#### Higher-order condensate corrections to ϒ masses, leptonic decay rates and sum rules

Revised: May
Higher-order condensate corrections to masses, leptonic decay rates and sum rules
T. Rauh 0 1 2
Quark Masses and SM Parameters
0 system. We demonstrate
1 DH1 3LE , United Kingdom
2 IPPP, Department of Physics, University of Durham
With the recent completion of NNNLO results, the perturbative description of the system has reached a very high level of sophistication. perturbative corrections as an expansion in terms of local condensates, following the approach pioneered by Voloshin and Leutwyler. The leading order corrections up to dimension eight and the potential NLO corrections at dimension four are computed and given in analytical form. We then study the convergence of the expansion for the masses, the leptonic decay rates and the non-relativistic moments of the that the condensate corrections to the (1S) mass exhibit a region with good convergence, which allows us to extract mb(mb) = 4214 37 (pert.) +2202 (non-pert.) MeV, and show that non-perturbative contributions to the moments with n
E ective Field Theories; Heavy Quark Physics; Nonperturbative E ects
1 Introduction 2 3 4
Leading order condensate corrections of dimensions four, six and eight
Dimension four contribution at NLO: potential contributions
Phenomenology of condensate corrections
4.1
4.2
4.3
4.4
The
The
The
(1S) ! l+l decay width
The non-relativistic moments
5
Conclusions
A Condensate corrections to the energy levels and wave functions
trum [1, 2],1 the leptonic decay rate of the
(1S) [4] and the non-relativistic moments
of the total e+e
! bbX cross section [5] have been determined and perturbation theory
is well behaved. This has important phenomenological implications. For instance, some
of the most precise determinations of the bottom-quark mass rely on the comparison of
the perturbative expressions for the non-relativistic moments [5{8] or the masses of the
(1S) and
b(1S) resonances [9{11] and recently also the n = 2 states [12] with their
experimental values.
Bottomonium can be treated as a non-relativistic system where the bottom-quark
velocity v is of the order of the strong-coupling constant: v
s(mbv)
1. There is a
large hierarchy between the dynamical scales mb (hard), mbv (soft) and mbv2 (ultrasoft) of
the system. The perturbative calculations [1{5] were performed using the e ective theory
potential non-relativistic QCD (PNRQCD) [13{16], where the hard and the soft scale have
been integrated out and the only dynamical modes left are potential bottom quarks
and
anti-bottom quarks , with energy and momentum of the order mbv2 and mbv, respectively,
as well as ultrasoft gluons and light quarks. The Lagrangian for perturbative calculations
1See also [3] for the case of unequal masses.
{ 1 {
up to NNNLO takes the form
LPNRQCD =
+
Z
+ Lultrasoft;
8m3
4
8m3
d
d 1r h ay bi (x + r)Vab;cd(r; @) h cy di (x)
where the coupling to the ultrasoft gluon eld in the bottom-quark bilinear parts has been
multipole expanded in the spatial components [17], the third line describes the interactions
through spatially non-local potentials, which are given in [4, 16, 18{24], and the ultrasoft
Lagrangian is a copy of the QCD Lagrangian which only contains the ultrasoft gluon and
light quark elds.
Purely perturbative calculations within PNRQCD are valid when the ultrasoft scale
mbv2 is much larger than the QCD scale
QCD. This is certainly the case for top quarks,
which are studied in [25, 26],2 but is questionable in the bottomonium sector. Assuming the
hierarchy holds, non-perturbative corrections can be incorporated in terms of local vacuum
condensates as a power series in ( QCD=(mbv2))2, following the approach of Voloshin and
Leutwyler [29{32]. In this work, we compute higher-order corrections in this approach and
assess the convergence of the series.
In the limit
QCD
mbv2, the gluon eld in the PNRQCD Lagrangian can be split
into two parts
A (t; x) = Aus(t; x) + Anp(t; x):
The superscripts denote the ultrasoft and the non-perturbative gluon eld with momentum
of the order mbv2 and
QCD, respectively. All couplings of the non-perturbative component
to other modes must be multipole-expanded because the non-perturbative eld with a large
wavelength of the order 1= QCD cannot resolve the dynamics of the potential bottom quarks
or of the ultrasoft gluons. A convenient gauge choice for the non-perturbative gluon eld
is given by Fock-Schwinger gauge
x Anp(t; x) = 0;
A0np(t; 0) = 0;
which removes the coupling of the bottom quarks to the Anp
0
eld. The leading
nonperturbative contribution in the PNRQCD Lagrangian then takes the form of a
chromo
Lnon-perturbative =
y ( gsx Enp(0; 0) + : : : )
+ y ( gsx Enp(0; 0) + : : : ) ;
(1.4)
and is of the order m2v2 2
QCD because Enp
b
2
QCD and the strong coupling at the QCD
scale is counted as order one. This implies that the chromoelectric dipole coupling to
the non-perturbative gluon eld is suppressed by v( QCD=(mbv2))2 with respect to the
2Furthermore, the sizeable top-quark decay width provides a cuto on non-perturbative e ects [27, 28].
{ 2 {
(1.1)
(1.2)
(1.3)
expanded couplings between the non-perturbative and ultrasoft modes in Lultrasoft are not
required for the leading order condensate corrections. Their relevance at higher orders is
assessed in section 3, where we discuss the NLO QCD corrections to the leading term in
the Voloshin-Leutwyler approach.
