Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents

TL;DR

This paper computes high-order perturbative renormalization group functions for $O(n)$-symmetric $^4$ theory, providing precise estimates of critical exponents in lower dimensions through advanced loop calculations and resummation techniques.

Contribution

It presents the sixth loop order renormalization functions and estimates for critical exponents, extending previous calculations to higher loops and including non-subdivergent diagrams up to 11 loops.

Findings

Sixth loop order renormalization group functions computed.

Critical exponents estimated in three and two dimensions.

Comparison of asymptotic beta function behaviour with diagram estimates.

Abstract

We present the perturbative renormalization group functions of $O(n)$-symmetric $\phi^4$ theory in $4-2\varepsilon$ dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

Figures

Tables

Table 1. Table 1 : The number of (isomorphism classes of) one-particle irreducible ϕ 4 superscript italic-ϕ 4 \phi^{4} Feynman graphs, excluding graphs with tadpoles (as those vanish in massless DimReg). We include the column for 7 7 7 loops just to illustrate the growth in complexity.

loops	1	2	3	4	5	6	7
2-point graphs (${\Gamma }_2$)	0	1	1	4	11	50	209
4-point graphs (${\Gamma }_4$)	1	2	8	26	124	627	3794
primitive 4-point graphs (${\Gamma }_4$)	1	0	1	1	3	10	44

Table 2. Table 2 : Numerical values for the 6 6 6 -loop field anomalous dimension.

$n$	${\gamma }_{\varphi }^{\mathrm{MS}}\left(g\right)$
$0$	$0.05556g^2-0.03704g^3+0.1929g^4-1.006g^5+7.095g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$1$	$0.08333g^2-0.06250g^3+0.3385g^4-1.926g^5+14.38g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$2$	$0.11111g^2-0.09259g^3+0.5093g^4-3.148g^5+24.71g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$3$	$0.13889g^2-0.12731g^3+0.6993g^4-4.689g^5+38.44g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$4$	$0.16667g^2-0.16667g^3+0.9028g^4-6.563g^5+55.93g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$

Table 3. Table 3 : Numerical values for the 6 6 6 -loop mass anomalous dimension.

$n$	${\gamma }_{m^2}^{\mathrm{MS}}\left(g\right)$
$0$	$-0.6667g+0.5556g^2-2.056g^3+10.76g^4-75.70g^5+636.7g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$1$	$-1.0000g+0.8333g^2-3.500g^3+19.96g^4-150.8g^5+1355g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$2$	$-1.3333g+1.1111g^2-5.222g^3+31.87g^4-255.8g^5+2434g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$3$	$-1.6667g+1.3889g^2-7.222g^3+46.64g^4-394.9g^5+3950g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$
$4$	$-2.0000g+1.6667g^2-9.500g^3+64.39g^4-571.9g^5+5983g^6+𝒪\Leftarrow {\}}^{\mapsto }\Rightarrow$

Table 4. Table 4 : Numerical values for the 6 6 6 -loop beta function.

$n$	${\beta }^{\mathrm{MS}}\left(g\right)$
$0$	$-2\epsilon g+2.667g^2-4.667g^3+25.46g^4-200.9g^5+2004g^6-23315g^7+𝒪\Leftarrow {\}}^{\forall }\Rightarrow$
$1$	$-2\epsilon g+3.000g^2-5.667g^3+32.55g^4-271.6g^5+2849g^6-34776g^7+𝒪\Leftarrow {\}}^{\forall }\Rightarrow$
$2$	$-2\epsilon g+3.333g^2-6.667g^3+39.95g^4-350.5g^5+3845g^6-48999g^7+𝒪\Leftarrow {\}}^{\forall }\Rightarrow$
$3$	$-2\epsilon g+3.667g^2-7.667g^3+47.65g^4-437.6g^5+4999g^6-66243g^7+𝒪\Leftarrow {\}}^{\forall }\Rightarrow$
$4$	$-2\epsilon g+4.000g^2-8.667g^3+55.66g^4-533.0g^5+6318g^6-86768g^7+𝒪\Leftarrow {\}}^{\forall }\Rightarrow$

Table 5. Table 5 : Our 6 6 6 -loop coefficients of the RG functions, compared to asymptotic Padé-approximant predictions from EllisKarlinerSamuel:betaQCDprediction ; ChishtieEliasSteele:MassivePhi4 . The errors are given as ( APAP − exact ) / exact APAP exact exact (\text{APAP}-\text{exact})/\text{exact} .

$n$	0	1	2	3	4	5
${\beta }_7^{\mathrm{MS}}$	23315	34776	48999	66243	86768	110840
APAP EllisKarlinerSamuel:betaQCDprediction	23656	35374	49916	67604	88660
error	1.46%	1.72%	1.87%	2.06 %	2.18%
APAP ChishtieEliasSteele:MassivePhi4		35233	49381	66426	86636	110292
error		1.31%	0.78%	0.28%	-0.15%	-0.50%
${\gamma }_{m^2}^{\mathrm{MS}}$		1355	2434	3950	5983	8618
APAP ChishtieEliasSteele:MassivePhi4		1478	2740	4803	9476	-3374
error		9%	13%	22%	58%	-139%
${\gamma }_{\varphi }^{\mathrm{MS}}$		14.4	24.7	38.4	55.9	77.5
APAP ChishtieEliasSteele:MassivePhi4		11.2	20.7	35.0	56.2	87.3
error		-22.0%	-16.3%	-8.9%	0.5%	12.6%

Table 6. Table 6 : The upper part of the table shows the ℓ ℓ \ell -loop coefficients β ℓ + 1 MS subscript superscript 𝛽 MS ℓ 1 \beta^{\mathrm{MS}}_{\ell+1} of the ϕ 4 superscript italic-ϕ 4 \phi^{4} beta function β MS = ∑ k β k MS ​ ( − g ) k superscript 𝛽 MS subscript 𝑘 subscript superscript 𝛽 MS 𝑘 superscript 𝑔 𝑘 \beta^{\mathrm{MS}}=\sum_{k}\beta^{\mathrm{MS}}_{k}(-g)^{k} (with n = 1 𝑛 1 n=1 ) and their ratios to the asymptotic formulas ( 20 ) and ( 23 ). In the lower two rows we show the contribution β ℓ + 1 prim subscript superscript 𝛽 prim ℓ 1 \beta^{\mathrm{prim}}_{\ell+1} containing only the primitive graphs, and their proportion of the full ℓ ℓ \ell -loop coefficient.

loop order $\ell$	1	2	3	4	5	6
${\beta }_{\ell +1}^{\mathrm{MS}}/{\overline{\beta }}_{\ell +1}$ in %	548	83.5	43.8	33.5	30.9	31.4
${\beta }_{\ell +1}^{\mathrm{MS}}/{\tilde{\beta }}_{\ell +1}$ in %	43.1	12.5	9.58	9.41	10.4	12.1
${\beta }_{\ell +1}^{\mathrm{MS}}$	3	5.67	32.5	272	2849	34776
${\beta }_{\ell +1}^{\mathrm{prim}}$	3	0	14.4	124	1698	24130
${\beta }_{\ell +1}^{\mathrm{prim}}/{\beta }_{\ell +1}^{\mathrm{MS}}$ in %	100	0	44.3	45.8	59.6	69.4

Table 7. Table 7 : The first zeros of the exactly known perturbative coefficients β ℓ + 1 MS subscript superscript 𝛽 MS ℓ 1 \beta^{\mathrm{MS}}_{\ell+1} from ( 19 ) as functions of n 𝑛 n , for ℓ ≤ 6 ℓ 6 \ell\leq 6 loops, and similarly for our estimates β ℓ + 1 prim subscript superscript 𝛽 prim ℓ 1 \beta^{\mathrm{prim}}_{\ell+1} for the primitive contributions from Appendix  B (up to 11 11 11 loops).

	loop order $\ell$	first zero	second zero	third zero
\ldelim{6*[${\beta }_{\ell +1}^{\mathrm{MS}}\left(n\right)$]	1	-8
	2	-4.67
	3	-4.025	-41.4
	4	-4.020	-12.1	3219
	5	-4.0017	-8.76	-44.0
	6	-4.00044	-7.52	-20.0
\ldelim{6*[${\beta }_{\ell +1}^{\mathrm{prim}}\left(n\right)$]	6	-3.99754	-7.22	-35.6
	7	-3.99982	-6.58	-15.1
	8	-3.99994	-6.31	-10.8
	9	-3.999997	-6.18	-9.24
	10	-3.99999991	-6.10	-8.55
	11	-4.000000095	-6.05	-8.21

Table 8. Table 8 : Numerical 6 6 6 -loop ε 𝜀 \varepsilon -expansion of the critical exponent η 𝜂 \eta in D = 4 − 2 ​ ε 𝐷 4 2 𝜀 D=4-2\varepsilon dimensions.

$n$	$\eta \left(\epsilon \right)$
$0$	$0.062500{\epsilon }^2+0.13281{\epsilon }^3-0.13388{\epsilon }^4+0.84815{\epsilon }^5-5.8067{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$1$	$0.074074{\epsilon }^2+0.14952{\epsilon }^3-0.13326{\epsilon }^4+0.82101{\epsilon }^5-5.2014{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$2$	$0.080000{\epsilon }^2+0.15200{\epsilon }^3-0.12630{\epsilon }^4+0.74269{\epsilon }^5-4.3921{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$3$	$0.082645{\epsilon }^2+0.14719{\epsilon }^3-0.11919{\epsilon }^4+0.65225{\epsilon }^5-3.6495{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$4$	$0.083333{\epsilon }^2+0.13889{\epsilon }^3-0.11336{\epsilon }^4+0.56421{\epsilon }^5-3.0312{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$

Table 9. Table 9 : Numerical 6 6 6 -loop ε 𝜀 \varepsilon -expansion of the critical exponent 1 / ν 1 𝜈 1/\nu in D = 4 − 2 ​ ε 𝐷 4 2 𝜀 D=4-2\varepsilon dimensions.

$n$	${\nu }^{-1}\left(\epsilon \right)$
$0$	$2.0000-0.50000\epsilon -0.34375{\epsilon }^2+0.91540{\epsilon }^3-4.6002{\epsilon }^4+30.596{\epsilon }^5-246.77{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$1$	$2.0000-0.66667\epsilon -0.46914{\epsilon }^2+0.99622{\epsilon }^3-4.9096{\epsilon }^4+30.440{\epsilon }^5-228.65{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$2$	$2.0000-0.80000\epsilon -0.56000{\epsilon }^2+0.97949{\epsilon }^3-4.8756{\epsilon }^4+28.136{\epsilon }^5-198.59{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$3$	$2.0000-0.90909\epsilon -0.62359{\epsilon }^2+0.92057{\epsilon }^3-4.6976{\epsilon }^4+25.278{\epsilon }^5-168.91{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$4$	$2.0000-1.00000\epsilon -0.66667{\epsilon }^2+0.84684{\epsilon }^3-4.4586{\epsilon }^4+22.469{\epsilon }^5-142.96{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$

Table 10. Table 10 : Numerical 6 6 6 -loop ε 𝜀 \varepsilon -expansion of the correction to scaling ω 𝜔 \omega in D = 4 − 2 ​ ε 𝐷 4 2 𝜀 D=4-2\varepsilon dimensions.

$n$	$\omega \left(\epsilon \right)$
$0$	$2.0000\epsilon -2.6250{\epsilon }^2+14.589{\epsilon }^3-100.57{\epsilon }^4+859.94{\epsilon }^5-8320.5{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$1$	$2.0000\epsilon -2.5185{\epsilon }^2+12.946{\epsilon }^3-83.762{\epsilon }^4+663.99{\epsilon }^5-5959.1{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$2$	$2.0000\epsilon -2.4000{\epsilon }^2+11.497{\epsilon }^3-70.724{\epsilon }^4+523.96{\epsilon }^5-4401.7{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$3$	$2.0000\epsilon -2.2810{\epsilon }^2+10.263{\epsilon }^3-60.498{\epsilon }^4+421.82{\epsilon }^5-3341.1{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$
$4$	$2.0000\epsilon -2.1667{\epsilon }^2+9.2207{\epsilon }^3-52.351{\epsilon }^4+345.65{\epsilon }^5-2596.3{\epsilon }^6+𝒪\Leftarrow {\epsilon }^{\mapsto }\Rightarrow$

Table 11. Table 11 : Estimates for critical exponents in D = 3 𝐷 3 D=3 dimensions of the O ​ ( n ) 𝑂 𝑛 O(n) vector model. Results from the conformal bootstrap and Monte Carlo techniques are listed first (we tried to collect the most accurate predictions in each case). Our estimates from the 5 5 5 - and 6 6 6 -loop ε 𝜀 \varepsilon -expansions are shown next. For comparison of the resummation methods, we display the 5 5 5 -loop results (from ε 𝜀 \varepsilon -expansion without D = 2 𝐷 2 D=2 boundary conditions) according to GuidaZinnJustin:CriticalON .

		$n=0$	$n=1$	$n=2$	$n=3$	$n=4$
\ldelim{4*[$\eta$]		0.031043(3)191919From $\gamma =1.156953\left(1\right)$ Clisby:ScaleFreeSAW and $\nu =0.5875970\left(4\right)$ ClisbyDuenweg:Hydrodynamic via $\gamma =\nu \left(2-\eta \right)$ in (10).	0.036298(2)KosPolandDuffinVichi:Islands	0.0381(2)CampostriniHasenbuschPelissettoVicari:4He	0.0378(3)HasenbuschVicari:Anisotropic	0.0360(3)202020Given in HasenbuschVicari:Anisotropic and compatible with $0.0365\left(10\right)$ Hasenbusch:EliminatingN34 and $y_h=\left(5-\eta \right)/2=2.4820\left(2\right)$ in Deng:O(4) .
	${\epsilon }^6$	0.0310(7)	0.0362(6)	0.0380(6)	0.0378(5)	0.0366(4)
	${\epsilon }^5$	0.0314(11)	0.0366(11)	0.0384(10)	0.0382(10)	0.0370(9)
	GuidaZinnJustin:CriticalON	0.0300(50)	0.0360(50)	0.0380(50)	0.0375(45)	0.036(4)
\ldelim{4*[$\nu$]		0.5875970(4)ClisbyDuenweg:Hydrodynamic	0.629971(4)KosPolandDuffinVichi:Islands	0.6717(1)CampostriniHasenbuschPelissettoVicari:4He	0.7112(5)CampostriniHasenbuschPelissettoRossiVicari:Heisenberg	0.7477(8)212121From $y_t=1/\nu =1.3375\left(15\right)$ in Deng:O(4) , compatible with $\nu =0.749\left(2\right)$ Hasenbusch:EliminatingN34 and $0.750\left(2\right)$ HasenbuschVicari:Anisotropic .
	${\epsilon }^6$	0.5874(3)	0.6292(5)	0.6690(10)	0.7059(20)	0.7397(35)
	${\epsilon }^5$	0.5873(13)	0.6290(20)	0.6687(13)	0.7056(16)	0.7389(24)
	GuidaZinnJustin:CriticalON	0.5875(25)	0.6290(25)	0.6680(35)	0.7045(55)	0.737(8)
\ldelim{4*[$\omega$]		0.904(5)222222Computed from $\omega \nu =\Delta =0.531\left(3\right)$ according to Belohorec:PhD and $\nu =0.5875970\left(4\right)$ in ClisbyDuenweg:Hydrodynamic .	0.830(2)3dIsingBootstrapII	0.811(10)e	0.791(22)e	0.817(30)232323These are the results given as ${\Delta }_{S^{\prime }}=3+\omega$ in (EcheverriHarlingSerone:EffectiveBootstrap, , Table 2).
	${\epsilon }^6$	0.841(13)	0.820(7)	0.804(3)	0.795(7)	0.794(9)
	${\epsilon }^5$	0.835(11)	0.818(8)	0.803(6)	0.797(7)	0.795(6)
	GuidaZinnJustin:CriticalON	0.828(23)	0.814(18)	0.802(18)	0.794(18)	0.795(30)

Table 12. Table 12 : Previous estimates for D = 2 𝐷 2 D=2 dimensions from the 5 5 5 -loop ε 𝜀 \varepsilon -expansion according to GuillouZinnJustin:Accurate , our results for the resummation of the same series and also for six loops. The first row for the exponents η 𝜂 \eta and ν 𝜈 \nu shows the exact values for the Ising model (column n = 1 𝑛 1 n=1 ) due to Onsager Onsager:CrystalStatisticsI and the conjectures of Nienhuis Nienhuis:ExactO(n) in the cases n = − 1 𝑛 1 n=-1 and n = 0 𝑛 0 n=0 . We also report theoretical expectations for the correction to scaling exponent ω 𝜔 \omega .

