Table Of ContentHiggs boson decay into bottom quarks and
uncertainties of perturbative QCD
predictions
A.L. Kataev
9
0
0
Institute for Nuclear Research of Russian Academy of Sciences
2
117312, Moscow, Russia
n
a
J
V.T. Kim
6
2
St.-Petersburg Nuclear Physics Institute of Russian Academy of Sciences
]
h 188300, Gatchina, Russia
p
-
p
e
Abstract
h
[
Different methods for treating the results of higher-order perturbative
2
QCDcalculationsofthedecaywidthoftheStandardModelHiggsboson
v
2 into bottom quarks are discussed. Special attention is paid to the anal-
9 ysis of the MH dependence of the decay width Γ(H b¯b) in the cases
9 →
3 when the mass of b-quark is defined as the running parameter in the
. MS-scheme and as the quark pole mass. The relation between running
4
4
0 and pole masses is taken into account in the order αs-approximation.
8
Some special features of applications of Analytical PerturbationTheory
0
(APT) are commented.
:
v
i
X
r 1 Introduction
a
The study of the Higgs boson decay width into bottom quarks is rather impor-
tant for calculations of the branching ratios of this important ingredient of the
Standard Model and its various extensions. Current LEP and Tevatron fits of
theStandardModelparameters yield thevaluefor theHiggs boson massaround
M = 76+33 GeV C.L. 68%. With the direct LEP search limit M 115 GeV
H −24 H ≥
the fits provide at C.L. 95% M 182 GeV. In the case, if the scalar Higgs
H
≤
particle has the mass in the region 115 GeV M 2M , its decay width
H W
to bottom quarks Γ(H b¯b) dominates over ≤other c≤hannels. In particular, it
→
is determining the branching ratio of H γγ process which is considered to be
→
one of the most promising channel for searches of Higgs particles at LHC in the
1
mass region specified above (for reviews see [1]–[4]). There is also a possibil-
ity that the signal for H b¯b process may be seen at Tevatron [5] through
→
WH- and ZH- channels and at CMS-TOTEM [6] or/and FP420 [7] experimen-
tal proposals at LHC, aimed at searches of central exclusive H production, as
discussed from theoretical point of view, e.g., in Refs. [8,9]. The mentioned
experimentally-oriented motivation is pushing ahead the intention to study in
more detail the dominant theoretical effects to Γ(H b¯b) in the region of rel-
→
atively light Higgs boson. These effects are related to high-order perturbative
QCD predictions with their intrinsic uncertainties. Moreover, the comparison of
various representations for Γ(H → b¯b) = ΓHb¯b is rather important for planning
the experimental program of high energy linear e+e−-colliders for measuring
Higgs boson couplings [4].
2 QCD expressions for Γ
Hb¯b
Let us first consider QCD theoretical prediction for ΓHb¯b expressed in terms of
running b-quark mass and the QCD coupling constant in the MS-scheme as
m2(M )
Γ = Γb b H 1+ ∆Γ ai(M ) . (1)
Hb¯b 0 m2 (cid:20) i s H (cid:21)
b Xi≥1
Here Γb = 3√2/8πG M m2, m and M are the pole b-quark and Higgs boson
0 F H b b H
masses, a (M ) = α (M )/π and m (M ) are the QCD running parameters,
s H s H b H
defined in the MS-scheme. The coefficients ∆Γ are known analytically up to
i
4-th order correction of perturbation theory [10]. They consist of the posi-
tive contributions dE, calculated directly in the Euclidean region, and from the
i
proportional to π2 kinematic effects, which appear as the result of analytical
continuation from the Euclidean space-like to the Minkowskian time-like region.
This π2-term arises first in Eq. (1) at the a2-correction [11]. Its coefficient was
s
corrected later in [12], [13], but the kinematic π2-term remained unaffected.