The condensate corrections to the considered observables can be extracted from the
non-relativistic Green function at the origin
2mb is the non-relativistic energy of the system and the Hamiltonian has
with the color-singlet and color-octet projectors
(P1)abcd =
bc da;
(P8)abcd = 2 TbAc TdAa;
where the color indices are assigned in the same way as in the potential term in (1.1). The
LO Hamiltonian H^bb;0 is of the order mbv2. The non-perturbative dynamics at the scale
QCD are described by the Hamiltonian H^np which is of the order
QCD. The leading
interaction between the bottomonium and non-perturbative sector is given by the
chromoelectric dipole term
^
HD =
2
gs A x Enp;A(0; 0);
with aAbcd = TaAb cd + abTcAd when the color indices are again assigned in the same way as in
the potential term in (1.1), which is of the order 2
QCD=(mbv). Assuming
QCD
mbv2 the
interaction H^D and the non-perturbative Hamiltonian H^np can therefore both be treated
as perturbations and the physical state
j0i
j0ibb
j0inp
1 "
n=0
factorizes into the product of a bottom-antibottom state j0ibb at zero spatial separation
and the non-perturbative vacuum state j0inp. The expansion of the Green function (1.5)
in powers of
QCD then takes the form
G(E) = h0jG^bb(E)j0ibb h0j0inp +X
h0jG^bb(E) Ax^i hG^bb(E)i1+2n Bx^j G^bb(E)j0ibb
h j 4
2
(Enp)jB j0inp +: : :
(1.11)
1
n=0
= h0jG^(b1b)(E)j0ibb +X
h0jG^(b1b)(E)x^i hG^(8)(E)i1+2n x^i G^(1)(E)j0ibb On +: : : ;
bb bb
{ 3 {
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
gsEi
gsEj
gsEi
gsEj
Hnp
Hnp
single and double lines denote the LO color-singlet and color-octet Green functions, respectively.
Higher-dimensional corrections are obtained by inserting additional pairs of the non-perturbative
Hamiltonian H^np in between the two insertions of the chromoelectric dipole H^D.
is the perturbative part of the Green function and we adopted the notation of [33]:
s (Enp)iA hH^np
i2n
(Enp)iA j0inp :
The properties H^npj0inp = 0 and h0jgs (Enp)iA j0inp = 0 have been used to remove insertions
of H^np that are not in between insertions of H^D and single insertions of H^D. Terms with
an odd number of H^np insertions between the two H^D insertions vanish, because they can
be related to the vacuum expectation values of operators with odd numbers of Lorentz
indices by using Lorentz invariance, see [29].
The rst term in (1.11) is the purely perturbative part. The sum contains the leading
non-perturbative contributions, which are proportional to vacuum expectation values of
operators of even dimensions and are suppressed by v2( QCD=(mbv2))4;6;8;::: with respect
to the perturbative expression. The contributions of dimension four and six are shown
in
gure 1. The extra suppression factor v2 is present because terms without at least
two insertions of H^D vanish. The dimension-four correction contains the gluon
condensate h s G2i and has originally been studied in [29{32] and more recently in [4, 5]. The
dimension-six correction to the masses and leptonic decay rates of the
(N S) resonances
has been calculated in [33].
Due to the extra suppression factor v2, smallness of the dimension-four contribution
is not su cient to demonstrate the convergence of the expansion in
QCD=(mbv2) and the
calculation of higher-order condensate corrections is necessary to gain more insight. We
compute the leading corrections up to dimension eight in section 2. The NLO potential
corrections to the dimension-four condensate contribution are determined in section 3. The
size of the condensate corrections to observables in the
system is discussed in section 4.
We conclude in section 5.
2
Leading order condensate corrections of dimensions four, six and eight
The leading order condensate corrections are nite and can be computed in four dimensions
in position space. Inserting spatial integrations, the dimension-four contribution in (1.11)
(1.12)
(1.13)
(0)
4QCD
G(E) = O0
Z
d3r1
Z
0
The integrals can be evaluated using the known representations of the LO Green function
G(1;8), where the superscript indicates whether the bottomonium state is in a color singlet
(1) or octet (8) con guration. It is convenient to decompose the Green function in terms
of partial waves
where l is the quantum number of the angular momentum of the bottom pair and Pl(z)
are the Legendre polynomials. We use an integral representation from [34],
G[(l1];8)(r; r0; E) =
mbp
2
(2pr)l(2pr0)l
(l + 1 + (1;8)) (l + 1
(1;8))
1
Z
0
1
Z
0
du
dt [ut]l (1;8)
[(1 + t)(1
u)]l+ (1;8) exp
p [r0(1
2u) + r(1 + 2t)] ;
valid for r0 < r and a sum representation from [30, 35],
G[(l1];8)(r; r0; E) =
2
mbp (2pr)l(2pr0)le p(r+r0) X1
s!L(s2l+1)(2pr)L(s2l+1)(2pr0)
s=0 (s + 2l + 1)!(s + l + 1
(1;8)) ;
with the Laguerre polynomials
We have de ned the variables
L(s )(z) =
ezz
s!
d
dz
s
e zzs+
:
p = p
mbE;
(1) =
representation (2.3) for the color-singlet Green functions and the sum representation (2.4)
for the color-octet Green function. The angular integrals in (2.1) project out the S-wave
component of the color-singlet Green functions and the P-wave component of the
coloroctet Green function. We obtain
where
(0)
4QCD
and 1 are the polygamma functions of order 0 and 1, respectively. The condensate
corrections to the S-wave energy levels EN and the wave functions at the origin j N (0)j2 can
be obtained from the expansion of (2.10) for
near positive integer values N as described
e.g. in [26, 36, 37]. The results are given in appendix A.
The same strategy can be applied for the calculation of the dimension six and eight
condensate corrections. Again, the angular integrals project out the S-wave component
of the color-singlet Green functions and the P-wave components of the color-octet Green
functions. We nd
(0)
6QCD
G(E) = R6
4
mb2 sCF 9 X
1
1
X
1
X
{ 6 {
δV
1 the multiple sums in (2.11) and (2.12)
are reduced to a single sum, which can be solved in terms of polygamma functions. The
lengthy results are available as an ancillary Mathematica le. The dimension six and eight
contributions to the energy levels and wave functions are given in appendix A.
