper limit allowed by asymptotic freedom, denoted $N_{u}$ (given by Eq. (7) below), the resultant value of $\alpha_{IR}$ approaches zero. This means that one can equivalently express a physical quantity in a scheme-independent manner as a series in powers of the variable

[TABLE]

where $C_{A}$ is the quadratic Casimir invariant for the adjoint representation, and $T_{f}$ is the trace invariant for the fermion representation $R$ casimir . Here, $\alpha_{IR}\to 0\ \Longleftrightarrow\ \Delta_{f}\to 0$. Hence, for $N_{f}$ less than, but close to $N_{u}$, this expansion variable $\Delta_{f}$ is reasonably small, and one can envision reliable perturbative calculations of physical quantities at this IR fixed point in powers of $\Delta_{f}$. Following the original calculations of the one- and two-loop coefficients of the beta function b1 -gw2 , some early work on this was reported in bz ; grunberg92 .

In this paper we consider a vectorial, asymptotically free gauge theory and present scheme-independent calculations, for a general gauge group $G$ and fermion representation $R$, of two physical quantities in the IR theory at $\alpha_{IR}$ of considerable importance, namely (i) the anomalous dimension, denoted $\gamma_{\bar{\psi}\psi,IR}$, of the (gauge-invariant) fermion bilinear $\bar{\psi}\psi=\sum_{j=1}^{N_{f}}\bar{\psi}_{j}\psi_{j}$ to $O(\Delta_{f}^{4})$ and (ii) the derivative $\beta^{\prime}_{IR}=d\beta/d\alpha$ to $O(\Delta_{f}^{5})$, both evaluated at $\alpha=\alpha_{IR}$. These are the highest orders in powers of $\Delta_{f}$ to which these quantities have been calculated. We give explicit expressions for these quantities in the special cases where $G={\rm SU}(N_{c})$ and the fermion representation $R$ is the fundamental ($F$), adjoint ($adj$), and symmetric and antisymmetric rank-2 tensors, ($S_{2}$,  $A_{2}$). Our results extend our previous scheme-independent calculations of $\gamma_{\bar{\psi}\psi,IR}$ to $O(\Delta_{f}^{3})$ in gtr and of the derivative $\beta^{\prime}_{IR}$ to $O(\Delta_{f}^{4})$ in dex for general $G$ and $R$, and our scheme-independent calculation of $\gamma_{\bar{\psi}\psi,IR}$ to $O(\Delta_{f}^{4})$ for $G={\rm SU}(3)$ and $R=F$ in gsi (see also flir ). A brief report on some of our results was given in dexs .

Scheme-independent series expansions of $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ can be written as

[TABLE]

and

[TABLE]

where $d_{1}=0$ for all $G$ and $R$ gtr ; gsi ; dex . In general, the calculation of the coefficient $\kappa_{j}$ in Eq. (2) requires, as inputs, the values of the $b_{\ell}$ for $1\leq\ell\leq j+1$ and the $c_{\ell}$ for $1\leq\ell\leq j$. The calculation of the coefficient $d_{j}$ in Eq. (3) requires, as inputs the values of the $b_{\ell}$ for $1\leq\ell\leq j$. We refer the reader to gtr and dex for discussions of the procedure for calculating the coefficients $\kappa_{j}$ and $d_{j}$. We denote the truncation of these series to maximal power $j=p$ as $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$ and $\beta^{\prime}_{IR,\Delta_{f}^{p}}$, respectively. Where it is necessary for clarity, we will also indicate the fermion representation $R$ in the subscript.

Our main new results here include the general expressions, for arbitrary gauge group $G$ and fermion representation $R$, for the coefficient, $\kappa_{4}$ in Eq. (48) below, and for the coefficient $d_{5}$, given in Eq. (252) below, as well as reductions of these formulas for special cases and, for $R=F$, calculations in the LNN limit (90). As will be discussed further below, the derivative $\beta^{\prime}_{IR}$ is equivalent to the anomalous dimension of the non-Abelian field strength squared, ${\rm Tr}(F_{\mu\nu}F^{\mu\nu})$. Our present calculations make use of the newly computed five-loop coefficient in the beta function for this gauge theory for general $G$ and $R$ in b5 , as our work in flir ; gsi made use of the calculation of this five-loop coefficient for the case $G={\rm SU}(3)$ and $R=F$ in b5su3 .

In addition to being of interest and value in their own right, our new scheme-independent calculations, performed to the highest order yet achieved, are useful in several ways. First, we will compare our results for $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ for various $G$ and $R$ with the values that we obtained at comparable order with the conventional $n$-loop approach in bvh -lnn . Our new results have the merit of being scheme-independent at each order in $\Delta_{f}$, in contrast to scheme-dependent series expansions of $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ in powers of the IR coupling. Second, there is, at present, an intensive program to study this IR behavior on the lattice lgtreviews . Thus, it is of considerable interest to compare our scheme-independent results for $\gamma_{\bar{\psi}\psi,IR}$ for various theories with values measured in lattice simulations of these theories. We have done this in gsi ; dex ; dexs (as well as in our work on conventional $n$-loop calculations bvh ; flir ), and we will expand upon this comparison here. Third, we believe that our scheme-independent expansions for these physical quantities are of interest in the context of the great current resurgence of research activity on conformal field theories (CFT). Much of this current activity makes use of operator-product expansions and the associated bootstrap approach cft_bootstrap . Our method of scheme-independent series expansions for physical quantities at an IR fixed point is complementary to this bootstrap approach in yielding information about a conformal field theory.

Our calculations rely on $\alpha_{IR}$ being an exact zero of the beta function and thus an exact IR fixed point of the renormalization group, and this property holds in the non-Abelian Couloumb phase (conformal window). In this phase, the chiral symmetry associated with the massless fermions is preserved in the presence of the gauge interaction. However, there has also been interest in vectorial asymptotically free gauge theories that exhibit quasi-conformal behavior associated with an approximate IRFP in the phase with broken chiral symmetry, which could feature a substantial value of an effective $\gamma_{\bar{\psi}\psi,IR}\sim O(1)$ wtc . Our scheme-independent calculations are also relevant to this area of research in two ways: (i) if N_{f}\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}N_{f,cr}, then the effective values of quantities such as $\gamma_{\bar{\psi}\psi,IR}$ may be close to the values calculated via the $\Delta_{f}$ expansion from within the NACP; (ii) combining our calculations of $\gamma_{\bar{\psi}\psi,IR}$ with an upper bound on this anomalous dimension from conformal invariance and an assumption that this bound is saturated as $N_{f}\searrow N_{f,cr}$ yields an estimate of the value of $N_{f,cr}$. This is useful, since the value of $N_{f,cr}$ for a given $G$ and $R$ is not known exactly at present and is the subject of current investigation, including lattice studies, as discussed further below.

Although most of our paper deals with new scheme-independent results for physical quantities, one of the ouputs of our calculations is a new type of series expansion for a scheme-dependent quantity, namely $\alpha_{IR}$. The conventional procedure for calculating the IR zero of a beta function at the $n$-loop order, which we have applied in earlier work to four-loop order for arbitrary $G$ and $R$ bvh -lnn (see also ps ) is to examine the $n$-loop beta function, which has the form of $\alpha^{2}$ times a polynomial of degree $n-1$ in $\alpha$, and then determine the $n$-loop value $\alpha_{IR,n\ell}$ as the (real, positive) root of this polynomial closest to the origin. However, in flir , we investigated the five-loop beta function for $G={\rm SU}(3)$ and $R=F$, as calculated in the standard $\overline{\rm MS}$ scheme, and found that, over a substantial range of values of $N_{f}$ in the non-Abelian Coulomb phase, it does not have any positive real root. We were able to circumvent this problem in flir by the use of Padé approximants, but nevertheless, it is a complication for this conventional approach to calculating $\alpha_{IR}$. The new calculation of $\alpha_{IR}$ as an expansion in powers of $\Delta_{f}$ up to $O(\Delta_{f}^{4})$ for general $G$ and $R$ that we present here has the advantage that it always yields a physical value, in contrast to the situation with the $n$-loop beta function.

The paper is organized as follows. Some relevant background and methods are discussed in Section II. We present our calculation of $\kappa_{4}$ in the scheme-independent expansion of $\gamma_{\bar{\psi}\psi,IR}$ for general $G$ and $R$ in Section III, together with evaluations for $G={\rm SU}(N_{c})$ and $R=F,\ adj,\ S_{2}$, and $A_{2}$. These are compared with values from $n$-loop calculations and with lattice measurements. In this section we also present results for case $R=F$ in the limit $N_{c}\to\infty,\ N_{f}\to\infty$, with $N_{f}/N_{c}$ fixed, which we call the LNN limit. In Section IV we present our calculation of the coefficient $d_{5}$ in the scheme-independent expansion of $\beta^{\prime}_{IR}$ for general $G$ and $R$, with evaluations for the above-mentioned specific representations. Section V gives an analysis of the five-loop rescaled beta function in the LNN limit and a determination of the interval over which it exhibits a physical IR zero. Section VI is devoted to the calculation of the coefficients in an expansion of $\alpha_{IR}$ in powers of $\Delta_{f}$ up to $O(\Delta_{f}^{4})$. Our conclusions are given in Section VII, and some auxiliary formulas are listed in an appendix.

II Background and Methods

In this section we review some background and methods relevant for our calculations. The series expansion of $\beta$ in powers of $\alpha$ is

[TABLE]

where $b_{\ell}$ is the $\ell$-loop coefficient. For a general operator ${\cal O}$, we denote the full scaling dimension as $D_{\cal O}$ and its free-field value as $D_{{\cal O},free}$. The anomalous dimension of this operator, denoted $\gamma_{\cal O}$, is defined via the relation gammaconvention

[TABLE]

An operator of particular interest is the (gauge-invariant) fermion bilinear, $\bar{\psi}\psi$. The expansion of the anomalous dimension of this operator, $\gamma_{\bar{\psi}\psi}$, in powers of $\alpha$ is

[TABLE]

where $c_{\ell}$ is the $\ell$-loop coefficient. As noted above, the coefficients $b_{1}$, $b_{2}$, and $c_{1}$ are scheme-independent, while the $b_{\ell}$ with $\ell\geq 3$ and the $c_{\ell}$ with $\ell\geq 2$ are scheme-dependent gross75 . For a general gauge group $G$ and fermion representation $R$, the coefficients $b_{1}$ and $b_{2}$ were calculated in b1 and b2 , and $b_{3}$ and $b_{4}$ were calculated in b3 and b4 (and checked in b4p ) in the commonly used $\overline{\rm MS}$ scheme msbar . For $G={\rm SU}(3)$ and $R=F$, $b_{5}$ was calculated in b5su3 and recently, an impressive calculation of $b_{5}$ for general gauge group $G$ and fermion representation $R$ was presented in b5 , again in the $\overline{\rm MS}$ scheme. We also make use of the $c_{\ell}$ up to loop order $\ell=4$, calculated in c4 . Although we use these coefficients as calculated in the $\overline{\rm MS}$ scheme below, we emphasize that the main results of this paper are calculations of the quantities $\kappa_{4}$ and $d_{5}$ which, like all of the $\kappa_{j}$ and $d_{j}$, are scheme-independent. We denote the $n$-loop $\beta$, $\beta^{\prime}$, and $\gamma_{\bar{\psi}\psi}$ as $\beta_{n\ell}$, $\beta^{\prime}_{n\ell}$, and $\gamma_{\bar{\psi}\psi,n\ell}$. As discussed above, we denote the IR zero of $\beta_{n\ell}$ as $\alpha_{IR,n\ell}$, and the corresponding evaluations of $\beta^{\prime}_{n\ell}$ and $\gamma_{\bar{\psi}\psi,n\ell}$ at $\alpha_{IR,n\ell}$ as $\beta^{\prime}_{IR,n\ell}$ and $\gamma_{\bar{\psi}\psi,IR,n\ell}$. The symbols $\alpha_{IR}$, $\gamma_{\bar{\psi}\psi,IR}$, and $\beta^{\prime}_{IR}$ refer to the exact values of these quantities.

For a given $G$ and $R$, as $N_{f}$ increases, $b_{1}$ decreases through positive values and vanishes with sign reversal at $N_{f}=N_{u}$, with

[TABLE]

where $C_{A}$ and $T_{f}$ are group invariants casimir ; nfintegral . Hence, the asymptotic freedom condition yields the upper bound $N_{f}<N_{u}$.

There is a range of $N_{f}<N_{u}$ where $b_{2}<0$, so the two-loop beta function has an IR zero, at the value

[TABLE]

The $n$-loop beta function has a double UV zero at $\alpha=0$ and $n-1$ zeros away from the origin. Among the latter zeros of the beta function, the smallest (real, positive) zero, if there is such a zero, is the physical IR zero, $\alpha_{IR,n\ell}$, of $\beta_{n\ell}$. As $N_{f}$ decreases below $N_{u}$, $b_{2}$ passes through zero to positive values as $N_{f}$ decreases through

[TABLE]

Hence, with $N_{f}$ formally extended from nonnegative integers to nonnegative real numbers nfintegral , $\beta_{2\ell}$ has an IR zero (IRZ) for $N_{f}$ in the interval

[TABLE]

Thus, $N_{\ell}$ is the lower ($\ell$) end of this interval phasenote

As $N_{f}$ decreases in this interval, $\alpha_{IR,2\ell}$ increases. Therefore, in order to investigate the IR zero of the beta function for $N_{f}$ toward the middle and lower part of $I_{IRZ}$ with reasonable accuracy, one requires higher-loop calculations. These were performed in gkgg ; elias , bvh -lnn , ps ,flir for $\alpha_{IR,n\ell}$ and for the anomalous dimension of the fermion bilinear operator (see also bfs ; bfss ). Since the $b_{\ell}$ with $\ell\geq 3$ are scheme-dependent, it is necessary to determine the degree of sensitivity of the value obtained for $\alpha_{IR,n\ell}$ for $n\geq 3$ to the scheme used for the calculation. This was done in sch -gracey2015 .

