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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 891 (2015) 42–72

www.elsevier.com/locate/nuclphysb

The bottom-quark mass from non-relativistic

sum rules

at NNNLO

M. Beneke

, A. Maier

, J. Piclum

a,b,c

, T. Rauh

Physik Department T31, James-Franck-Straße 1, Technische Universität München, D-85748 Garching, Germany

Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany

Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, Sidlerstrasse 5,

CH-3012

Bern, Switzerland

Received 26

November 2014; accepted 1 December 2014

Available

online 4 December 2014

Editor: Tommy

Ohlsson

Abstract

We determine the mass of the bottom quark from high moments of the bb production cross section

in e

−

annihilation, which are dominated by the threshold region. On the theory side next-to-next-to-

next-to-leading

order (NNNLO) calculations both for the resonances and the continuum cross section are

used for the ﬁrst time. We ﬁnd m

(2GeV) = 4.532

+0.013

−0.039

GeV for the potential-subtracted mass and

) =4.193

+0.022

−0.035

GeV for the MS bottom-quark mass.

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

The bottom-quark mass is one of only a handful of fundamental QCD parameters, and thus its

precise knowledge is of considerable interest by itself. Furthermore, there are also phenomeno-

logical applications which beneﬁt from a small uncertainty in the value of the bottom-quark

mass. Examples include Higgs decays to bottom quarks and decays of B mesons.

Sum rules pro

vide a well-established method for the determination of heavy-quark masses [1,2].

Considering the normalised bottom production cross section

E-mail address: a.maier@tum.de (A. Maier).

http://dx.doi.org/10.1016/j.nuclphysb.2014.12.001

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

M. Beneke et al. / Nuclear Physics B 891 (2015) 42–72 43

(s) =

σ(e

−

→bb +X)

σ(e

−

→μ

−

)

, (1.1)

where s is the square of the e

−

center-of-mass energy, we can deﬁne its moments M

∞



(s)

n+1

=−6πi



(s)

n+1

12π











. (1.2)

denotes the contribution from bottom quarks to the vacuum polarisation function. C is an

arbitrary closed contour that encloses the origin and does not cross the branch cut, i.e. the part

of the positive real half-axis where R

(s) = 12π Im Π

(s + i) > 0. Quark–hadron duality now

permits to determine the bottom-quark mass by equating moments obtained from the measured

hadronic cross section with moments calculated from derivatives of the theoretically predicted

polarisation function, which can themselves be obtained from the theoretically predicted partonic

cross section by the above dispersion relation (1.2).

For small v

alues of n it is most convenient to calculate moments directly from the derivatives

of the polarisation function in conventional perturbation theory. As n increases the theory un-

certainty grows and for n  10 the perturbation series shows no signs of convergence anymore

(cf. [3,4]).

For larger values of n the integral over R

in Eq. (1.2) is increasingly dominated by small

kinetic energies E =

√

s − 2m

∼ m

/n [5,6], where m

is the bottom-quark mass. Thus, the

quarks are non-relativistic with a small velocity v ∼ 1/

√

n  1. At the same time the strong

coupling constant α

is of the same order as the quark velocity. For this reason terms scaling with

powers of α

/v, which originate from Coulomb interaction, have to be summed to all orders. This

is achieved in the effective theory of potential non-relativistic QCD (PNRQCD) [7]. In summary,

for large-n, “non-relativistic” moments the power counting for the cross section up to NNNLO

is given by

∼v







⎧

⎪

⎨

⎪

⎩

1LO,

,v NLO,

,α

v,v

NNLO,

,α

v,α

NNNLO.

(1.3)

The admissible values of n are limited by the requirement that the ultrasoft scale m

/n remains

larger than the intrinsic strong interaction scale of a few hundred MeV [8].

In this wo

rk we present the ﬁrst determination of the bottom-quark mass from large-n sum

rules at next-to-next-to-next-to-leading order (NNNLO). The corresponding NNLO calculations

have been done about ﬁfteen years ago [8–10]. A partial NNNLO result that uses NNNLO accu-

racy for the bound-state contribution and NNLO accuracy for the continuum contribution to the

moments appeared recently [11]. The outline of this paper is as follo

ws. First, in Section 2, we

derive the values of the relevant moments from experimental data. We proceed in Section 3 with

an outline of the corresponding PNRQCD calculation and a discussion of suitable mass schemes.