3
Dimension four contribution at NLO: potential contributions
is of the order
that order:
The NLO corrections to the dimension-four condensate contribution involve an insertion of
the NLO Coulomb potential as shown in
gure 2 and ultrasoft loops as shown in
gure 3.
The upper panel of gure 3 shows the diagrams with ultrasoft gluon loops, where the
gluon coupling to the color-octet state originates from the leading term gsA0us(t; 0) in the
multipole expansion. The equivalent coupling to the color singlet state vanishes because
the ultrasoft gluons cannot resolve the spatial separation of the bottom-antibottom state
and the net color charge vanishes in the singlet state.3 The diagram in the lower panel
of gure 3 shows the contribution from the light-quark condensate hqqi with q = u; d; s
which is also counted as dimension four because, due to chirality suppression, the quark
condensate hqqi only appears together with one power of the light quark mass mq which
QCD. There is a number of other e ects that could possibly contribute at
An s correction to the Wilson coe cient of the chromoelectric dipole operator (1.4).
The Wilson coe cient was found to be trivial up to O( s2) in [39, 40].
O(mb2v3 2
QCD) terms in the multipole expansion (1.4) of the gluon coupling to bottom
quarks in the spatial components. They are identical to the multipole expansion of
the coupling to the ultrasoft gluon eld and were determined in [41], where they are
denoted as h(S1O;0). There is no NLO contribution from these terms because they either
have vanishing tree level Wilson coe cients and are thus suppressed by an additional
power of s
v or involve the chromomagnetic instead of the chromoelectric eld,
which only yields a vanishing condensate 0 EiABjB 0
= 0 at NLO.
E
D
Contrary to the ultrasoft gluon-bottom coupling, the interactions of the
nonperturbative gluon
eld must also be multipole expanded in the time component.
The expansion of the A0 component is trivial due to our gauge choice (1.3) and
already the linear term t(@0Ai)(0; 0) in the expansion of the spatial component is only
3See also [38] for a more formal argument based on a eld transformation.
{ 7 {
HJEP05(218)
ing ultrasoft loops. Lines that carry ultrasoft momentum are drawn in red.
relevant at higher powers. Thus, no contributions of this type need to be considered
at NLO.
The potential corrections are determined below, whereas the ultrasoft contribution is
postponed to future work. The NLO correction to the Coulomb potential is given by
(1)V (1;8)(q) =
s2 C(1;8)
q2
2
q2
1
0 +
2
q2
a1( ) ;
(3.1)
where the color factors are given by C(1) =
CF and C(8) = CA=2
CF and
a1( ) =
CA[11
where nf is the number of massless quarks. Denoting the contribution from the left (right)
diagram in gure 2 by DVD (DDV), we nd
4QCD
(1); potG(E) = (1); DVDG(E) + 2 (1); DDVG(E)
=
O0 s2 a1 + 0 du
d
4QCD
4QCD
CA
2
=
6
2u
{ 8 {
The rst triple insertion function takes the form
KV (u; s1; s2) (s2 + 3)!(s2 + 2 + =8)
;
s2!Hc(s2)
(3.2)
(3.3)
u=0
:
(3.4)
HJEP05(218)
The second triple-insertion function yields
(4 )2mb2( sCF )6 s1=0 s2=0 (s1 + 3)!(s1 + 2 + =8)
1
X
1
X
s1!Hc(s1)
KD(s1; s2) (s2 + 1)(s2 + 1
H(u; s2 + 1)
)
;
where
( =2p)2u
The full u-dependence of (3.5) is not needed here. To evaluate (3.3) we only need the value
and the rst derivative at u = 0. We obtain
(3.5)
(3.6)
(3.8)
(3.9)
(3.10)
(3.11)
where
k
2k
(s1+3)! (s2+1)!
( 1)s1+s2 4! s2!(s1+3 s2)! s1!(s2+1 s1)! ;
1
else,
s2
s1
3
and H(u; k) is de ned as in [36]. Also here, we only need the value and the rst derivative
L
E
(k
) +
( (1
k
)
(k + 1
)) :
(3.12)
{ 9 {
The in nite sums in (3.4) and (3.9) converge quickly and can be truncated with negligible
uncertainty at si
30 for the numerical evaluation of the Green function. The
contributions to the energy levels and wave functions from the potential corrections can be extracted
by expanding (3.4) and (3.9) for
near positive integer values N . The results are given in
appendix A.
4
Phenomenology of condensate corrections
The size of non-perturbative corrections to the moments and to the properties of the
resonances has been strongly disputed for various reasons. First, the assumption
mbv2 is questionable and is certainly only valid for a limited number of observables in
the
system. Here, we perform an unbiased analysis of the expansion in terms of local
QCD
condensates and assess the validity based on its convergence.
The breakdown of this
expansion is a clear indication that the above assumption is inappropriate.
Furthermore, the numerical values of the local condensates are very uncertain. The
condensate O0 is proportional to the gluon condensate and we will use the standard value
h s G2iSVZ = 0:012 GeV4 from [42] below, unless indicated otherwise. We note however,
that signi cantly larger values have also been obtained in the literature, see e.g. [43{
Clearly, the situation is even more uncertain for the higher-dimensional
condensates. Since our main objective is the assessment of the convergence properties, we rely on
naive rescaling
O0SVZ =
(285 MeV)4;
O1naive = (285 MeV)6;
O2naive =
(285 MeV)8:
(4.1)
The value of O0 is scale independent since the gluon condensate h s G2i is not renormalized.