The nonanomalous global flavor symmetry of the theory is

[TABLE]

This $G_{fl}$ symmetry is preserved in the (deconfined) non-Abelian Coulomb phase. As in gtr ; flir ; gsi ; dex ; dexs , we focus on this phase in the present work, since both the expansion in a small $\alpha_{IR}$ and the scheme-independent expansion in powers of $\Delta_{f}$ start from the upper end of the interval $I_{IRZ}$ in this phase. In contrast, in the phase with confinement and spontaneous chiral symmetry breaking, the gauge interaction produces a bilinear fermion condensate, $\langle\bar{\psi}\psi\rangle$, and this breaks $G_{fl}$ to ${\rm SU}(N_{f})_{V}\otimes{\rm U}(1)_{V}$, where ${\rm SU}(N_{f})_{V}$ is the diagonal subgroup of ${\rm SU}(N_{f})_{L}\otimes{\rm SU}(N_{f})_{R}$.

We will consider the flavor-nonsinglet ($fns$) and flavor-singlet ($fs$) bilinear fermion operators $\sum_{j,k=1}^{N_{f}}\bar{\psi}_{j}(T_{a})_{jk}\psi_{k}$ and $\sum_{j=1}^{N_{f}}\bar{\psi}_{j}\psi_{j}$, where here $T_{a}$ with $a=1,...,N_{f}^{2}-1$ is an generator of the global flavor group SU($N_{f}$). We will usually suppress the explicit flavor indices and thus write these operators as $\bar{\psi}T_{a}\psi$ and $\bar{\psi}\psi$. These have the same anomalous dimension (e.g., gracey_gammatensor ), which we denote simply as the anomalous dimension for the flavor-singlet operator, $\gamma_{\bar{\psi}\psi}$. In vectorial gauge theories of the type considered here, these fermion bilinear operators are gauge-invariant, and hence the anomalous dimension $\gamma_{\bar{\psi}\psi}$ and its IR value, $\gamma_{\bar{\psi}\psi,IR}$, are physical. (In contrast, in a chiral gauge theory, fermion bilinears are generically not gauge-invariant, and hence neither are their anomalous dimensions.)

Since $\alpha_{IR}$ vanishes (linearly) with $\Delta_{f}$ as $\Delta_{f}\to 0$, we can express it as a series expansion in this variable, $\Delta_{f}$. We thus write

[TABLE]

The calculation of the $a_{j}$ requires, as input, the $b_{\ell}$ with $1\leq\ell\leq j+1$ gtr ; dex .

A basic question concerns the part of the interval $I_{IRZ}$ in which the series expansions for $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ in Eqs. (2) and (3) are reliable. We analyzed this question in gtr ; gsi ; dex ; dexs and concluded that these expansions for $\gamma_{IR}$ and $\beta^{\prime}_{IR}$ should be reasonably reliable throughout much of the interval $I_{IRZ}$ and non-Abelian Coulomb phase. We will use our higher-order calculations in this paper to extend this analysis here. We recall that the properties of the theory change qualitatively as $N_{f}$ decreases through the value $N_{f,cr}$ and spontaneous chiral symmetry breaking occurs, with the fermions gaining dynamical masses. The (chirally symmetric) non-Abelian Coulomb phase with $N_{f,cr}<N_{f}<N_{u}$ is clearly qualitatively different from the confined phase with spontaneous chiral symmetry breaking at smaller $N_{f}$ below $N_{f,cr}$. Therefore, one does not, in general, expect the small-$\Delta_{f}$ series expansion to hold below $N_{f,cr}$. Estimating the range of applicability of this expansion is thus connected with estimating the value of $N_{f,cr}$. For general $G$ and $R$, as $N_{f}$, formally continued from the nonnegative integers to the nonnegative real numbers, decreases from the upper end of the interval $I_{IRZ}$ at $N_{u}$ to the lower end of this interval at $N_{f}=N_{\ell}$, $\Delta_{f}$ increases from 0 to the maximal value

[TABLE]

Recall that for a function $f(z)$ that is analytic about $z=0$ and has a Taylor series expansion

[TABLE]

the radius of convergence of this series, $z_{c}$, can be determined by the ratio test

[TABLE]

Of course, we cannot apply the full ratio test here, since we have only calculated the $\kappa_{j}$ and $d_{j}$ to finite order. However, we can get a rough measure of the range of applicability of the series expansions in $\Delta_{f}$ (and also $\Delta_{r}$ in the LNN limit lnn discussed below) by computing the ratios $\kappa_{j-1}/\kappa_{j}$ and $d_{j-1}/d_{j}$ for the values of $j$ for which we have calculated these coefficients.

The series expansion (2) for $\gamma_{IR}$ starts at $\Delta_{f}=0$, i.e., at the upper end of the non-Abelian Coulomb phase, and extends downward through this phase. Given that the theory at $\alpha_{IR}$ in this phase is conformal, there is an upper bound from conformal invariance, namely gammabound

[TABLE]

We have used this in our earlier work bvh ; gtr ; flir ; gsi ; dex ; dexs and we will apply it with our higher-order calculations here. As discussed in bvh , in the phase with spontaneous chiral symmetry breaking (S$\chi$SB), there is a similar upper bound, $\gamma_{\bar{\psi}\psi,IR}<2$. This follows from the requirement that if $m(k)$ is the momentum-dependent running dynamical mass generated in association with the S$\chi$SB, then $\lim_{k\to\infty}m(k)=0$ (see Eqs. (4.1)-(4.2) of bvh ). Thus, if the approximate calculation of the anomalous dimension of a given quantity at a fixed value of $\Delta_{f}$, computed up to order $\Delta_{f}^{p}$, yields a value greater than 2, then we can infer that the perturbative calculation is not applicable at this value of $\Delta_{f}$ or equivalently, $N_{f}$.

In particular, this can give information on the extent of the non-Abelian Coulomb phase and the value of $N_{f,cr}$. The application of this bound is particularly powerful in the context of our present scheme-independent calculations because we find that the $\kappa_{j}$ in Eq. (2) are positive for all of the representations considered here, and hence, for a given $p$, $\gamma_{IR,\Delta_{f}^{p}}$ is a monotonically increasing function of $\Delta_{f}$ or equivalently it increases monotonically as $N_{f}$ decreases from its upper limit, $N_{u}$. If one assumes that $\gamma_{IR}$ saturates its upper bound, (18) and if a calculation of $\gamma_{IR}$ is reliable in the regime where it is approaching 2 from below, then one can, in principle, determine the value of $N_{f,cr}$, where $\gamma_{IR}$ reaches this upper bound after approaching it from below. In this context, it should be mentioned that in a supersymmetric (vectorial) gauge theory (SGT) with $N_{f}$ pairs of massless chiral superfields transforming according the representations $R$ and $\bar{R}$ of a gauge group $G$, the exact expression for $\gamma_{IR}$ is known nsvz ; seiberg , and (i) it increases monotonically with decreasing $N_{f}$ in the NACP; and (ii) it saturates its upper bound (which, in the SGT case is $\gamma_{IR,SGT}\leq 1$) at the lower end of the non-Abelian Coulomb phase. Specifically, in this supersymmetri gauge theory, the upper and lower ends of the NACP occur at nfintegral

[TABLE]

and

[TABLE]

and

[TABLE]

Thus, $\gamma_{\bar{\psi}\psi,IR,SGT}$ increases from 0 to 1 as $N_{f}$ decreases from $N_{u,SGT}$ to $N_{\ell,SGT}$. However, it is not known if this saturation occurs in the non-supersymmetric case. In practice, we are only able to apply this test in an approximate manner because for a given $G$ and $R$, as $N_{f}$ decreases toward the lower part of $I_{IRZ}$, the ratio test already shows that higher-order terms in the $\Delta_{f}$ expansion are becoming increasingly non-negligible, so that the truncation of the infinite series (2) to maximal power $p=4$ involves an increasingly great uncertainty, as does an extrapolation to $p=\infty$.

For some perspective, we note that in order to asses the accuracy of the $\Delta_{f}$ expansion, the coefficients $\kappa_{j,SGT}$ were calculated for $j=1,\ 2$ in gtr and were found to be in perfect agreement with the corresponding Taylor series expansion of the exact expression (23). This check was carried to one higher order in dexs for the case $G={\rm SU}(N_{c})$ and $R=F$ with a calculation of $\gamma_{IR,SGT,\Delta_{f}^{3}}$, and again, perfect agreement was found with the exact result. This agreement explicitly demonstrated the scheme independence of the $\kappa_{j,SGT}$, since the calculations were carried out using inputs computed in the $\overline{DR}$ scheme, while (23) was derived in the NSVZ scheme nsvz . Furthermore, as a consequence of electric-magnetic duality seiberg , as $N_{f}\searrow N_{\ell,SGT}$ in the non-Abelian Coulomb phase, the physics is described by a magnetic theory with coupling strength going to zero, or equivalently, by an electric theory with divergent $\alpha_{IR}$. Therefore, this perfect agreement, order-by-order, between the $\kappa_{j,SGT}$ and the expansion of the exact expression (23) for $\gamma_{IR,SGT}$ in powers of $\Delta_{f}$, showed that the $\Delta_{f}$ expansion in this supersymmetric gauge theory is able to treat situations with strong, as well as weak, coupling. This could not be done with conventional perturbative series expansions in powers of $\alpha$ bfs ; bfss .

III Calculation of $\gamma_{\bar{\psi}\psi,IR}$ to $O(\Delta_{f}^{4})$

III.1 General $G$ and $R$

The coefficients $\kappa_{j}$ in the scheme-independent expansion of $\gamma_{\bar{\psi}\psi,IR}$ in powers of $\Delta_{f}$, Eq. (2), contain important information about the theory. For a general asymptotically free vectorial gauge theory with gauge group $G$ and $N_{f}$ massless fermions in a representation $R$, the coefficients $\kappa_{j}$ were given in gtr up to order $j=3$, yielding the expansion of $\gamma_{\bar{\psi}\psi,IR}$ to order $\Delta_{f}^{3}$. It is convenient to define

[TABLE]

since this factor occurs repeatedly in denominators of various expressions. For reference, we list the $\kappa_{j}$ for $1\leq j\leq 3$ below:

[TABLE]

[TABLE]

and

[TABLE]

Here, $\zeta_{s}=\sum_{n=1}^{\infty}n^{-s}$ is the Riemann zeta function, the quantities $C_{A}$, $C_{f}$, and $T_{f}$ are group invariants, the contractions $d_{A}^{abcd}d_{A}^{abcd}$, $d_{R}^{abcd}d_{A}^{abcd}$, $d_{R}^{abcd}d_{R}^{abcd}$ are additional group-theoretic quantities given in b4 , and $d_{A}$ is the dimension of the adjoint representation of $G$. In gtr ; dex , the expression for $\kappa_{3}$ was given with terms written in order of descending powers of $C_{A}$. It is also useful to express this coefficient $\kappa_{3}$ in an equivalent form that renders certain factors of $D$ explicit and shows the simple factorization of terms multiplying $\zeta_{3}$, and we have done this in Eq. (31).

Our new result here for $\kappa_{4}$ for a general gauge group $G$ and fermion representation $R$ is

[TABLE]

Here, $d_{R}$ is the dimension of the fermion representation $R$. As before, we have indicated the simple factors in the prefactor and, for sufficiently simple cases, also factorizations of numbers in numerator terms. We will follow the same format for indicating numerical factorizations below. We proceed to evaluate this general expression for the gauge group $G={\rm SU}(N_{c})$ and several specific fermion representations $R$, namely the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor. As stated in the introduction, we will use the abbreviations $F$, $adj$, $S_{2}$, and $A_{2}$ to refer to these representations. It is also worthwhile to evaluate our general formulas for other gauge groups and their representations, including orthogonal, symplectic, and exceptional groups. We will report these evaluations for other groups and their representations elsewhere. There has, indeed, been interest in conformal phases for theories with these other gauge groups othergroups .

The coefficients $\kappa_{1}$ and $\kappa_{2}$ are manifestly positive for all $G$ and $R$. For $G={\rm SU}(N_{c})$ with all physical $N_{c}$, and for representations $R=F,\ adj,\ S_{2}$, we have found that $\kappa_{3}$ and $\kappa_{4}$ are also positive gtr -dexs . As one of the results in the present paper, we generalize this further to include $R=A_{2}$. That is, for all physical $N_{c}$ and for all of these representations, we find that $\kappa_{j}>0$ for $j=3,\ 4$ as well as the manifestly positive cases $j=1,\ 2$. Thus, extending our previous discussion in gtr -dexs , the property that, for all of these representations $R$, $\kappa_{j}>0$ for $1\leq j\leq 4$ and for all $N_{c}$ implies two important monotonicity results: (i) for these $R$, and with a fixed $p$ in the interval $1\leq p\leq 4$, $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$ is a monotonically increasing function of $\Delta_{f}$, i.e., it increases monotonically with decreasing $N_{f}$; and (ii) for these $R$, and with a fixed $N_{f}\in I_{IRZ}$, $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$ is a monotonically increasing function of $p$ in the range $1\leq p\leq 4$. In addition to the manifestly positive $\kappa_{1}$ and $\kappa_{2}$, a plausible conjecture is that, for these $R$, $\kappa_{j}>0$ for all $j\geq 3$. Assuming that this conjecture is valid, then three consequences are that for these representations $R$, (iii) for fixed $N_{f}$, $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$ is a monotonically increasing function of $p$ for all $p$; (iv) $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$ is a monotonically increasing function of $\Delta_{f}$, i.e. it increases with decreasing $N_{f}$, for all $p$; and hence (v) (assuming that the infinite series (2) converges), the quantity $\gamma_{\bar{\psi}\psi,IR}$ defined by this infinite series, and equivalent to $\lim_{p\to\infty}\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$, is a monotonically increasing function of $\Delta_{f}$, i.e., it increases monotonically with decreasing $N_{f}$.