In Section 4 we summarise our results for R

and for the bottom-quark mass and compare them

to other recent high-precision sum rule analyses. For the comparison we consider the works by

Chetyrkin et al. [12], Hoang et al. [13], and Penin and Zerf [11]. We conclude in Section 5. Since

we are aiming at high precision, we include the ef

fects of the non-zero charm-quark mass. The

details of this computation are given in four appendices.

44 M. Beneke et al. / Nuclear Physics B 891 (2015) 42–72

Table 1

Contributions

to the experimental moments M

exp

in (10 GeV)

−2n

n 678910

Resonances 0.1861(20) 0.2004(22) 0.2166(24) 0.2351(27) 0.2560(29)

Continuum 0.0240(85) 0.0182(58) 0.0140(41) 0.0110(29) 0.0088(21)

Total 0.2101(88) 0.2185(62) 0.2307(47) 0.2461(39) 0.2648(36)

n 11 12 13 14 15

Resonances 0.2797(33) 0.3064(36) 0.3364(40) 0.3702(45) 0.4081(50)

Continuum 0.0070(15) 0.0057(11) 0.0046(8) 0.0038(6) 0.0031(4)

Total 0.2867(36) 0.3120(38) 0.3410(41) 0.3740(45) 0.4112(50)

2. Experimental moments

For sufﬁciently large values of n the experimental moments M

exp

are dominated by Υ bound

states. In the narrow-width approximation for the resonances Υ(1S) to Υ(4S) we obtain

exp

=9π



N=1

α(M

Υ(NS)

)

Υ(NS)→l

−

2n+1

Υ(NS)

∞



cont

(s)

n+1

, (2.1)

where we take the current PDG values [14] for the masses M

Υ(NS)

and the leptonic widths

Υ(NS)→l

−

of the Υ resonances. In the energy region of interest we approximate the running

QED coupling by a constant value, which is given in terms of the ﬁne structure constant by

α(2m

) ≈ 1.036 α [15].

For the remaining continuum integral we use experimental data [16] corrected for initial state

radiation between

√

cont

= 10.6178 GeV and 11.2062 GeV (see [12] for details). In the ab-

sence of data for higher center-of-mass energies we assume R

to stay roughly constant with

=0.3 ±0.2. For n = 6, 10, 15 the unknown high-energy part constitutes approximately 53%,

35%, 21% of the total continuum contribution, respectively. The large uncertainty of this part is

therefore not expected to be important. The resulting values and uncertainties for the experimen-

tal moments are shown in T

able 1.

3. Theory moments

3.1. The cross section in PNRQCD

The normalised cross section for bottom production at NNNLO in PNRQCD has the form

=12πe







−





G(E) +···



, (3.1)

see e.g. [17]. e

and m

are the bottom quark’s fractional electric charge and pole mass, respec-

tively. E is the kinetic energy with the usual relation E =

√

s −2m

to the center-of-mass energy

√

s. c

and d

are the matching coefﬁcients of the non-relativistic vector current, i.e.

†

χ +

†

χ +···, (3.2)

where j

are the spatial components of the relativistic current

bγ

b, and ψ (χ ) the non-

relativistic quark (antiquark) spinors. Finally, G(E) is the non-relativistic current correlator

M. Beneke et al. / Nuclear Physics B 891 (2015) 42–72 45

G(E) =

(d − 1)



iEx

0|T



†



(x)



†



(0)|0



PNRQCD

(3.3)

in d space–time dimensions.

Below threshold, i.e. for E<0, G(E

) develops inﬁnitely many poles which can be interpreted

as S-wave, spin-1 b

b bound states. In the vicinity of such a bound state G(E) behaves as

G(E)

E→E

−−−−→

|ψ

(0)|

−E −i

, (3.4)

where ψ

(0) is the bound state wave function at the origin while E

is the binding energy.

Splitting off the bound-state contribution we can thus write the moments as

12π

∞



N=1

(2m

)

2n+1

∞



(s)

n+1

(3.5)

with the residues



−







(0)



+···. (3.6)

Note that Z

is a physical quantity, while the wave function ψ

(0) and the matching coefﬁcients

separately depend on the factorisation scale and scheme that separates short- and long-distances.

