Five-Particle Phase-Space Integrals in QCD

TL;DR

This paper derives analytical formulas for five-particle phase-space integrals in massless QCD, providing both exact epsilon-series expressions and numerical tools for cross-checking in various dimensions.

Contribution

It offers the first complete set of analytical five-particle phase-space master integrals in massless QCD with explicit epsilon-expansions and accompanying Monte Carlo code for numerical validation.

Findings

Analytical epsilon-series expressions for 31 master integrals.

Numerical cross-checks in 4, 6, and 8 dimensions.

Implementation of Monte Carlo code based on RAMBO algorithm.

Abstract

We present analytical expressions for the 31 five-particle phase-space master integrals in massless QCD as an $\epsilon$-series with coefficients being multiple zeta values of weight up to 12. In addition, we provide computer code for the Monte-Carlo integration in higher dimensions, based on the RAMBO algorithm, that has been used to numerically cross-check the obtained results in 4, 6, and 8 dimensions.

Tables

Table 1. Table 1: Cut diagrams for five-particle phase-space master integrals in QCD. Dashed lines represent cut propagators and carry final-state momenta p 1 , … , p 5 subscript 𝑝 1 … subscript 𝑝 5 p_{1},\dots,p_{5} . Labels represent propagators, so that "123" corresponds to p 1 + p 2 + p 3 subscript 𝑝 1 subscript 𝑝 2 subscript 𝑝 3 p_{1}+p_{2}+p_{3} and "012" to q − p 1 − p 2 𝑞 subscript 𝑝 1 subscript 𝑝 2 q-p_{1}-p_{2} (where q 𝑞 q is the initial-state momentum, i.e., q = p 1 + ⋯ + p 5 𝑞 subscript 𝑝 1 ⋯ subscript 𝑝 5 q=p_{1}+\dots+p_{5} ).

$F_i=S_{\Gamma }\int {\text{d}\text{PS}}_5\frac{1}{D_1^{\left(i\right)}\dots D_n^{\left(i\right)}},$

(1)

Table 2. Table 2: Numerical results for the ratio F i / F 1 subscript 𝐹 𝑖 subscript 𝐹 1 F_{i}/F_{1} with the corresponding uncertainties (standard deviations) indicated in the parenthesis. Missing entries correspond to divergent integrals.

$i$	Numerical results			Analytic results
	$D=4$	$D=6$	$D=8$	$D=4$	$D=6$	$D=8$
2	–	1708(2)00	4699(1)0	–	171085.62	47000.531
3	3.7823(4)	3.1704(2)	3.0221(1)	3.7823736	3.1704486	3.0221118
4	–	1504.7(8)	725.3(1)	–	1504.4507	725.26806
5	–	1007.4(5)	580.80(9)	–	1007.5235	580.76347
6	–	6191(5)00	14496(6)0	–	619633.25	144975.32
7	46.46(4)	18.533(2)	15.205(1)	46.435253	18.532303	15.205538
8	–	2031(2)0	5357(2)	–	20297.189	5355.3611
9	–	4313(3)	2406.7(4)	–	4312.8823	2406.7943
10	10.436(2)	7.1508(5)	6.5093(3)	10.435253	7.1507477	6.5092878
11	228.8(1)	62.67(1)	47.663(4)	229.11836	62.667046	47.663194
12	–	157.34(4)	102.26(1)	–	157.33521	102.26408
13	–	13729(8)	4000(1)	–	13732.166	4000.2779
14	–	268.46(8)	172.80(2)	–	268.45969	172.79805
15	–	6322(6)0	16048(5)	–	63316.356	16049.857
16	–	4414(3)0	12952(4)	–	44117.898	12951.443
17	–	1243.4(7)	709.6(1)	–	1243.1369	709.52840
18	–	3002(2)00	5899(3)0	–	300402.99	58965.517
19	–	4982(4)00	10637(4)0	–	498329.79	106357.81
20	–	2360(2)000	5777(2)00	–	2362594.9	577686.64
21	–	6312(5)0	20642(5)	–	63147.876	20642.071
22	–	8402(7)00	24407(8)0	–	840453.94	244075.75
23	–	1443(1)000	4556(1)00	–	1443198.3	455543.43
24	–	1391(1)00	3997(1)0	–	139263.92	39966.878
25	–	3347(3)00	8526(3)0	–	335128.10	85254.217
26	25.563(6)	15.376(1)	13.6042(7)	25.564747	15.376404	13.604247
27	–	697.3(3)	397.84(6)	–	697.18948	397.83514
28	143.9(1)	52.855(7)	42.917(3)	143.97886	52.853837	42.917424
29	–	4409(3)0	13702(3)	–	44117.898	13700.597
30	–	6327(6)0	16181(5)	–	63316.356	16178.566
31	–	8955(8)0	19055(8)	–	89611.062	19051.115