Using the notations of Ref. [14] one can write down the following relations:
17
∆Γ = dE = ; (2)
1 1 3
∆Γ = dE γ (β +2γ )π2/3; (3)
2 2 − 0 0 0
∆Γ = dE dE(β +γ )(β +2γ )+β γ +2γ (β +2γ ) π2/3; (4)
3 3 − 1 0 0 0 0 1 0 1 0 0
∆Γ = dE (cid:2)dE(β +γ )(3β +2γ )+dEβ (5β +6γ )/2 (cid:3)
4 4 − 2 0 0 0 0 1 1 0 0
+ 4dEγ(cid:2)(β +γ )+β γ +2γ (β +γ )+γ (3β +4γ ) π2/3
1 1 0 0 2 0 1 1 1 2 0 0
+ γ (β +γ )(β +2γ )(3β +2γ )π4/30, (cid:3) (5)
0 0 0 0 0 0 0
2
where the n -dependence of dE (i > 2) read
f i
10801 39 65 2
dE = ζ n ζ
2 (cid:20) 144 − 2 3(cid:21)− f(cid:20)24 − 3 3(cid:21)
51.567 1.907n 42.032 (n = 5); (6)
f f
≈ − ≈
163613 109735 815 46147 262 5 25
dE = ζ + ζ n ζ + ζ + ζ
3 (cid:20) 5184 − 216 3 12 5(cid:21)− f(cid:20) 486 − 9 3 6 4 9 5(cid:21)
15511 1
+ n2 ζ
f (cid:20)11664 − 3 3(cid:21)
648.71 63.742n +0.92913n2 353.23 (n = 5); (7)
≈ − f f ≈ f
10811054729 3887351 458425 265 373975
dE = ζ + ζ2+ ζ + ζ
4 (cid:20) 497664 − 324 3 432 3 18 4 432 5
1375 178045
ζ ζ
6 7
− 32 − 768 (cid:21)
1045811915 5747185 955 9131 41215
+ n + ζ ζ2 ζ + ζ
f(cid:20)− 373248 5184 3− 16 3 − 576 4 432 5
2875 665
+ ζ + ζ
6 7
288 72 (cid:21)
220313525 11875 5 25 5015
+ n2 ζ + ζ2+ ζ ζ
f (cid:20) 2239488 − 432 3 6 3 96 4− 432 5(cid:21)
520771 65 1 5
+ n3 + ζ + ζ + ζ
f (cid:20)−559872 432 3 144 4 18 5(cid:21)
9470.8 1454.3n +54.783n2 0.45374n3 3512.2 (n = 5). (8)
≈ − f f − f ≈ f
The term of Eq. (2) was evaluated in Ref. [11]. It is in agreement with the
expressed in other ways results of previous studies, performed in [15–17]. The
second coefficient was corrected in [12,13]. The result of Eq. (7) was obtained
in Ref. [18]. The exact value of the Euclidean coefficient of Eq. (8), analytically
calculated in [10], turned out to be in reasonable agreement with the estimates,
obtained within the used in Ref. [14] a variant of the effective charge approach
(ECH) and the principle of minimal sensitivity (PMS) approach. The variant
of the two approaches was developed in Ref. [19]. The coefficients β and γ
i i
enter the expansions of the QCD renormalization group (RG) β-function and
anomalous dimension of mass function γ . The QCD β-function can be defined
m
as
da
s
= β(a ) (9)
dlnµ2 s
= β a2 β a3 β a4 β a5 β a6+O(a7).
− 0 s − 1 s − 2 s − 3 s − 4 s s
3
Its MS-scheme expressions were calculated analytically up to 4-loop correc-
tions [20], confirmed recently in the work [21]. We present here the results
of analytical evaluation of the coefficients of Eq. (9) (β and β are scheme-
0 1
independent) in the MS-scheme, supplemented by the numerical expressions,
related to n = 5 number of active flavours:
f
1 2
β = 11 n
0 f
4(cid:20) − 3 (cid:21)
2.75 0.1667 n 1.9167 (n = 5) (10)
f f
≈ − ≈
1 38
β = 102 n
1 f
16(cid:20) − 3 (cid:21)
6.375 0.7917n 2.4167 (n = 5) (11)
f f
≈ − ≈
1 2857 5033 325
β = n + n2
2 64(cid:20) 2 − 18 f 54 f(cid:21)
22.32 4.3689n +0.09404n2 2.8267 (n = 5) (12)
≈ − f f ≈ f
1 149753 1078361 6508
β = +3564ζ + ζ n
3 3 3 f
256(cid:20)(cid:18) 6 (cid:19)−(cid:18) 162 27 (cid:19)
50065 6472 1093
+ + ζ n2+ n3
(cid:18) 162 81 3(cid:19) f 729 f(cid:21)
114.23 27.134n +1.5824n2 +0.0059n3 18.852 (n = 5). (13)
≈ − f f f ≈ f
Throughout this work we fix n = 5 and neglect the contribution of lighter four
f
quarkstotherelationbetweenrunningmassm (M )andpole(oron-shell)mass
b H
m inEq.(1)(moredetailswillbegiven below). Thisisdoneforself-consistency
b
offurtheranalysis. Indeed,thesimilarcontributionstothecoefficientfunctionof
Eq.(1)arestillunknownandareexpectedtobesmall. Noticeaninterestingfact:
the growth of the coefficients of β-function at n = 5 is starting to manifest itself
f
from the four-loop only (on the contrary to the case with n = 3 whenthe values
f
of the coefficients of perturbative series for the β-function are monotonically
increasing from the one-loop order). As to the five-loop coefficient β in Eq.