(3.1) and Z

(3.6) are understood to be strictly expanded up to NNNLO according to the

PNRQCD power counting (1.3). For example, terms of order α

and higher that are generated

through the product of lower-order terms are to be dropped. Contrarily, we will leave the factor

(2m

)

−(2n+1)

in Eq. (3.5) unexpanded for reasons discussed in Section 3.3. Not expanding

this factor is equivalent to resumming the poles back into the polarisation function (cf. [18]), i.e.

replacing

→Π

∞



N=1



]

expanded

−E]

unexpanded

−



−E



expanded



(3.7)

in the contour integral that deﬁnes the moments (1.2).

3.2. Technical implementation

We require the following ingredients for the NNNLO analysis of non-relativistic moments:

• The hard W

ilson coefﬁcient c

at order α

[19] and d

at order α

[20].

• The third-order corrections to the S-w

ave energy levels E

[21–23], to the wave func-

tions ψ

at the origin [22–24] and to G(E) [23,25] from non-relativistic potentials up to

NNNLO [17,26–30].

• Ultrasoft corrections to these quantities [31–33].

For an e

xtensive overview of all third-order corrections see [17,25].

Note that the results in [25,33] are calculated for the case of top quarks, that is in the comple

energy plane for ﬁnite imaginary part Γ corresponding to the top-quark width. The application of

these results to bottom quarks requires to take the limit Γ →0, which is non-trivial, since the an-

alytic e

xpressions cannot be straightforwardly evaluated numerically for vanishing width. In [33]

46 M. Beneke et al. / Nuclear Physics B 891 (2015) 42–72

the Γ =0 result has been constructed for the ultrasoft contribution by extrapolation. Evaluating

the third-order potential corrections given in [25] for real energy E requires a substantial amount

of extra work, which we brieﬂy discuss in the following.

The higher-order potential corrections to the Green function are expressed in terms of nested

harmonic sums, sums over gamma and polygamma functions, and generalised hypergeometric

functions [23,25]. The complex variable λ = (α

/2)

√

−m/(E +iΓ ) appears for example

in the argument of (poly-)gamma functions or as one of the parameters of the hypergeometric

functions. In our application, we have to evaluate the Green function for positive values of the

energy E, starting at E = 0. For vanishing width Γ , λ tends to +i∞ as E tends to zero. Thus,

we ha

ve to ensure that all expressions are well-deﬁned in this limit and that their numerical

evaluation is possible.

In most cases it is possible to express the correction to the Green function in terms of harmonic

sums. This is in particular the case for most of the generalised hypergeometric functions, which

can be treated as described in Appendix A.1 of [34]. The harmonic sums can then be analytically

continued with the methods of [35,36]. Ho

wever, in some cases the correction to the Green

function is expressed in terms of single or even double sums which could not be expressed as

nested harmonic sums. For such sums it was often necessary to truncate the summation at some

(λ-dependent) value and construct suitable asymptotic e

xpansions to approximate the remainder.

In all cases we have checked that the numerical precision is sufﬁcient for our extraction of the

bottom-quark mass, such that the numerical uncertainty can be neglected.

A relati

vely simple example of this procedure is given by the sum

∞



k=1

[(k −λ)(ψ(k −λ) −ψ(k)) + kλψ

(1)

(k)]

, (3.8)

which appears in the insertion of the Darwin term. Here ψ is the logarithmic derivative of the

gamma function and ψ

(1)

the ﬁrst derivative of ψ. The sum converges only slowly when λ is

large, which makes the numerical evaluation difﬁcult. Therefore, we introduce a cut-off Λ for

the summation and explicitly sum all terms up to this cut-off. Choosing Λ to be much larger than

|λ|, we can approximate the remainder by e

xpanding the summand in the limit k →∞. Note that

for ψ(k − λ) this is not simply an expansion in |λ|/k, but rather a double expansion for k  1

and k |λ|. In the ﬁrst step the entire argument of the ψ function is considered large and the

terms of the resulting asymptotic series are further e

xpanded for |λ|/k → 0in the second step,

yielding

ψ(k −λ) = ln(k −λ) −

2(k −λ)

−

12(k −λ)

120(k −λ)



(k −λ)



=ln k − (1 +2λ)

−



1 + 6λ +6λ



12k

−



λ + 3λ

+2λ





1 − 30λ

−60λ

−30λ



120k





. (3.9)

After expanding the summand in (3.8), the sum over k from Λ +1to inﬁnity can be evaluated

in terms of Hurwitz zeta functions (in more complicated cases we also encounter derivatives of

this function). The ﬁrst three terms are

ζ(3,Λ+ 1) +

ζ(4,Λ+ 1) +



−3 + 4λ



ζ(5,Λ+ 1), (3.10)

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