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JHEP04(2018)080

Published for SISSA by Springer

Received: January 14, 2018

Revised: April 8, 2018

Accepted: April 9, 2018

Published: April 13, 2018

Two-loop integrals for CP-even heavy quarkonium

production and decays: elliptic sectors

Long-Bin Chen,

Jun Jiang

and Cong-Feng Qiao

b,c,1

School of Physics & Electronic Engineering, Guangzhou University,

Guangzhou 510006, China

School of Physics, University of Chinese Academy of Sciences,

YuQuan Road 19A, Beijing 100049, China

CAS Center for Excellence in Particle Physics,

Beijing 100049, China

E-mail: [email protected], [email protected],

[email protected]

Abstract: By employing the diﬀerential equations, we compute analytically the elliptic

sectors of two-loop master integrals appearing in the NNLO QCD corrections to CP-even

heavy quarkonium exclusive production and decays, which turns out to be the last and

toughest part in the relevant calculation. The integrals are found can be expressed as Gon-

charov polylogarithms and iterative integrals over elliptic functions. The master integrals

may be applied to some other NNLO QCD calculations about heavy quarkonium exclusive

production, like γ

∗

γ → Q

Q, e

−

→ γ + Q

Q, and H/Z

→ γ + Q

Q, heavy quarkonium

exclusive decays, and also the CP-even heavy quarkonium inclusive production and decays.

Keywords: Heavy Quark Physics, Perturbative QCD

ArXiv ePrint: 1712.03516

Corresp onding author.

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP04(2018)080

JHEP04(2018)080

Contents

1 Introduction 1

2 Notation and kinematics 2

3 Iterated integrals and complete elliptic integrals 3

4 Elliptic integral sectors 4

4.1 Sector I: integrals with massive sunrise integrals as subtopology 5

4.2 Sector II: non-planar two-loop three-point integrals 12

4.3 Analytic continuation and discussions 14

5 Conclusions and outlooks 14

A The deﬁnition for integrals 15

B The typical analytical results 16

1 Introduction

Precision physics in colliders requires more higher-order corrections in perturbation theory.

Unravelling the mathematical structure of Feynman integrals in multiloop calculation is

somehow critical to handle the complexity of higher order calculations and may help us to

obtain a better control of the perturbative expansion. In recent years, the corresponding

research achieved some breakthroughs and becomes now one of the hot topics in physics

and mathematics.

One of the powerful methods to evaluate the master integrals analytically attributes to

the diﬀerential equation [1–5]. With recent developments [6–10], this method becomes now

a prevailing one in tackling those integrals unsolvable before. It was noticed by Henn that

generically in multi-loop calculation, choosing a set of suitable basis for master integrals can

greatly simplify the corresponding diﬀerential equations [6], which can be calculated iter-

atively in dimensional regularization scheme. In light of this proposal, many of multi-loop

Feynman integrals for various phenomenological processes have been calculated [11–23].

Note, some Feynman integrals in two-loop or higher order possess new mathematical struc-

tures [24–31], which cannot be expressed as multiple polylogarithms and ask for diﬀerent

technique to deal with. A typical example is the massive two-loop sunrise integral, which

has been studied intensively [32–40].

The heavy quarkonium production and decay are one of the hot topics in particle

physics ever since the ﬁrst discovery in 1974, especially with the advent of Nonrelativistic

Quantum Chromodynamics (NRQCD) factorization formalism [41]. Up to date there still

– 1 –

JHEP04(2018)080

∗

Figure 1. Typical two-loop Feynman diagrams for CP-even heavy quarkonium production.

exist some discrepancies between experimental data and theoretical expectations [42–45],

which appeal for precision calculations. In one of our previous works [46] we gave out a set

of 86 two-loop master integrals about heavy quarkonium production and decay, which can

be cast into the canonical form and expressed in terms of multiple polylogarithms. However,

for those Feynman integrals with functions beyond the realm of multiple polylogarithms the

calculation is not done yet. In fact, to date, only a limited number of similar calculations

have been performed in the literature.