		$n=-1$	$n=0$	$n=1$
\ldelim{4*[$\eta$]	Nienhuis:ExactO(n)	$0.15$	$0.208333\dots$	$0.25$
	${\epsilon }^6$	0.130(17)	0.201(25)	0.237(27)
	${\epsilon }^5$	0.137(23)	0.215(35)	0.249(38)
	GuillouZinnJustin:Accurate		0.21(5)	0.26(5)
\ldelim{4*[$\nu$]	Nienhuis:ExactO(n)	$0.625$	$0.75$	$1$
	${\epsilon }^6$	0.6036(23)	0.741(4)	0.952(14)
	${\epsilon }^5$	0.6025(27)	0.747(20)	0.944(48)
	GuillouZinnJustin:Accurate		0.76(3)	0.99(4)
\ldelim{4*[$\omega$]			2Nienhuis:ExactO(n) ; CGJPRS:Scaling2dSAW	1.75CalabreseCaselleCeliPelissettoVicari:NonAnalyticity
	${\epsilon }^6$	1.95(28)	1.90(25)	1.71(9)
	${\epsilon }^5$	1.88(30)	1.83(25)	1.66(11)
	GuillouZinnJustin:Accurate		1.7(2)	1.6(2)

Table 13. Table 13 : The upper part shows the counts of completed primitive ϕ 4 superscript italic-ϕ 4 \phi^{4} graphs and the estimates for the primitive contributions β ℓ + 1 prim superscript subscript 𝛽 ℓ 1 prim \beta_{\ell+1}^{\mathrm{prim}} to the beta function of the ϕ 4 superscript italic-ϕ 4 \phi^{4} model ( n = 1 𝑛 1 n=1 ) at ℓ ℓ \ell -loop order obtained with the Hepp bounds ( 55 ). The lower part of the table shows the fitting parameters used in the approximation ( 56 ).

loop order $\ell$	6	7	8	9	10	11
completions	5	14	49	227	1354	9722
${\beta }_{\ell +1}^{\mathrm{prim}}$ estimate	$2.41\cdot {10}^4$	$3.71\cdot {10}^5$	$6.06\cdot {10}^6$	$1.05\cdot {10}^8$	$1.89\cdot {10}^9$	$3.57\cdot {10}^{10}$
${\beta }_{\ell +1}^{\mathrm{prim}}/{\overline{\beta }}_{\ell +1}^{\mathrm{MS}}$	21.8%	26.2%	31.6%	37.6%	44.3%	51.5%
$a_{\ell }$	$9.78/{10}^5$	$2.44/{10}^5$	$5.22/{10}^6$	$1.16/{10}^6$	$2.34/{10}^7$	$5.11/{10}^8$
$b_{\ell }$	1.419	1.395	1.389	1.382	1.382	1.378
$c_{\ell }$	$2.93/{10}^6$	$3.48/{10}^7$	$4.98/{10}^8$	$6.87/{10}^9$	$1.04/{10}^9$	$1.46/{10}^{10}$

Equations

$$\mathcal{L}=\frac{1}{2}{m}^2{Z}_1{\varphi }^2+\frac{1}{2}{Z}_2{\left(\partial \varphi \right)}^2+\frac{16{\pi }^2}{4!}{Z}_{4}g{\mu }^{2\epsilon }{\varphi }^4$$

$$Z_{\varphi }=\frac{{\varphi }_0}{\varphi }=\sqrt{Z_2},Z_{m^2}=\frac{m_0^2}{m^2}=\frac{Z_1}{Z_2}\text{and}{Z}_g=\frac{g_0}{{\mu }^{2\epsilon }g}=\frac{Z_4}{Z_2^2}.$$

$$Z_i=Z_i(g,\epsilon )=1+\sum _{k=1}^{\infty }\frac{Z_{i,k}(g)}{{\epsilon }^k},$$

$$\beta (g,\epsilon ):={\mu \frac{\partial g}{\partial \mu }|}_{g_0}=-2\epsilon {\left(\frac{\partial \log (g{Z}_g)}{\partial g}\right)}^{-1}=-2\epsilon g+2g^2\frac{\partial Z_{g,1}(g)}{\partial g},$$

$${\gamma }_i(g):={\mu \frac{\partial \log Z_i}{\partial \mu }|}_{g_0,m_0,{\varphi }_0}=\beta (g)\frac{\partial \log Z_i(g)}{\partial g}=-2g\frac{\partial Z_{i,1}(g)}{\partial g}\text{for}i=m^2,\varphi .$$

$$\beta (g,\epsilon )=-2\epsilon g+\frac{n+8}{3}{g}^2+\text{oldcal}O\left(g^3\right)$$

$$g_{\star }(\epsilon )=\frac{6}{n+8}\epsilon +\text{oldcal}O\left({\epsilon }^2\right)\text{such that}\beta (g_{\star }(\epsilon ),\epsilon )=0.$$

$$\omega (\epsilon ):={\beta }^{\prime }(g_{\star }(\epsilon ),\epsilon )={\frac{\partial }{\partial g}|}_{g=g_{\star }(\epsilon )}\beta (g,\epsilon )=2\epsilon +\text{oldcal}O\left({\epsilon }^2\right)$$

$$\eta (\epsilon ):=2{\gamma }_{\varphi }(g_{\star }(\epsilon ))\text{and}\nu (\epsilon ):={\left[2+{\gamma }_{m^2}(g_{\star }(\epsilon ))\right]}^{-1},$$

$${\gamma }_{\varphi }=\frac{n+2}{36}{g}^2+\text{oldcal}O\left(g^3\right)\text{and}{\gamma }_{m^2}=-\frac{n+2}{3}g+\text{oldcal}O\left(g^2\right),$$

$$\eta (\epsilon )=\frac{2(n+2){\epsilon }^2}{(n+8{)}^2}+\text{oldcal}O\left({\epsilon }^3\right)\text{and}\nu (\epsilon )=\frac{1}{2}+\frac{(n+2)\epsilon }{2(n+8)}+\text{oldcal}O\left({\epsilon }^2\right).$$

$$\gamma =\nu (2-\eta ),D\nu =2-\alpha ,\beta \delta =\beta +\gamma \text{and}\alpha +2\beta +\gamma =2.$$

$$\begin{array}{rl}{Z}_1& =1+{\partial }_{m^2}\text{oldcal}K\text{oldcal}{R}^{\prime }{\Gamma }_2(p,m^2,g,\mu ),\\ Z_2& =1+{\partial }_{p^2}\text{oldcal}K\text{oldcal}{R}^{\prime }{\Gamma }_2(p,m^2,g,\mu )\text{and}\\ Z_4& =1+\text{oldcal}K\text{oldcal}{R}^{\prime }{\Gamma }_4(p,m^2,g,\mu )/g.\end{array}$$

$$\text{oldcal}K\left(\sum _n{c}_n{\epsilon }^n\right):=\sum _{n<0}{c}_n{\epsilon }^n.$$

$${\mathcal{C}}_G=\sum _{i_1,\dots ,i_{E(G)}=1}^n\prod _{v\in V(G)}\lambda \left(i_{v^{[1]}},\dots i_{v^{[4]}}\right)$$

$$\lambda (a,b,c,d):=\frac{{\delta }_{a,b}{\delta }_{c,d}+{\delta }_{a,c}{\delta }_{b,d}+{\delta }_{a,d}{\delta }_{b,c}}{3}.$$

$${\mathcal{C}}_G={\mathcal{C}}_{\hat{G}}\cdot \{\begin{array}{ll}\frac{1}{n}& \text{if G is a propagator graph, and}\\ \frac{3}{n(n+2)}& \text{if G is a vertex graph.}\end{array}$$

$$\begin{split}\gamma^{\mathrm{MS}}_{\phi}(g)&=\frac{n+2}{36}g^{2}-\frac{(n+8)(n+2)}{432}g^{3}-\frac{5(n^{2}-18n-100)(n+2)}{\numprint{5184}}g^{4}\\ &-\Big{[}\numprint{1152}(5n+22)\zeta_{4}-\numprint{48}(n^{3}-6n^{2}+64n+184)\zeta_{3}\\ &\quad+(39n^{3}+296n^{2}+\numprint{22752}n+\numprint{77056})\Big{]}\frac{(n+2)g^{5}}{\numprint{186624}}\\ &-\Big{[}\numprint{512}(2n^{2}+55n+186)\zeta^{2}_{3}-\numprint{6400}(2n^{2}+55n+186)\zeta_{6}\\ &\quad+\numprint{4736}(n+8)(5n+22)\zeta_{5}-\numprint{48}(n^{4}+2n^{3}+328n^{2}+\numprint{4496}n+\numprint{12912})\zeta_{4}\\ &\quad+\numprint{16}(n^{4}-936n^{2}-\numprint{4368}n-\numprint{18592})\zeta_{3}\\ &\quad+(29n^{4}+794n^{3}-\numprint{30184}n^{2}-\numprint{549104}n-\numprint{1410544})\Big{]}\frac{(n+2)g^{6}}{\numprint{746496}}+\oldcal{O}\left(g^{7}\right)\end{split}$$

$${\zeta }_{3,5}:=\sum _{1\le n<m}\frac{1}{n^3{m}^5}\approx 0.037707673,$$

$$\begin{split}\gamma^{\mathrm{MS}}_{m^{2}}(g)&=-\frac{(n+2)g}{3}+\frac{5(n+2)g^{2}}{18}-\frac{(5n+37)(n+2)g^{3}}{36}\\ &+\Big{[}\numprint{288}(5n+22)\zeta_{4}+\numprint{48}(3n^{2}+10n+68)\zeta_{3}-(n^{2}-\numprint{7578}n-\numprint{31060})\Big{]}\frac{(n+2)g^{4}}{\numprint{7776}}\\ &-\Big{[}\numprint{9600}(2n^{2}+55n+186)\zeta_{6}-\numprint{768}(2n^{2}+145n+582)\zeta^{2}_{3}-\numprint{768}(5n^{2}-14n-72)\zeta_{5}\\ &\quad-\numprint{288}(3n^{3}-29n^{2}-816n-\numprint{2668})\zeta_{4}+\numprint{48}(17n^{3}+940n^{2}+\numprint{8208}n+\numprint{31848})\zeta_{3}\\ &\quad+(21n^{3}+\numprint{45254}n^{2}+\numprint{1077120}n+\numprint{3166528})\Big{]}\frac{(n+2)g^{5}}{\numprint{186624}}\\ &+\Big{[}\numprint{1152}(14n^{2}+189n+526)(2063\zeta_{8}-144\zeta_{3,5})-\numprint{921600}(15n^{2}+239n+718)\zeta_{3}\zeta_{5}\\ &\quad-\numprint{640}(\numprint{1080}n^{3}+\numprint{25180}n^{2}+\numprint{284525}n+\numprint{814062})\zeta_{7}\\ &\quad-\numprint{15360}(2n^{3}-157n^{2}-\numprint{2512}n-\numprint{8268})\zeta_{3}\zeta_{4}\\ &\quad+\numprint{5120}(27n^{3}+\numprint{1082}n^{2}+\numprint{13072}n+\numprint{40008})\zeta^{2}_{3}\\ &\quad+\numprint{3200}(285n^{3}+\numprint{7178}n^{2}+\numprint{73768}n+\numprint{196032})\zeta_{6}\\ &\quad+\numprint{320}(45n^{4}+\numprint{3622}n^{3}+\numprint{12202}n^{2}+\numprint{207708}n+\numprint{753040})\zeta_{5}\\ &\quad-\numprint{240}(47n^{4}+\numprint{2606}n^{3}-\numprint{5480}n^{2}-\numprint{194320}n-\numprint{489328})\zeta_{4}\\ &\quad-\numprint{80}(51n^{4}-\numprint{9208}n^{3}-\numprint{419076}n^{2}-\numprint{3342688}n-\numprint{8997136})\zeta_{3}\\ &\quad-\numprint{5}(43n^{4}+\numprint{48234}n^{3}-\numprint{5154216}n^{2}-\numprint{63140784}n-\numprint{145482928})\Big{]}\frac{(n+2)g^{6}}{\numprint{9331200}}\\ &+\oldcal{O}\left(g^{7}\right)\end{split}$$