Equations

$$s_{kl\dots q}={\left(p_k+p_l+\cdots +p_q\right)}^2,$$

$$\textrm{d}\textrm{PS}_{N}=\left(\prod_{i=1}^{N}\textrm{d}^{D}p_{i}\,\delta^{+}\big{(}p_{i}^{2}\big{)}\right)\delta^{(D)}\Big{(}q-p_{1}-\ldots-p_{N}\Big{)},$$

$$S_{\Gamma}=\left(q^{2}\right)^{5-2D}\frac{\big{(}2\pi\big{)}^{4}}{\pi^{2D}}\Gamma\left(\frac{D}{2}-1\right)\Gamma\left(3\frac{D}{2}-3\right).$$

$$\int\textrm{d}\textrm{PS}_{N}=\left(q^{2}\right)^{\frac{D}{2}(N-1)-N}\,\frac{\pi^{\frac{D}{2}(N-1)}}{\big{(}2\pi\big{)}^{N-1}}\frac{\Gamma\left(\frac{D}{2}-1\right)^{N}}{\Gamma\Big{(}\left(\frac{D}{2}-1\right)\big{(}N-1\big{)}\Big{)}\,\Gamma\Big{(}\left(\frac{D}{2}-1\right)N\Big{)}}$$

$$F_{1}=S_{\Gamma}\int\textrm{d}\textrm{PS}_{5}=\frac{\Gamma\big{(}\frac{D}{2}-1\big{)}^{6}\,\Gamma\big{(}3\frac{D}{2}-3\big{)}}{\Gamma\big{(}4\frac{D}{2}-4\big{)}\,\Gamma\big{(}5\frac{D}{2}-5\big{)}}.$$

$$F_i(D+2)=M_{ij}(D)F_j(D).$$

$$F_i(D+2)=M_{ii}(D)F_i(D)+\sum _{k\in S_i}{M}_{ik}(D)F_k(D)$$

$$F_i(D)={\omega }_i(D)H_i(D)+R_i(D),$$

$$F_{i}=S_{\Gamma}\left(\prod_{k=1}^{N-1}\Omega_{D-k}\right)\int\left(\prod_{l<m}\textrm{d}s_{lm}\right)\big{(}{\Delta_{N}}\big{)}^{\frac{D-N-1}{2}}\Theta\big{(}\Delta_{N}\big{)}\,\delta\left(1-s_{1\dots N}\right)\frac{1}{D^{(i)}_{1}~{}\dots~{}D^{(i)}_{n}},$$

$${\Delta }_N=\frac{(-1{)}^{N+1}}{2^N}\left|\begin{array}{cccc}{s}_{11}& s_{12}& \cdots & s_{1N}\\ s_{12}& s_{22}& \cdots & s_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ s_{1N}& s_{2N}& \cdots & s_{NN}\end{array}\right|,$$

$${\Omega }_k=2{\pi }^{\frac{k}{2}}\Gamma {\left(\frac{k}{2}\right)}^{-1}.$$

$$F_i(D\to \infty )=S_{\Gamma }\left(\prod _{k=1}^{N-1}{\Omega }_{D-k}\right){\left({\Delta }_N^{max}\right)}^{\frac{D}{2}}{\left(\frac{2\pi }{D}\right)}^{\frac{1}{2}\left(\frac{N(N-1)}{2}-1\right)}\left({\mathcal{C}}_i+\mathcal{O}\left(D^{-1}\right)\right),$$

$${\Delta }_N^{max}=\frac{1}{N^N(N-1{)}^{N-1}}$$

$$F_i(D\to \infty )={\pi }^{\frac{7}{2}}{2}^{\frac{25}{2}}\frac{{\left(4^4{5}^5\right)}^{-\frac{D}{2}}\Gamma \left(\frac{3D}{2}-3\right)}{D^{\frac{9}{2}}\Gamma \left(\frac{D-4}{2}\right)\Gamma \left(\frac{D-3}{2}\right)\Gamma \left(\frac{D-1}{2}\right)}\left({\mathcal{C}}_i+\mathcal{O}\left(D^{-1}\right)\right).$$