III.2 $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{4}}$ for

$G={\rm SU}(N_{c})$ and $R=F$

An important special case is $G={\rm SU}(N_{c})$ with $R$ being the fundamental representation. For this case, the general expression for the interval $I_{IRZ}$, Eq. (10), is nfintegral

[TABLE]

The factor $D$ in Eq. (24) has the explicit form

[TABLE]

The general results for $\kappa_{p}$ with $1\leq p\leq 3$ in (25)-(31) from gtr take the following forms given in dex :

[TABLE]

[TABLE]

and

[TABLE]

For $\kappa_{4,F}$, we have dexs

[TABLE]

We have checked that when we substitute the value $N_{c}=3$ in our expression for $\kappa_{4,F}$ in Eq. (62), the result agrees with our previous calculation of $\kappa_{4,F}$ for this case in Eq. (9) of Ref. gsi .

The explicit numerical expressions for the scheme-independent series expansions of $\gamma_{\bar{\psi}\psi,IR}$ to order $\Delta_{f}^{4}$ for $R=F$ and $N_{c}=2,\ 3,\ 4$ are as follows:

[TABLE]

[TABLE]

and

[TABLE]

In these equations,

[TABLE]

Plots of $\gamma_{\bar{\psi}\psi,IR,F,\Delta_{f}^{p}}$ for $N_{c}=2$ and $N_{c}=3$ and $1\leq p\leq 4$ were given in dexs . These showed the two monotonicity properties mentioned above. For an extended comparison, we show the plots of $\gamma_{\bar{\psi}\psi,IR,F,\Delta_{f}^{p}}$ for $2\leq N_{c}\leq 4$ and $1\leq p\leq 4$ in Figs. 1-3.

In Table 1 we list the values of $\gamma_{\bar{\psi}\psi,IR,F,\Delta_{f}^{p}}$ for $1\leq p\leq 4$ for the SU(2), SU(3), and SU(4) theories, with $N_{f}$ in the respective interval $I_{IRZ}$ for each. For comparison, we also include the values of $\gamma_{\bar{\psi}\psi,IR,n\ell}$ obtained with our earlier $n$-loop calculations in bvh , using series expansions in powers of $\alpha$ evaluated at $\alpha=\alpha_{IR,n\ell}$ for $1\leq n\leq 4$ with $b_{3}$ and $b_{4}$ and $c_{n}$, $2\leq n\leq 4$ calculated in the $\overline{\rm MS}$ scheme. (See Table VI in bvh for a list of numerical values of values of $\gamma_{\bar{\psi}\psi,IR,n\ell}$.) As discussed above, if, for a given $N_{c}$ and $N_{f}$, a calculated value of $\gamma_{\bar{\psi}\psi,IR}$ violates the upper bound $\gamma_{\bar{\psi}\psi,IR}\leq 2$ in (18), this is unphysical (marked with a symbol “u” in Table 1) and indicates that the perturbative calculation is unreliable and hence not applicable for this $N_{f}$. In the case of the $n$-loop values $\gamma_{IR,n\ell}$, if this occurs at the two-loop level, it also leads to caution concerning $\gamma_{IR,n\ell}$ for $n=3,\ 4$, and this is similarly indicated with a “u”. The computations of $\gamma_{IR,n\ell}$ in bvh ; ps made use of the $b_{n}$ and $c_{n}$ up to the $n=4$ loop level, where the scheme-dependent $b_{3}$, $b_{4}$, and $c_{n}$ with $2\leq n\leq 4$ had been calculated in the widely used $\overline{\rm MS}$ scheme b3 ; b4 ; b4p ; c4 . As we pointed out in flir , the five-loop beta function in the $\overline{\rm MS}$ scheme does not exhibit a physical IR zero over a substantial lower part of $I_{IRZ}$. We discuss this further below. For compact notation, we will often leave the subscript $\bar{\psi}\psi$ implicit on these and other quantities and thus write $\gamma_{\bar{\psi}\psi,IR}\equiv\gamma_{IR}$, $\gamma_{\bar{\psi}\psi,IR,n\ell}\equiv\gamma_{IR,n\ell}$, etc. From Eqs. (7) and (9) it follows that the respective lower and upper ends of the intervals $I_{IRZ}$ for these theories are $(N_{u},\,N_{\ell})=(5.55,\,11)$, (8.05,   16.5), and (10.61,   22) for SU(2), SU(3), and SU(4), and hence the physical intervals $I_{IRZ}$ are $6\leq N_{f}\leq 10$ for SU(2), $9\leq N_{f}\leq 16$ for SU(3), and $11\leq N_{f}\leq 21$ for SU(4).

Since the calculation of $\kappa_{j}$ and the resultant $\gamma_{IR,\Delta_{f}^{j}}$ uses information from the $(j+1)$-loop beta function from (4) and the $j$-loop expansion of $\gamma_{\bar{\psi}\psi}$ in (6), it is natural to compare the (SI) $\gamma_{IR,\Delta_{f}^{p}}$ with the (SD) $\gamma_{IR,p^{\prime}\ell}$ for $p^{\prime}=p$ and $p^{\prime}=p+1$. In the upper and middle part of the interval $I_{IRZ}$ for a given $N_{c}$, we find that $\gamma_{IR,\Delta_{f}^{4}}$ is slightly larger than $\gamma_{IR,4\ell}$, with the difference increasing as $N_{f}$ decreases below $N_{u}$, i.e., as $\Delta_{f}$ increases.

It is important to assess the range of applicability and reliability of these results from the $\Delta_{f}$ expansion. We did this in gtr ; gsi ; dex and extend our analysis here, using our new result for $\kappa_{4}$. Following our discussion above on the ratio test for the determination of the radius of convergence of a Taylor series, the ratios of successive coefficients, $\kappa_{j-1}/\kappa_{j}$, give an approximate measure of the range of applicability of the $\Delta_{f}$ expansion for $\gamma_{IR}$. For a given $G$ and $R$, this range may be compared with the maximum size of $\Delta_{f}$ in the interval $I_{IRZ}$ where the scheme-independent two-loop beta function $\beta_{2\ell}$ has an IR zero. For the present case of $G={\rm SU}(N_{c})$ and $R=F$, the general formula (15) takes the form

[TABLE]

This has the respective values

[TABLE]

We begin by reviewing the SU(3) theory, for which

[TABLE]

As discussed in gtr ; gsi ; dex , these results suggest that for the SU(3) theory with $R=F$, the $\Delta_{f}$ expansion calculated to this order should be reasonably reliable over a substantial part, including the upper and middle portions, of the interval $I_{IRZ}$ and the non-Abelian Coulomb phase.

Using our new results, we now extend this analysis to the SU(2) and SU(4) theories (and will give a further analysis in the LNN limit of Eq. (90)). We find

[TABLE]

and

[TABLE]

Since $(\Delta_{f})_{max}$ has the respective values 5.45 and 11.39 for the SU(2) and SU(4) theories, we are led to the same conclusion for these theories that we reached for the SU(3) theory, namely that the $\Delta_{f}$ expansion should be reasonably reliable over a substantial portion of the respective intervals $I_{IRZ}$.

As discussed above, another way to assess the range of applicability of the $\Delta_{f}$ expansion is to check to see whether the resultant values of $\gamma_{IR,\Delta_{f}^{p}}$ obey the upper bound $\gamma_{IR}\leq 2$ in (18). As is evident from Table 1, all of our values of $\gamma_{IR,\Delta_{f}^{p}}$ listed there obey this bound. This again shows the advantages of the scheme-independent $\Delta_{f}$ expansion as a way of calculating $\gamma_{IR}$ to a given order, as compared with the conventional $n$-loop calculation of $\gamma_{IR,n\ell}$. As is also evident from Table 1 for each of the cases listed there, namely $N_{c}=2,\ 3,\ 4$, one finds unphysically large values of $\gamma_{IR,n\ell}$ for values of $N_{f}$ in the lower portions of the respective intervals $I_{IRZ}$. In bvh and later works we explained this as a consequence of the fact that, for a given $G$ and $R$, as $N_{f}$ decreases toward $N_{\ell}$ in the interval $I_{IRZ}$, the coupling $\alpha_{IR}$ increases from weak toward strong coupling. Thus, toward the lower end of the respective intervals $I_{IRZ}$, the IR coupling $\alpha_{IR,n\ell}$ become too large for the perturbative $n$-loop calculations of $\gamma_{IR,n\ell}$ to be applicable. In contrast, the $\Delta_{f}$ expansion can be applied over a considerably greater portion of the interval $I_{IRZ}$ to yield results for $\gamma_{IR,\Delta_{f}^{p}}$ that obey the upper bound (18). We will show this further below for the LNN limit (90). This also demonstrates that the $\Delta_{f}$ expansion for $\gamma_{IR}$ is able to be used in situations with substantially stronger IR coupling than is the case with the conventional expansion in powers of this coupling yielding the $n$-loop value $\gamma_{IR,n\ell}$.

We proceed to compare our values in Table 1 with lattice measurements. The SU(3) theory with $R=F$ and $N_{f}=12$ has been the subject of many lattice measurements. In gsi , we compared our results for this theory with lattice measurements, so we only briefly review that discussion here. We recall that there is not, at present, a consensus among all lattice groups as to whether this theory is in an IR-conformal phase or is in a chirally broken phase lgtreviews . There is a considerable spread of values of $\gamma_{IR}$ in published papers, including the values (where uncertainties in the last digits are indicated in parentheses) $\gamma_{IR}\sim 0.414(16)$ lsd , $\gamma_{IR}\simeq 0.35$ degrand , $\gamma_{IR}\simeq 0.4$ latkmi , $\gamma_{IR}=0.27(3)$ ah1 , $\gamma_{IR}\simeq 0.25$ ah2 (see also ah3 ), $\gamma_{IR}=0.235(46)$ lmnp , and 0.2\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}\gamma_{IR}\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}0.4 kuti . We refer the reader to lgtreviews and lsd -kuti for discussions of estimates of overall uncertaintites in these measurements. Our value $\gamma_{IR,\Delta_{f}^{4}}=0.338$ and our extrapolated value for $\lim_{p\to\infty}\gamma_{IR,\Delta_{f}^{p}}=\gamma_{IR}$, namely $\gamma_{IR}=0.40$, are consistent with this range of lattice measurements and are somewhat higher than our five-loop value $\gamma_{IR,5\ell}=0.255$ from the conventional $\alpha$ series that we obtained in flir . It is hoped that further work by lattice groups will lead to a consensus concerning whether this theory is IR conformal or not and concerning the value of $\gamma_{IR}$.

The SU(3) theory with $N_{f}=10$ has been investigated on the lattice in lsdnf10 , with the result $\gamma_{IR}\sim 1$. While our highest-order $n$-loop values, namely our four-loop result, $\gamma_{IR,4\ell}=0.156$ bvh , and our five-loop result, $\gamma_{IR,5\ell}=0.211$ obtained using Padé methods flir , are smaller than this lattice value, our extrapolated scheme-independent value, $\gamma_{IR}=0.95\pm 0.06$ gsi , is consistent with it.

There have also been a number of lattice studies of the SU(3) theory with $N_{f}=8$ latkminf8 ; lsdnf8 ; appelquist_eft , which have yielded the estimate $\gamma_{IR}\simeq 1$. As is evident from Fig. 2, if we were to continue the curve for $\gamma_{IR,\Delta_{f}^{4}}$ plotted there downward further to $N_{f}=8$, the resultant value would be compatible with $\gamma_{IR}\sim 1$. We note that this theory may well be in the chirally broken phase, and there is not yet a clear consensus as to whether it is in this phase or possibly near the lower end of the IR-conformal non-Abelian Coulomb phase. In this context, one may recall that if, for a given $G$ and $R$, $N_{f}<N_{f,cr}$, so that there is spontaneous chiral symmetry breaking, then the IR zero of the beta function is only approximate, since the theory flows away from this value as the fermions gain dynamical mass and are integrated out, leaving a pure gluonic low-energy effective field theory. For such a theory, the quantity extracted from either continuum or lattice analyses as $\gamma_{IR}$ is only an effective anomalous dimension that describes the renormalization-group behavior as the theory is flowing near to the approximate zero of the beta function. A general comment is that the determination of $N_{f,cr}$ relies upon effective methods to analyze the lattice data lgtreviews ; progress on this continues lsd -fleming_eft .

Theories with an SU(2) gauge group and $N_{f}=8$ have been of interest in the context of certain ideas for physics beyond the Standard Model (SM) ckm , in which the number of Dirac fermions is $N_{f}=N_{wk}(N_{c}+1)=8$, where $N_{wk}=2$, corresponding to the SU(2) factor group in the SM and $N_{c}=3$ colors. There have been several lattice of this SU(2) theory with $N_{f}=8$, including su2nf8 ; lgtreviews ; tuominen2017 . These are consistent with this theory being IR-conformal, and the recent study tuominen2017 has reported the measurement $\gamma_{IR}=0.15\pm 0.02$. For comparison, as listed in Table 1, our previous higher $n$-loop values were $\gamma_{IR,3\ell}=0.272$ and $\gamma_{IR,4\ell}=0.204$ bvh , and our current highest-order scheme-independent value is $\gamma_{IR,\Delta_{f}^{4}}=0.298$. These are somewhat higher than this lattice result.

There have also been a number of lattice studies of the SU(2) theory with $N_{f}=6$ su2nf6 ; yamada_su2nf6 ; tuominen_su2nf6 ; lgtreviews . From this work, it is not yet clear if this theory is IR-conformal or chirally broken. Ref. yamada_su2nf6 obtained the range $0.26<\gamma_{IR}<0.74$, while Ref. tuominen_su2nf6 found $\gamma_{IR}\simeq 0.275$. Our higher-order scheme-independent values, as listed in Table 1, in particular, $\gamma_{IR,\Delta_{f}^{4}}=0.698$, are in agreement with the range given in yamada_su2nf6 and are somewhat higher than the value from tuominen_su2nf6 .