We neglect the scale dependence of the higher-dimensional condensates which is very weak
compared to that of the coe cients which contain large powers of s. The estimate O1naive
is in good agreement with the result of [33], where an expression for O1 in terms of the
dimension-six gluon condensate hG3i and the quark-condensate hqqi has been derived based
on the factorization hypothesis. The analysis of [33] also shows that O1 is only weakly
scale dependent.
In addition, the corrections to the masses and leptonic decay rate depend strongly on
the renormalization scale, because large powers of s appear in the ratios (2.8) and (2.13).
The fact that di erent powers of s appear in the contributions of di erent dimensions
also complicates the assessment of the convergence and di erent conclusions have been
drawn based on di erent scale choices. We distinguish the scale c, used in the condensate
corrections, from the renormalization scale
in the perturbative contribution. The main
motivation for the calculation of the potential corrections to the dimension-four
contribution has been to gain more insight into the appropriate scale choice for c by considering
the convergence of the perturbative series. We note that the potential corrections contain
all logarithms ln( c) that are required to cancel the c dependence of the dimension-four
contribution at NLO. The ultrasoft correction must therefore be free of logarithms ln( c)
and is less scale dependent, which justi es performing this analysis based on incomplete
NLO corrections. Scales below 0.8 GeV are not considered below, because the value of s
and perturbation theory in general become unreliable in this regime.
First, we brie y review the status of the purely perturbative prediction for the mass of the
(1S) resonance. We use QQbar Threshold [26, 46] in the PS mass scheme [47] with the
input value mPS = 4:532+00::001339 GeV from [5, 6]. The e ects of a non-zero charm-quark mass
b
are included up to NNLO [5] using the mass mc(3 GeV) = 993 MeV from [48, 49]. The
default values of QQbar Threshold are taken for the strong coupling
s(mZ ) = 0:1184
0:0010 and all other parameters, and QED corrections are taken into account with NNLO
accuracy. The result is shown in the top panel of gure 4. We observe that the convergence
is best for scales that are considerably larger than the soft scale
mb s( )CF . This has
motivated the authors of [11] to choose a central scale of 5.35 GeV, which is signi cantly
larger than that of [9, 10, 12] (2.5 and 1.9 GeV, respectively) and leads to a much smaller
estimate for the perturbative uncertainty. Here, we choose a central scale of 3 GeV such that
a variation by factors 1/2 and 2 covers the choices of [9{12]. The perturbative expansion
takes the form
M pe(1rtS) (3 GeV) = (9 366 + 82 + 4
27) MeV:
(4.2)
In addition to the perturbative uncertainty from scale variation, we also take into account
the parametric uncertainty from the bottom-quark PS mass and we use the size of the
charm-quark mass e ects up to NNLO as an estimate for the missing NNNLO correction.
The parametric uncertainty from the strong coupling is small in the PS mass scheme and
is neglected.
The condensate corrections with the values of (4.1) are shown in the lower panel of
gure 4. At the considered orders, the mass scheme is ambiguous and we use the one-loop
pole mass in the condensate contribution. In the PS scheme, the condensate contributions
are slightly enhanced and the convergence is slightly worsened but the overall conclusions
are unchanged. We observe that the potential corrections to the dimension-four condensate
contribution stabilize the behaviour under scale variation and show a clear preference
for rather small scales around
c = 1:2 GeV, which we take as the central value. The
condensate contribution takes the form
M co(n1dS) (1:2 GeV) = (17
3)
O0
OSVZ
0
4
O1
1
Onaive + 1
O2
The grey band in gure 4 is obtained by varying the value of the condensate O0 between
0 GeV4 and 3O0SVZ. In our analysis the condensates of dimension six and eight are varied
between 0 GeV6 and 33=2O1naive, and 0 GeV8 and 32O2naive, respectively. We use this variation
at the central scale as an estimate for the uncertainty from the value of the condensates.
The condensate expansion becomes unstable near c
2 GeV where the LO dimen
sion four, six and eight contributions are all of the same size. The variation of c between
0.8 GeV and 2 GeV yields an uncertainty of +336 MeV. We take
36 MeV as an estimate for
HJEP05(218)
2
4
6
μ [GeV]
V
(1 - 0.02
S
Υ
- 0.04
- 0.06
HJEP05(218)
(4,0)
The curves in the bottom panel show the e ects of cumulatively adding the condensate contributions
(i; j) where i denotes the dimension and j the order in perturbation theory. The gray band is
spanned by variation of O0 by factors of 0 and 3, while O1 and O2 are unchanged.
4.5
4.4
[) 4.3
m4.2
recent results from the masses of bb bound states (ACP'14 [9, 10], KMS'15 [11], MO'17 [12])
and non-relativistic sum rules (HRS'12 [8], BMPR'14 [5, 6]). The bottom-quark masses obtained
from lower orders in pure perturbation theory, while retaining all known condensate contributions,
are shown as well. The order of the PS-MS mass relation has been correlated with the order in
perturbation theory.
the perturbative uncertainty in order to also account for the unknown ultrasoft NLO
correction. Combining the perturbative and condensate contributions we nd
M (1S) = 9 437 +61114 MeV
36 ( c) +2194 (O0) +418 (O1) +110 (O2) MeV;
(4.4)
which is in good agreement with the experimental value M ex(p1S) = 9 460:30
0:26 MeV.
The stable behaviour of the condensate corrections in the range 0:8 GeV . c . 2 GeV
facilitates the determination of the bottom-quark mass from the experimental value of the
(1S) mass. We obtain
mbPS(2 GeV) = 4544
39 (pert.) +2225 (non-pert.) MeV = 4544 +4446 MeV;
(4.5)
where we have symmetrized the uncertainty from variation of the renormalization scale ,
by taking the maximum of the positive and negative error. The perturbative uncertainty
is obtained by adding the errors from variation of
and
s as well as our estimate of
higher-order charm-quark mass e ects in quadrature. The variation of the scale
c and
of the values of the condensates is combined into the non-perturbative uncertainty. The
result (4.5) is converted to the MS scheme at NNNLO [50, 51] using QQbar Threshold.