4
(9), it was estimated in Ref. [22] by means of Pad´e approximations approach.
The input information, used in these estimates, is the analytical result for the
n 4-contribution to β calculated in [23]. In our normalization conditions it has
f 4
the following form
1 1205 152
β[4] = ζ n4 = 0.0017969n4 (14)
4 1024(cid:20)2916 − 81 3(cid:21) f − f
The approximation of Ref. [22], which is taking it into account, reads:
105
β 7.59 2.19n +20.5n2 49.8 10−5n3 1.84 10−5n4 (15)
4 ≈ 1024(cid:20) − f f − f − f(cid:21)
741.2 213.87n +20.02n2 0.0483n3 0.0018n4 165.2 (n = 5).
≈ − f f − f − f ≈ f
4
We will use this estimate in our studies. The related to the MS-scheme anoma-
lous mass dimension function is defined as
dlnm
b
= γ (a ) (16)
dlnµ2 m s
= γ a γ a2 γ a3 γ a4 γ a5+O(a6).
− 0 s− 1 s − 2 s − 3 s − 4 s s
The four-loop correction was calculated independently in Ref. [24] and in
Ref. [25]. Let us present the explicitly known coefficients:
γ = 1; (17)
0
1 202 20
γ = n 4.2083 0.13889n 3.5139 (n = 5); (18)
1 f f f
16(cid:20) 3 − 9 (cid:21) ≈ − ≈
1 2216 160 140
γ = 1249 + ζ n n2
2 64(cid:20) −(cid:18) 27 3 3(cid:19) f − 81 f(cid:21)
19.516 2.2841n 0.027006n2 7.42 (n = 5); (19)
≈ − f − f ≈ f
1 4603055 135680
γ = + ζ 8800ζ
3 3 5
256(cid:20) 162 27 −
91723 34192 18400
+ ζ 880ζ ζ n
3 4 5 f
−(cid:18) 27 9 − − 9 (cid:19)
5242 800 160 332 64
+ + ζ ζ n2 ζ n3 (20)
(cid:18) 243 9 3− 3 4(cid:19) f −(cid:18)243 − 27 3(cid:19) f(cid:21)
98.933 19.108n +0.27616n2 +0.005793n3 11.034 (n = 5).
≈ − f f f ≈ f
In the same work [22] the following model for the five-loop coefficient of γ was
m
proposed:
γ 530 143n +6.67n2+0.037n3 8.54 10−5n4 13.68 (n = 5) (21)
4 ≈ − f f f − × f ≈ − f
It is based on application of the variant of the Pad´e approximation approach,
used for getting Eq. (15) discussed above. The explicit analytical expression
for the n4 contribution to γ , extracted from the QED results of Ref. [26] and
f 4
confirmed later on in [23], namely
1 65 5ζ ζ
γ[4] = + 3 4 n4 (22)
4 12(cid:18)5184 324 − 36(cid:19) f
was used. Notice that the numerical value of Eq. (21) is negative. This means
thatthe uncertainties of this Pad´e estimate are notsmall. Indeed, theanalytical
calculations of n3 contribution to Eq. (21), performed in Ref. [27], gave
f
1 331865 803 7 20
γ[3] = + ζ + ζ ζ n3 = 0.10832n3. (23)
4 12(cid:20)124416 432 3 12 4− 9 5(cid:21) f f
5
It does not agree with the similar coefficient in Eq. (21). Substituting Eq. (23)
into Eq. (21) one can see that for n = 5 the estimate for γ is still negative,
f 4
but rather small, namely γ 4.76595. We will incorporate this value in our
4
≈ −
further analysis just by fixing some parts of existing five-loop ambiguities. We
hope that this expression, obtained by matching the explicit results of Eq. (22)
and Eq. (23) and the Pad´e resummation technique, may be improved in the
future.