$$\begin{split}\beta^{\mathrm{MS}}(g)&=-2\varepsilon g+\frac{n+8}{3}g^{2}-\frac{3n+14}{3}g^{3}+\frac{96(5n+22)\zeta_{3}+33n^{2}+922n+\numprint{2960}}{216}g^{4}\\ &-\Big{[}\numprint{1920}(2n^{2}+55n+186)\zeta_{5}-288(n+8)(5n+22)\zeta_{4}\\ &\quad+96(63n^{2}+764n+\numprint{2332})\zeta_{3}-(5n^{3}-\numprint{6320}n^{2}-\numprint{80456}n-\numprint{196648})\Big{]}\frac{g^{5}}{\numprint{3888}}\\ &+\Big{[}\numprint{112896}(14n^{2}+189n+526)\zeta_{7}-768(6n^{3}+59n^{2}-446n-\numprint{3264})\zeta^{2}_{3}\\ &\quad-\numprint{9600}(n+8)(2n^{2}+55n+186)\zeta_{6}+256(305n^{3}+\numprint{7466}n^{2}+\numprint{66986}n+\numprint{165084})\zeta_{5}\\ &\quad-288(63n^{3}+\numprint{1388}n^{2}+\numprint{9532}n+\numprint{21120})\zeta_{4}\\ &\quad-16(9n^{4}-\numprint{1248}n^{3}-\numprint{67640}n^{2}-\numprint{552280}n-\numprint{1314336})\zeta_{3}\\ &\quad+(13n^{4}+\numprint{12578}n^{3}+\numprint{808496}n^{2}+\numprint{6646336}n+\numprint{13177344})\Big{]}\frac{g^{6}}{\numprint{62208}}\\ &-\Big{[}\numprint{204800}(\numprint{1819}n^{3}+\numprint{97823}n^{2}+\numprint{901051}n+\numprint{2150774})\zeta_{9}\\ &\quad+\numprint{14745600}(n^{3}+65n^{2}+619n+\numprint{1502})\zeta^{3}_{3}\\ &\quad+\numprint{995328}(42n^{3}+\numprint{2623}n^{2}+\numprint{25074}n+\numprint{59984})\zeta_{3,5}\\ &\quad-\numprint{20736}(\numprint{28882}n^{3}+\numprint{820483}n^{2}+\numprint{6403754}n+\numprint{14174864})\zeta_{8}\\ &\quad-\numprint{5529600}(8n^{3}-635n^{2}-\numprint{9150}n-\numprint{25944})\zeta_{3}\zeta_{5}\\ &\quad+\numprint{11520}(440n^{4}+\numprint{126695}n^{3}+\numprint{2181660}n^{2}+\numprint{14313152}n+\numprint{29762136})\zeta_{7}\\ &\quad+\numprint{207360}(n+8)(6n^{3}+59n^{2}-446n-\numprint{3264})\zeta_{3}\zeta_{4}\\ &\quad-\numprint{23040}(188n^{4}+132n^{3}-\numprint{93363}n^{2}-\numprint{862604}n-\numprint{2207484})\zeta^{2}_{3}\\ &\quad-\numprint{28800}(595n^{4}+\numprint{20286}n^{3}+\numprint{277914}n^{2}+\numprint{1580792}n+\numprint{2998152})\zeta_{6}\\ &\quad+\numprint{5760}(\numprint{4698}n^{4}+\numprint{131827}n^{3}+\numprint{2250906}n^{2}+\numprint{14657556}n+\numprint{29409080})\zeta_{5}\\ &\quad+\numprint{2160}(9n^{5}-\numprint{1176}n^{4}-\numprint{88964}n^{3}-\numprint{1283840}n^{2}-\numprint{6794096}n-\numprint{12473568})\zeta_{4}\\ &\quad-720(33n^{5}+\numprint{2970}n^{4}-\numprint{477740}n^{3}-\numprint{10084168}n^{2}-\numprint{61017200}n-\numprint{117867424})\zeta_{3}\\ &\quad-45(29n^{5}+\numprint{22644}n^{4}-\numprint{3225892}n^{3}-\numprint{88418816}n^{2}-\numprint{536820560}n-\numprint{897712992})\Big{]}\frac{g^{7}}{\numprint{41990400}}\\ &+\oldcal{O}\left(g^{8}\right)\end{split}$$

$${\beta }_k^{\mathrm{MS}}\sim {\overline{\beta }}_k:=k!\cdot k^{3+n/2}\cdot C_{\beta }\text{as k→∞}$$

$$C_{\beta }=\frac{36\cdot 3^{(n+1)/2}}{\pi \Gamma (2+n/2)A^{2n+4}}\exp \left[-\frac{3}{2}-\frac{n+8}{3}\left({\gamma }_{\text{E}}+\frac{3}{4}\right)\right].$$

$$\ln A=\frac{1}{12}-{\zeta }^{\prime }(-1)=\frac{1}{12}\left({\gamma }_{\text{E}}+\ln (2\pi )-\frac{{\zeta }^{\prime }(2)}{\zeta (2)}\right).$$

$${\tilde{\beta }}_k:=\Gamma (k+4+n/2)\cdot C_{\beta },$$

$${\beta }_k^{\mathrm{MS}}\sim k!\cdot k^{3+n/2}\cdot C_{\beta }\left(1+\sum _{j=1}^{\infty }\frac{a_j}{k^j}\right)\text{as k→∞}$$

$$f_k\sim C_f\cdot k!\cdot a^k\cdot k^{b_f}\text{as}k\to \infty .$$

$$a=\frac{3}{n+8}\text{and}{b}_f=\{\begin{array}{ll}3+n/2& \text{for f=η,}\\ 4+n/2& \text{for f=ν−1 and}\\ 5+n/2& \text{for f=ω.}\end{array}$$

$${\mathfrak{B}}_f^b(x):=\sum _{k=0}^{\infty }\frac{f_k}{\Gamma (k+b+1)}(-x{)}^k.$$

$$\tilde{f}(\epsilon ):={\int }_0^{\infty }{t}^b{e}^{-t}{\mathfrak{B}}_f^b(2\epsilon t)\mathrm{d}t$$

Full text

I Introduction

The field-theoretic renormalization group approach Vasilev ; ZinnJustin ; KleinertSchulteFrohlinde:CriticalPhi4 has a long and successful history in the study of critical phenomena, going back to the famous $\varepsilon$-expansion WilsonFisher:3.99 . In particular, it predicts critical exponents of second order phase transitions with high accuracy LeGuillouZinnJustin:n3D when combined with resummation methods LeGuillouZinnJustin:CriticalFromField . More specifically, one can extract approximate exponents for three dimensional $O(n)$ universality classes from the renormalization group functions of $\phi^{4}$ theory in $4-2\varepsilon$ dimensions.111Another approach renormalizes the theory directly in three dimensions LeGuillouZinnJustin:n3D ; BakerNickelGreenMeiron:Ising3 .

Considerable effort has thus been invested in the calculation of the latter to increasingly high orders in perturbation theory.

After the results BrezinLeGuillouZinnJustinNickel:HigherOrder ; VladimirovKazakovTarasov:Calculation for three and four loops, the computation ChetyrkinKataevTkachov:5loopPhi4 ; ChetyrkinGorishnyLarinTkachov:5loopPhi4 ; Kazakov:MethodOfUniqueness of the fifth order yielded highly accurate critical exponents GuillouZinnJustin:Accurate . The subsequent correction KNFCL:5loopPhi4 of the perturbative result, which only very recently was confirmed by numeric methods AdzhemyanKompaniets:5loopNumerical , affected the resummed exponents only marginally GuidaZinnJustin:CriticalON .

For many years, the renormalization group method in $3$ dimensions provided the most accurate theoretical predictions for critical exponents, consistent with the only slightly less precise results from the fifth order $\varepsilon$-expansion GuidaZinnJustin:CriticalON . However, considerable progress of other techniques has by now produced a multitude of much more refined results. Among those, we like to point out the particularly astonishing performance of the conformal bootstrap program EcheverriHarlingSerone:EffectiveBootstrap ; KosPolandDuffinVichi:Islands and Monte Carlo methods ClisbyDuenweg:Hydrodynamic ; Clisby:ScaleFreeSAW , which reached unprecedented accuracy in some cases.

It was therefore overdue to improve on the $\varepsilon$-expansion, which had been stuck at five loops for 25 years. Finally, the 6-loop result for the field anomalous dimension was published in BatkovichKompanietsChetyrkin:6loop , and we provided the complete set of renormalization group functions for $\phi^{4}$ theory ($n=1$) in KompanietsPanzer:LL2016 . These were obtained in a Feynman diagram computation, that became feasible through the automatization of new techniques Panzer:HyperIntAlgorithms ; Brown:TwoPoint ; BrownKreimer:AnglesScales to calculate Feynman integrals, very briefly summarized in section III.

Here, we present the six loop renormalization group functions for arbitrary values of $n$, in the minimal subtraction scheme. The exact (and slightly unwieldy) expressions are given in section IV, together with tables of numeric values for the most interesting cases $n\in\left\{0,1,2,3,4\right\}$. We then discuss numerous checks of our result and like to stress in particular the confirmation of the beta function and the field anomalous dimension, at $n=1$, by Oliver Schnetz Schnetz:NumbersAndFunctions . Furthermore, we compare the coefficients of the beta function, supplemented by estimates up to 11 loops, with the expected asymptotic behaviour. This analysis confirms the known fact that the convergence is rather slow in absolute terms, but it also shows that the qualitative trend is correct and we observe a striking pattern of zeros in the dependence on $n$, in agreement with the asymptotic prediction.

In section V, we give the $\varepsilon$-expansions for the critical exponents and recall the Borel resummation method with conformal mapping, which was employed to great effect in VladimirovKazakovTarasov:Calculation ; GuillouZinnJustin:Accurate . Since this basic idea can be implemented in many different ways and incorporates several arbitrary parameters, we include a rather detailed discussion of its characteristics. Our resummation algorithm is accurately defined in section V.3, followed by a discussion of how we estimate the errors.

Finally, our resummed results for the critical exponents in three and two dimensions are summarized and discussed in sections VI and VII. The reader only interested in these results will find them in tables 11 and 12. In short, we find increased accuracy (in comparison to the five loop resummation) and good agreement with results from other methods. While the record precision for $\eta$ and $\nu$ in the cases $n=0$ and $n=1$ from bootstrap and Monte Carlo methods is clearly out of reach, the $\varepsilon$-expansion seems superior for the correction to scaling exponent $\omega$ and is on par for the Fisher exponent $\eta$ when $n\geq 2$. Values for $\nu$ tend to be low in comparison with simulations in $3$ dimensions and the theoretical predictions in $2$ dimensions.

In our conclusion VIII, we anticipate that the upcoming $7$-loop renormalization Schnetz:NumbersAndFunctions is very likely to result in estimates with smaller uncertainties, which will provide even stronger tests of the compatibility of different theoretical approaches.

We hope that our results will be useful for further analyses. In particular, it would be interesting to compare our critical exponents with other resummation methods applied to the six loop series. Another application might be to probe the asymptotic behaviour of the renormalization group functions, as in Shrock:6loopPhi4 ; Shrock:BetaZeroPhi4 . Also, the $\varepsilon$-expansions and $Z$-factor contributions of individual diagrams should suffice to study other universality classes like the $O(n)$ model with cubic anisotropy MudrovVarnashev:ModifiedBorel ; KleinertSchulteFrohlinde:CubicEps5 or even more complicated cases like CalabreseParrucciniSolokov:Chiral ; KalagovKompanietsNalimov:U(r) .

Therefore we provide an extensive set of data with this article as described in Appendix A and available under DOI 10.5287/bodleian:pvx4nMyQr. We added the computed Feynman integrals to the Loopedia Loopedia:Release .

II Field theory and renormalization

We consider the theory of $n$ scalar fields $\phi=(\phi_{1},\ldots,\phi_{n})$ with an $O(n)$ symmetric interaction $\phi^{4}\mathrel{\mathop{:}}=(\phi^{2})^{2}=(\phi_{1}^{2}+\cdots+\phi_{n}^{2})^{2}$. In $D=4-2\varepsilon$ Euclidean dimensions, the corresponding renormalized Lagrangian is

[TABLE]

and contains an arbitrary mass scale $\mu$, such that $g$ stays dimensionless. The $Z$-factors relate the renormalized field $\phi$, mass $m$ and coupling $g$ to the bare field $\phi_{0}$, bare mass $m_{0}$ and bare coupling $g_{0}$ via

[TABLE]

In dimensional regularization tHooftVeltman:RegularizationGaugeFields and minimal subtraction, these $Z$-factors depend only on $\varepsilon$ and $g$ and admit expansions into formal Laurent series

[TABLE]

where each $Z_{i,k}(g)$ is a formal power series in the coupling $g$. The renormalization group (RG) functions can be read off from the residues (at $\varepsilon=0$) of the $Z$-factors tHooft:DimRegRG ; Collins:CountertermsDimReg . In particular, the beta function can be computed as

[TABLE]

whereas the anomalous dimensions for the field and mass are given by

[TABLE]

These RG functions are formal power series in the coupling $g$. The first terms

[TABLE]

show that, at least for small $\varepsilon>0$, the beta function admits a non-trivial zero

[TABLE]

This critical coupling is a formal power series in $\varepsilon$ and determines a fixed point of the renormalization group flow. This fixed point is IR-attractive, meaning that the correction to scaling exponent

[TABLE]

is positive. The anomalous dimensions at the critical point define the critical exponents

[TABLE]

which we compute as formal power series in $\varepsilon$. According to the leading terms

[TABLE]

the first terms of their $\varepsilon$-expansions are

[TABLE]

We note that there are more critical exponents, but those are related to $\eta$ and $\nu$ via the following scaling222Relations that explicitly involve the dimension $D$ are often distinguished and called hyperscaling relations. For simplicity, we will refer to all of (10) just as scaling relations. relations KleinertSchulteFrohlinde:CriticalPhi4 , which, for the purpose of this paper, we simply take as definitions of $\alpha$, $\beta$, $\gamma$ and $\delta$:

[TABLE]

The critical exponents and the correction to scaling exponent $\omega$ are independent of the renormalization scheme and they conjecturally describe phase transitions of numerous physical systems in several universality classes. In section V we describe how we resummed the $\varepsilon$-expansions to arrive at the estimates for these quantities in $D=3$ and $D=2$ dimensions as presented in sections VI and VII.

III Calculational techniques

We compute the $Z$-factors (2) as the counterterms for the one-particle irreducible correlation functions $\Gamma_{N}$ of $N=2$ and $N=4$ fields, by expanding them as Feynman diagrams. The ultraviolet (UV) subdivergences are subtracted with the Bogoliubov-Parasiuk $\oldcal{R}^{\prime}$-operation BogoliubovParasiuk:Kausalfunktionen ; BogShirk , such that

[TABLE]

In the minimal subtraction ($\mathrm{MS}$) scheme, the form (3) of the $Z$-factors is obtained by projecting onto the pole part with respect to the regulator $\varepsilon=(4-D)/2$,

[TABLE]

We use standard techniques Vasilev ; ZinnJustin ; KleinertSchulteFrohlinde:CriticalPhi4 to simplify the computation:

• •

Acting with $-\partial_{m^{2}}$ squares a propagator, which is equivalent to a $4$-point graph with two vanishing external momenta. This way, $Z_{1}$ can be expressed in terms of a subset of the graphs contributing to $Z_{4}$ (KleinertSchulteFrohlinde:CriticalPhi4, , section 11.7).

• •

Using infrared rearrangement (IRR) Vladimirov:ManyLoopPhi4 ; ChetyrkinKataevTkachov:Gegenbauer , we can set all internal masses to zero and nullify some external momenta such that only massless propagators remain to be computed.