We distinguish the scale
m used in the conversion, which is set to mbPS, and estimate
the uncertainty through variation of
m by factors of 1/2 and 2 and symmetrization as
described above. We nd
mb(mb) = 4214
37 (pert.) +2202 (non-pert.) MeV = 4214 +4423 MeV:
(4.6)
The result shows good convergence and agrees with other recent determinations of mb from
the data on the
system as shown in
gure 5. In conclusion, our analysis demonstrates
that the determination of the bottom-quark mass from the
(1S) mass is possible with a
total uncertainty of the order of
45 MeV. It should however be noted that this approach
to the determination of the bottom-quark mass is on a less sound footing theoretically
than the extraction based on non-relativistic moments with n
10, which are discussed in
section 4.4.
(2S) mass. The scale dependence of the
perturbative result is shown in
gure 6. Since the soft scale is lower for the n = 2 states, we
reduce the central scale to 2 GeV, where the perturbative series takes the form
M pe(2rtS) (2 GeV) = (9 534 + 198 + 154 + 116) MeV:
(4.7)
As the plot shows, the convergence is rather slow, independently of the choice of scale. We
also note that the charm-mass e ects at NNLO are +39 MeV and signi cantly larger than
for the
(1S) mass (+8 MeV). As we argued in [5], the charm-mass e ects are a measure
for the IR sensitivity of an observable. Thus, the signi cantly larger value is an indication
that the non-perturbative correction should be considerably larger and less convergent for
the
(2S) mass than for the
(1S) mass.
Turning to the condensate corrections, which are shown in the lower panel of gure 6,
we can con rm this expectation. The expansion already breaks down for c = 0:8 GeV,
where the individual contributions are
M co(n2dS) (0:8 GeV) = (258
267)
O0
OSVZ
0
293
O1
1
Onaive + 365
O2
At lower scales, the use of perturbation theory cannot be justi ed. Thus, while we cannot
rule out the convergence of the local condensate expansion unambiguously due to the large
uncertainties of the Oi, clearly no reliable prediction for the non-perturbative contribution
can be obtained like this.
A more promising approach to the (2S) mass is to assume the hierarchy
QCD
mbv2
mbv. Then, the ultrasoft contribution takes the form of a non-local
condensate instead of a perturbative correction [15, 52, 53]. This implies that the leading
nonperturbative correction is of the order
M no(2nS-p)erturbative
mb s2
QCD
mb s
2
QCD
mb s2
mb s4;
(4.9)
which is formerly of NNLO, and the conclusion that the local condensate expansion breaks
down is equivalent to the statement that the
(2S) system is outside the radius of
convergence for the presently unknown function . In this scenario, the perturbative NNNLO
results, which contain the perturbative evaluation of the ultrasoft contribution cannot be
used and we have to resort to the NNLO expressions. The result for the
(2S) mass reads
M (2S) = 9 886 +119252 ( ) +2756 (mb) +2286 ( s)
O(100) (non-pert.) MeV;
(4.10)
10.2
9.8
9.6
S
2
(
Υ
M- 0.2
Δ
Pole scheme
6
μ [GeV]
NNNLO
NNLO
NLO
LO
8
10
(4,0)
The curves in the bottom panel show the e ects of cumulatively adding the condensate contributions
(i; j) where i denotes the dimension and j the order in perturbation theory. The gray band is
spanned by variation of O0 by factors of 0 and 3, while O1 and O2 are unchanged.
2.0
k
0.0
1.5
1.0
(1S) resonance. The curves in the bottom panel show the e ects of cumulatively adding
the condensate contributions (i; j) where i denotes the dimension and j the order in perturbation
theory. The gray band is spanned by variation of O0 by factors of 0 and 3, while O1 and O2
are unchanged. In this
gure the gray band does not contain the potential corrections to the
dimension-four contribution.
where the estimate for the non-perturbative contributions follows from the assumption that
the function
in (4.9) is of order one. Within the large uncertainty, the experimental value
M ex(p2S) = 10023:26
0:31 MeV can be reproduced.
4.3
The
(1S) ! l+l
decay width
The perturbative NNNLO result for the leptonic decay width of the (1S) resonance has been obtained in [4]. Here, we repeat their analysis including charm-mass e ects up to { 16 {
NNLO, which increase the leptonic width by 0.03 keV. The scale dependence is shown
in gure 7 and we adopt 3.5 GeV as the central scale. The perturbative series stabilizes
at NNNLO
pert( (1S) ! l+l ) (3:5 GeV) =
2
4
9mb2 cv cv
cv +
dv
3
E1
mb
but falls short of the experimental value
about 20%. Following [4] we determine the scale uncertainty from variation between 3 and
10 GeV. The other input parameters are varied as above.
The condensate contributions are shown in the lower panel of gure 7. Using the same
central scale c = 1:2 GeV as for the
(1S) mass, we obtain
(4.11)
eV:
(4.12)
(4.13)
(4.14)
cond( (1S) ! l+l ) (1:2 GeV) = (352
862)
O0
OSVZ
0
149
O1
1
Onaive + 64
O2
Focusing rst on the leading-order contributions, we see that the expansion converges and
yields a contribution of 0.27 keV that closes the di erence between the perturbative and
the experimental value. Compared to the
(1S) mass the expansion breaks down at a
smaller scale around 1.6 GeV.