III.3 LNN Limit for $G={\rm SU}(N_{c})$ and $R=F$

For $G={\rm SU}(N_{c})$ and $R=F$, it is of interest to consider the limit

[TABLE]

We will use the symbol $\lim_{LNN}$ for this limit, where “LNN” stands for “large $N_{c}$ and $N_{f}$” with the constraints in Eq. (90) imposed. This is also called the ’t Hooft-Veneziano limit. Anticipating our later discussion of theories with fermions in two-index representations (adjoint and symmetric and antisymmetric rank-2 tensor), we will use the symbol $\lim_{LN}$, where “LN” stands for “large $N_{c}$”, to denote the original ’t Hooft limit

[TABLE]

and $N_{f}$ fixed and finite.

Continuing our discussion of the LNN limit, as relevant to theories with fermions in the fundamental represention, we define the following quantities in this limit:

[TABLE]

[TABLE]

and

[TABLE]

with values

[TABLE]

and

[TABLE]

(to the indicated floating-point accuracy). With $I_{IRZ}:\ N_{\ell}<N_{f}<N_{u}$, it follows that the corresponding interval in the ratio $r$ is

[TABLE]

The critical value of $r$ such that for $r>r_{cr}$, the LNN theory is IR-conformal and for $r<r_{cr}$, it exhibits spontaneous chiral symmetry breaking is denoted $r_{cr}$ and is defined as

[TABLE]

We define the scaled scheme-independent expansion parameter for the LNN limit

[TABLE]

As $r$ decreases from $r_{u}$ to $r_{\ell}$ in the interval $I_{IRZ,r}$, $\Delta_{r}$ increases from 0 to a maximal value

[TABLE]

We define rescaled coefficients $\hat{\kappa}_{j,F}$

[TABLE]

that are finite in this LNN limit. The anomalous dimension $\gamma_{IR}$ is also finite in this limit and is given by

[TABLE]

From the results for $\kappa_{j}$, $j=1,\ 2,\ 3$ in gtr or the special cases given above for $G={\rm SU}(N_{c})$ and $R=F$ in Eqs. (51)-(55), we have

[TABLE]

[TABLE]

and

[TABLE]

where, as above, we indicate the factorizations of the denominators. (The numerators do not, in general, have such simple factorizations; for example, in $\kappa_{3,F}$, $2193944=2^{3}\cdot 274243$.) From our new expression for $\kappa_{4}$, we calculate

[TABLE]

Hence, numerically, to order $O(\Delta_{r}^{4})$,

[TABLE]

Using these results for $\gamma_{IR,F,\Delta_{r}^{p}}$ with $1\leq p\leq 4$ for $R=F$ in the LNN limit, we can now carry out a polynomial extrapolation to $p=\infty$. To do this, we fit an expression for $\gamma_{IR,F,\Delta_{r}^{p}}$ with some subset of the $p$ terms to a polynomial in $1/p$. We denote the resultant value generically as $\gamma_{IR,F,s}$, where here $s$ denotes the subset of the $p$ terms used for the extrapolation. We shall use, as a necessary condition for $\gamma_{IR,F,s}$ to be reliable, the requirement that it not differ too much from the highest-order value, $\gamma_{IR,F,\Delta_{r}^{4}}$. Quantitatively, we require that for the given subset $s$, $\gamma_{IR,F,s}/\gamma_{IR,F,\Delta_{r}^{4}}<1.5$. We find that this condition is satisfied if $r\in I_{IRZ,r}$ is r\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle>}}{{\sim}}}}3.5, but that it is not satisfied as $r$ decreases below this value toward the lower end of the interval $I_{IRZ,r}$ at $r_{\ell}=2.615$. As an example, at $r=4.0$, depending on the subset of terms used for the extrapolation, we obtain $\gamma_{IR,F,s}/\gamma_{IR,F,\Delta_{r}^{4}}\simeq 1.2$, while at $r=3.6$, this ratio increases to $\simeq 1.4$. We remark that the value $r=4.0$ corresponds to $N_{f}=12$ for the SU(3) theory and $N_{f}=8$ for the SU(2) theory.

Previously, in gsi we performed this analysis for the special case $G={\rm SU}(3)$ and $R=F$ and, for that work, we studied how the extrapolated value depends on the subset of terms that one includes for the fit. We perform the corresponding analysis here for this LNN case. We study three sets of terms:

[TABLE]

[TABLE]

[TABLE]

There are countervailing advantages of these sets of terms. The two-term set (118) has the advantage of using the two highest-order terms, while the three-term and four-term sets have the advantage of using progressively more terms in the fit. The fits to the sets (118)-(120) yield polynomials in the variable $p^{-1}$ of the respective forms

[TABLE]

[TABLE]

and

[TABLE]

The extrapolated values in the limit $p\to\infty$ given by these fits are, respectively, as

[TABLE]

[TABLE]

and

[TABLE]

We have calculated these quantities analytically. Below, we list the corresponding expressions with coefficients given to the indicated floating-point precision:

[TABLE]

[TABLE]

and

[TABLE]

Note that there are strong cancellations between individual terms for relevant values of $r\in I_{IRZ,r}$. Some examples will show the range of resultant values of extrapolations for these different choices of sets of terms used in the fits. As anticipated, for values of $r$ in the upper part of the interval $I_{IRZ,r}$, all of the different types of extrapolation give quite similar results. For example,

[TABLE]

As $r$ decreases in the interval $I_{IRZ,r}$, the differences between the extrapolations using the different sets of terms increase slightly, e.g., for a value roughly in the middle of this interval, namely $r=4.0$, we find

[TABLE]

Toward the lower part of the interval $I_{IRZ,r}$, these differences increase further, but also, as discussed above, for a given $r$, all of the different types of extrapolations involve greater uncertainties, since each of the extrapolated values differs more from the value of highest-order explicitly calculated quantity, $\gamma_{IR,\Delta_{r}^{4}}$. For example, for $r=3.0$,

[TABLE]

The ratios of these values divided by the highest-order explicitly calculated value, $\gamma_{IR,F,\Delta_{r}^{4}}$, are

[TABLE]

Given our fiducial requirement that the ratio of the extrapolated value for $p\to\infty$ divided by the highest-order explicitly calculated value, should not be greater than 1.5 for the extrapolation to be considered reasonably reliable, it follows that we would not consider the latter two extrapolations in Eq. (148) to be sufficiently reliable to meet this requirement.

It is interesting to compare these scheme-independent calculations of $\gamma_{IR,F,\Delta_{r}^{p}}$ to order $1\leq p\leq 4$ with the results from the conventional $n$-loop calculations as truncated expansions in $\alpha_{IR,F,n\ell}$, denoted $\gamma_{IR,F,n\ell}$ from Table V of lnn up to $n=4$ loop order. We list our scheme-independent values together with these $n$-loop values in Table 2. For each value of $r$, we also include the extrapolated value, $\gamma_{IR,F,ex234}$ for the $p\to\infty$ limit, and the ratio $\gamma_{IR,F,ex234}/\gamma_{IR,\Delta_{r}^{4}}$. We do not include the results from the $n=5$ loop conventional calculation, because of the absence of a physical IR zero in the five-loop beta function for $2.615<r<4.323$ in $I_{IRZ,r}$. Although the extrapolated values $\gamma_{IR,F,ex234}$ for $r$ values below $r=3.5$ are included, we caution that these do not satisfy our fiducial criterion for sufficient reliability of extrapolation, since they differ by too much from our highest-order calculated values, $\gamma_{IR,\Delta_{r}^{4}}$. For this reason, although we can roughly apply the method discussed in Section II to use the extrapolated value of $\gamma_{IR}$ to estimate the lower end, $r_{cr}$, of the IR-conformal non-Abelian Coulomb phase (defined in Eq. (367)), this involves a substantial degree of uncertainty. Bearing this caveat in mind, the resulting estimate would be that $r_{cr}\sim 2.7$. If one were to pull back from the LNN limit and multiply this value of $r_{cr}$ by a specific finite value of $N_{c}$ to get an estimate of the corresponding $N_{f,cr}$, then, for example, for $N_{c}=3$, i.e., $G={\rm SU}(3)$, this would yield $N_{f,cr}\sim 8$. This estimate is consistent with the estimate 8\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}N_{f,cr}\mathrel{\raisebox{-2.58334pt}{\stackrel{{\scriptstyle\textstyle<}}{{\sim}}}}9 that we derived from our calculation of $\gamma_{IR,F,\Delta_{f}^{4}}$ for this theory and extrapolation to obtain $\lim_{p\to\infty}\gamma_{IR,F,\Delta_{f}^{p}}$ in gsi . Clearly, the lower that one goes in $N_{c}$ away from the LNN limit, the greater is the error in performing this conversion from a specific $r$ value in the LNN limit to a corresponding ratio $N_{f}/N_{c}$ with finite $N_{f}$ and $N_{c}$, so we do not perform this conversion for $N_{c}=2$.

In Fig. 4 we plot $\gamma_{IR,F,\Delta_{r}^{p}}$, i.e., the value of $\gamma_{IR}$ for $R=F$, calculated to order $\Delta_{r}^{p}$ with $1\leq p\leq 4$, in the scheme-independent expansion, as a function of $r\in I_{IRZ,r}$. As a consequence of the positivity of the $\hat{\kappa}_{p,F}$ in Eqs. (109)-(111), for a fixed $r$, $\gamma_{IR,F,\Delta_{r}^{p}}$ is a monotonically increasing function of the order of calculation, $p$. As $r$ decreases toward the lower end of the interval $I_{IRZ,r}$ at $r=r_{\ell}=2.615$, the value of $\gamma_{IR}$ calculated to the highest order in this LNN limit, namely $O(\Delta_{r}^{4})$, is slightly greater than 1.

As we did for specific SU($N_{c}$) theories above, here we proceed to investigate the range of applicability of the scheme-independent series expansion for $\gamma_{IR}$ in the LNN limit. As is evident from Table 2, all of our values of $\gamma_{IR,F,\Delta_{r}^{p}}$ for $1\leq p\leq 4$ satisfy the bound $\gamma_{IR}\leq 2$. This is also true for all of our extrapolated values, $\gamma_{IR,F,ex234}$, except for the lowest value of $r$ listed, namely $r=2.8$, for which $\gamma_{IR,F,ex234}=2.09$, slightly above this bound. Thus, these results in the LNN limit again demonstrate the advantage of the scheme-independent expansions, since they enable us to calculate self-consistent values of $\gamma_{IR,F,\Delta_{r}}$ over a greater range of the interval $I_{IRZ,r}$ than is the case with the conventional $n$-loop calculations. To show the latter in detail, we have explicitly listed the values of $\gamma_{IR,F,3\ell}$ and $\gamma_{IR,F,4\ell}$ for values of $r$ where $\gamma_{IR,F,2\ell}$ was unphysically large.

To investigate the range of applicability of the scheme-independent expansions further, it is worthwhile to obtain an estimate of this range from ratios of successive coefficients. From the coefficients $\hat{\kappa}_{j,F}$ that we have calculated with $1\leq n\leq 3$, we compute the ratios

[TABLE]

[TABLE]

and

[TABLE]

Recalling that the maximal value of $\Delta_{r}$ in the interval $I_{IRZ,r}$ is 2.885 (Eq. (106), these ratios are consistent with the inference that the small-$\Delta_{r}$ series expansion may be reasonably accurate throughout most of this interval $I_{IRZ,r}$.

III.4 $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{4}}$ for

$G={\rm SU}(N_{c})$ and $R=adj$

Here we present our results for the $\kappa_{j}$ coefficients and thus $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{j}}$ with $1\leq j\leq 4$ for $G={\rm SU}(N_{c})$ and $N_{f}$ fermions in the adjoint representation, $R=adj$. We will usually denote these as $\kappa_{j,adj}$ and $\gamma_{\bar{\psi}\psi,IR,adj,\Delta_{f}^{j}}$ but sometimes, when no confusion will result, we will omit this $adj$ subscript for brevity of notation.

In this theory, Eqs. (9) and (9) yield, for the upper and lower ends of the interval $I_{IRZ}$, the values

[TABLE]

and

[TABLE]

so this interval includes only one integral value of $N_{f}$, namely $N_{f}=2$. We note that since the adjoint representation is self-conjugate, a theory with $N_{f}$ Dirac fermions with $R=adj$ is equivalent to a theory with $N_{f,Maj}=2N_{f}$ Majorana fermions. Hence, here, one may also allow the half-integral values $N_{f}=3/2,\ 5/2$ corresponding to $N_{f,Maj}=3,\ 5$. We have

[TABLE]

For this case, the factor $D$ in Eq. (24) is simply $D=18$. In dex we gave the coefficients $\kappa_{j,adj}$ for $1\leq n\leq 3$. These are as follows:

[TABLE]

[TABLE]

and

[TABLE]

where, as before, we indicate the simple factorizations of the denominators. The coefficient $\kappa_{4,adj}$ is

[TABLE]

The coefficients $\kappa_{1,adj}$ and $\kappa_{2,adj}$ are manifestly positive, and we find that for all physical $N_{c}$, the coefficients $\kappa_{3,adj}$ and $\kappa_{4,adj}$ are also positive. Although $\kappa_{1,adj}$ and $\kappa_{2,adj}$ are independent of $N_{c}$, the coefficients $\kappa_{j,adj}$ for $j=3,\ 4$ do depend on $N_{c}$. We find that $\kappa_{3,adj}$ and $\kappa_{4,adj}$ are, respectively, monotonically increasing and monotonically decreasing functions of $N_{c}$. The $N_{c}\to\infty$ limits of $\kappa_{3,adj}$ and $\kappa_{4,adj}$ are given by the respective first terms in Eqs. (162) and (167).