These express all $Z$-factors in terms of $p$-integrals (massless propagators) without infrared divergences, and our task is thus reduced to the computation of the $\varepsilon$-expansion of these integrals. The number of $\phi^{4}$ Feynman graphs contributing to $\Gamma_{2}$ and $\Gamma_{4}$ is summarized in table 1.333The $\phi^{4}$ graphs with $\leq 6$ loops were already enumerated and tabulated in NickelMeironBaker:Compilation24 . Asymptotically, the number of graphs grows factorially with the number of loops and precise higher order expansions were recently presented in Borinsky:RenormalizedAsymptoticEnumeration .

A few of them are primitive (free of subdivergences) and those were computed, up to $\leq 6$ loops, already long ago in Broadhurst:5loopsbeyond .444Some of the results in Broadhurst:5loopsbeyond in terms of zeta values were based on numeric techniques, and one value in particular was only later identified in BroadhurstKreimer:KnotsNumbers as the double zeta value (17). Analytic proofs of these $6$-loop periods were later given in Schnetz:K34 ; Panzer:MasslessPropagators ; Schnetz:GraphicalFunctions .

In fact, the partial results BroadhurstKreimer:KnotsNumbers ; Schnetz:Census at higher loop orders have recently been augmented significantly, including in particular the complete set of primitive $\phi^{4}$ Feynman integrals with $7$ loops PanzerSchnetz:Phi4Coaction .

In order to calculate the missing integrals with (UV) subdivergences, which was the main technical challenge, we construct auxiliary counterterms using the $\oldcal{R}^{\prime}$ operation of the BPHZ-like one-scale scheme introduced in BrownKreimer:AnglesScales . The resulting linear combinations of integrals are convergent in $D=4$ dimensions and can thus be computed exactly, term-by-term after expanding in $\varepsilon$, with the program HyperInt Panzer:HyperIntAlgorithms based on the algorithm proposed in Brown:TwoPoint .

We gave a detailed account of this new method in KompanietsPanzer:LL2016 . The entire computation is automated with programs written in Maple™ and Python, using the GraphState/Graphine library to manipulate Feynman graphs BatkovichKirienkoKompanietsNovikov:GraphState ; BatkovichKompaniets:Toolbox , which will be published separately.555Maple is a trademark of Waterloo Maple Inc.

The only addition to be made to the exposition in KompanietsPanzer:LL2016 , is that each Feynman graph $G$ is now not only weighted with the usual combinatorial symmetry factor $1/\left|\operatorname{Aut}(G)\right|$, but also with an additional $O(n)$-group factor $\mathscr{C}_{G}$ which is a polynomial in $n$. It equals the number of ways one can assign a component $1\leq i_{e}\leq n$ of the field $\phi$ to each of the edges $e$ of the graph, in such a way that at each vertex $v$, the flavours of the four edges $v^{[1]},\ldots,v^{[4]}$ meeting at $v$ can be grouped into two equal pairs—according to the interaction term $(\phi^{2})^{2}$. Hence,

[TABLE]

gives the group factor for vacuum (i.e. 4-regular) graphs, with

[TABLE]

For a propagator (2-point) graph $G$, let $\widehat{G}$ denote the vacuum graph obtained by gluing the external legs together, and for a vertex (4-point) graph $G$, let $\widehat{G}$ denote the graph obtained by attaching all external legs to an additional vertex (this is known as the completed graph Schnetz:Census ). Then it is easy to check that

[TABLE]

IV Results for the RG functions

We now present our results for the $6$-loop renormalization group functions, computed in the minimal subtraction (MS) scheme in $D=4-2\varepsilon$ dimensions. Among those, the anomalous dimension of the field takes the simplest form, because it only involves Riemann zeta values $\zeta_{k}=\sum_{n=1}^{\infty}1/n^{k}$:

[TABLE]

Note that this result was already obtained in BatkovichKompanietsChetyrkin:6loop with different methods; so our computation provides a non-trivial check. Numeric values for $n\in\left\{0,1,2,3,4\right\}$ are given in table 2. The following results contain also the double zeta value

[TABLE]

which (conjecturally) cannot be written as a polynomial in Riemann zeta values with rational coefficients Broadhurst:1440 ; Broadhurst:5loopsbeyond . Our new results are the coefficient of $g^{6}$ in the anomalous dimension of the mass (see table 3 for numeric expansions),

[TABLE]

and the coefficient of $g^{7}$ in the beta function

[TABLE]

with numeric values given in table 4. The expansions (16), (18) and (19) are available in computer readable form in the attached files (see Appendix A).

IV.1 Checks

Our computation exactly reproduced the full $5$-loop results ChetyrkinKataevTkachov:5loopPhi4 ; ChetyrkinGorishnyLarinTkachov:5loopPhi4 ; Kazakov:MethodOfUniqueness ; KNFCL:5loopPhi4 and the 6-loop field anomalous dimension BatkovichKompanietsChetyrkin:6loop , which in turn is consistent with the first three terms in the large $n$ expansion of the critical exponent $\eta$ from VasilevPismakHonkonen:SimpleMethod ; VasilevPismakHonkonen:largeNeta ; VasilevPismakHonkonen:1n3 . The leading and subleading terms in the large $n$ expansion of the $6$-loop beta function, computed almost $20$ years ago in Gracey:LargeNf ; BroadhurstGraceyKreimer:PositiveKnots , provided a further successful check.

In addition, we confirmed David Broadhurst’s results for the $\mathrm{MS}$-renormalized 5-loop propagator, obtained in $1993$. In fact, in Broadhurst:WithoutSubtractions he obtained the $\varepsilon$-expansions of all $5$-loop propagator integrals with $\leq 5$ loops, including the finite parts.666We are very thankful to David Broadhurst for making his notes available to us.

These results agree perfectly with our $\varepsilon$-expansions for those graphs.

A particularly strong check of our results is due to Oliver Schnetz Schnetz:NumbersAndFunctions , who computed $\beta$ and $\gamma_{\phi}$ to $6$ loops for the case $n=1$, using completely different techniques including single-valued integration and graphical functions Schnetz:GraphicalFunctions .

Also, we initially computed all necessary integrals with a combination of:

• •

integration by parts (IBP) ChetyrkinTkachov:IBP ,

• •

$\varepsilon$-expansions of $4$-loop massless propagators BaikovChetyrkin:FourLoopPropagatorsAlgebraic ; SmirnovTentyukov:FourLoopPropagatorsNumeric ; LeeSmirnov:FourLoopPropagatorsWeightTwelve ,

• •

infrared rearrangement (IRR) extended by the $\oldcal{R}^{\ast}$-operation Chetyrkin:RRR ; ChetyrkinGorishnyLarinTkachov:Analytical5loop ; ChetyrkinTkachov:InfraredR ; ChetyrkinSmirnov:Rcorrected ; BatkovichKompaniets:6loop-Rstar and

• •

parametric integration, using hyperlogarithms, of primitive (free of subdivergences) linear combinations of graphs; examples of this technique are given in (Panzer:MasslessPropagators, , graphs $M$, $M_{3,5}$ and $M_{5,1}$) and (Panzer:PhD, , section 5.3.2).

Together with our more recent strategy KompanietsPanzer:LL2016 of parametric integration with the one-scale scheme, we have in fact computed all diagrams with at least two different exact methods ourselves. In addition, the most complicated diagrams were also checked numerically using sector decomposition BinothHeinrich:SectorDecomposition to at least 3 significant digits, using a computer program by the first author. We furthermore cross-checked our generation of the Feynman graphs and their symmetry factors with GraphState by comparison with the output of the program feyngen from Borinsky:feyngen . Our numbers of $2$- and $4$-point graphs in table 1 agree with the lists in NickelMeironBaker:Compilation24 (after discarding the tadpole contributions). The $O(n)$ group factors were confirmed independently by evaluating (13) with a simple Form Vermaseren:NewFORM program.

Also, we verified that, after explicitly expanding $\beta(g,\varepsilon)=-2\varepsilon/[\partial_{g}\log(gZ_{g})]$ from (4) and $\gamma_{i}(g)=\beta(g)\partial_{g}\log Z_{i}(g)$ from (5) as series in $g$, all poles in $\varepsilon$ cancel and the results indeed coincide with the final expressions in (4) and (5) in terms of the residues $Z_{i,1}$. This shows that all higher order poles of the $Z$-factors are consistent with the first order poles as dictated by the renormalization group.

Finally, we find our results for the $6$-loop coefficients of the RG functions to be in good agreement with various past predictions based on the $\leq 5$-loop coefficients. For example, $\beta^{\mathrm{MS}}_{7}\approx 34400$ ChetyrkinGorishnyLarinTkachov:5loopPhi4 and $\beta^{\mathrm{MS}}_{7}\approx 34393$ Kompaniets:ACAT2016 for the case $n=1$ are very close to our value $34776$, where $\beta^{\mathrm{MS}}=\sum_{k}(-g)^{k}\beta^{\mathrm{MS}}_{k}$. Even more interestingly, asymptotic Padé-approximant predictions (APAPs) were provided for various values of $n$ in ChishtieEliasSteele:MassivePhi4 ; EllisKarlinerSamuel:betaQCDprediction , as summarized in table 5. In the case of the beta function, we see indeed only very small deviations ($\lesssim 2\%$) from our exact results. It seems plausible to expect that the APAP method might provide potentially even more accurate predictions for the $7$ loop coefficient.

For the anomalous dimensions, the APAP forecasts are still reasonable though significantly less precise, as was already noted and discussed in ChishtieEliasSteele:MassivePhi4 at the five loop level.777The outlyingly large deviations in $\gamma_{m^{2}}$ for $n=4$ and $n=5$ at six loops are caused by a pole of the Padé approximant at $n\approx 4.7$; the agreement becomes much better again for $n=6$ and $n=7$. We are grateful to the authors of ChishtieEliasSteele:MassivePhi4 for their correspondence and investigation of this matter. In conclusion, the discrepancy at $n=5$ is very likely an artifact of the Padé method than an indicator of an error in our result for $\gamma_{m}^{2}$.

In this context let us point out that a qualitatively similar situation occurs for predictions based on the conformal Borel technique: the $5$- and $6$-loop predictions Kompaniets:ACAT2016 (at $n=1$) for the beta function are good within $1\%$, whereas the error of the predictions BatkovichKompanietsChetyrkin:6loop for $\eta$ increases from $0.5\%$ for $6$ loops to $2.5\%$ for $7$ loops (according to the $7$-loop result for $\eta$ given in Schnetz:NumbersAndFunctions ).

In the following, we discuss the dependence of the RG function coefficients on the perturbative order and on the internal degrees of freedom. We will see that our results are consistent with the expected behaviour.

IV.2 Asymptotics

It has long been known that the renormalization group functions are asymptotic series in the coupling $g$, with factorially growing coefficients Lipatov:DivergencePT ; BrezinGuillouZinnJustin:phi2N . For the minimal subtraction scheme, the precise leading asymptotic behaviour was first computed in McKaneWallaceBonfim using $4-2\varepsilon$ dimensional instantons McKaneWallace:Instanton . Namely, if we denote the coefficients of the beta function by $\beta^{\mathrm{MS}}(g)=\sum_{k}\beta^{\mathrm{MS}}_{k}(-g)^{k}$, then

[TABLE]

where $C_{\beta}$ is a constant that only depends on the number $n$ of field components:

[TABLE]

In this expression, $\gamma_{\text{{E}}}=-\Gamma^{\prime}(1)\approx 0.577$ denotes the Euler-Mascheroni constant and $A\approx 1.282$ is the Glaisher-Kinkelin constant defined by

[TABLE]

It is well-known that the perturbative coefficients reach their asymptotics rather slowly in the $\phi^{4}$ model KazakovTarasovShirkov:AnalyticContinuation ; DittesKubyshinTarasov:4loop , which is illustrated in figure 1 and table 6. We see that, even at $6$ loops, the ratio $\beta^{\mathrm{MS}}_{7}/\overline{\beta}_{7}\approx 0.31$ is still far from one. One also should bear in mind that, at such low perturbative orders, this kind of ratio depends significantly on the function used to model the asymptotic behaviour (KazakovTarasovShirkov:AnalyticContinuation, , Fig. 4). For illustration, we include in figure 1 a comparison of $\beta^{\mathrm{MS}}_{k}$ to

[TABLE]

which is another reasonable function with the same asymptotics as (20).

In order to probe the asymptotic regime further, we studied the primitive contributions $\beta^{\mathrm{prim}}_{k}$ to $\beta^{\mathrm{MS}}_{k}$, that is, the sum of the contributions of all $4$-point $\phi^{4}$ graphs that are free of subdivergences. These dominate the leading asymptotics of $\beta^{\mathrm{MS}}_{k}$ as $k\rightarrow\infty$, according to (McKaneWallaceBonfim, , page 1865):888See also (BrezinGuillouZinnJustin:phi2N, , page 15): “Finally, the leading diagrams at order $K$ give a single power of $\ln\Lambda$, they are those which do not involve any divergent subgraph; i.e., they are the completely irreducible diagrams.”

In the context of high-order estimates in the perturbative series, we interpret the extra pole in $\varepsilon$ as the one produced by the totally irreducible diagrams at high orders. These diagrams are known to be the dominant ones at $K$th order for $K$ large for $d<4$ and moreover diverge only like $1/\varepsilon$.

Indeed, the last row of table 6 shows how the primitive graphs become more and more relevant as the loop number increases.999Such a comparison was already made in evaluation of the four loop calculation, see KazakovTarasovShirkov:AnalyticContinuation .

At $6$ loops, the primitive graphs $\beta^{\mathrm{prim}}_{7}$ already constitute $69\%$ of the coefficient $\beta^{\mathrm{MS}}_{7}$. The primitive contributions at $7$ loops are known exactly PanzerSchnetz:Phi4Coaction and included in figure 1.

Furthermore, we were able to obtain accurate numeric estimates for all primitive graphs with up to $11$ loops, using a new method (based on the so-called Hepp bound) recently introduced by the second author. A brief sketch of this technique is provided in Appendix B. We see in figure 1 that even at $11$ loops, the primitive contributions (which are expected to be close to the full $\beta^{\mathrm{MS}}_{12}$) reach merely about half of the value predicted by the asymptotic formula (20). Details are given in table 13.