However, with the addition of the potential corrections to the dimension four
contribution, the agreement is destroyed. The potential correction already exceeds the LO term at
the scale 0.7 GeV and becomes twice as large at 0.9 GeV. This apparent breakdown of the
perturbative series makes it impossible to give a reliable estimate of the non-perturbative
contribution. However, it is conceivable that the large potential corrections are
compensated by the missing ultrasoft correction, thus stabilizing the perturbative expansion of the
dimension-four contribution. Therefore, no de nite conclusions about the validity of the
local condensate expansion for ( (1S) ! l+l ) can be drawn without a calculation of
the full NLO corrections to the dimension-four contribution.
4.4
The non-relativistic moments
The moments Mn of the normalized inclusive bb production cross section
Rb(s) =
(e+e
(e+e
!
! bb + X)
+
)
;
in e+e collisions with the center-of-mass energy s, are de ned as
Mn
Z 1
0
ds
Rb(s)
sn+1 =
6 i
ds
I
C
b(s)
sn+1 =
The normalized cross section is related to the bottom-quark contribution
b to the photon
vacuum polarization by the optical theorem Rb(s) = 12 Im
must be closed around s = 0 without crossing the branch cut for real s
b(s + i ). The contour C
M 2 (1S). The
perturbative contributions to the moments up to NNNLO have been discussed in detail
in [5]. The non-perturbative corrections can be determined by inserting the condensate
contribution to the cross section
cond(s) =
b
2Nceb2 "
s
(0)
4QCD
+ (0)
6QCD
G(E) +
2c(v1) 4 s (0)
4QCD
G(E) + (1);potG(E)
4QCD
G(E) + (0)
8QCD
#
G(E) + : : : ;
(4.15)
8CF is the hard matching coe cient of the vector
current, into (4.14). Following the discussion in [5] we choose not to expand the
prefactor 1=s around 1=(4mb2). Contrary to the perturbative contribution, we cannot split the
condensate corrections into a resonance and continuum part, since both are separately
divergent [5]. The total corrections to the moments are however well-de ned and can be
computed numerically using the representation of the moments (4.14) involving contour
integration or, in principle, analytically by taking derivatives at q2 = 0. Our numerical
results for the leading order dimension-four condensate contribution are in good agreement
with the approximate result presented in [32].
The scale dependence of the dimension-four contribution are shown in gure 8. Results
are given in the pole mass scheme using the same inputs as given above. We refrain from
using the PS or other threshold mass schemes, because the perturbative expansion in these
schemes becomes unstable in large regions of the scale c. This can be traced back to the
appearance of large powers of
in the expression for the Green function (2.10), which are
expanded in the PS mass scheme as
0
mPS sCF
b
1k
A
b
1 +
k mPS; (1)
b
EPS
!
+ : : : ;
(4.16)
where EPS = p
s
2mbPS
2 mPS; (0) and mPS; (i) is the NiLO contribution to the PS-pole
mass relation.4 This is reminiscent of the destabilization of the NLO correction to the
gluon condensate contribution [54] to the relativistic moments in the MS scheme [55].
The top panel of gure 8 shows the leading order result. Although the contribution is
proportional to R4 /
s 6( c), its absolute value decreases for larger scales c. Given that
the condensate corrections to the
(1S) mass and leptonic decay rate become unstable for
scales larger than about 2 GeV and 1.6 GeV, respectively, this behaviour must be caused
by very pronounced cancellations between the contribution from the
(1S) resonance and
the remaining resonances and the continuum (rest), which was pointed out in [5]. For the
tenth, sixteenth and twenty-fourth moment, this cancellation is e ective at the level of one
part in 139, 52 and 20 at the scale c = 2 GeV and at one part in 1530, 659 and 297 for
c = 10 GeV and the growth of the degree of the cancellation for higher scales dominates
4However, taken at face value, the dimension-four contributions in the PS scheme are smaller than in
the pole scheme.
0
(4 n - 6
,
- 10
2
0
- 2
- 6
2
0
1
,
(4 n
) /
ℳ- 4
ℳ- 8
- 10
2
2
imental moments from [5]. The upper panel shows the leading order contribution and the lower
panel the leading order contribution plus the potential corrections. The relative corrections have
been rescaled by a factor of 100.
4
4
μc[GeV]
6
6
μc[GeV]
Pole scheme
HJEP05(218)
Pole scheme
8
8
8
10
12
16
20
24
8
10
12
16
20
24
10
10
over the growth of the factor s 6( c). While this qualitative behaviour is expected due to
the reduced infrared sensitivity of the moments compared to the properties of the
(1S)
resonance, the extent of the cancellations and the resulting smallness of the corrections is
rather surprising, especially for larger values of n & 16 where power counting predicts a
breakdown of the expansion in powers of (n QCD=mb).
The results including the potential NLO corrections are shown in the lower panel of
gure 8. Above 3 GeV the corrections do not exceed the size of about
20% for the
considered moments. This is due to even more pronounced cancellations within the potential
corrections which are e ective up to about one part in 104 (n = 10), 5 104 (n = 16) and
2 103 (n = 24) at c = mb. The corrections mainly have the e ect of stabilizing the
scale dependence at lower scales
c . 2 GeV, such that we
nd good behaviour of the
dimension-four contribution at partial NLO over the considered range of scales between 1
and 10 GeV.
We can try to assess the convergence of the condensate expansion based only on the
dimension-four results. Compared to the perturbative result, they are of the relative order
1=n
(n QCD=mb)4, where the extra factor of 1=n accounts for the v2-suppression from
the two insertions of the dipole operator. Thus, we expect a breakdown of the condensate
expansion in n QCD=mb when the dimension-four contribution is of the relative size 1=n.
From the lower plot in gure 8 we deduce that this point is reached in the ballpark of n
20,
where the condensate contribution is of the size of -4% of the experimental moment at its
peak, which is compatible with the expectation from the power counting argument.