$$R_6$$

$$\displaystyle+\epsilon^{3}\Bigg{(}-558+161\zeta_{2}+197\zeta_{3}+61\zeta_{2}^{2}-80\zeta_{3}\zeta_{2}+207\zeta_{5}\Bigg{)}$$

$$\displaystyle+\epsilon^{4}\Bigg{(}-3025+939\zeta_{2}+897\zeta_{3}+\frac{1157}{5}\zeta_{2}^{2}-400\zeta_{3}\zeta_{2}+1035\zeta_{5}+\frac{288}{5}\zeta_{2}^{3}-153\zeta_{3}^{2}\Bigg{)},$$

$$R_{8,a}$$

$$R_{8,b}$$

$$F_1$$

$$\displaystyle+\epsilon^{4}\Bigg{(}\frac{96102601}{1492992}-\frac{207493}{10368}\zeta_{2}-\frac{689}{72}\zeta_{3}+\frac{17}{240}\zeta_{2}^{2}\Bigg{)}+\epsilon^{5}\Bigg{(}\frac{2206279853}{5971968}-\frac{5668169}{41472}\zeta_{2}$$

$$\displaystyle-\frac{207493}{2592}\zeta_{3}+\frac{901}{960}\zeta_{2}^{2}+\frac{169}{18}\zeta_{3}\zeta_{2}-\frac{65}{6}\zeta_{5}\Bigg{)}+\epsilon^{6}\Bigg{(}\frac{437728233961}{214990848}-\frac{1249333813}{1492992}\zeta_{2}$$

$$\displaystyle-\frac{5668169}{10368}\zeta_{3}+\frac{271337}{34560}\zeta_{2}^{2}+\frac{8957}{72}\zeta_{3}\zeta_{2}-\frac{3445}{24}\zeta_{5}-\frac{4007}{5040}\zeta_{2}^{3}+\frac{169}{9}\zeta_{3}^{2}\Bigg{)}$$

$$F_2$$

$$\displaystyle+\frac{1}{\epsilon}\Bigg{(}\frac{1450}{3}-\frac{2080}{3}\zeta_{2}+\frac{2816}{3}\zeta_{3}+25\zeta_{2}^{2}\Bigg{)}-\frac{4390}{3}+\frac{6370}{3}\zeta_{2}-\frac{8192}{3}\zeta_{3}-\frac{275}{2}\zeta_{2}^{2}$$

$$\displaystyle+\frac{6656}{3}\zeta_{3}\zeta_{2}-2504\zeta_{5}+\epsilon\Bigg{(}\frac{13090}{3}-\frac{18850}{3}\zeta_{2}+\frac{25088}{3}\zeta_{3}+400\zeta_{2}^{2}-\frac{36608}{3}\zeta_{3}\zeta_{2}$$

$$\displaystyle+13772\zeta_{5}-211\zeta_{2}^{3}+\frac{13136}{3}\zeta_{3}^{2}\Bigg{)}$$

$$F_3$$

$$\displaystyle+\epsilon^{3}\Bigg{(}\frac{65347}{64}-\frac{3999}{16}\zeta_{2}-\frac{499}{2}\zeta_{3}-\frac{777}{10}\zeta_{2}^{2}+86\zeta_{3}\zeta_{2}-237\zeta_{5}\Bigg{)}+\epsilon^{4}\Bigg{(}\frac{887695}{128}$$

$$\displaystyle-\frac{64219}{32}\zeta_{2}-\frac{6303}{4}\zeta_{3}-\frac{5967}{16}\zeta_{2}^{2}+602\zeta_{3}\zeta_{2}-1659\zeta_{5}-\frac{1827}{20}\zeta_{2}^{3}+204\zeta_{3}^{2}\Bigg{)}$$

$$F_4$$

$$\displaystyle+191\zeta_{3}\zeta_{2}-\frac{1129}{2}\zeta_{5}+\epsilon\Bigg{(}10\zeta_{2}+52\zeta_{3}+\frac{801}{20}\zeta_{2}^{2}-\frac{191}{2}\zeta_{3}\zeta_{2}+\frac{1129}{4}\zeta_{5}-\frac{16547}{70}\zeta_{2}^{3}$$

$$\displaystyle+487\zeta_{3}^{2}\Bigg{)}$$

$$F_5$$

$$F_6$$

$$\displaystyle+\frac{1}{\epsilon}\Bigg{(}-\frac{7793}{3}+\frac{10450}{9}\zeta_{2}+\frac{2386}{3}\zeta_{3}-\frac{4927}{18}\zeta_{2}^{2}\Bigg{)}+\frac{31259}{3}-\frac{67705}{9}\zeta_{2}+4296\zeta_{3}$$