3 Analytical continuation effects and APT approach
Consider now in more detail the contributions to Γ from the Minkowskian
Hb¯b
coefficients, definedinEqs.(3)–(5). Letusremind,thattheyarecomposed from
the Euclidean terms (see Eqs. (6)–(8)) and the combinations of the coefficients
of the QCD RG functions β(a ) and γ (a ), which are defining proportional to
s m s
π2 analytical continuation effects in Eqs. (3)–(5). These kinematic effects turn
out to be negative and quite sizable. Thus, the coefficients ∆Γ (i 2) are
i
≥
smaller, than their Euclidean “analogs” dE. Indeed, for n =5 we have [12]
i f
∆Γ 29.147. (24)
2
≈
The numerical values of other terms obey the same pattern [10]:
∆Γ 41.758; (25)
3
≈
∆Γ 825.75. (26)
4
≈ −
This means that it is of interest how the effects of analytical continuation influ-
ence other results of perturbative QCD predictions. In the beginning of 80s this
problem was discussed in the numberof works on the subject (see, e.g., [28,29]).
At that time the problem of resummation of the π2-contributions to Γ was
Hb¯b
also considered in Ref. [11]. However, the real interest to resummations of the
analyticalcontinuationeffectswasattractedlateronafterappearanceofContour
Improved Technique (CIT) [30,31] and Analytic Perturbation Theory (APT),
in particular. This method was proposed and developed by D. V. Shirkov and
I. L. Solovtsov in the process of common investigations in Refs. [32]–[34] and in
the separate publications as well (see works done with the decisive contribution
of I. L. Solovtsov, [35,36], and the ones created by inspiration of D.V.Shirkov
[37,38]). This QCD approach was already used in various applications (see,
e.g., [39,40]). Among them are the studies of Higgs boson decay into b¯b-pair
with the help of Fractional Analytical Perturbation Theory (FAPT) [41], which
are complementary to definite considerations of Ref. [42]. Quite recently some
new [43] and even a bit corrected [44] discussions of applications of FAPT to
Γ and their comparisons with the results of Ref. [42], appeared in the liter-
Hb¯b
ature. In view of this, we will focus ourselves here on the brief discussions of
6
several ideas of Ref. [42]. It is worth to stress, that this work was motivated in
part by the desire to understand whether the bridge may be built between the
renormalon approach (for a review see, e.g., [45]) and the resummation of the
proportional to β -effects within Shirkov–Solovtsov APT. Note, that one of the
0
main cornerstones of renormalon approach isβ -resummation procedureas well.
0
Today, thanks to the works of Refs. [41,44] it is understood, that this bridge
does exist. Theessential point in thestudies of Ref. [42] is that theCIT[30] and
the concept of b-quark invariant mass are playing an important role. In view of
the fact that the positive features of the invariant mass are sometimes not taken
used to the total extent, let us remind the basic steps of definitions of this QCD