In gure 9, the relative contributions of dimension six (upper panel) and eight (lower
panel) are shown. Both are signi cantly smaller than our expectation based on the putative
breakdown of the expansion around n
20, which would imply that the dimension six
and eight corrections are both of the order 1=n
0:05. This smallness is the result of
cancellations between the contribution from the
(1S) resonance and the rest that are
even stronger than at dimension four. Explicitly, they are at the level of about one part
in 3 105 (n = 10), 5 104 (n = 16) and 104 (n = 24) at dimension six and about one part
in 108 (n = 10), 107 (n = 16) and 2 106 (n = 24) at dimension eight. We believe that the
reason for this behaviour is the o -shellness of the moments which are de ned as derivatives
of the vacuum polarization function at q2 = 0, far away from the physical cut at s
M 2 (1S). This o -shellness e ectively acts as an IR cuto and suppresses higher-dimensional
corrections, which probe the IR regime. On the other hand, the properties of the Upsilon
resonances, that we discussed above, are on-shell quantities and the higher-dimensional
condensate contributions do not appear suppressed with respect to our expectations from
power counting.
From the point of view of the convergence of the condensate expansion, it appears that
the moments can be described reliably up to values of n much larger than 20. However,
as pointed out in [5], the validity of quark-hadron duality must be questioned when the
moment is completely saturated by lowest state. This is the case for the higher values
considered here, where the relative contribution of the
(1S) to the experimental moments
amounts to 95% for n = 20 and 97% for n = 24 [5]. By the term `violation of
quarkhadron duality' we refer to contributions which have a trivial Taylor expansion and are,
HJEP05(218)
⨯ -4
) ℳ/
0
,
(8 n
-6
ℳ-8
-10
Pole scheme
8
10
12
16
20
24
8
10
12
16
20
24
6
of dimension six (upper panel) and eight (lower panel). The relative corrections have been rescaled
by factors of 103 and 105, respectively.
therefore, not captured by the condensate expansion, like e.g. exponential terms of the form
exp(
mb=(n QCD)). Behaviour that is consistent with the presence of such contributions
has been observed in the 't Hooft model [56].5 However, the size of these contributions in
four-dimensional QCD is di cult to quantify and we do not attempt this here. We note,
however, that the exponential terms originate from coherent soft
uctuations [57], e.g.
from contributions where the o -shellness is distributed among many soft lines carrying
momenta of the order
QCD, which pushes the bottom pair close to its mass shell. It is
conceivable that such an e ect does not experience a similar suppression from the e ective
IR cuto as the higher-dimensional condensate contributions.
5Ref. [56] considers observables in the Minkowski domain, where the exponential terms must be
analytically continued and manifest as oscillations.
In the range n QCD
mb the exponential exp( mb=(n QCD)) is of order one and we
cannot exclude that duality violation e ects are relevant at the high accuracy we require for
reliable determinations of the bottom-quark mass. We conclude that, in practice, the range
of moments is limited by our knowledge of the validity of quark-hadron duality and not by
the convergence of the condensate expansion and advise that moments with n & 16 are not
used for determinations of the bottom-quark mass. On the other hand, for n
10
dualityviolating e ects are exponentially suppressed and the condensate expansion provides a
reliable determination of the non-perturbative e ects. Our results given in
gure 8 and 9
show that the condensate contributions in this region are in the subpercent range and can
safely be neglected compared to the perturbative uncertainties.
5
We have determined the leading order condensate corrections to the
(N S) masses,
leptonic decay rates and sum rules up to and including dimension eight. In addition the
potential NLO corrections to the dimension-four contribution have been computed, which
allows us to assess the preferred scale choice in the condensate corrections. Our results
suggest that the expansion is well behaved for the mass of the
(1S), but breaks down for
the higher states. The former observation has been used to determine the bottom-quark
mass with the results given in (4.5) and (4.6).
The leading order condensate corrections to ( (1S) ! l+l ) have a small window
of convergence for 0:8 GeV . c . 1:6 GeV, where they lead to good agreement with the
experimental value, but the partial NLO corrections to the dimension-four contribution
exceed the leading order correction and cause us to question the perturbative stability.
Thus, a nal verdict for the leptonic decay rate of the (1S) is only possible once the missing ultrasoft correction has been calculated.
Last but not least, we have considered the non-relativistic moments (4.14). We nd
extremely good convergence of the higher-dimensional condensate contributions which clearly
shows that non-perturbative contributions to moments with n
10 are negligible. On
the other hand, we cannot unambiguously exclude the possibility of relevant violations of
quark-hadron duality for n & 16 despite the surprising smallness of the dimension six and
eight corrections. Thus, the non-relativistic moments with n
10 remain the theoretically
cleanest approach for determinations of the bottom-quark mass from the
system.
Acknowledgments
I am grateful to M. Beneke, A. Maier and M. Stahlhofen for helpful discussions, to A. Maier
for comments on the manuscript and to V. Mateu and P. Ortega for communication
regarding [12]. I wish to thank the Erwin Schrodinger International Institute for Mathematics
and Physics (ESI) in Vienna for hospitality during the programme Challenges and
Concepts for Field Theory and Applications in the Era of LHC Run-2 where part of this work
was done.
HJEP05(218)
Condensate corrections to the energy levels and wave functions
We give the results for the condensate corrections to the energy levels and the wave
functions at the origin of the S-wave bottomonium states.