$$\displaystyle+\frac{209681}{180}\zeta_{2}^{2}+\frac{14708}{3}\zeta_{3}\zeta_{2}-\frac{28148}{3}\zeta_{5}+\epsilon\Bigg{(}-\frac{117869}{3}+\frac{317353}{9}\zeta_{2}-31444\zeta_{3}$$

$$\displaystyle-\frac{66809}{45}\zeta_{2}^{2}-\frac{101858}{9}\zeta_{3}\zeta_{2}+\frac{84602}{3}\zeta_{5}-\frac{357871}{126}\zeta_{2}^{3}+\frac{96472}{9}\zeta_{3}^{2}\Bigg{)}$$

$$F_7$$

$$\displaystyle+\epsilon^{3}\Bigg{(}-\frac{5709}{4}+\frac{2495}{4}\zeta_{2}+\frac{1337}{2}\zeta_{3}+72\zeta_{2}^{2}-151\zeta_{3}\zeta_{2}+207\zeta_{5}\Bigg{)}+\epsilon^{4}\Bigg{(}-\frac{80649}{8}$$

$$\displaystyle+\frac{35823}{8}\zeta_{2}+\frac{18035}{4}\zeta_{3}+\frac{4881}{10}\zeta_{2}^{2}-1510\zeta_{3}\zeta_{2}+2070\zeta_{5}-\frac{387}{10}\zeta_{2}^{3}-387\zeta_{3}^{2}\Bigg{)}$$

$$F_8$$

$$\displaystyle+\frac{1397}{6}\zeta_{3}-\frac{823}{60}\zeta_{2}^{2}\Bigg{)}-\frac{439}{2}+\frac{2401}{6}\zeta_{2}-\frac{2032}{3}\zeta_{3}+\frac{9053}{120}\zeta_{2}^{2}+\frac{1829}{3}\zeta_{3}\zeta_{2}$$

$$\displaystyle-\frac{2381}{3}\zeta_{5}+\epsilon\Bigg{(}\frac{1309}{2}-\frac{7105}{6}\zeta_{2}+\frac{6223}{3}\zeta_{3}-\frac{3292}{15}\zeta_{2}^{2}-\frac{20119}{6}\zeta_{3}\zeta_{2}+\frac{26191}{6}\zeta_{5}$$

Full text

1 Introduction

Nowadays, perturbative calculations play the key role in describing data from high-energy particle colliders, such as the LHC, as well as in improving the precision of numerical parameters in Standard Model and other models. It is clear now that higher-order calculations will play an even more crucial role in processing data from future high-luminosity colliders, like the FCC or the ILC, where theoretical errors will dominate over experimental statistical errors. These arguments motivate us to make one step forward beyond available fully- inclusive phase-space integrals for a four-particle decay [GGH04] and calculate a set of yet unknown integrals that corresponds to a five-particle decay of a color-neutral off-shell particle in Quantum Chromodynamics. These results, among others, are necessary ingredients in calculating various exclusive quantities with the method of differential equations where they are needed to determine integration constants, as for example discussed in [Git16, GM15] for the three-loop time-like splitting functions [AMV12].

In this paper, we focus on the calculation of master integrals that can be used to express any other integral of the corresponding topology provided a set of integration-by-parts rules (IBP) [CT81] is known. Our approach is based on techniques for solving dimensional recurrence relations (DRR) [Tar96] described in [LM17a, LM17b]. In particular, we use DREAM package [LM17a] to obtain numerical results for desired integrals with 2000-digit precision and restore analytical form in terms of multiple zeta values (MZV) [Fur03, BBV10, LM16] up to weight 12 using the PSLQ method [FBA99] as implemented in Mathematica. We also present a Monte-Carlo code, based on the RAMBO algorithm [KSE86], for numerical integration of the phase-space integrals in arbitrary (integer) number of dimensions that has been used to check consistency of the obtained results.

This paper is organized as following. In Section 2 we introduce our notation and describe our calculational method in more details. In Section 3 we provide complete results for four-particle integrals and discuss numerical cross-checks using Monte-Carlo integration. In Section 4 we make final remarks. In Appendix A we provide the complete list of master integrals.

Additionally we provide auxiliary files on arXiv222https://arxiv.org containing the complete master integrals with MZV weight up to 12 as well as the Monte-Carlo integration routines with the corresponding results.

2 The Method