Thus, to order $\Delta_{f}^{4}$, we have

[TABLE]

In Fig. 5 we show $\gamma_{\bar{\psi}\psi,IR,adj,\Delta_{f}^{p}}$ with $1\leq p\leq 4$ for the SU(2) theory, as a function of $N_{f}$, formally generalized from the nonnegative integers to the real numbers. In Table 3 we list values of $\gamma_{\bar{\psi}\psi,IR,adj,\Delta_{f}^{p}}$ with $1\leq p\leq 4$ for $N_{f}=2$ and $N_{c}=2$ and $N_{c}=3$. For comparison, we also include our $n$-loop values $\gamma_{\bar{\psi}\psi,IR,adj,n\ell}$ calculated in the conventional manner via power series in the coupling (in the $\overline{\rm MS}$ scheme), from Table VIII of bvh .

Among SU($N_{c}$) theories with fermions in the adjoint representation, the SU(2) theory with $N_{f}=2$ (Dirac) fermions has been of particular interest sannino_su2adj . In the following, for notational brevity, the subscript $adj$ is understood implicitly. For this theory, as listed in Table 3 we obtain the values $\gamma_{IR,\Delta_{f}^{2}}=0.465$, $\gamma_{IR,\Delta_{f}^{3}}=0.511$, and $\gamma_{IR,\Delta_{f}^{4}}=0.556$, which are close to our earlier higher-order $n$-loop calculations in bvh , namely $\gamma_{IR,3\ell}=0.543$ and $\gamma_{IR,4\ell}=0.500$. It is of interest to compare these values with the results of lattice studies. There have been a number of such studies, and these are consistent with the conclusion that this theory is conformal in the infrared catterall2010 -montvay ,lgtreviews . These studies have yielded a rather large range of measured values for $\gamma_{IR}$, including the following (where the published estimated uncertainties in the last digits are indicated in parentheses): $\gamma_{IR}=0.49(13)$ catterall2010 , $\gamma_{IR}=0.22(6)$ deldebbio2010 , $\gamma_{IR}=0.31(6)$ degrand2011 , $\gamma_{IR}=0.17(5)$ lsd2011 , $\gamma_{IR}=0.37(2)$ deldebbio2016 , $\gamma_{IR}=0.20(3)$ tuominen2016 , and $\gamma_{IR}=0.50(26)$ giedt2016 . (See these references and montvay for additional discussion of estimates of overall uncertainties.) Our scheme-independent calculation of $\gamma_{IR}$ to $O(\Delta_{f}^{4})$ and our earlier $n$-loop calculations of $\gamma_{IR,n\ell}$ up to $n=4$ loops are clearly consistent with the larger among these lattice values. Before carrying out a comparison of our results with the full set of lattice values, it will be necessary to narrow the current wide range of lattice measurements.

It is of interest to investigate the $N_{c}\to\infty$ limit for an SU($N_{c}$) gauge theory with fermions in the adjoint representation. Since in this case, the upper and lower ends of the interval $I_{IRZ}$, given by $N_{u}=11/4$ in Eq. (155) and $N_{\ell}=17/16$ in Eq. (9) are independent of $N_{c}$, it follows that $\Delta_{f}$ is also independent of $N_{c}$. Hence, for $R=adj$,

[TABLE]

where

[TABLE]

The values of $\hat{\kappa}_{j,adj}$ are evident from the full expressions for $\kappa_{j,adj}$ that we have given above in Eqs. (158)-(167); for example, $\hat{\kappa}_{3,adj}=61873/(2^{3}\cdot 3^{10})$.

III.5 $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{4}}$ for

$G={\rm SU}(N_{c})$ and $R=S_{2},\ A_{2}$

Here we present our results for the $\kappa_{j}$ coefficients and thus $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{j}}$ with $1\leq j\leq 4$ for $G={\rm SU}(N_{c})$ and $N_{f}$ fermions in the symmetric and antisymmetric rank-2 tensor representations of SU($N_{c}$), $S_{2}$ and $A_{2}$. Since many formulas for these two cases are simply related to each other by sign reversals in certain terms, it is convenient to treat these cases together. As before bvh , we shall use the symbol $T_{2}$ (rank-2 tensor) to refer to these cases together. (Do not confuse this use of $T$ with our use of the symbol $T$ in Section VII of Ref. dex for the anomalous dimension of the operators $\bar{\psi}\sigma_{\mu\nu}\psi$ and operators $\bar{\psi}T_{a}\sigma_{\mu\nu}\psi$, where it referred to the antisymmetric Dirac tensor $\sigma_{\mu\nu}=(i/2)[\gamma_{\mu},\gamma_{\nu}]$.)

The values of $N_{u}$ and $N_{\ell}$ for $R=T_{2}$ are bvh

[TABLE]

and

[TABLE]

so that

[TABLE]

The factor $D$ in Eq. (24) takes the explicit form

[TABLE]

whence

[TABLE]

Both $F_{+}$ and $F_{-}$ are positive-definite for the physical range $N_{c}\geq 2$. At the lower end of the interval $I_{IRZ}$, $\Delta_{f}$ takes on the maximum value

[TABLE]

If $N_{c}=2$, then $S_{2}$ is the same as the adjoint representation, so we focus on $N_{c}\geq 3$ here. For this $R=S_{2}$ theory, the illustrative values $N_{c}=3$ and $N_{c}=4$ yield the respective intervals $I_{IRZ}$ $1.22<N_{f}<3.30$ and $1.35<N_{f}<3.67$. Hence, the physical integral values of $N_{f}$ in these respective intervals $I_{IRZ}$ are $N_{f}=2,\ 3$ for both $N_{c}=3$ and $N_{c}=4$. Furthermore, the $A_{2}$ representation is the singlet if $N_{c}=2$ and is the same as the conjugate fundamental, $\bar{F}$ if $N_{c}=3$, so in the case of $A_{2}$, we restrict to $N_{c}\geq 3$ and focus mainly on $N_{c}\geq 4$. In the SU(4) theory with $R=A_{2}$, the interval $I_{IRZ}$ is $4.945<N_{f}<11$, including the integral values $5\leq N_{f}\leq 10$.

Here, using our general results (25)-(48), we give explicit expressions for the $\kappa_{j}$ with $1\leq j\leq 4$ for the case $G={\rm SU}(N_{c})$ and fermion representation $R=T_{2}$. From the general expressions for $\kappa_{j}$ with $1\leq j\leq 4$, Eqs. (25)-(48), we calculate the following. In each expression, the $+$ and $-$ signs refer to the $S_{2}$ and $A_{2}$ special cases of $T_{2}$, respectively:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We comment on some factors in these $\kappa_{j,T_{2}}$ expressions. The property that the $\kappa_{j,A_{2}}$ coefficients contain an overall factor of $(N_{c}-2)$ (possibly raised to a power higher than 1), and hence vanish for $N_{c}=2$, is a consequence of the fact that for $N_{c}=2$, the $A_{2}$ representation is a singlet, so for SU(2), fermions in the $A_{2}=$ singlet representation have no gauge interactions and hence no anomalous dimensions. Clearly, this property holds in general; i.e., the coefficients $\kappa_{j,A_{2}}$ for all $j$ contain an overall factor of $(N_{c}-2)$ (as well as possible additional factors of $(N_{c}-2)$).

As noted above, if $N_{c}=2$, then the $S_{2}$ representation is the same as the adjoint representation, so the coefficients must satisfy the equality $\kappa_{j,S_{2}}=\kappa_{j,adj}$ for this SU(2) case, and we have checked that they do. Note that this equality requires (i) that the term proportional to $\zeta_{3}$ in $\kappa_{3,S_{2}}$ must be absent if $N_{c}=2$, since $\kappa_{3,adj}$ does not contain any $\zeta_{3}$ term, and, indeed, this is accomplished by the factor $(N_{c}-2)$ multiplying the $\zeta_{3}$ term in $\kappa_{3,S_{2}}$; and (ii) the term proportional to $\zeta_{5}$ in $\kappa_{4,S_{2}}$ must be absent if $N_{c}=2$, since $\kappa_{4,adj}$ does not contain any $\zeta_{5}$ term, and this is accomplished by the factor $(N_{c}-2)$ multiplying this $\zeta_{5}$ term in $\kappa_{4,S_{2}}$. Similarly, as we observed above, if $N_{c}=3$, then the $A_{2}$ representation is the same as the conjugate fundamental representation, $\bar{F}$, so the coefficients must satisfy the equality $\kappa_{j,A_{2}}=\kappa_{j,F}$ for this SU(3) case, and we have checked that they do.

The resultant $\Delta_{f}$ expansions for $\gamma_{\bar{\psi}\psi,IR,S_{2},\Delta_{f}^{4}}$ with $2\leq N_{c}\leq 4$ are

[TABLE]

[TABLE]

and

[TABLE]

For $R=A_{2}$, we give illustrative results for the $\Delta_{f}$ expansion of $\gamma_{\bar{\psi}\psi,IR}$ for $N_{c}=4,\ 5$:

[TABLE]

and

[TABLE]

In Fig. 6 we present a plot of $\gamma_{\bar{\psi}\psi,S_{2},IR,\Delta_{f}^{p}}$ for $G={\rm SU}(3)$, $R=S_{2}$, and $1\leq p\leq 4$, as a function of $N_{f}$. We list values of the $\gamma_{IR,S_{2},\Delta_{f}^{p}}$ with $1\leq p\leq 4$ for the SU(3) and SU(4) theories with $R=S_{2}$ in Table 4. In both of these theories, the interval $I_{IRZ}$ includes the two integer values $N_{f}=2,\ 3$. For comparison, we also include the values $\gamma_{IR,S_{2},n\ell}$ for $2\leq n\leq 4$ calculated via the conventional power series expansion to $n$-loop order and evaluated at $\alpha=\alpha_{IR,n\ell}$ from Table XI in our previous work, Ref. bvh . As is evident from this table, for a given $N_{c}$ and $N_{f}$, there is reasonable agreement between the $n=4$ loop values $\gamma_{IR,S_{2},\Delta_{f}^{4}}$ and $\gamma_{IR,S_{2},4\ell}$. For example, for SU(3) and $N_{f}=2$, $\gamma_{IR,S_{2},4\ell}=1.12$ while $\gamma_{IR,S_{2},\Delta_{f}^{4}}=1.13$.

We next compare our calculation of $\gamma_{\bar{\psi}\psi,IR,S_{2},\Delta_{f}^{p}}$ to order $p=4$ with lattice measurements. A theory of particular interest is the SU(3) gauge theory with $N_{f}=2$ flavors of fermions in the $S_{2}$ representation, and lattice studies of this theory include degrand_sextet and kuti_sextet (see also lgtreviews ). As indicated in Table 4, our higher-order scheme-independent results are $\gamma_{IR,\Delta_{f}^{3}}=0.960$, and $\gamma_{IR,\Delta_{f}^{4}}=1.132$. By comparison, our $n$-loop results from bvh for this theory are $\gamma_{IR,3\ell}=0.500$ and $\gamma_{IR,4\ell}=0.470$. The lattice study degrand_sextet concluded that this theory is IR-conformal and obtained $\gamma_{IR}<0.45$ degrand_sextet , while Ref. kuti_sextet concluded that it is not IR-conformal and got an effective $\gamma_{IR}\sim 1$ kuti_sextet . One hopes that further work by lattice groups will lead to a consensus concerning whether this theory is IR conformal or not and concerning the value of $\gamma_{IR}$.

Regarding the range of applicability of the $\Delta_{f}$ expansion for these cases, we compute the following ratios of successive coefficients for the $G={\rm SU}(3)$, $R=S_{2}$ case:

[TABLE]

[TABLE]

and

[TABLE]

The first two ratios, (200) and (201), are slightly larger than $(\Delta_{f})_{max,S_{2}}=519/250=2.076$ in $I_{IRZ}$ for this theory. However, the third ratio is about 40 % less than this maximal value of $\Delta_{f,S_{2}}$. This suggests that because of slow convergence, one must use the $\Delta_{f}$ expansion with caution in the lower part of the interval $I_{IRZ}$ in this theory.

We list values of the $\gamma_{IR,A_{2},\Delta_{f}^{p}}$ with $1\leq p\leq 4$ for the SU(4) theory with $R=A_{2}$ and $N_{f}\in I_{IRZ}$ for this theory in Table 5. Again, for comparison, we include the values $\gamma_{IR,A_{2},n\ell}$ for $2\leq n\leq 4$ calculated via the conventional power series expansion to $n$-loop order and evaluated at $\alpha=\alpha_{IR,n\ell}$ from Table XII in our previous work bvh . As expected, the agreement between the two methods of calculation is best at the upper end of the interval $I_{IRZ}$, where the IRFP occurs at weak coupling. For example, for $N_{f}=9$, $\gamma_{IR,A_{2},\Delta_{f}^{4}}=0.242$, while $\gamma_{IR,4\ell}=0.232$.

It is of interest to consider the $N_{c}\to\infty$ (LN) limit of Eq. (95) for these theories with $R=S_{2}$ and $A_{2}$. In this LN limit, the upper ends of the interval $I_{IRZ}$ for the $S_{2}$ and $A_{2}$ representations approach the same limit, and similarly for the lower ends:

[TABLE]

[TABLE]

Hence, in this $N_{c}\to\infty$ limit, the interval $I_{IRZ}$ is formally $2.125<N_{f}<5.5$, including the physical integer values $3\leq N_{f}\leq 5$. Similarly, in this limit, the variable $\Delta_{f}$ is given by $\Delta_{f}=(11/2)-N_{f}$ and reaches a maximum value, at $N_{f}=N_{\ell,T_{2}}$, of

[TABLE]

This the $N_{c}\to\infty$ limit of (176).