We conclude that it is not clear how the knowledge of the leading asymptotic behavior of the perturbative coefficients might be used to accurately predict perturbative coefficients at higher orders. It is thus interesting to note that, in principle, corrections to the leading asymptotic behaviour of the form

[TABLE]

can indeed be calculated. In fact, the first correction $a_{1}$ was computed long ago in Kubyshin:Corrections for the MOM scheme and much more recently in KomarovaNalimov:FirstCorrectionO(N) for the $\mathrm{MS}$ scheme, using a method developed in KomarovaNalimov:HigherOrdersO(N) . Unfortunately, these results need to be adjusted101010In particular, in both of these papers, the value of $U_{\text{reg}}$ needs to be corrected to $\frac{n+8}{6}(\gamma_{\text{{E}}}+5/3+\ln\pi)$, as was kindly pointed out to us by M. Nalimov. Note that, with this correction, the result for the leading asymptotics computed in KomarovaNalimov:FirstCorrectionO(N) coincides with (20) and (21) from McKaneWallaceBonfim .

and we therefore cannot presently discuss if the term $a_{1}/k$ narrows the gap between the exactly known low-order coefficients $\beta^{\mathrm{MS}}_{k}$ and the asymptotic predictions. This correction could also inflate the gap, because the expansion (24) is itself only asymptotic Suslov:HigherOrderLipatov ; LobaskinSuslov:HigherLipatov and as such only guarantees an improved fit for very large perturbative orders $k$.

In particular, we like to stress that the idea to truncate (24), fit the coefficients $a_{j}$ to make this polynomial in $1/k$ match the known low order perturbative terms and then use this polynomial to predict (extrapolate) higher order perturbative coefficients, is not justified.111111Because (24) is asymptotic, it should be expected that longer truncations with the correct values of $a_{j}$ yield increasingly worse predictions for the low order perturbative coefficients. Hence, forcing these to be matched requires the fitted $a_{j}$ to deviate from their exact values, so there is no reason why those should result in good predictions for higher orders $k$.

For a detailed discussion and criticism of such a procedure, see KazakovPopov:QMandQFT and KazakovPopov:GellMannLow .

IV.3 Dependence on $n$

The coefficients $\beta^{\mathrm{MS}}_{k}=\beta^{\mathrm{MS}}_{k}(n)$ are functions of the number $n$ of fields, because the contribution of each Feynman graph is multiplied with a corresponding group factor (15). More precisely, $\beta^{\mathrm{MS}}_{k}(n)$ is a polynomial in $n$ for each order $k$, as seen in (19). Once normalized by the asymptotic growth $k!\cdot k^{3+n/2}$ from (20), these coefficients approach $C_{\beta}$ in the limit $k\rightarrow\infty$. Since this limit is dominated by the primitive graphs, our estimates for $\beta^{\mathrm{prim}}_{k}$ should exhibit the same behaviour.

Figure 2 shows that these expectations are indeed fulfilled. First, we note that the observation (from figure 1 at $n=1$), that even the $11$th perturbative order is far from the asymptotic regime, extends to all $n\gtrsim-3$. However, all curves share a zero at $n\approx-4$, and the primitive contributions with $11$ loops vanish also near to the next zero of $C_{\beta}$ at $n=-6$. We note that the asymptotic coefficient $C_{\beta}$ is approximated rather well by the primitive contributions $\beta^{\mathrm{prim}}_{12}$ in the intermediate region where $-6<n<-4$.

This phenomenon of zeros at certain values of $n$ was observed in Pobylitsa:Superfast and follows simply from the factor $\Gamma(2+n/2)$ in the denominator of (21), since it implies that $C_{\beta}$, which governs the asymptotic behaviour of $\beta_{k}^{\mathrm{MS}}$ according to (20), vanishes at even values of $n\leq-4$.

Indeed, the zero of $\beta^{\mathrm{MS}}_{\ell+1}(n)$ that is closest to the origin converges rapidly towards $-4$ as $\ell$ increases, which was checked for $\ell\leq 5$ loops in Pobylitsa:Superfast . In table 7, we see that this trend continues at $\ell=6$ loops and, with an impressive rate of convergence, the same phenomenon continues in our estimates of the primitive contributions $\beta^{\mathrm{prim}}_{\ell+1}$ up to $\ell=11$ loops. Furthermore, at these higher loop orders, we can see the convergence of the next zeros to the expected values $n=-6$ and $n=-8$.

Note that the group factors $\mathscr{C}_{G}$ can be negative for such values of $n$, and in fact only because of these opposite signs is the cancellation of $\beta^{\mathrm{prim}}_{\ell+1}(n)$ possible at all. As a collective phenomenon, sensitive to all Feynman periods at loop order $\ell$, we thus interpret the convergence of zeros in table 7 towards even values of $n\leq-4$ as a strong consistency check of our exact $6$-loop results and also of our estimates for the primitive contributions up to $11$ loops.

V Resummation of critical exponents

From the renormalization group functions $\gamma_{\phi}$, $\gamma_{m^{2}}$ and $\beta$ in section IV, it is straightforward to work out the $\varepsilon$-expansions of critical exponents. We focus on $\eta$, $\nu^{-1}$ and $\omega$, for which (8) and (7) yield the results shown in tables 8–10.

The $\varepsilon$-expansion $f(\varepsilon)=\sum_{k=0}^{\infty}f_{k}(-2\varepsilon)^{k}$ of a critical exponent $f$ around $D=4-2\varepsilon$ dimensions is a formal power series with factorially growing coefficients

[TABLE]

In fact, this leading asymptotic behaviour is completely determined by the asymptotics (20) of the beta function, the leading terms (9) of the perturbation series and the defining equations (6), (7) and (8).121212The equations (6)–(8) furthermore relate corrections to the leading asymptotics, as for example computed in KomarovaNalimov:FirstCorrectionO(N) . If one encodes the full asymptotic expansions as generating functions, these relations can be computed elegantly in an algebraic way Borinsky:GeneratingAsymptotics .

In particular, all the coefficients $C_{f}$ can be expressed as multiples of $C_{\beta}$ from (21) and were all computed in McKaneWallaceBonfim . The values of $a$ and the exponents $b_{f}$ were already obtained in BrezinGuillouZinnJustin:phi2N :

[TABLE]

In order to obtain estimates for the critical exponents in $D=3$ dimensions, we must resum the divergent series $f(\varepsilon)$ at $\varepsilon=1/2$.

This problem of resummation is a huge subject, see for example the review Caliceti:UsefulAlgorithms , and many different approaches have been put forward. Unfortunately, no consensus has so far been reached on the optimal method to resum $\varepsilon$-expansions. We therefore think that a careful comparison of the various methods, based on the $6$-loop perturbation series presented here, would be very valuable. In particular in view of the potential for further higher order perturbative computations in the near future, like the $7$-loop $\varepsilon$-expansions in the $\phi^{4}$ model Schnetz:NumbersAndFunctions , such further insight into the resummation problem is very desirable.

However, such an extensive analysis would exceed the scope of this article and we decided to discuss only the method of Borel resummation with conformal mapping. It would be very interesting to see how other approaches, including order-dependent mapping ZinnJustin:ODM ; SeznecZinnJustin:OrderDependent , large-coupling expansions Kleinert:ConvertingWeakStrong ; KleinertFrohlinde:5loopStrong ; Kleinert:7loopPhi4strong and self-similar factor approximants YukalovYukalova:CriticalSelfSimilar , fare with the $6$-loop perturbative input.

V.1 Borel resummation with conformal mapping

We will describe the method introduced first in KazakovTarasovShirkov:AnalyticContinuation for the resummation of series in the coupling $g$ and then also applied to $\varepsilon$-expansions VladimirovKazakovTarasov:Calculation . This technique has a successful history in the resummation of critical exponents LeGuillouZinnJustin:CriticalFromField ; GuillouZinnJustin:Accurate ; GuidaZinnJustin:CriticalON ; GorishnyLarinTkachov:O(eps5) and is explained in detail for example in (KleinertSchulteFrohlinde:CriticalPhi4, , chapter 16).

To begin with, we denote the Borel transform of $f=\sum_{k=0}^{\infty}f_{k}(-2\varepsilon)^{k}$ as

[TABLE]

According to (25), it defines an analytic function in the domain $\left|x\right|<1/a$ with a singularity at $x=-1/a$ of the form $(1+ax)^{b-b_{f}-1}$. We assume131313While the Borel summability with respect to the coupling $g$ has been established in the fixed dimensions $D=2$ EckmannMagnenSeneor:Borel and $D=3$ MagnenSeneor:phi4_3 , it remains an open question for the $\varepsilon$-expansion.

that $f$ is Borel summable, which means that $\mathfrak{B}^{b}_{f}(x)$ admits an analytic continuation to the positive real axis $x>0$ and also includes that the Borel sum

[TABLE]

converges and gives the correct value of the critical exponent at $\varepsilon=1/2$, which is our case of interest ($D=3$). By construction, this Borel sum $\tilde{f}(\varepsilon)$ has the perturbative expansion $f(\varepsilon)$ as required. In order to compute the integral (28), we must analytically continue the Borel transform $\mathfrak{B}^{b}_{f}(x)$ from the circle $\left|x\right|<1/a$ of convergence to the positive real line. This is achieved with the conformal transformation

[TABLE]

as it maps the integration domain $x\in(0,\infty)$ to the interval $w\in(0,1)$. Assuming that all singularities of $\mathfrak{B}^{b}_{f}(x)$ lie on the cut $(-\infty,-1/a]$, which is mapped onto the unit circle $\left|w\right|=1$, the expansion of the Borel transform $\mathfrak{B}^{b}_{f}(x(w))$ into a series in $w$ converges in the full integration domain and thus provides the sought-after analytic continuation.

Because we only know the first few expansion coefficients $f_{k}$ with $k\leq 6$, we can only approximate the Borel transform. Following KazakovTarasovShirkov:AnalyticContinuation ; VladimirovKazakovTarasov:Calculation , we introduce a second parameter $\lambda$ and the truncation order $\ell$ to write it as

[TABLE]

The coefficients $B_{f,k}^{b,\lambda}$ are functions of $b$ and $\lambda$, determined by matching the coefficients of $x^{0}$ through $x^{\ell}$ in the expansion of $\mathfrak{B}^{b,\lambda,\ell}_{f}(x)$ with the perturbative constraints (27). This ensures that $\mathfrak{B}^{b}_{f}(x)$ is approximated well for small values of $x$. Crucially, the parameter $\lambda$ allows us to also adjust the growth $\mathfrak{B}^{b,\lambda,L}_{f}(x)\sim x^{\lambda}$ for large $x$ to better match the behaviour of the actual Borel transform. Without this degree of freedom, our approximations $\mathfrak{B}^{b,0,\ell}_{f}(x)\rightarrow\sum_{k=0}^{\ell}B_{f,k}^{b,0}$ would always approach a constant value at $x\rightarrow\infty$. This appears to model the Borel transform only very poorly, and a careful choice of $\lambda$ is essential to significantly improve the quality of the resummation VladimirovKazakovTarasov:Calculation ; GorishnyLarinTkachov:O(eps5) ; Kompaniets:ACAT2016 .

Our approximate result for the Borel sum (28) is then given by

[TABLE]

Finally, a third parameter was introduced in GuillouZinnJustin:Accurate to improve the results even further. Namely, we consider a homographic transformation

[TABLE]

to re-expand the original $\varepsilon$-expansions as series in $\varepsilon^{\prime}$ (for $q=0$, nothing changes). We then proceed as above, taking this new series as input:

[TABLE]

The motivation for (32) is that it allows us to map potential singularities of critical exponents as functions of $\varepsilon$ further away from the point $\varepsilon=1/2$, in order to diminish their possibly detrimental influence on the resummation GuillouZinnJustin:Accurate .

In closing, let us stress that there are several aspects in which this basic scheme might be adjusted. For example, we could choose other conformal mappings than (29), replace (32) by a different transformation and instead of (30) we could use another class of functions to approximate the Borel transform.141414For example, hypergeometric functions were proposed in MeraPedersenNikolic:Nonperturbative and (KleinertSchulteFrohlinde:CriticalPhi4, , chapter 19). Furthermore, Padé approximants have been suggested as a replacement for the Taylor series (30); see JentschuraSoff:ImprovedConformal ; GaddahJwan:ConformalBorel .

V.2 Dependence on resummation parameters

Our resummation procedure is formulated in terms of three parameters: $b$, $\lambda$ and $q$. If we knew the full perturbation series of a critical exponent $f(\varepsilon)$, the resummed value $\tilde{f}(\varepsilon)=\lim_{\ell\rightarrow\infty}\tilde{f}^{b,\lambda,q}_{\ell}(\varepsilon)$ would not depend on these choices. But since we only have the first few terms of the $\varepsilon$-expansions at hand, we are forced to consider the truncations $\tilde{f}^{b,\lambda,q}_{\ell}(\varepsilon)$ with $\ell\leq 6$. These do depend on the parameters, which therefore have to be chosen carefully.

Let us first comment on $b$. The asymptotic behaviour of the coefficients of our approximations of the Borel transform, $\mathfrak{B}^{b,\lambda,\ell}_{f}(x)$ from (30), is given by

[TABLE]

It was noted in KazakovTarasovShirkov:AnalyticContinuation that we can therefore match the leading asymptotics (25) of the perturbation series and our model for the Borel transform by setting $b=b_{f}+3/2$, according to (27) and $\Gamma(k+b+1)\sim k!\cdot k^{b}$. This fixed value was indeed used in KazakovTarasovShirkov:AnalyticContinuation ; VladimirovKazakovTarasov:Calculation ; GorishnyLarinTkachov:O(eps5) ; BatkovichKompanietsChetyrkin:6loop ; Kompaniets:ACAT2016 , with the idea that it incorporates the contributions from very high order perturbation theory. We do not follow this strategy, for the following reasons:

For an exact resummation of the high order contributions, we would actually also have to enforce a precise matching of the constant151515This was pointed out in KazakovTarasovShirkov:AnalyticContinuation . We note that in this reference, the overall sign in equation (14) seems wrong, and in equations (13) and (14), $a_{0}$ should not appear, according to our (34).

[TABLE] 2. 2.

In section IV.2 we saw that even six loops remain far away from the asymptotic regime (figure 1). The contribution of the resummed asymptotic higher orders is thus likely outweighed by the deviation of the $7$-loop contribution from its asymptotic estimate (25). 3. 3.

Variation of the parameter $b$, as first suggested in LeGuillouZinnJustin:n3D , can improve the resummation and also provides a hint towards the uncertainty of the result.

Instead, let us investigate how the resummation depends on $b$. According to (27), the Taylor coefficients of the Borel transform become smaller with increasing $b$. Since the non-truncated Borel sum does not depend on $b$, this implies that the dominant contribution to the integral (28) must come from larger values of $x$. Hence, with increasing $b$, our model for the large-$x$ behaviour of the Borel transform (encoded in the parameter $\lambda$) should become more relevant.

Figure 3 demonstrates this behaviour, by plotting the resummations $\tilde{\eta}^{b,\lambda,q}_{\ell}(1/2)$ of the critical exponent $\eta$ (with various truncation orders $\ell$ of the $\varepsilon$-expansion) as a function of $b$ for several values of $\lambda$. First note that, as expected, in each case the dependence on $b$ decreases if we take more terms of the $\varepsilon$-expansion into account (it will disappear completely only in the limit $\ell\rightarrow\infty$). Furthermore, the choice of $\lambda$ has a strong influence on the $b$-dependence of the resummation.