parameter:
1. Define the running quark mass through the solution of the following RG
equation for the anomalous dimension term (AD):
as(MH) γ (x)
m2(M ) =m2(m )exp 2 m dx ; (27)
b H b b (cid:20)− Z β(x) (cid:21)
as(mb)
2. Take the integral in the r.h.s. of Eq. (27):
a (M ) 2γ0/β0 AD(a (M )) 2
m2(M ) = m2(m ) s H s H ; (28)
b H b b (cid:18)a (m )(cid:19) (cid:18)AD(a (m ))(cid:19)
s b s b
3. Define
a2 1 3 a3
AD(a ) = 1+P a + P2+P s + P3+ P P +P s
s (cid:20) 1 s (cid:18) 1 2(cid:19) 2 (cid:18)2 1 2 1 2 3(cid:19) 3
1 4 a4
+ P4+ P P +P2P +P s ; (29)
(cid:18)6 1 3 1 3 1 2 4(cid:19) 4 (cid:21)
4. Calculate its coefficients, expressed through
β γ γ γ β2 β γ γ
P = 1 0+ 1 1.17549, P = 0 1 β 1 1+ 2 1.16196;
1 − β2 β ≈ 2 β2(cid:18)β − 2(cid:19)− β2 β ≈
0 0 0 0 0 0
β β β β2 γ
P = 1 2 1 1 β β 0
3 (cid:20) β − β (cid:18)β − 2(cid:19)− 3(cid:21) β2
0 0 0 0
γ β2 β γ γ
+ 1 1 β 1 2 + 3 3.1505 (30)
β2(cid:18)β − 2(cid:19)− β2 β ≈ −
0 0 0 0
γ β2 β2 β2 2β β β
P = 0 1 1 β + 2 1 1 2 β β
4 β4(cid:20)β2 (cid:18)β − 2(cid:19) β − β (cid:18) β − 3(cid:19)− 4(cid:21)
0 0 0 0 0 0
γ β β β β2
+ 1 1 2 1 1 β β
β2(cid:20) β − β (cid:18)β − 2(cid:19)− 3(cid:21)
0 0 0 0
γ β2 γ β γ
+ 2 1 β 3 1 + 4 33.2389; (31)
β2(cid:18)β − 2(cid:19)− β2 β ≈ −
0 0 0 0
7
5. Define the invariant mass
−1
γ0
mˆb = mb(mb) as(mb)β0AD(as(mb)) . (32)
(cid:20) (cid:21)
It should be stressed, that this definition seems to be simpler than one,
introduced in Ref. [15], namely
−1
γ0
mˆb = mb(mb) (2β0as(mb))β0AD(as(mb)) , (33)
(cid:20) (cid:21)
which is often used in the literature.
In the large-β approximation the expression for Γ can be expressed as [42]
0 Hb¯b
mˆ2(M )
ΓHb¯b = Γb0 bm2H (as(MH))ν0(cid:20)A0+ dnAn(as(MH))(cid:21) (34)
b nX≥1
A = 1 1+β2π2a2 −δn/2(a )n−1sin δ arctan(β πa ) , (35)
n β δ π 0 s s n 0 s
0 n (cid:0) (cid:1) (cid:2) (cid:3)
where δ = n + ν 1, ν = 2γ /β . Taking now into account that within
n 0 0 0 0
−
large-β approximation, one has the following LO-expression
0
1
a (M ) = (36)
s H LO β ln(M2/Λ2)
0 H
fixing n = 0 and expanding A to first order in a we are getting the function:
0 s
1 sin(b arctan(π/L )
A = MH , (37)
0 bLb (1+π2/L2 )b/2
MH MH
where b = ν 1, L = ln(M2/Λ2). This expression was first derived in
0 − MH H
80s in Ref. [11]. Unfortunately, its usefulness was not understood at this time.
At the new stage of the development of QCD the analogs of this formula, are
forming the basis of FAPT method [41], which is allowing to resum not only the
terms proportional to γ and β , but higher order corrections of RG functions
0 0
as well. Thus, the extension of the Shirkov–Solovtsov method to the case of
fractional powers [41] may be considered in part as the generalization for CIT
resummation of “large-β ” contributions, which also arises within renormalon
0
approach [42]. In view of the appearance of the works of Refs. [41,44], [43], it
may be interesting to compare in the future the results of these two approaches
in more detail.