As with the adjoint representation, we define

[TABLE]

We find that

[TABLE]

From our general expressions for $\kappa_{j,T_{2}}$ with $1\leq j\leq 4$, we calculate

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Hence,

[TABLE]

and, in the limit $p\to\infty$,

[TABLE]

Thus, for both $R=S_{2}$ and $R=A_{2}$,

[TABLE]

We observe that for all of the cases we have calculated, namely $1\leq j\leq 4$,

[TABLE]

One can understand this relation from the structure of the relevant group invariants, including the fact that the trace invariant $T(R)$ satisfies

[TABLE]

We thus infer more generally that the relation (217) holds for all $j$. In Table 6 we list the resultant common values of $\gamma_{IR,T_{2},\Delta_{f}^{p}}$ for $1\leq p\leq 4$ and $N_{f}\in I_{IRZ}$ in the LN limit. As noted above, in this LN limit, this interval consists of the integral values $N_{f}=3,\ 4,\ 5$.

Concerning the range of applicability of the $\Delta_{f}$ expansion in this LN limit, we compute the ratios

[TABLE]

[TABLE]

and

[TABLE]

The first two ratios, (219) and (220), are slightly greater than the maximum value $(\Delta_{f})_{max,T_{2}}=3.375$, but the third ratio, (223), is smaller than this maximum value, suggesting that in this limit, for these tensor representations, because of slow convergence, one must use caution in applying the $\Delta_{f}$ expansion in the lower part of the interval $I_{IRZ}$. This is similar to what we found for the $S_{2}$ representation in the SU(3) theory.

IV Calculation of $\beta^{\prime}_{IR}$ to $O(\Delta_{f}^{5})$

IV.1 General $G$ and $R$

The derivative $\beta^{\prime}_{IR}$ is an important physical quantity characterizing the conformal field theory at $\alpha_{IR}$. We denote the gauge field of the theory as $A_{\mu}^{a}$ (where $a$ is a group index), the field strength-tensor as $F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gc_{abc}A_{\mu}^{b}A_{\nu}^{c}$ (where $c_{abc}$ is the structure constant of the Lie algebra of $G$) and the rescaled field-strength tensor as $F_{\mu\nu,r}^{a}=gF_{\mu\nu}^{a}$, so that the gauge field kinetic term in the Lagrangian is ${\cal L}_{g}=-[1/(4g^{2})]F_{\mu\nu,r}^{a}F_{r}^{a\ \mu\nu}$. The trace anomaly states that the trace of the energy-momentum tensor $T^{\mu}_{\nu}$ satisfies the relation traceanomaly

[TABLE]

Therefore, the full scaling dimension of the operator $F_{r,\mu\nu}F_{r}^{a\ \mu\nu}$, which we denote as $D_{F^{2}}$, satisfies tracerel

[TABLE]

where we use the shorthand notation $F^{2}\equiv F_{r,\mu\nu}^{a}F_{r}^{a\ \mu\nu}$. We denote the anomalous dimension of $F^{2}$, $\gamma_{F^{2}}$ via the equation gammaconvention

[TABLE]

and its evaluation at $\alpha=\alpha_{IR}$ as $\gamma_{{}_{F^{2},IR}}$. From Eq. (225), it follows that at a zero of the beta function away from the origin, in particular, at $\alpha_{IR}$, the derivative $\beta^{\prime}_{IR}$ is equivalent to the anomalous dimension of the operator $F^{a}_{r,\mu\nu}F_{r}^{a\ \mu\nu}$:

[TABLE]

In dex we calculated the expansion coefficients $d_{j}$ of $\beta^{\prime}_{IR}$ in Eq. (3) to order $\Delta_{f}^{4}$ for general $G$ and $R$, and to order $\Delta_{f}^{5}$ for the special case $G={\rm SU}(3)$ and fermion representation $R=F$, the fundamental. Here we calculate the next higher-order coefficient, namely $d_{5}$, for general $G$ and $R$. For this purpose, we make use of the recent computation of the five-loop beta function coefficient, $b_{5}$, in b5 . The computation in b5 was performed in the $\overline{\rm MS}$ scheme, so that we can combine it with the scheme-independent $b_{1}$ and $b_{2}$ b1 ; b2 and the results for $b_{3}$ and $b_{4}$ that have also been calculated in the $\overline{\rm MS}$ scheme b3 ; b4 . However, we again stress that since the $d_{n}$ coefficients are scheme-independent, it does not matter which scheme one uses to calculate them. We first recall our previous results from Ref. dex :

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

In Ref. dex we presented the expression for $d_{4}$ with terms written in order of descending powers of $C_{A}$. It is also useful to express this coefficient $d_{4}$ in an equivalent form that renders certain factors of $D$ explicit and shows the simple factorization of terms multiplying $\zeta_{3}$, and we have done this in Eq. (LABEL:d4).

Here we present our calculation of $d_{5}$ for arbitrary $G$ and $R$:

[TABLE]

We proceed to evaluate these coefficients $d_{j}$ up to $j=5$, and hence the derivative $\beta^{\prime}_{IR}$ up to $O(\Delta_{f}^{5})$ below for $G={\rm SU}(N_{c})$ and several specific representations. The coefficients $d_{2}$ and $d_{3}$ are manifestly positive for arbitrary $G$ and $R$. These signs are indicated in Table 7. We discuss the signs of $d_{4}$ and $d_{5}$ for various representations below.

IV.2 $\beta^{\prime}_{IR,\Delta_{f}^{4}}$ for $G={\rm SU}(N_{c})$ and $R=F$

Here we present the evaluation of our general result (252) for the case $G={\rm SU}(N_{c})$ and $R=F$. For reference, we first recall our results from dex for $d_{j}$ with $2\leq j\leq 4$ (and also recall that $d_{1}=0$ for all $G$ and $R$):

[TABLE]

[TABLE]

and

[TABLE]

This coefficient can be written equivalently in a form that shows the simple factorization of the terms multiplying $\zeta_{3}$:

[TABLE]

In dexs we presented the expression for $d_{5,F}$ with terms ordered as descending powers of $N_{c}$. As with $d_{4,F}$, it is also useful to display this coefficient in an equivalent form that shows the simple factorizations of the terms multiplying $\zeta_{3}$ and $\zeta_{5}$:

[TABLE]

We have checked that when we set $N_{c}=3$ in our general result for $d_{5,F}$ in Eq. (267), the result agrees with our earlier calculation of $d_{5,F}$ in Eq. (5.20) of Ref. dex .

As observed above, the coefficients $d_{2}$ and $d_{3}$ are manifestly positive for any $G$ and $R$. We find that $d_{4,F}$ and $d_{5,F}$ are negative-definite for $G={\rm SU}(N_{c})$ and all physical values of $N_{c}\geq 2$. These results are summarized in Table 7.

We list below the explicit numerical expressions for $\beta^{\prime}_{IR}$ to order $\Delta_{f}^{5}$, denoted $\beta^{\prime}_{IR,{\rm SU}(N_{c}),F,\Delta_{f}^{5}}$, for the gauge groups SU($N_{c}$) with $N_{c}=2,\ 3,\ 4$, with fermions in the fundamental representation, to the indicated floating-point precision:

[TABLE]

[TABLE]

and

[TABLE]

In Table 8 we list the (scheme-independent) values that we calculate for $\beta^{\prime}_{IR,F,\Delta_{f}^{p}}$ with $2\leq p\leq 4$ for the illustrative gauge groups $G={\rm SU}(2)$, SU(3), and SU(4), as functions of $N_{f}$ in the respective intervals $I_{IRZ}$ given in Eq. (10). For comparison, we list the $n$-loop values of $\beta^{\prime}_{IR,F,n\ell}$ with the $2\leq n\leq 4$ from bc ; dex , where $\beta^{\prime}_{IR,F,3\ell}$ and $\beta^{\prime}_{IR,F,4\ell}$ are computed in the $\overline{\rm MS}$ scheme. Although, for completeness, we list values of $\beta^{\prime}_{IR,F,2\ell}$ with $N_{f}$ extending down to the lower end of the respective intervals $I_{IRZ}$ for each value of $N_{c}$, we caution that in a number of cases, including $N_{f}=6$ for SU(2), $N_{f}=9$ for SU(3), and $10\leq N_{f}\leq 12$ for SU(4), the corresponding values of $\alpha_{IR,2\ell}$ (discussed further below) are too large for the perturbative $n$-loop calculations to be applicable. Moreover, since for a considerable range of values of $N_{f}\in I_{IRZ}$ for each $N_{c}$, the five-loop beta function $\beta_{5\ell}$ calculated via the conventional power series expansion has no physical IR zero, we restrict the resultant $\beta^{\prime}_{IR,F,n\ell}$ evalulations to $1\leq n\leq 4$ loops.

In Figs. 7-9 we plot the values of $\beta^{\prime}_{IR}$, calculated to order $\Delta_{f}^{p}$ with $2\leq p\leq 5$, for $R=F$ for the gauge groups SU(2), SU(3), and SU(4). In the general calculations of $\gamma_{IR}$ as a series in powers of $\Delta_{f}$ to maximal power $p=3$ (i.e., order $\Delta_{f}^{3}$) in gtr and, for $G={\rm SU}(3)$ and $R=F$, to maximal power $p=4$ in gsi , it was found that, for a fixed value of $N_{f}$, or equivalently, $\Delta_{f}$, in the interval $I_{IRZ}$, these anomalous dimensions increased monotonically as a function of $p$. This feature motivated our extrapolation to $p=\infty$ in gtr to obtain estimates for the exact $\gamma_{IR}$. In contrast, here we find that, for a fixed value of $N_{f}$, or equivalently, $\Delta_{f}$, in $I_{IRZ}$, as a consequence of the fact that different coefficients $d_{n}$ do not all have the same sign, $\beta^{\prime}_{IR,\Delta_{f}^{p}}$ is not a monotonic function of $p$. Because of this non-monotonicity, we do not attempt to extrapolate our series to $p=\infty$.

A lattice measurement of $\beta^{\prime}_{IR}$ has been reported in hasenfratz_betaprime for the SU(3) theory with $R=F$ and $N_{f}=12$, namely $\beta^{\prime}_{IR}=0.26(2)$. The earlier higher-order values calculated in bc via $n$-loop expansions in the coupling are $\beta^{\prime}_{IR,3\ell}=0.2955$ and $\beta^{\prime}_{IR,4\ell}=0.282$, which agree with this lattice measurement. As indicated in Table 8, our higher-order scheme-independent values for this theory are $\beta^{\prime}_{IR,\Delta_{f}^{3}}=0.258$, $\beta^{\prime}_{IR,\Delta_{f}^{4}}=0.239$, and $\beta^{\prime}_{IR,\Delta_{f}^{5}}=0.228$. Given the possible contributions of higher-order terms in the $\Delta_{f}$ expansion, we consider that our scheme-independent calculation of $\beta^{\prime}_{IR}$ to this order is also consistent with the lattice measurement from Ref. hasenfratz_betaprime .

To get a rough estimate of the range of accuracy and applicability of the series expansion for $\beta^{\prime}_{IR}$ for this $R=F$ case, we can compute ratios of coefficients, as discussed before. For the illustrative case of SU(3), we have

[TABLE]

[TABLE]

and

[TABLE]

Since $N_{u}=16.5$ and $N_{\ell}=153/19=8.053$ in this SU(3) theory, the maximal value of $\Delta_{f}$ in the interval $I_{IRZ}$, as given by (73), is

[TABLE]

Therefore, these ratios suggest that the small-$\Delta_{f}$ expansion may be reasonably reliable in most of this interval, $I_{IRZ}$ and the associated non-Abelian Coulomb phase.

IV.3 $\beta^{\prime}_{IR,\Delta_{f}^{5}}$ in LNN Limit

The appropriately rescaled beta function that is finite in the LNN limit is

[TABLE]

where $\xi=4\pi x=\lim_{LNN}\alpha N_{c}$ was defined in Eq. (90). This has the series expansion

[TABLE]

where

[TABLE]

and $\tilde{b}_{\ell}=\hat{b}_{\ell}/(4\pi)^{\ell}$. The $\hat{b}_{\ell}$ with $1\leq\ell\leq 4$ were analyzed in bc ; lnn and are listed for the reader’s convenience in the Appendix.

From the recent calculation of $b_{5}$ in b5 , for general $G$ and $R$, in the $\overline{\rm MS}$ scheme b5 , we calculate

[TABLE]

(In this expression although $\zeta_{4}$ could be expressed explicitly as $\zeta_{4}=\pi^{4}/90$, we leave it in abstract form to be parallel with the $\zeta_{3}$ and $\zeta_{5}$ terms.) We find that this coefficient $\hat{b}_{5}$ is positive throughout the entire asymptotically free interval $0\leq r<5.5$. (Considered formally as a function of $r\in{\mathbb{R}}$, $\hat{b}_{5}$ is negative for $r<-58.609$, positive for $-58.609<r<14.336$, and negative for $r>14.336$, where the numbers are quoted to the given floating-point accuracy.)

Since the derivative $d\beta_{\xi}/d\xi$ satisfies the relation

[TABLE]

it follows that $\beta^{\prime}$ is finite in the LNN limit (90). In terms of the variable $x$ defined in Eq. (96), we have

[TABLE]

Because $\beta^{\prime}_{IR}$ is scheme-independent and is finite in the LNN limit, one is motivated to calculate the LNN limit of the scheme-independent expansion (3). For this purpose, in addition to the rescaled quantities $\Delta_{r}$ defined in Eq. (103), we define the rescaled coefficient

[TABLE]

which is finite. Then each term

[TABLE]

is finite in this limit. Thus, writing $\lim_{LNN}\beta^{\prime}_{IR}$ as $\beta^{\prime}_{IR,LNN}$ for this $R=F$ case, we have

[TABLE]

We denote the value of $\beta^{\prime}_{IR,LNN}$ obtained from this series calculated to order $O(\Delta_{f}^{p})$ as $\beta^{\prime}_{IR,LNN,\Delta_{f}^{p}}$.