In fact, for most $\lambda$’s, the dependence on $b$ is very strong. However, we find that for a rather small range of $\lambda$, the $5$- and $6$-loop resummations become almost insensitive to $b$ as illustrated by the plateau in the second plot of figure 3. This stability (with respect to $b$) deteriorates quickly if we resum fewer terms of the $\varepsilon$-expansion. As expected from our discussion earlier, we also see that for larger values of $b$, the resummation depends more strongly on the tuning of $\lambda$. Finally note that with (26), the value $b=b_{\eta}+3/2=5$ chosen in VladimirovKazakovTarasov:Calculation ; GorishnyLarinTkachov:O(eps5) seems too small and misses the plateaus.

In conclusion, we take the sensitivity with respect to $b$ both as an indicator of the uncertainty of the resummation and as a criterion to choose $\lambda$ and $q$. This is a common approach in general, and the existence of wide plateaus in this case was for example pointed out in MudrovVarnashev:ModifiedBorel .161616In MudrovVarnashev:ModifiedBorel ; DelamotteDudkaHolovatchMouhanna:Frustrated , the same approach was used to optimize the value for $a$ in (29). It was observed that the dependence on $a$ is very weak and that the best choices of $a$ lie very close to the value from (26) in the asymptotic growth (25). We therefore keep $a$ fixed at this value.

The power of studying the sensitivity with respect to the resummation parameters was also demonstrated in DelamotteDudkaHolovatchMouhanna:Frustrated , and indeed we will also apply this criterion with respect to variations of $\lambda$.

In figure 4 we see the dependence of the resummation on $\lambda$ for various $b$. As we expect from figure 3, very small values like $b=5$ give a very unstable picture. For suitable larger values like $b=15$ we find $\lambda$-intervals where the six loop resummation $\tilde{\eta}_{6}$ (and to a lesser degree also the $5$-loop resummation $\tilde{\eta}_{5}$) varies only very little. If we further increase $b$, the curves become more sensitive to $\lambda$ again. However, even in the plot for $b=25$, the value at the near-optimal $\lambda=2.5$ from figure 3 remains essentially unchanged.

Finally, it is important to stress the role of the homographic transformation, indexed by $q$. The re-expansion in $\varepsilon^{\prime}$ after the substitution (32) adjusts the coefficients in a non-trivial way and thereby has a potential to alter the apparent convergence of the resummation procedure. In figure 5 we show the dependence on $b$ for various choices of $q$, where in each case $\lambda$ was tuned to minimize the sensitivity to $b$.

The wide plateau existent for $q=0.2$ (already shown in figure 3) gets shorter for larger $q$ and the dependence on $b$ becomes much stronger away from a short range of $q$ near $0.2$. Also note that the level of the (shortened) plateaus shifts with $q$ (in figure 5, the plateau moves down when $q$ is raised above $0.2$). Clearly, such a strong correlation of the resummation result with a free parameter is not desirable. But we see that the stability criterion with respect to $b$ nevertheless clearly singles out a preferred range of $q$. The same qualitative observation applies to the dependence on $\lambda$, that is, for $q$ far from $0.2$, the dependence on $\lambda$ (as in the plots like the ones shown in figure 4) becomes stronger.

In summary, we confirm the observation of GuillouZinnJustin:Accurate that the homographic transformation (32), with a suitable choice of $q$, can significantly improve the apparent stability of the resummation with respect to variations in $b$ and $\lambda$. Incidentally, note that the nearly optimal choice (with respect to these stabilities) of setting $(b,\lambda,q)$ to $(15,2.5,0.2)$ yields a result of $\tilde{\eta}^{b,\lambda,q}_{6}\approx 0.03611$, which agrees well with earlier resummations and estimates from other methods (see table 11). Without the homographic transformation, that is $q=0$, the stability and the agreement with other results would be much worse (see the left-most plot in figure 5).

V.3 Resummation algorithm

In order to quantify the sensitivity of a function $F(x)$ with respect to a resummation parameter $x$, we pick a scale $\Delta_{x}$ and define

[TABLE]

as the minimum spread of the values $F(x^{\prime})$ around $F(x)$ inside an interval $x^{\prime}\in[a,a+\Delta_{x}]$ of width $\Delta_{x}$ that contains $x$ (so $a$ runs from $x-\Delta_{x}$ to $x$). A smaller value of this quantity corresponds to an increasingly flat plateau (of width $\Delta_{x}$) in the kind of plots we show above. Following our discussion in section V.2, we want to pick the resummation parameters such that these spreads, and in particular $\mathrm{Var}_{b}(\tilde{f}_{\ell}^{b,\lambda,q})$, become as small as possible. But this is not the only desirable criterion.

A further indicator for the uncertainty of the resummation is given by the size of the corrections going from one loop order to the next. In fact, in VladimirovKazakovTarasov:Calculation ; GorishnyLarinTkachov:O(eps5) , $\lambda$ was determined exclusively by minimizing the relative differences

[TABLE]

of the last few loop orders. Pictorially, this amounts to searching for intersections of the curves labelled $\varepsilon^{\leq 6}$, $\varepsilon^{\leq 5}$ and $\varepsilon^{\leq 4}$ in the plots as shown in figures 3 and 4.

These two very general approaches are called principle of minimum sensitivity (PMS) Stevenson:ResolutionAmbiguity and principle of fastest apparent convergence (PFAC) Grunberg:RGimprovedQCD . In our context of critical exponents, they are discussed for example in DelamotteDudkaHolovatchMouhanna:Frustrated , and they can be applied in many different ways. In fact, we found that several works on resummations leave out some of the details of the employed procedure, making it difficult to reproduce the results.171717Some notable exceptions are CarmonaPelissettoVicari:GinzburgLandau and (ButtiToldin:N>4, , Appendix A), where the scanned parameter ranges are discussed in detail.

Therefore, let us state our method precisely.

We combine both, the PMS and the PFAC, into the error estimate

[TABLE]

The first term accounts for the uncertainty due to the unknown corrections from higher perturbative orders (estimated by the differences of the largest two loop orders), whereas the spreads in the second line take care of the arbitrariness in the choice of the parameters $b$, $\lambda$ and $q$. We pick the scales as follows:

• •

$\Delta_{b}=20$, because we indeed find such strikingly wide plateaus (with only minute variations of the resummation result), as seen in figure 3 and noticed in MudrovVarnashev:ModifiedBorel , for all critical exponents and values of $n$ that we considered.

• •

$\Delta_{\lambda}=1$, since the dependence on $\lambda$ is stronger and even at six loops the plateaus do not grow much longer (see figure 4).

• •

$\Delta_{q}=0.02$, as the dependence on $q$ is very strong (for fixed $\lambda$); higher values of $\Delta_{q}$ would yield unrealistically high error estimates.

Our resummation of a critical exponent $f$ at order $\ell$ then proceeds as follows:

Sample $\tilde{f}^{b,\lambda,q}_{\ell}$ as defined in (33) for parameters in the cube

[TABLE]

where $b$ runs over half-integers and $\lambda$ and $q$ are probed in steps of $0.02$. 2. 2.

For each such point $(b,\lambda,q)$, compute the error estimate (36). 3. 3.

Find the (global) minimum $\overline{E}^{f}_{\ell}$ of these values for $E^{f}_{\ell}(b,\lambda,q)$.

For the example of $\eta$ (in the $3$-dimensional Ising case $n=1$), the “apparently best” resummation, according to our error estimate, occurs at $(b,\lambda,q)=(11,2.56,0.2)$ and yields $\overline{E}^{\eta}_{6}\approx 0.0002$. These resummation parameters are very close to the second plots in figures 3 and 4. The actual value of the resummed critical exponent is $\tilde{\eta}_{6}^{11,2.56,0.2}\approx 0.03615$ and agrees (within the error estimate $\overline{E}^{\eta}_{6}$) with results from completely different theoretical approaches (see table 11).

A comment is due on our choice for the function $E^{f}_{\ell}(b,\lambda,q)$, which we use as a quantitative measure for the “quality” of a resummation. Obviously, there are many different reasonable definitions of such a measure.181818For example, one could incorporate more low order corrections $|\tilde{f}_{\ell}-\tilde{f}_{\ell-k}|$ with $k\geq 2$, or disregard the stability $\mathrm{Var}_{b}(\tilde{f}_{\ell-1})$, or take further stabilities of lower loop orders into account.

And even if we stay with our definition (36), it still depends on the somewhat arbitrary parameters $\Delta_{b}$, $\Delta_{\lambda}$ and $\Delta_{q}$. However, we tested numerous such variations and found that their effect on the selection of the “apparently best” resummation parameters $(b,\lambda,q)$ only results in very small shifts of the critical exponents. David Broadhurst attributes a fitting quote to Jean Zinn-Justin (paraphrased):

In work on resummation, there is always an undeclared parameter: the number of methods tried and rejected before the paper was written.

We stopped counting.

V.4 Error estimates

The estimation of resummation errors is a notoriously difficult undertaking, for mainly two reasons:

• •

We do not know the perturbative coefficients of the next loop order.

• •

The free parameters (in our case: $b$, $\lambda$ and $q$) can in principle be tuned to reproduce any value for the critical exponents.

Therefore, we can only hope to get a good guess of the error if we assume:

1. (A)

The next perturbative correction is not much larger than the last known correction. 2. (B)

The exact critical exponent is close to the resummed critical exponent $\tilde{f}_{\ell}^{b,\lambda,q}$ when the parameters $(b,\lambda,q)$ are chosen in order to minimize $E^{f}_{\ell}(b,\lambda,q)$.

We chose the widths $\Delta_{b}$, $\Delta_{\lambda}$ and $\Delta_{q}$ as given above such that $E^{f}_{\ell}(b,\lambda,q)$ from (36) should be considered as a lower bound on the error that is inherent to the resummation $\tilde{f}_{\ell}^{b,\lambda,q}$. To verify that this guess of the error is self-consistent, we consider plots as shown in figure 6, i.e. the critical exponent $\eta$ of the $3$-dimensional Ising model ($n=1$). Each point $(\tilde{\eta}_{6}^{\theta},E^{\eta}_{6}(\theta))$ represents a set $\theta=(b,\lambda,q)$ of resummation parameters with an error estimate $\leq 0.0006\approx 3\overline{E}^{\eta}_{6}$. We notice that the optimal resummation ($\tilde{E}^{\eta}_{6}\approx 0.0002$) is rather sharply localized. The spread of the resummation results around this optimum increases in line with the error estimate—this means that our error estimates are indeed consistent with the actual spread.

But there are also cases, like the critical exponent $\nu$ in the $O(4)$ model as illustrated in the right plot of figure 6, where there exist many resummation parameters that yield a close to minimal error estimate, but which produce results that are spread much more widely than the error estimate suggests. In such a situation, we do not pick the “apparently best” resummation (like we do for $\eta$), but instead we choose a mean value (e.g.  $1.351$ in the $O(4)$ example) as a more faithful representation of the distribution of resummations. This is how we compiled our resummation data in table 11 in the next section. Similar plots are provided for all exponents in an ancillary file (see appendix A).

In order to take the actual spreads of the best resummations (like shown in figure 6) into account, all errors reported in the following section consist of the minimum error estimate $\overline{E}^{f}_{\ell}$, plus an additional contribution given by two standard deviations of the set of all resummations with an error $\leq 3\overline{E}^{f}_{\ell}$. One might argue that this would overestimate the error, since we aimed to account for the spreads due to variations in the parameters $(b,\lambda,q)$ already in (36). However, given the arbitrariness in fixing the widths $\Delta_{x}$ and the unknown uncertainties neglected by assuming (A) and (B) above, we are more confident with stating these enlarged errors (in particular since underestimated errors are not uncommon in the literature (JaschKleinert:Fast3, , last paragraph)).

Let us point out in an explicit example how the error predicted by the principles PMS and PFAC, like (36), can underestimate the corrections from higher order contributions. We consider again the exponent $\nu$ in the 3-dimensional $O(4)$ model. The apparently best $5$-loop resummation gives $\nu^{-1}\approx 1.352$ at $\theta=(b,\lambda,q)=(10,0.82,0.4)$ with an error estimate of $\overline{E}^{1/\nu}_{5}\approx 0.002$. On the scale of figure 7, we see indeed only very small fluctuations of the $5$-loop resummation when $b$ and $\lambda$ are varied. Furthermore, the $3$-, $4$- and $5$-loop resummations essentially coincide at $\theta$; in other words, the $4$- and $5$-loop corrections almost vanish at this point. Nonetheless, we see that the $6$-loop correction is significant and much larger than the fluctuations of the $5$-loop result. This kind of behaviour appears to be linked with large spreads of the best resummations as shown in the right plot of figure 6, and explains why, in some rare cases, our error estimates for the $6$-loop resummations quoted in table 11 actually exceed the error estimates for the resummations of the 5-loop series.

In such a case one could say that the $5$-loop result seemed converged, but towards an erroneous value. This phenomenon is particularly well-known to occur in two dimensions, where it was called “anomalous apparent convergence” in DelamotteDudkaHolovatchMouhanna:Frustrated . Here we also like to stress that the large $x$ behaviour of the Borel transform $\mathfrak{B}^{b,\lambda,\ell}_{f}(x)$ of critical exponents $f$ is still unknown. Hence the power law model (30) is only justified because it seems to work very well in practice, but it remains unclear if $\lambda$ can actually be interpreted as the exponent of a power law asymptotic behaviour of $f(\varepsilon)$ for large $\varepsilon$. If the exact large $x$ behaviour is of a different form, then the PMS might miss the correct value (Kleinert:ConvertingWeakStrong, , Appendix A).

VI Estimates for critical exponents in $3$ dimensions

The critical phenomena of many interesting physical systems are described by the $O(n)$ universality classes. We refer to PelissettoVicari:CriticalPhenomena for a comprehensive discussion, and only recall some of the most interesting examples:

• $n=0$

(self-avoiding walks): polymers Gennes:ScalingPolymers ,

• $n=1$

(Ising universality class): liquid-vapour transitions, uniaxial magnets,

• $n=2$

(XY universality class): superfluid $\lambda$-transition of helium LipaNissenStrickerSwansonChui:SpecificHeat ,

• $n=3$

(Heisenberg universality class): isotropic ferromagnets,

• $n=4$:

finite temperature QCD with two light flavours PisarskiWilcez:ChiralChromo .

These are the systems that we consider below; larger values of $n$ were discussed for example in ButtiToldin:N>4 ; AntonenkoSokolov:O(n)>3 ; HoltmannSchulze:O(6) .

The critical behaviour of each universality class is governed by the critical exponents $\alpha$, $\beta$, $\gamma$, $\delta$, $\eta$ and $\nu$. However, in our field theoretic approach, only two of them are independent and determine all others through the scaling relations (10). So, while discussing agreement with other theoretical methods, we will only consider the critical exponents $\eta$, $\nu$ and the correction to scaling exponent $\omega$.