8
4 On-shell and RG-resummation approach
4.1 On-shell parameterization
In the previous section we derived the relation between runningand invariant b-
quarkmasses. However, thereisalso thepossibility touseon-shellapproach and
express the width Γ through the b-quark pole mass m and the MS-scheme
Hb¯b b
coupling constant α (M ) in different orders of perturbation theory and com-
s H
pare the results obtained with the running mass motivated RG-resummation
approach. This analysis was done at the α2-level in Refs. [46,47]. In these
s
studies the effects of mass dependent O(α m2/M2)-corrections, extracted from
s b H
the calculations of Ref. [48], were also included. Among most important results
of Refs.[46,47] were the explicit demonstration of the importance of taking into
account α2-corrections in theon-shell approach. Indeed,these effects turnedout
s
to be rather important for decreasing the difference between the Γ expres-
Hb¯b
sions, evaluated in the on-shell and RG-resummed approaches. The results of
our studies were confirmed later on by the considerations of Ref. [49], where the
expression of the O(α2m2/M2)-contribution was evaluated and included. Note,
s b H
however, that for the considered at present masses of Higgs boson these effects
are less important, than higher order perturbative QCD corrections, and can
be safely neglected. Keeping in mind the demands of Tevatron and LHC ex-
periments and the ongoing discussion of the scientific program for International
Linear Collider, in this section we will study the similar problem in more detail,
taking into account the information on available at present higher-order QCD
corrections to the RG-functions, coefficient function for Γ in Eq. (1) (see
Hb¯b
Sec.2) and the relation between running and on-shell masses which we present
in the following form:
m2(m ) 8
b b = 1 a (m ) 18.556a (m )2 175.76a (m )3 1892a (m )4. (38)
m2 −3 s b − s b − s b − s b
b
The a2-correction is the result of calculations of Ref. [50], confirmed later on
s
in [51]. The a3-term comes from the analytical calculations of Ref. [52] and is
s
confirming semi-analytical similar result, obtained in Ref. [53]. Note, that its
coefficient turned out to be in a good agreement with the ECH/PMS estimate
of Ref. [14]. This fact and the success of the ECH/PMS prediction for the value
of dE term [10], we are supplementing the estimates of Ref. [14] by definite
4
RG-inspired considerations and get our personal ECH-inspired number for the
coefficient of a4-term in Eq. (38). Proceeding further on with the help of the
s
derived in Ref. [14] RG equations for the transformation of m2(M ) to m2(m )
b H b b
and of a (m ) to a (M ), we get the following analog of Eq. (1):
s b s H
Γ = Γb 1+∆Γba (M )+∆Γba (M )2+∆Γba (M )3+∆Γba (M )4 , (39)
Hb¯b 0(cid:20) 1 s H 2 s H 3 s H 3 s H (cid:21)
9
where Γ = 3√2/(8π)G M m2 and
0 F H b
∆Γb = 3 2L; (40)
1 −
∆Γb = 4.5202 18.139L+0.08333L2; (41)
2 − −
∆Γb = 316.878 133.421L 1.15509L2 +0.050926L3; (42)
3 − − −
∆Γb = 4366.17 1094.62L 55.867L2 1.8065L3 +0.04774L4 (43)
4 − − − −
andL = ln(M2/m2). WewilldefinenowtheQCDcouplingconstant indifferent
H b
orders of perturbation theory as
1 β ln(Log)
1
a (M ) = 1 ; (44)
s H NLO β Log)(cid:20) − β2Log2 (cid:21)
0 0
as(MH)N2LO =as(MH)NLO +∆as(MH)N2LO; (45)
as(MH)N3LO =as(MH)N2LO+∆as(MH)N3LO; (46)
as(MH)N4LO =as(MH)N3LO+∆as(MH)N4LO; (47)
1
∆as(MH)N2LO = β5Log3(cid:18)β12ln2(Log)−β12ln(Log)+β2β0−β12(cid:19);
0
1 5 1
∆as(MH)N3LO = β7Log4(cid:20)β13(cid:18)−ln3(Log)+ 2ln2(Log)+2ln(Log)− 2(cid:19)
0
β
3β β β ln(Log)+β2 3 ;
− 0 1 2 0 2 (cid:21)
1 13 3
∆as(MH)N4LO = β9Log5(cid:20)β14(cid:18)ln4(Log)− 3 ln3(Log)− 2ln2(Log)+4ln(Log)
0
7
+ +3β2β 2ln2(Log) ln(Log) 1
6(cid:19) 1 2(cid:18) − − (cid:19)
1 5 β
β β 2ln(Log)+ + β2+ 4 ,
− 1 3(cid:18) 6(cid:19) 3 2 3 (cid:21)
where Log = 2 ln(MH/Λ(MnSf=5)) and the additional terms ∆as(MH)N3LO and
∆as(MH)N4LO were obtained in Refs. [54,55] with the corresponding N3LO and
N4LOmatchingconditions,whichareallowingtodeterminethevaluesofΛ(nf=5)
MS
from Λ(nf=4) by passing threshold of production of heavy flavor (in our case on-
MS
shellmassm ). TheanalyticalresultsofRef.[55]areincompleteagreementwith
b
the mixture of previous analogous analytical and semi-analytical calculations of
Refs. [56,57]. These conditions generalize to higher orders the NLO and N2LO
formulae, derived in Ref. [58] (the corresponding N2LO relation was corrected
in Ref. [59]). To save the space, we will not present here the explicit form of
these equations. An interested reader can consult Ref. [60], where the results
of Refs. [58,59] and [54] are presented for the case of considering b-quark pole
10