The contributions are parametrized as
j N (0)j2 = j N(0)(0)j2 @1 + X f N(i) +
X
EN = EN(0) @1 + X e(Ni) +
X
0
1
i=1
0
1
X e(Nk;l)A ;
1
k=4;6;8;::: l=0
1
i=1
1
X f (k;l)A ;
N
1
k=4;6;8;::: l=0
6The expression for the gluon condensate correction to the ground state wave function given in [31]
contains a mistake that was later corrected in [33]. I am grateful to A. Pineda for bringing this to my attention.
e(4;0) = R4
N
N
f (4;0) =
R4
32N 6 25515N 6
109935N 4 + 101216N 2
26624
112065638400N 4 + 50981371904N 2
where the leading order expressions are given by
En(0) =
mb s2CF2 ;
4n2
j n(0)(0)j2 =
1
the perturbative corrections of relative order si are e(i) and f N(i), and e(k;l); f (k;l) are the
N N N
condensate corrections of relative order ( QCD=(mb s2))k sl+2 to the N th energy level and
wave function, respectively. At dimension four, we obtain
in agreement with the results from [30, 31, 33].6 For the dimension-six corrections to the
energy levels and wave functions, we nd
e(6;0) = R6 81 (9N 2
N
4096N 10
64) (6561N 4
46158344158975776N 10 + 114216987240880128N 8
168309372752363520N 6 + 145600287615221760N 4
68153404341354496N 2 + 13295844358881280 ;
(A.1)
(A.2)
(A.3)
(A.6)
The correction to the energy levels is identical to the result of [33]. Our result for the wave
function correction however di ers from the one in [33] in the coe cients in the square
bracket that multiply powers of N , while the constant term is in agreement. Numerically
the di erence is tiny, dropping from 3 permille for N = 1 to 1.8 permille for N = 10.
The calculation has been repeated for an arbitrary number of colors Nc and we observe
that, while our result for the energy levels agrees with that of [33], the discrepancy in the
squared wave function persists in the large-Nc limit where the octet potential vanishes.
The full Nc-dependence of the dimension-six contribution is given in Mathematica format
in the ancillary le that is distributed with this article. Our dimension-eight results read
e(8;0) = R8 6561(64 9N2)3(1024 81N2)(16384 25920N2+6561N4)5
N
131072N14
513297061199674600970728035N 30
28809695301605440114072286106N 28
(A.8)
739783191218801346196996467082948404493005N 46
118424806386048034335763849263957780781041084N 44
+8614327589477425734307468706116579381193741895N 42
378783092051612274031903244647052432202630343536N 40
+11297942118461809329963818184791182730335090044672N 38
243143128750125652415373537739611048866722611757056N 36
+3920817963564181189981156819150310468833247662964736N 34
48564520997046676673717761034578880399807666580357120N 32
+469935568662507411174423052932263135345073787276099584N 30
3594124289377594431764429130697307804693677753786957824N 28
+21896983904235671174732114490714870006523720168378466304N 26
106786420612452063591697653002182767135747603942793543680N 24
+417824025967497761694434177318848448549510813159365017600N 22
1311624595310101793819336380085499993464121204574984863744N 20
6596089628312909075581437831841936847452213443718287458304N 16
12894168869874165524006895663064691969753810987628940492800N 12
+12244261212399551038854557341492532616026313680323426123776N 10
8720306429636174559372052451043628307310497364671742345216N 8
1572764930888520531444116515121465620114569888243055067136N 4
32577921122492492137783209690205922464480693174927360000 :
(A.9)
The NLO corrections to the dimension-four condensate contributions have the form
e(4;1) = R4
N
f (4;1) = R4
N
4
4
s h CA=2
CF
CF eeNDVD + eeNDDV + eeNusi ;
s h CA=2 CF feNDVD + feNDDV + feNusi ;
CF
(A.10)
(A.11)
where the potential terms are
eeNDVD =
128N6
9 (6561N4 25920N2 + 16384)2 n2 [a1 + 2 0 (S1 + LN )] 167403915N 10
1486558575N 8 + 4690934208N 6
6303780864N 4 + 3483631616N 2
536870912 + 0(9N
8) 94419351N 9 + 46727442N 8
800382951N 7
381105162N 6 + 2367378684N 5 + 1061906784N 4
2883647232N 3
1162401792N 2 + 1320550400N + 405274624 o;
(A.12)
HJEP05(218)
eeNDDV = [a1 + 2 0 (S1 + LN )]
256N6
(
+ 189115999977472 + 0 81 (6561N4 25920N2 + 16384)2
128N5
1
9 (6561N4 25920N2 + 16384)2
8888937845922281277N 19
+ 2436883078824792686592N 6 + 6290577008666263683072N 5
1065043454471757103104N 4
1829313053453482721280N 3
36NS1
3295011258945N 14 + 41202169005150N 12
204102832598208N 10
688295779762176N 6 + 498462848188416N 4
178391466639360N 2 + 21990232555520
33554432N
6561N4 25920N2 + 16384
32805N 6 + 2177280N 4
N
4993024N 2 + 1048576 S1 8
+
111537N 6 + 124416N 4
5386240N 2 + 7340032 S1
9N
8
36N 2 S2
2
6
167403915N 10 + 1486558575N 8
4690934208N 6
+ 6303780864N 4
3483631616N 2 + 536870912
(A.14)
)
;
81 (6561N 4
+ 6036677324559894970368N 4 + 609631074176867500032N 3
1054643390588414590976N 2 + 220638101144321654784N
18446744073709551616 +
4N S1
652412229271110N 14 + 7881876138460500N 12
13405824N 4 + 24866816N 2 + 1048576 S1
465831N 6 + 30824064N 4
64811008N 2 + 7340032 S1
2
6
18N 2 S2
5859137025N 10 + 48288205485N 8
136847786688N 6
+ 159273676800N 4
81050992640N 2 + 14629732352
(A.15)
16777216N
)
;
N
8
9N
8
rently unknown.
where LN = ln(N =(mb sCF )), S1(x) = Pkx=1 k 1 is the analytic continuation of the
harmonic number to non-integer values and Si = PN
k=1 k i without an explicit argument
is the N th harmonic number of rank i. The ultrasoft corrections eeNus and feNus are
cur
Open Access.
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HJEP05(218)
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