In table 11, we present the summary of our results for the six loop resummation of these exponents, in $D=3$ dimensions for $0\leq n\leq 4$. For comparison of the resummation methods, we also show the outcome of applying our resummation procedure to the five-loop $\varepsilon$-expansions, in comparison with the results given in GuidaZinnJustin:CriticalON for the summation of the same series. It should be noted that we do not consider the renormalization in fixed dimension $D\in\left\{2,3\right\}$, where the resummation problem is slightly different and has been approached, for example, with the pseudo-$\varepsilon$ expansion NikitinaSokolov:Pseudo2 ; SokolovNikitina:FisherPseudo going back to Nickel (LeGuillouZinnJustin:CriticalFromField, , reference 19).

The table furthermore includes a tiny selection of estimates obtained with other theoretical approaches (in particular Monte Carlo and the conformal bootstrap), which by no means can represent the vast literature on this subject. We merely tried to pick the most recent results of the highest apparent accuracy, in order to compare them against our field theoretic method.

Let us first state the following general observations:

When we apply our resummation algorithm to the five loop $\varepsilon$-expansions, we obtain values for the critical exponents that are compatible with the resummation of GuidaZinnJustin:CriticalON . This indicates that our resummation procedure is consistent with their method, though it differs from ours.242424Our method from section V is an implementation of the ideas lined out in GuillouZinnJustin:Accurate . In GuidaZinnJustin:CriticalON , however, the authors abandoned the parameter $\lambda$ from (30) and in its stead introduced a further parameter $r$ via a transformation $f(\varepsilon)\mapsto(1+r\varepsilon)f(\varepsilon)$. 2. 2.

Our error estimates (at 5 loops) are smaller than those given in GuidaZinnJustin:CriticalON . 3. 3.

The $6$-loop resummation results are consistent with the $5$-loop results (within the quoted errors), and in most cases the apparent errors of the $6$-loop results are significantly reduced compared to $5$ loops. 4. 4.

Overall the agreement of the $6$-loop resummation with predictions from other theoretical approaches is very good, in particular for $\eta$.

The largest apparent discrepancies between our $6$-loop resummation and other estimates occur for $\omega$ at $n=0$ and the exponent $\nu$ when $n\geq 1$. In fact, the trend that renormalization group based predictions for $\nu$ tend to be lower than results from statistical approaches has been observed long ago. Our six loop results narrow this gap only slightly, but due to the large error estimates in those cases we do not attach any significance to these differences yet. Once the $7$ loop perturbative results become available, it will be interesting to revisit these cases.

We will now briefly discuss the universality classes one-by-one and, for completeness, show the full set of critical exponents as obtained via the scaling relations (10) from our resummation results for $\eta$ and $\nu$. However, these derived exponents might be determined more accurately via direct resummations of the individual series, as in GuidaZinnJustin:CriticalON , or other resummation techniques.

Note that we resum the $\varepsilon$-expansions as explained in section V, without enforcing any boundary values of exactly known critical exponents in two dimensions. The latter technique is often used to improve the resummation results for three dimensions GuidaZinnJustin:CriticalON ; GuillouZinnJustin:Accurate ; Gracey:4loopPhi3 . However, the exact boundary values are not known in all cases, and it seems difficult to quantify the effect of this procedure on the error estimates. We therefore do not enforce any two-dimensional boundary values; instead, we test our method in section VII by comparing our resummation results in two dimensions with exact predictions.

VI.1 Self-avoiding walks ($n=0$)

Over the last decade, successive improvements of Monte Carlo methods significantly diminished the uncertainty of critical exponents ClisbyLiangSlade:SAWlace ; Clisby:SAWfastpivot ; Clisby:ScaleFreeSAW ; ClisbyDuenweg:Hydrodynamic . The latest and most accurate estimates are $\gamma=1.156953(1)$ Clisby:ScaleFreeSAW and $\nu=0.5875970(4)$ ClisbyDuenweg:Hydrodynamic . In contrast, the value $\omega\nu=\Delta=0.531(3)$, computed long ago in Belohorec:PhD and confirmed by the very recent result $0.528(8)$ of ClisbyDuenweg:Hydrodynamic , remains the most precise determination of the correction to scaling. In conclusion, we derive

[TABLE]

Applying the resummation procedure described in section V to the six loop $\varepsilon$-expansions of $\eta$, $\nu$ and $\omega$ (tables 8–10) yields

[TABLE]

The values of $\eta$ and $\nu$ are in good agreement with (37), but the correction to scaling exponent $\omega$ differs by $\approx 7\%$. Note that our error estimate for $\omega$ increases from $5$ to $6$ loops, which hints towards a badly convergent situation.

For completeness, we compute the other critical exponents via the scaling relations (10) from (38):

[TABLE]

VI.2 Ising universality class ($n=1$)

Experimental measurements in Ising systems, as discussed for example in SengersShanks:Fluids ; LytleJacobs:Turbidity ; PelissettoVicari:CriticalPhenomena , have rather larger uncertainties. Theoretical predictions are more accurate, like the Monte Carlo simulations Hasenbusch:ScalingLattice3dIsing with

[TABLE]

The most accurate values were obtained with the conformal bootstrap KosPolandDuffinVichi:Islands ; 3dIsingBootstrapII :

[TABLE]

Our resummations for $\eta$, $\nu$ and $\omega$ and the other exponents derived via (10) are

[TABLE]

VI.3 XY universality class ($n=2$)

Famous for the very precise measurement $\alpha=-0.0127(3)$ in the microgravity experiment at the $\lambda$-transition of liquid helium LipaNissenStrickerSwansonChui:SpecificHeat , this universality class also describes planar Heisenberg magnets. Theoretical predictions from a combination of Monte Carlo simulations and High-Temperature expansions in CampostriniHasenbuschPelissettoVicari:4He are

[TABLE]

The conformal bootstrap EcheverriHarlingSerone:EffectiveBootstrap provides a correction to scaling exponent $\omega=\Delta_{S^{\prime}}-3=0.811(10)$. Resumming the six loop $\varepsilon$-expansions, we obtain

[TABLE]

where the values in the second row are calculated with the scaling relations (10). The accuracy for $\alpha=2-3\nu$ is so small due to the vicinity of $\nu$ and $2/3$, and serves another motivation for the seven loop calculation of the $\varepsilon$-expansion.

VI.4 Heisenberg universality class ($n=3$)

For experimental results, we refer to GRRNEBDM:Nd . The most precise theoretical predictions stem from Monte Carlo simulations HasenbuschVicari:Anisotropic :

[TABLE]

Monte Carlo combined with High-Temperature expansion CampostriniHasenbuschPelissettoRossiVicari:Heisenberg :

[TABLE]

and the correction to scaling exponent $\omega=\Delta_{S^{\prime}}-3=0.791(22)$ from the conformal bootstrap EcheverriHarlingSerone:EffectiveBootstrap . Our resummations yield

[TABLE]

VI.5 The case $n=4$

The Monte Carlo results $\eta=0.0360(3)$, $\nu=0.750(2)$ from HasenbuschVicari:Anisotropic and $\eta=0.0365(10)$, $\nu=0.749(2)$ given in Hasenbusch:EliminatingN34 are consistent with each other and also with the values $\eta=0.0360(4)$, $\nu=0.7477(8)$ obtained via $y_{t}=1/\nu$ and $y_{h}=(5-\eta)/2$ from the results in Deng:O(4) . The correction to scaling exponent is $\omega=\Delta_{S^{\prime}}-3=0.817(30)$ according to the conformal bootstrap EcheverriHarlingSerone:EffectiveBootstrap .

Our resummation results and the scaling relations (10) lead to

[TABLE]

VII Critical exponents in two dimensions

In two dimensions, the resummation of critical exponents is known to be much less accurate, most likely due to non-analyticities in the beta function at the critical point CalabreseCaselleCeliPelissettoVicari:NonAnalyticity .252525 It seems, however, that the $\varepsilon$-expansion yields much more accurate predictions for critical exponents in two dimensions (see table 12) than the fixed dimension approach OrlovSokolov:Critical2dim5loop .

Indeed, our errors (determined automatically by the procedure from section V.4) reflect this expectation.

The results shown in table 12 are again compatible with the $5$-loop resummations from GuillouZinnJustin:Accurate . Furthermore, we can compare them with the following predictions:

• $n=1$

The exact critical exponents $\eta=1/4$ and $\nu=1$ of the Ising model were computed by Onsager Onsager:CrystalStatisticsI . The convergence of our perturbative results seems slow, and in particular $\nu$ seems to stabilize at a value slightly lower than expected. This phenomenon of “anomalous apparent convergence” was already discussed in detail in DelamotteDudkaHolovatchMouhanna:Frustrated at the $5$-loop level, and seems to persist at $6$ loops.

The correction to scaling, however, is in good agreement with the prediction $\omega=1.75$ from (CalabreseCaselleCeliPelissettoVicari:NonAnalyticity, , equation (21)).

• $n=0$

Our results are compatible with $\eta=5/24$, $\nu=3/4$ and $\omega=2$ as already conjectured by Nienhuis Nienhuis:ExactO(n) , though $\nu$ again seems slightly too small and the uncertainty of $\omega$ is large. The value of $\omega$ has been subject to extensive debate CGJPRS:Scaling2dSAW , so increased accuracy from the $7$-loop $\varepsilon$-expansion would be particularly desirable here.

• $n=-1$

The value of $\eta$ is roughly consistent with Nienhuis’ $\eta=3/20$, but for $\nu$ the prediction of $5/8$ is very far from our result (almost 10 times our error estimate).

VIII Summary and Outlook

After many years of work, new mathematical insights into the structure of Feynman integrals have matured into practically applicable techniques that overcome limitations of traditional approaches so far as to enable progress with the perturbative computation of renormalization group functions. Finally, after 25 years, we were thus able to improve on the five-loop results KNFCL:5loopPhi4 of the $O(n)$-symmetric $\phi^{4}$ model. We like to point out that the primitive graphs relevant to this computation have been essentially known for 30 years Broadhurst:5loopsbeyond . So the challenge was not in unknown transcendental numbers beyond zeta values, but the complexity introduced through subdivergences.

Our approach rests on a significantly improved understanding of the parametric representation of Feynman integrals Brown:PeriodsFeynmanIntegrals ; Brown:TwoPoint , symbolic integration algorithms based on hyperlogarithms Panzer:HyperIntAlgorithms and the Hopf algebra CK:RH1 of renormalization underlying the BPHZ scheme BrownKreimer:AnglesScales . But also other techniques, like graphical functions and single-valued integration Schnetz:GraphicalFunctions ; GolzPanzerSchnetz:GfParam , can be used for this kind of calculations, as demonstrated by the impressive, independent work of Oliver Schnetz Schnetz:NumbersAndFunctions . In fact, these methods are so powerful that even the $7$ loop computation seems now not only feasible but is already underway Schnetz:NumbersAndFunctions . Note that the contributions from $7$-loop graphs without subdivergences were investigated numerically long ago BroadhurstKreimer:KnotsNumbers ; Schnetz:Census and are by now already known exactly PanzerSchnetz:Phi4Coaction .

We are optimistic that these tools will also have further applications. They should be particularly amenable to $\phi^{3}$ theories, whose renormalization only very recently reached the 4-loop level Gracey:4loopPhi3 ; Gracey:F4at4loops . Fermions and gauge fields provide additional technical difficulties, but even in this very challenging domain, significant progress was achieved recently. Let us just mention the Gross-Neveu model GraceyLutheSchroeder:4loopGN and the particularly impressive 5-loop renormalization of QCD BaikovChetyrkinKuehn:5loopQCD ; LutheMaierMarquardSchroeder:CompleteQCD5 ; RuijlUedaVermaserenVogt:4loopQCDvanishing and generalizations BaikovChetyrkinKuehn:5loopFermionGeneral ; LutheMaierMarquardSchroeder:5loopGeneralDims ; HerzogRuijlUedaVermaserenVogt:YMFermions . Those computations drew on yet another set of recently improved methods, e.g. Baikov:BaikovMethod ; LutheSchroeder:5loopTadpoles ; HerzogRuijl:Rstar ; RUV:ForcerLL2016 .

It is our hope that explicit longer perturbation series, like the ones presented here, will lead to an improved understanding of how they approach their asymptotic behaviour, and provide a sufficiently robust and precise testing ground to evaluate and compare the myriad of resummation methods that have been proposed over time, which usually make various unproven assumptions. Such an analysis is an important task in order to turn a finite number of perturbative coefficients reliably into very precise estimates for physical quantities and necessary to harvest the predictive power of increasingly high order perturbation series. Ultimately, we hope that the deep theory of resurgence and trans-series DunneUensal:CPN-1 ; Dorigoni:AlienIntroduction , in combination with the structure of Dyson-Schwinger equations BellonClavier:DSEBorelSingularities , will provide superior tools for this task; but so far, its explicit practical lessons seem to restrict to the well-known insight that the analytic continuation of the Borel transform should be tailored to have the expected branch cuts ChermanKoroteevUnsal:ResurgenceWeakStrong .

As an application, we resummed the $6$-loop $\varepsilon$-expansions for the critical exponents in $3$ dimensions and found that, in many cases, the resulting reduction of their error estimates renders the renormalization group method again competitive in comparison with recently advanced bootstrap and Monte Carlo techniques. We expect that the $7$-loop renormalization will provide critical exponents with even higher accuracies and allow for a more stringent analysis of the compatibility of these very different methods. This is an important task in order to check various assumptions that might be inherent to a particular approach. For example, recent bootstrap results assume that the $O(n)$ models are realized at a “kink” on the boundary of the domain of allowed operator dimensions 3dIsingBootstrapII ; EcheverriHarlingSerone:EffectiveBootstrap .

Acknowledgements.