From Eqs. (228)-(LABEL:d4), we find that the approach to the LNN limits for $\hat{d}_{j,F}$ involves correction terms that vanish like $1/N_{c}^{2}$. This is the same property that was found in bc ; lnn and, in the same way, it means that the approach to the LNN limit for finite $N_{c}$ and $N_{f}$ with fixed $r=N_{f}/N_{c}$ is rather rapid, as discussed in lnn . In dex we gave the $\hat{d}_{j,F}$ for $1\leq n\leq 4$; in addition to $\hat{d}_{1}=0$ (which holds for any $G$ and $R$), these are

[TABLE]

[TABLE]

and

[TABLE]

Here we give the next higher coefficient:

[TABLE]

In these equations we have indicated the simple factorizations of the denominators that were already evident in the general analytic expressions (228)-(LABEL:d4). Although the numerical coefficients in the numerators of terms in Eq. (309) do not, in general, have simple factorizations, they do contain various powers of 2; for example, in $\hat{d}_{5,F}$, $1444864=2^{10}\cdot 17\cdot 83$, etc. Thus, numerically, to order $\Delta_{r}^{5}$, for the LNN limit of this theory with $R=F$, we have

[TABLE]

where the coefficients are given to the indicated floating-point precision. We may again calculate ratios of successive magnitudes of these coefficients to get a rough estimate of the range over which the small-$\Delta_{r}$ expansion is reliable in this LNN limit. We find

[TABLE]

[TABLE]

and

[TABLE]

For $r\in I_{IRZ,r}$, the maximal value of $\Delta_{r}$ is $(\Delta_{r})_{\rm max}=75/26=2.885$. The first two ratios, (315) and (316) suggest that the $\Delta_{r}$ expansion for $\beta^{\prime}_{IR}$ may be reasonably reliable over a reasonable fraction of the interval $I_{IRZ,r}$. From the third ratio, (317), we infer that the expansion is expected to be more accurate in the upper portion of the interval $I_{IRZ,r}$ than the lower portion.

In Ref. dex we presented a comparison of these scheme-independent calculations of $\beta^{\prime}_{IR,LNN}$ calculated up to the $\Delta_{r}^{4}$ order with the results of conventional $n$-loop calculations, denoted $\beta^{\prime}_{IR,n\ell,LNN}$, computed up to the $n=4$ loop order for which the $b_{n}$ were known at that time. We refer the reader to dex for details of this discussion. Here we shall extend this comparison to the $\Delta_{r}^{5}$ order. In Table 9 we list the numerical values of these conventional $n$-loop calculations up to $n=4$, in comparison with our scheme-independent results calculated to $O(\Delta_{r}^{p})$ with $p$ up to 5. (The conventional 4-loop values $\beta^{\prime}_{IR,4\ell}$ for some values of $r$ toward the lower part of $I_{IRZ,r}$ supersede the corresponding entries in Table II of dex .) Both $\beta^{\prime}_{IR,n\ell}$ and $\beta^{\prime}_{IR,\Delta_{r}^{n}}$ use, as inputs, the coefficients of the beta function up to loop order $n$, although $\beta^{\prime}_{IR,\Delta_{r}^{n}}$ does this in a scheme-independent manner. We see that, especially for $r$ values in the upper part of the interval $I_{IRZ,r}$, the results are rather close, and, furthermore, that, as expected, for a given $r$, the higher the loop level $n$ and the truncation order $p$ in the respective calculations of $\beta^{\prime}_{IR,n\ell}$ in the $\overline{\rm MS}$ scheme and the scheme-independent $\beta^{\prime}_{IR,\Delta_{r}^{p}}$, the better the agreement between these two results. Toward the lower end of the interval $I_{IRZ,r}$, both the conventional expansion of $\beta^{\prime}_{IR}$ and the scheme-independent expansion of $\beta^{\prime}_{IR}$ in powers of $\Delta_{r}$ become less reliable, and hence it is understandable that the results differ from each other in this lower part of $I_{IRZ,r}$.

IV.4 $\beta^{\prime}_{IR,\Delta_{f}^{5}}$ for $G={\rm SU}(N_{c})$ and $R=adj$

Here we calculate the $d_{j}$ and hence $\beta^{\prime}_{IR,\Delta_{f}^{j}}$ for $j$ up to $j=5$ in the SU($N_{c}$) gauge theory with fermion representation $R=adj$. As was discussed above, in this case, the interval $I_{IRZ}$ contains the single Dirac value, $N_{f}=2$. For this value of $N_{f}$, Eq. (157) yields $\Delta_{f}=3/4$. We recall that the $d_{j}$ for $2\leq j\leq 4$ are dex

[TABLE]

[TABLE]

and

[TABLE]

Here, from our new general result (252) for $d_{5}$, we obtain the next coefficient for this case of the adjoint representation:

[TABLE]

While the $d_{j,adj}$ with $2\leq j\leq 4$ are positive-definite, we thus find that $d_{5,adj}$ is negative-definite. These results on signs are listed in Table 7. In the $N_{c}\to\infty$ (LN) limit of Eq. (95), the values of $\hat{d}_{j,adj}$ can be read off directly from our general results in Eqs. (318)-(329); for example, $\hat{d}_{4,adj}=46871/(2^{2}\cdot 3^{12})$, etc.

With these coefficients, one can again compute ratios to obtain a crude idea of the region over which the small-$\Delta_{f}$ series expansion is reliable. We have

[TABLE]

and, taking the large-$N_{c}$ limit for simplicity,

[TABLE]

[TABLE]

Since $\Delta_{f}=0.75$ for $N_{f}=2$, these ratios indicate that the small-$\Delta_{f}$ expansion should be reasonably accurate here.

IV.5 $\beta^{\prime}_{IR,\Delta_{f}^{5}}$ for $G={\rm SU}(N_{c})$ and

$R=S_{2},\ A_{2}$

Here we present our results for the $d_{j}$ coefficients and hence $\beta^{\prime}_{IR,\Delta_{f}^{j}}$ with $j$ up to 5 for $G={\rm SU}(N_{c})$ and $N_{f}$ fermions in the symmetric and antisymmetric rank-2 tensor representations, $S_{2}$ and $A_{2}$. As before with $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$, since many formulas for these two cases are simply related to each other by sign reversals in certain terms, it is convenient to treat these two cases together, denoting them collectively as $T_{2}$. We recall that for $R=A_{2}$, we restrict to $N_{c}\geq 3$.

From our general formulas (228)-(252), we obtain the following, where the upper and lower signs refer to the $S_{2}$ and $A_{2}$ special cases of $T_{2}$, respectively, and $F_{\pm}$ was defined in Eq. (175):

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We find that, in addition to the manifestly positive $d_{2,T_{2}}$, the coefficient $d_{3,T_{2}}$ is also positive for all relevant $N_{c}$. Here, by “relevant $N_{c}$”, we mean $N_{c}\geq 2$ for $S_{2}$ and $N_{c}\geq 3$ for $A_{2}$. In contrast, while $d_{4,S_{2}}$ is positive for all relevant $N_{c}$, we find that $d_{4,A_{2}}$ is negative for $N_{c}=3,\ 4,\ 5$, passes through zero at $N_{c}=5.515$, and is positive for $N_{c}\geq 6$. Further, we find that $d_{5,S_{2}}$ and $d_{5,A_{2}}$ are both negative for their respective physical ranges, $N_{c}\geq 2$ and $N_{c}\geq 3$. These sign properties are listed in Table 7.

Some general comments are in order concerning these $d_{j,T_{2}}$ expressions. These are analogous to the comments that we made for the $\kappa_{j,T_{2}}$ coefficients. The property that all of the $d_{j,A_{2}}$ coefficients contain an overall factor of $(N_{c}-2)$ (possibly raised to a power higher than 1), and hence vanish for $N_{c}=2$, is a consequence of the fact that for $N_{c}=2$, the $A_{2}$ representation is a singlet, so for SU(2), fermions in the $A_{2}=$ singlet representation have no gauge interactions and do not contribute to the beta function or $\beta^{\prime}_{IR}$.

Furthermore, if $N_{c}=2$, then the $S_{2}$ representation is the same as the adjoint representation, so the coefficients must satisfy the equality $d_{j,S_{2}}=d_{j,adj}$ for this SU(2) case, and we have checked that they do. This equality requires (i) that the term proportional to $\zeta_{3}$ in $d_{4,S_{2}}$ must be absent if $N_{c}=2$, since $d_{4,adj}$ does not contain any $\zeta_{3}$ term, and this is accomplished by the factor of $(N_{c}-2)$ multiplying the $\zeta_{3}$ term in $d_{4,S_{2}}$; and (ii) the term proportional to $\zeta_{5}$ in $d_{5,S_{2}}$ must be absent if $N_{c}=2$, since $d_{5,adj}$ does not contain any $\zeta_{5}$ term, and this is accomplished by the factor $(N_{c}-2)$ multiplying this $\zeta_{5}$ term in $d_{5,S_{2}}$. Similarly, as observed before, if $N_{c}=3$, then the $A_{2}$ representation is the same as the conjugate fundamental representation, $\bar{F}$, so the coefficients must satisfy the equality $d_{j,A_{2}}=d_{j,F}$ for this SU(3) case, and we have checked that they do.

In the LN limit (95), as discussed above in the case of the anomalous dimension $\gamma_{IR,T_{2}}$, the upper ends of the interval $I_{IRZ}$ for the $S_{2}$ and $A_{2}$ theories approach the same value, $N_{u,T_{2}}$, given in Eq. (203), and similarly the lower ends of this interval for these $S_{2}$ and $A_{2}$ theories approach the same value, $N_{\ell,T_{2}}$, given in Eq. (204). We denote

[TABLE]

and we find

[TABLE]

which we denote simply as $\hat{d}_{j,T_{2}}$. Hence,

[TABLE]

Further, again in analogy with Eq. (217) and for the same reasons concerning group invariants in the LN limit, we have

[TABLE]

From our general expressions, we calculate

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

To estimate the region over which the $\Delta_{f}$ expansion converges, we calculate the ratios of adjacent coefficients. We have

[TABLE]

and similarly for the ratios $d_{j-1,T_{2}}/d_{j,T_{2}}$ for $j=4,\ 5$. For the LN limit,

[TABLE]

[TABLE]

and

[TABLE]

Since formally, $(\Delta_{f})_{max}=3.375$ from Eq. (205) and $\Delta_{f}=5.5$ for $N_{f}=2$, these ratios indicate that the $\Delta_{f}$ expansion for the LN limit of this $R=T_{2}$ case should be reasonably accurate in the interval $I_{IRZ}$ for this case.

V IR Zero of $\beta_{\xi}$ in the LNN Limit

In this section we analyze the zeros of the rescaled five-loop beta function in the LNN limit. This elucidates further the result that we first found for a finite value of $N_{c}$, namely $N_{c}=3$, in flir , namely that for SU(3), the five-loop beta function only has a physical IR zero in the upper range of the interval $I_{IRZ}$. We denote the $n$-loop rescaled beta function (281) in this LNN limit as $\beta_{\xi,n\ell}$, and its IR zero (if such a zero exists) as $\xi_{IR,n\ell}=4\pi x_{IR,n\ell}$. The analytic expressions of $\xi_{IR,2\ell}$ and $\xi_{IR,3\ell}$ were given in lnn , together with numerical values of $\xi_{IR,n\ell}$ for $1\leq n\leq 4$. Here we extend these results to the five-loop level, using the coefficient $\hat{b}_{5}$ in Eq. (296). As noted before, we use the $\hat{b}_{n}$ with $3\leq n\leq 5$ calculated in the $\overline{\rm MS}$ scheme. The reader is referred to lnn for analysis of these zeros up to the four-loop level.

In general, the IR zero of the $n$-loop beta function, $\beta_{\xi,n\ell}$, is the positive real root closest to the origin (if such a root exists) of the equation

[TABLE]

of degree $n-1$ in the variable $x$. The roots of Eq. (365) depend on the $n-1$ ratios $\hat{b}_{\ell}/{\hat{b}_{1}}$ for $2\leq\ell\leq n$. In particular, at the five-loop level, Eq. (365) is the quartic equation

[TABLE]

To analyze the roots of this equation, it is natural to start with $r$ in the vicinity of $r_{u}=11/2$, where $\hat{b}_{1}\to 0$ and hence one solution of Eq. (366) approaches zero, matching the behavior of $x_{IR,n\ell}$ for $2\leq n\leq 4$ in this limit. As we reduce $r$ from the value $r_{u}$ in the interval $I_{IRZ,r}$, we can thus calculate how the physical IR root, $x_{IR,5\ell}=\xi_{IR,5\ell}/(4\pi)$, changes. We find that, in contrast to the behavior of the IR zero of the lower-loop beta functions $\beta_{\xi,n\ell}$ with $2\leq n\leq 4$, here at the five-loop level, as $r$ decreases past a certain value $r_{cx}$, Eq. (366) (with $\hat{b}_{n}$, $n=3,\ 4,\ 5$ calculated in the $\overline{\rm MS}$ scheme) ceases to have a physical IR zero. We find that the value of $r_{cx}$ is

[TABLE]

to the indicated floating-point accuracy. This is determined as the relevant root of the discriminant of Eq. (366), which is a polynomial of degree 15 in the variable $r$. (The discriminants of the corresponding equations at loop levels 3 and 4 are polynomials of degree 3 and 8 in $r$.) For example, for the illustrative value $r=5$, near to the upper end of the interval $I_{IRZ,r}$, Eq. (366) has the solutions in $x$, expressed in terms of $\xi=4\pi x$: $\xi=0.36300$, $1.69540$, and $-1.48884\pm 1.08446i$. Of these, we identify the first as the IR zero, $\xi_{IR,5\ell}$. As $r$ decreases and approaches $r_{cx}$ from above, the two real roots approach a common value, $\xi\simeq 1.312$ and as $r$ decreases below $r_{cx}$, Eq. (366) has only two complex-conjugate pairs of solutions, roots, but no real positive solution. In Table 10 we list our new results for $\xi_{IR,5\ell}$, in comparison with the previously calculated values of $\xi_{IR,n\ell}$ in the LNN limit with $2\leq n\leq 4$ from Table III of lnn . Although we list $\xi_{IR,n\ell}$ values extending to the lower part of the interval $I_{IRZ,r}$ for completeness, it is clear that a number of these values are too large for the perturbative calculations to be reliable. For values of $r$ where the five-loop beta function (calculated in the $\overline{\rm MS}$ scheme) has no physical IR zero, we denote this as unphysical (u).