In this work, we calculate analytically all remaining integrals with diﬀerent mathemat-

ical structures from multiple polylogarithms in CP-even heavy quarkonium production and

decays. The master integrals will be classiﬁed into two sectors, one with integrals contain-

ing sub-topologies related to the two-loop massive sunrise integrals and the other involving

non-planar two-loop three-point integrals. Following the strategy suggested in ref. [39] and

with properly chosen basis, we cast the diﬀerential equations of those integrals in the ﬁrst

sector into a proper form that can be solved recursively. Of the second sector, the key

point is to ﬁnd the homogeneous solutions for the second-order diﬀerential equations of the

two-loop non-planar three-point massive integrals, with that the full solutions can then be

obtained by constant variation.

The paper is organized as follows. In section 2, the kinematics is discussed and the

derivatives with respect to kinematic variables will be given. In section 3, the iterative

integrals and complete elliptic integrals are introduced. In section 4, the elliptic type

integrals will be separated into two sectors, and the calculation procedure for them will

be elucidated respectively. For illustration, speciﬁc examples will be given. Section 5

is remained for conclusions and outlooks. The deﬁnition of master integrals is given in

appendix A, and several simple but typical analytical results are presented in appendix B.

2 Notation and kinematics

The heavy quarkonium exclusive production in electron-positron collision has a relatively

low background, and has played an important role in the study of quarkonium production

mechanism. Here we calculate the CP-even quarkonium production in two correlated

processes, that is in γ

∗

γ collision and in electron-position annihilation associated with

a photon,

∗

) + γ(k

) → Q(k

)

Q(k

¯q

) , (2.1)

∗

) → Q(k

)

Q(k

¯q

) + γ(k

) , (2.2)

– 2 –

JHEP04(2018)080

where k

= 2ss, k

= 0 and k

= k

¯q

= m

. The typical Feynman diagrams are showed in

ﬁgure 1. The process (2.1) is in Euclidean region with ss < 0, and the momenta satisfy

the following relations

+ k

)

= (k

+ k

¯q

)

= 4m

. (2.3)

Whereas, the process (2.2) is in Minkowski region with 2ss > 4m

, and

− k

)

= (k

+ k

¯q

)

= 4m

. (2.4)

Note, in the threshold expansion approach, quark and anti-quark momenta are taken to

be equal, i.e. k

= k

¯q

In order to express the results compactly, here we introduce three dimensionless vari-

ables x, y and z as follows:

= −

(1 − x)

= (y + 2) = (z + 1) . (2.5)

The NNLO QCD corrections to processes (2.1) and (2.2) are calculated in light of

Feynman diagrams. As a routine, with some algebraic manipulations, the amplitudes can

be reduced to a set of scalar integrals. We use the Mathematica package FIRE [47–49]

to reduce the scalar integrals to a minimum set of independent master integrals. The

calculation of these master integrals is the central issue, and normally turns out to be

a nontrivial work. In our calculation, we apply the method of diﬀerential equations to

calculate the master integrals.

The ﬁrst step of deriving diﬀerential equations is taking derivatives of the Lorentz in-

variant kinematic variables, and expressing them as linear combinations of master integrals.

The FIRE is also employed in the derivation of diﬀerential equations. The derivatives of

the external momenta can be expressed as the derivatives of ss and m

, like

∂

∂k

= k

∂ss

∂k

∂

∂ss

+ k

∂m

∂k

∂

∂m

(2.6)

with i(j) = 1 or 2. And in reverse, the derivative

∂

∂ss

can be expressed as a linear combi-

nation of derivatives k

∂

∂k

, i.e.,

2ss

∂

∂ss

= k

∂

∂k

ss + 2m

ss − 2m

∂

∂k

. (2.7)

The derivative transform can be readily obtained according to equation (2.5). With the

variables chosen in above, analytical results of the integrals can then be formulated in a

compact form, in terms of iterative integrals and elliptic integrals.

3 Iterated integrals and complete elliptic integrals

The Goncharov polylogarithms (GPLs) [50] are deﬁned as

,...,a

(x) ≡

t − a

,...,a

(x) , (3.1)

−→

(x) ≡

log

x , (3.2)

– 3 –

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