Both authors are indebted to Kostja Chetyrkin for hospitality at KIT in 2014, where this collaboration was started, and for encouraging us to undertake this $6$-loop project. We furthermore thank Oliver Schnetz for early access to the results of his independent computation, which reassured us on the correctness of our own calculations. The second author is also very grateful for an invitation to FAU Erlangen and very kind hospitality during this visit. Dmitrii Batkovich provided valuable checks of many $6$-loop integrals using the IBP/IRR/$\oldcal{R}^{\ast}$-method BatkovichKompaniets:6loop-Rstar ; BatkovichKompaniets:Toolbox . Also we thank John Gracey for reminding us of the large $n$-expansion results Gracey:LargeNf ; BroadhurstGraceyKreimer:PositiveKnots for the $\beta$ function and for discussions on the resummation of critical exponents. In particular the second author is grateful for hospitality at the University of Liverpool. David Broadhurst very kindly provided us with a copy of his notes Broadhurst:WithoutSubtractions , where he computed the $\mathrm{MS}$-renormalized $5$-loop propagator with extreme economy. The finite parts of these integrals form a subset of the $6$-loop counterterms and provided a valuable check of our results. Furthermore, we are grateful for Mikhail Nalimov’s kind and helpful correspondence regarding the work KomarovaNalimov:FirstCorrectionO(N) ; KomarovaNalimov:HigherOrdersO(N) on the asymptotic expansions in dimensional regularization. Farrukh Chishtie and Tom Steele very kindly explained to us their work ChishtieEliasSteele:MassivePhi4 on asymptotic Padé approximants and investigated the origin of the huge deviations of the six-loop predictions for $\gamma_{m^{2}}$ at $n=5$ (see footnote 7). We are also grateful to Andrey Kataev for fruitful discussions. In this manuscript, we incorporated several suggestions by John, David, Kostja, Mikhail, Oliver and an anonymous referee, who kindly read and commented on earlier drafts. We thank Simon Liebing for the precise translation of the title of ChetyrkinGorishnyLarinTkachov:Analytical5loop . Also we are grateful to Marc Bellon for discussions on the application of trans-series and resurgence to perturbative quantum field theory and hospitality at LPTHE Paris. Further thanks go to Christopher Beem for an illuminating explanation of the conformal bootstrap program. The work of the first author was supported by RFBR grant 17-02-00872-a. The second author’s work on this project was supported by the ESI Vienna through the workshop “The interrelation between mathematical physics, number theory and noncommutative geometry” and the Mainz Institute for Theoretical Physics (MITP) program “Amplitudes: Practical and Theoretical Developments”. He thanks both institutions for their hospitality during these inspiring meetings. This project was started while the second author was at the IHÉS with support from the ERC grant 257638 via the CNRS. The first author is grateful to All Souls College and the Mathematical Institute of the University of Oxford for hospitality and support during a visit in 2016. The calculations of the $Z$-factors and the resummations of the critical exponents were performed on computers of the mathematical institutes of the Humboldt-Universität zu Berlin, the University of Oxford and the Resource Center “Computer Center of SPbU”. Figures of Feynman graphs in this article were created with JaxoDraw BinosiTheussl:JaxoDraw and Axodraw Vermaseren:Axodraw .

Appendix A Description of ancillary files

This article is accompanied by a comprehensive data set in form of human readable text files. We provide these in two formats that are compatible with popular computer algebra software: Maple Maple (files ending with .mpl) and Mathematica Mathematica:11 (files ending on .m). Concretely, these include:

• •

$\varepsilon$-expansions of the renormalization group functions and critical exponents,

• •

individual counterterms ($Z$-factor contributions), symmetry and $O(n)$ factors of all $\leq 6$ loop $\phi^{4}$ graphs and

• •

$\varepsilon$-expansions of the massless propagators that we computed to obtain those.

Below we explain in detail the content and format of these files (referring only to the Maple files, since the Mathematica versions are built in complete analogy).

In addition, the attached document resummation.pdf provides detailed information on our resummations. Namely, for each exponent $f\in\left\{\eta,\nu^{-1},\omega\right\}$, dimension $D\in\left\{2,3\right\}$ and the corresponding values of $n$ considered in tables 11 and 12, it lists the parameters $(b,\lambda,q)$ with least apparent error (36) together with plots like in figures 3 and 4, showing the nearby dependence on $b$ and $\lambda$, and including the distribution of resummation results as in figure 6.

A.1 RG functions and critical exponents

Our six loop expansions of the renormalization group functions $\beta,\gamma_{\phi}$ and $\gamma_{m^{2}}$ defined in (4) and (5) are provided in the file expanded.mpl (in the MS scheme). It also contains the expansion of the critical coupling $g_{\star}(\varepsilon)$ from (6) and the resulting expansions for the universal critical exponents $\eta$, $\nu^{-1}$ and $\omega$ defined in (7) and (8), plus the critical exponents $\alpha$, $\beta$, $\gamma$ and $\delta$ computed via the scaling relations (10).

Each expansion is given symbolically with full $n$-dependence, followed by numeric evaluations for $n\in\left\{0,1,2,3,4\right\}$ like in the tables in sections IV and V.

A.2 counterterms of individual graphs

We computed the $Z$-factors defined in (2) via their expansions (11) in counterterms. For each $i\in\left\{1,2,4\right\}$, the file Z.mpl contains a list of graphs contributing to $Z_{i}$, similar to the table in (BatkovichKompanietsChetyrkin:6loop, , Appendix A) for $Z_{2}$. Each entry is of the form

[TABLE]

and indexed by the loop number $\ell$ and an integer $j$. A graph $G$ is specified by its Nickel index $\mathrm{NI}\left(G\right)$ as defined in (NickelMeironBaker:Compilation24, , section II), see also BatkovichKirienkoKompanietsNovikov:GraphState . This is an intuitive notation for an adjacency list with respect to a certain labelling of the vertices, illustrated in figure 8. Note that we consider graphs without fixed external labels (“non-leg-fixed” in the terminology of Borinsky:feyngen ), i.e. the symmetry factor is $\mathrm{Sym}\left(G\right)=(E_{G})!/\operatorname{Aut}(G)$ where $E_{G}\in\left\{2,4\right\}$ denotes the number of external legs and the automorphisms are allowed to permute them. Furthermore, the list contains the $O(n)$ group factor $\mathscr{C}_{G}$ from (15) and the actual counterterm contribution $z_{G}$, which, according to (11), equals $\partial_{p^{2}}\oldcal{K}\oldcal{R}^{\prime}G$ for $Z_{2}$ and $\oldcal{K}\oldcal{R}^{\prime}G$ for $Z_{4}$. The counterterm $Z_{1}$ is expressed as a linear combination of a subset of graphs contributing to $Z_{4}$, as explained in section III.

For example, the entry $\texttt{Z4}[6,458]$ in Z.mpl corresponds to the graph $G$ depicted in figure 8 with $\mathrm{NI}\left(G\right)=\texttt{ee12|345|346|45|5|6|ee|}$. After minimal subtraction of subdivergences in the $G$-scheme (50), its pole part is

[TABLE]

Multiplied with the symmetry and group factors $\mathrm{Sym}\left(G\right)=3/2$ and $\mathscr{C}_{G}=(5n+22)(3n^{2}+22n+56)/2187$, this gives a contribution to $Z_{4}$. The counterterm (49) also contributes to $Z_{1}$, but with symmetry factor $-1/2$ and group factor $(n+2)^{2}(5n+22)/243$, as dictated by entry $\texttt{Z1}[6,473]$ in Z.mpl. Note that this data can also be looked up in NickelMeironBaker:Compilation24 , where the graph is called 603-U7. Figure 1 ibid. also provides drawings of all relevant graphs.

The full $Z$-factor is obtained by summing $(-g)^{\ell}\mathrm{Sym}\left(G\right)\mathscr{C}_{G}z_{G}$ over all entries of the corresponding list. The file rg.mpl demonstrates this computation and furthermore generates the expansions of RG functions and critical exponents from these $Z$-factors (it outputs the contents of expanded.mpl mentioned earlier).

A.3 $\varepsilon$-expansions of massless propagators

We obtained the counterterms from massless propagators, which are also called “$p$-integrals” ChetyrkinTkachov:IBP , via the intermediate use of the BPHZ-like one-scale scheme BrownKreimer:AnglesScales as explained in detail in KompanietsPanzer:LL2016 . Since these $p$-integrals might be valuable for other applications, we include our results for their $\varepsilon$-expansions in the files p_int_g$i$.mpl, where $i\in\left\{2,4\right\}$. In the case $i=2$, we list all 1PI propagator graphs of $\phi^{4}$ theory, whereas the $p$-integrals given for $i=4$ arose from nullifying some external momenta and rerouting external legs of some subdivergences, according to BrownKreimer:AnglesScales ; KompanietsPanzer:LL2016 , in order to make those subdivergences single-scale. In particular, this means that the $p$-integrals listed for $i=4$ are usually not Feynman graphs of $\phi^{4}$ theory, due to vertices of valency greater than four.

The $\varepsilon$-expansions are given for external momentum squared $p^{2}=1$ in the $G$-scheme ChetyrkinKataevTkachov:Gegenbauer : each integration over a loop momentum $\vec{k}$ carries the measure

[TABLE]

which in particular normalizes the bubble \mathrm{NI}\left(\vbox{\hbox{\includegraphics[scale={0.3}]{Graphs/e11_e}}}\right)=\texttt{e11|e|}, which is the first entry $\texttt{pInt}[1,1]$ in p_int_4.mpl, to be exactly $1/\varepsilon$. For example, the entry $\texttt{pInt}[6,163]$ with Nickel index e123|e24|35|66|56|6|| gives the $\varepsilon$-expansion

[TABLE]

Appendix B Estimates for primitive diagrams up to 11 loops

In the discussion of the asymptotic behaviour in section IV.2, we included the contributions of primitive (subdivergence free) graphs to the $\beta$ function with up to $11$ loops in figure 1. These graphs had been enumerated in Schnetz:Census , but exact results are currently complete only up to $7$ loops PanzerSchnetz:Phi4Coaction . The Feynman integral of such a primitive $4$-point graph $G$ with $\ell$ loops (and $\ell+1$ vertices) has a simple pole $\mathcal{P}\left(G\right)/(\ell\varepsilon)$ and thus contributes $-(-g)^{\ell}\mathcal{P}\left(G\right)/\ell$ to $Z_{4}$ and $Z_{g}$. Taking the symmetry factors into account, the resulting contributions $\beta^{\mathrm{prim}}(g)=\sum_{k}\beta^{\mathrm{prim}}_{k}(-g)^{k}$ to the beta function defined in (4) are

[TABLE]

The residue $\mathcal{P}\left(G\right)$ is known as the Feynman period BEK and can be written in Schwinger parameters $x_{e}$ (one for each edge $e\in E(G)$ of the graph) as

[TABLE]

Here, the Symanzik polynomial $\psi_{G}$ is given by a sum over spanning trees $T$ of $G$,

[TABLE]

Beyond $7$ loops, not all Feynman periods are currently known PanzerSchnetz:Phi4Coaction . Standard numerical techniques for the evaluation of (53) are based on sector decomposition and Monte Carlo integration BinothHeinrich:SectorDecomposition . Unfortunately, these methods are not applicable in practice to graphs as complicated as the $\phi^{4}$ graphs with $11$ loops that we are facing here. However, we find that a decent numerical estimate for the integrals (53) can be obtained from a rather simple graph invariant, constructed by approximating the Symanzik polynomial (54) with its dominant (maximal) monomial.262626The integration domain can be subdivided into Hepp sectors $\left\{x\colon x_{\sigma(1)}<\cdots<x_{\sigma(E(G))}\right\}$, which are indexed by a permutation $\sigma$ of the edges $E(G)$. Inside each Hepp sector, a particular monomial of $\psi_{G}$ dominates all other monomials.

Definition 1**.**

The Hepp bound of a primitive $\phi^{4}$ graph $G$ is

[TABLE]

Note that $\mathcal{H}\left(G\right)\geq\mathcal{P}\left(G\right)$ is indeed an upper bound. It is much easier to compute than the actual period (53); in particular, it is just a rational number and the calculation of the Hepp bounds for all primitive graphs with up to $11$ loops is possible without much difficulty. The Hepp bound has many more interesting properties and will be explored in detail in a paper by the second author Panzer:HeppBound .

What is relevant for our purposes here is that, surprisingly, these easily obtainable numbers correlate strongly with the complicated Feynman period. For example, figure 9 shows the period $\mathcal{P}\left(G\right)$ as a function of the Hepp bound $\mathcal{H}\left(G\right)$ at $7$ loops, where all periods are known exactly due to PanzerSchnetz:Phi4Coaction .

This relationship between $\mathcal{P}\left(G\right)$ and $\mathcal{H}\left(G\right)$ is well approximated by a power law, and even better when we allow for one further parameter as in

[TABLE]

We fit the parameters $a_{\ell}$, $b_{\ell}$ and $c_{\ell}$ to the known periods at loop order $\ell$. The resulting coefficients are given in table 13 and the plots of $f_{\ell}$ are shown in figure 9 for $\ell=7$ and $\ell=11$ loops. In all cases, the approximation by the very simple fit curves (56) reproduces the known periods within $2\%$ accuracy.272727This is a bound on the error for an individual period $\mathcal{P}\left(G\right)$ as approximated by $f_{\ell}\left(\mathcal{H}\left(G\right)\right)$. On average, the error is much smaller.

We note though that, with growing loop number, only very few periods are known exactly, as current integration techniques only apply to graphs with certain special combinatorial structures PanzerSchnetz:Phi4Coaction . Our fits are thus biased—however, for the purpose of our discussion here, we expect this systematic error to be negligible.

To estimate the primitive contributions $\beta^{\mathrm{prim}}_{\ell+1}$ to $\beta^{\mathrm{MS}}_{\ell+1}$ at loop order $\ell$, we substitute $f_{\ell}\left(\mathcal{H}\left(G\right)\right)$ for $\mathcal{P}\left(G\right)$ in (52). The calculation can be economized due to the fact that primitive graphs $G$ with the same completion $F$ (the completion is the 4-regular graph obtained by adding a vertex $v$ and connecting it to the external legs, such that $G=F\setminus v$) have equal periods Schnetz:Census and Hepp bounds. Therefore, $\mathcal{H}\left(F\right)\mathrel{\mathop{:}}=\mathcal{H}\left(F\setminus v\right)$ is independent of the choice of the vertex $v$. Table 13 summarizes the number of (isomorphism classes of) such completions, which were given also in (Schnetz:Census, , Table 1). We used nauty McKayPiperno:II to generate these graphs and to count their automorphisms.

Using the orbit-stabilizer theorem, $\left|\operatorname{Aut}(F\setminus v)\right|=\left|\mathrm{Stab}_{\operatorname{Aut}(F)}(v)\right|=\left|\operatorname{Aut}(F)\right|/\left|\operatorname{Aut}(F)\cdot v\right|$, we can express the primitive contributions (52) as a sum over the completed graphs as282828The factor $4!$ accounts for the different labellings of the external legs of a decompletion, and the $\ell+2$ vertices $V(F)=\mathbin{\dot{\bigcup}}_{\text{orbits}}\operatorname{Aut}(F)\cdot v$ are partioned into the orbits corresponding to inequivalent uncompletions $G=F\setminus v$.

[TABLE]

where we also made the group factor $\mathscr{C}_{F}$ from (13) explicit, using (15). In table 13 we show the results for $n=1$ and see, for example, that at $11$ loops, the primitive contributions amount to $51.5\%$ of the asymptotic prediction $\overline{\beta}^{\mathrm{MS}}_{12}$ from (20). Based on a comparison in the $7$ loop case, where we know $\beta^{\mathrm{prim}}_{8}$ exactly, we expect our estimates for $\beta^{\mathrm{prim}}_{\ell+1}$ to be accurate within 1‰.