We note that the absence of a physical IR zero in the five-loop beta function (calculated in the $\overline{\rm MS}$ scheme) for $N_{f}$ values in the lower portion of the interval $I_{IRZ}$ does not necessarily imply that higher-loop calculations would yield similarly unphysical results. We gave an example of this in Section VIII of the second paper in sch , using an illustrative exact beta function. In this example, it was shown that a certain order of truncation of the Taylor series expansion in powers of $\alpha$ for this beta function did not yield any physical IR zero, but higher orders did converge toward this zero.

VI $\Delta_{f}$ Expansion for $\alpha_{IR}$ to $O(\Delta_{f}^{4})$

VI.1 General $G$ and $R$

Since the exact $\alpha_{IR}$ (and also the $n$-loop approximation to this exact $\alpha_{IR}$) vanishes as functions of $\Delta_{f}$, it follows that one can expand it as a power series in this variable. This expansion was given above as Eq. (12), and it was noted that the calculation of the coefficient $a_{j}$ requires, as input, the $\ell$-loop beta function coefficients $b_{\ell}$ with $1\leq\ell\leq j+1$. We denote the truncation of this infinite series (12) to maximal power $j=p$ as $\alpha_{IR,\Delta_{f}^{p}}$. Here we present a calculation of this series to $O(\Delta_{f}^{4})$, which is the highest order to which it has been calculated. Since $\alpha_{IR}$ is scheme-dependent, it follows that the $a_{j}$ coefficients in Eq. (12) are also scheme-dependent, in contrast to the scheme-independent coefficients $\kappa_{j}$ and $d_{j}$ in Eqs. (2) and (3). Nevertheless, it is still worthwhile to calculate these coefficients $a_{j}$ and the resultant finite-order approximations $\alpha_{IR,\Delta_{f}^{p}}$, for several reasons. First, this method has the advantage that $\alpha_{IR,\Delta_{f}^{p}}$ is always physical and thus avoids the problem that we found in flir and have further studied above, that the five-loop beta function calculated in the $\overline{\rm MS}$ scheme does not have a physical IR zero in the lower part of the interval $I_{IRZ}$. In gsi , for the special case $G={\rm SU}(3)$ and $R=F$, we presented the $a_{j}$ (denoted $\tilde{a}_{j}$ there) for $1\leq j\leq 4$.

Here, as a new result, we present the expressions for the $a_{j}$ for arbitrary $G$ and $R$, for $1\leq j\leq 4$. For this purpose, we use the $n$-loop beta function coefficients $b_{n}$ with $3\leq n\leq 5$ calculated in the $\overline{\rm MS}$ scheme. In particular, our result for $a_{4}$ makes use of the recently calculated five-loop beta function for general $G$ and $R$ b5 .

For general $G$ and $R$, recalling the definition of the denominator factor $D=7C_{A}+11C_{f}$ in Eq. (24), we find

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We next specialize to the case $G={\rm SU}(N_{c})$ and give explicit reductions of these general formulas for the representations of interest here.

VI.2 $R=F$

For $R=F$, our general results (368)-(393) reduce to the following expressions:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We have checked that setting $N_{c}=3$ in our new $a_{4}$ coefficient in Eq. (405) yields agreement with the value that we obtained previous for this special case in (Eq. (14) of) Ref. gsi .

We comment next on the signs of these coefficients. The coefficient $a_{1}$ is manifestly positive for arbitrary group $G$ and fermion representation $R$. We find that $a_{2,F}$ and $a_{3,F}$ are also positive for all physical $N_{c}\geq 2$. In contrast, we find that $a_{4,F}$ is negative for $N_{c}=2$ and positive for $N_{c}\geq 3$. With $N_{c}$ generalized from positive integers to positive real numbers in the range $N_{c}\geq 2$, we calculate that as $N_{c}$ increases through the value $N_{c}=2.1184$ (given to the indicated accuracy), $a_{4,F}$ passes through zero with positive slope.

We list below the explicit numerical expressions for $\alpha_{IR}$ to order $\Delta_{f}^{4}$, for $N_{c}=2,\ 3,\ 4$ and $R=F$, denoted , the indicated floating-point precision:

[TABLE]

[TABLE]

and

[TABLE]

In Figs. 10-12 we show $\alpha_{IR,F,\Delta_{f}^{p}}$ for $N_{c}=2,\ 3,\ 4$ and $1\leq p\leq 4$ as a function of $N_{f}$. Note that in Fig. 10 the curves for $p=3$ and $p=4$ are so close as to be indistinguishable for this this range of $N_{f}$.

In Table 11 we compare the values of the IR zero of the $n$-loop beta function for $1\leq n\leq 4$ from bvh with our values of $\alpha_{IR,F,\Delta_{f}^{p}}$ for $1\leq p\leq 4$ and $N_{c}=2,\ 3,\ 4$. Since the calculation of $\alpha_{IR,n\ell}$ uses the $\ell$-loop beta function coefficients $b_{\ell}$ with $1\leq\ell\leq n$, while the calculation of $\alpha_{IR,\Delta_{f}^{p}}$ uses the $b_{\ell}$ for $1\leq\ell\leq p+1$, the closest comparison is of $\alpha_{IR,n\ell}$ with $\alpha_{IR,\Delta_{f}^{n-1}}$, which both use $n$-loop information from the beta function. Although, for completeness, we include values of $\alpha_{IR,2\ell}$ for $N_{f}$ extending down to the lower end of the respective intervals $I_{IRZ}$ for each value of $N_{c}$, we caution that in a number of cases, including $N_{f}=6$ for SU(2), $N_{f}=9$ for SU(3), and $10\leq N_{f}\leq 12$ for SU(4), these values of $\alpha_{IR,2\ell}$ are too large for the perturbative $n$-loop calculations to be reliable. Concerning the comparison of the higher-order $n$-loop values of $\alpha_{IR,n\ell}$ with our values of $\alpha_{IR,F,\Delta_{f}^{p}}$, we see that for a given $N_{c}$ and $N_{f}$, at the upper end of the non-Abelian Coulomb phase, the values of $\alpha_{IR,\Delta_{f}^{n-1}}$ and $\alpha_{IR,n\ell}$ are quite close to each other, but $N_{f}$ decreases in this NACP in in the interval $I_{IRZ}$, $\alpha_{IR,\Delta_{f}^{n-1}}$ becomes slightly larger than $\alpha_{IR,n\ell}$.

In the LNN limit, for the IR zero of the rescaled beta function, we write

[TABLE]

where

[TABLE]

From our results for $a_{j,F}$, we calculate

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Thus, in the LNN limit, the expansion of $\xi_{IR}$, to $O(\Delta_{r}^{4})$, is

[TABLE]

VI.3 $R=adj$

For $R=adj$, our general results (368)-(393) reduce to the following expressions:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The coefficients $a_{j,adj}$ with $j=1,\ 2,\ 4$ are manifestly positive, and we find that $a_{3,adj}$ is also positive for all $N_{c}\geq 2$.

Since for the adjoint representation, $R=adj$, the upper and lower boundaries of the interval $I_{IRZ}$, $N_{u,T_{2}}=11/2$ in Eq. (155) and $N_{\ell,adj}=17/16$ in (156), are independent of $N_{f}$, it follows that $\Delta_{f}=N_{u}-N_{f}$ is also independent of $N_{c}$. From the general formula (12), in the LN limit of a theory with fermions in a two-index representation $R_{2}$, including the adjoint and symmetric and antisymmetric tensors, we can write

[TABLE]

where

[TABLE]

From our calculations above, setting $R_{2}=adj$, we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

VI.4 $R=S_{2},\ A_{2}$

For $R$ equal to the symmetric or antisymmetric rank-2 tensor representations, $S_{2}$ and $A_{2}$, we give the reductions of our general results (368)-(393) next. As before, it is convenient to consider these together, since many terms differ only by sign reversal. As above, the upper and lower signs refer to the $S_{2}$ and $A_{2}$ representations, respectively. Also, as before, for $A_{2}$, we require that $N_{c}\geq 3$. Recalling the definition of the denominator factor $F_{\pm}$ in Eq. (175), we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The same general comments that we made before concerning factors in the $\kappa_{j,T_{2}}$ and $d_{j,T_{2}}$ coefficients also apply here. Thus, for arbitrary $j$, the $a_{j,A_{2}}$ coefficients contain at least one overall factor of $(N_{c}-2)$ and hence vanish for $N_{c}=2$, as a result of the fact that for $N_{c}=2$, the $A_{2}$ representation is a singlet, so for SU(2), fermions in the $A_{2}=$ singlet representation are free fields and hence make no contribution to the beta function. Moreover, if $N_{c}=2$, then the $S_{2}$ representation is the same as the adjoint representation, so the $a_{j}$ coefficients must satisfy the equality $a_{j,S_{2}}=a_{j,adj}$ for this SU(2) case, and we have checked that they do. Similarly, if $N_{c}=3$, then the $A_{2}$ representation is the same as the conjugate fundamental representation, $\bar{F}$, so these coefficients must satisfy the equality $a_{j,A_{2}}=a_{j,F}$ for this SU(3) case, and we have checked that they do.

We next consider the LN limit of the theory with fermions in the $S_{2}$ or $A_{2}$ representations. Using the definition (439) with $R_{2}=S_{2}$ and $R_{2}=A_{2}$, we find that

[TABLE]

so we denote these simply as $\hat{a}_{j,T_{2}}$. In general, for the same group-theoretical reasons as led to the LN relation $\hat{\kappa}_{j,T_{2}}=2^{-j}\hat{\kappa}_{j,adj}$ in Eq. (217) and the LN relation $\hat{d}_{j,T_{2}}=2^{-j}\hat{d}_{j,adj}$ in Eq. (354),we have, in the LN limit,

[TABLE]

Explicitly, we calculate

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

VII Conclusions

In conclusion, in this paper we have presented a number of new results on scheme-independent calculations of various quantities in an asymptotically free vectorial gauge theory having an IR zero of the beta function. We have presented scheme-independent series expansions of the anomalous dimension $\gamma_{\bar{\psi}\psi,IR}$ to $O(\Delta_{f}^{4})$ and the derivative of the beta function, $\beta^{\prime}_{IR}$, to $O(\Delta_{f}^{5})$ for a theory with a general gauge group $G$ and $N_{f}$ fermions in a representation $R$ of $G$. We have given reductions of our general formulas for theories with $G={\rm SU}(N_{c})$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. We have compared our scheme-independent calculations of $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ with previous $n$-loop values of these quantities calculated via series expansions in powers of the coupling. For a number of specific theories we have also compared our new scheme-independent calculations of $\gamma_{\bar{\psi}\psi,IR}$ and $\beta^{\prime}_{IR}$ with lattice measurements. We have shown that for all of the representations we have studied, and for the full range $1\leq p\leq 4$ for which we have performed calculations, $\gamma_{\bar{\psi}\psi,IR}$ calculated to $O(\Delta_{f}^{p})$, denoted $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$, increases monotonically with decreasing $N_{f}$ (i.e., increasing $\Delta_{f}$) and, for a fixed $N_{f}$, $\gamma_{\bar{\psi}\psi,IR,\Delta_{f}^{p}}$, increases monotonically with the order $p$. For the representation $R=F$, we have presented results for the limit $N_{c}\to\infty$ and $N_{f}\to\infty$ with $N_{f}/N_{c}$ fixed. These higher-order results have been applied to obtain estimates of the lower end of the (IR-conformal) non-Abelian Coulomb phase. We have confirmed and extended our earlier finding that our expansions in powers of $\Delta_{f}$ should be reasonably accurate throughout a substantial portion of the non-Abelian Coulomb phase. We have also given expansions for $\alpha_{IR}$ calculated to $O(\Delta_{f}^{4})$ which provide a useful complementary approach to calculating $\alpha_{IR}$. Our scheme-independent calculations of physical quantities at a conformal IR fixed point yield new information about the properties of a conformal field theory.

Acknowledgements.

This research was supported in part by the Danish National Research Foundation grant DNRF90 to CP3-Origins at SDU (T.A.R.) and by the U.S. NSF Grant NSF-PHY-16-1620628 (R.S.)

Appendix A Series Coefficients for $\beta_{\xi}$ and

$\gamma_{\bar{\psi}\psi}$ in the LNN Limit

For reference, we list here the rescaled series coefficients for $\beta_{\xi}$ and $\gamma_{\bar{\psi}\psi}$ in the LNN limit (90). From the (scheme-independent) one-loop and two-loop coefficients in the beta function b1 ; b2 , it follows that in the LNN limit the $\hat{b}_{\ell}$ with $\ell=1,2$ are

[TABLE]

and

[TABLE]

The coefficients $b_{3}$ and $b_{4}$ have been calculated in the $\overline{\rm MS}$ scheme b3 ; b4 . With these inputs, one has lnn

[TABLE]

and

[TABLE]

The behavior of these coefficients $\hat{b}_{\ell}$ as functions of $r$ was discussed in lnn for $1\leq\ell\leq 4$. The positivity of $\hat{b}_{1}$ is equivalent to the asymptotic freedom of the theory, and requires $r$ to lie in the interval $0\leq r<11/2$. The existence of an IR zero in the two-loop beta function is equivalent to the condition that $\hat{b}_{2}<0$, which, in turn, is equivalent to the condition that $r\in I_{IRZ,r}$ as given in Eq. (101) . In this interval, $\hat{b}_{3}$ is negative-definite, while $\hat{b}_{4}$ is negative for for $2.615<r<3.119$ and positive for $3.119<r<5.5$ lnn .

For the coefficients $\hat{c}_{\ell}$ in Eq. (108), from c4 and references therein, one has lnn

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]