#### Inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) : complete angular analysis and a thorough study of collinear photons

Received: March
and a thorough study of collinear photons
Tobias Huber 0 1 2
Tobias Hurth 0 1 2 3
Enrico Lunghi 0 1 2 4
0 Bloomington , IN 47405 , U.S.A
1 Johannes Gutenberg University , D-55099 Mainz , Germany
2 Universit ̈at Siegen , D-57068 Siegen , Germany
3 PRISMA Cluster of Excellence and Institute for Physics , THEP
4 Physics Department, Indiana University
We investigate logarithmically enhanced electromagnetic corrections of all angular observables in inclusive B¯ supplemented by a dedicated Monte Carlo study on the treatment of collinear photons in order to determine the size of the electromagnetic logarithms. We then give the Standard Model predictions of all observables, considering all available NNLO QCD, NLO QED and power corrections, and investigate their sensitivity to New Physics. Since the structure of the double differential decay rate is modified in the presence of QED corrections, we also propose new observables which vanish if only QCD corrections are taken into account. Moreover, we study the experimental sensitivity to these new observables at Belle II.
aTheoretische Physik 1; Naturwissenschaftlich-Technische Fakult¨at
1 Introduction
2 Definition of the observables
3 Log-enhanced QED corrections to the double differential decay rate 5.1 5.2 5.3
Master formulas for the observables
5 Phenomenological results
Branching ratio, low-q2 region
Branching ratio, high-q2 region
5.6 The ratio R(s0)
6 New physics sensitivities
7 On the connection between theory and experiments Various experimental settings Validation
Monte Carlo estimate of QED corrections to HT and HL
8 Conclusion
A QED and QCD functions
A.1 QED functions for the double differential rate
A.2 Functions for the QCD corrections to the HI
A.3 Functions for the QED corrections to the HI
B New physics formulas
Introduction
By now the LHC experiment has not discovered any new degrees of freedom beyond the
Standard Model (SM). In particular, the measurements of the LHCb experiment and the
B-physics experiments of ATLAS and CMS have confirmed the simple
Cabibbo-KobayashiMaskawa (CKM) theory of the SM [1–3]. This corresponds to the general result of the
B-factories [4, 5] and of the Tevatron B-physics experiments [6, 7] which have not
indicated any sizable discrepancy from SM predictions in the B-meson sector (for reviews see
However, recently the first measurement of new angular observables in the exclusive
tainties it is not clear if this anomaly is a first sign for new physics beyond the SM, or a
consequence of hadronic power corrections; but of course, it could turn out to just be a
staeagerly awaited to clarify the situation. More recently, another slight discrepancy occurred.
is affected by unknown power corrections, the ratio RK is theoretically rather clean. This
might be a sign for lepton non-universality (see e.g. refs. [22–31]).
clean modes of the indirect search for new physics via flavour observables (for a review
and updates see refs. [32–34]); especially it allows for a nontrivial crosscheck of the recent
LHCb data on the exclusive mode [18, 35].
The observables within this inclusive mode are dominated by perturbative
contributions if the cc¯ resonances that show up as large peaks in the dilepton
invariant mass spectrum are removed by appropriate kinematic cuts — leading to so-called
‘perturbative di-lepton invariant mass windows’, namely the low di-lepton mass region
1 GeV2 < s = q
2 =
m`2` < 6 GeV2, and also the high dilepton mass region with
2 > 14.4 GeV2 (or q2 > 14.2 GeV2). In these regions a theoretical precision of order
10% is in principle possible.
By now the branching fraction has been measured by Belle and BaBar using the
sumof-exclusive technique only. The latest published measurement of Belle [36] is based on a
sample of 152 × 106 BB¯ events only, which corresponds to less than 30% of the dataset
available at the end of the Belle experiment. Babar has just recently presented an analysis
based on the whole dataset of Babar using a sample of 471 × 106 BB¯ events [37] which
updated the former analysis of 2004 [38].
In the low- and high-dilepton invariant mass region the weighted averages of the
experimental results read
B(B¯ → Xs`+`−)leoxwp = (1.58 ± 0.37) × 10−6 ,
B(B¯ → Xs`+`−)ehxigph = (0.48 ± 0.10) × 10−6 .
All the measurements are still dominated by the statistical error. The expectation is that
In addition, Belle has presented a first measurement of the forward-backward
asymmetry [39] and Babar a measurement of the CP violation in this channel [37].
The super flavour factory Belle II at KEK will accumulate two orders of magnitude
larger data samples [40]. Such data will push experimental precision to its limit. This is the
main motivation for the present study to decrease the theoretical uncertainties accordingly.
The theoretical precision has already reached a highly sophisticated level. Let us briefly
review the previous analyses.
contributions are calculated up to NNLL precision. The complete NLL QCD
contributions have been presented [41, 42]. For the NNLL calculation, many components
components for the NNLL QCD precision have been calculated in refs. [43–55].
smin = qm2in/mb2 [53].
• If only the leading operator of the electroweak hamiltonian is considered, one is led to
a local operator product expansion (OPE). In this case, the leading hadronic power
already been analysed [56–61]. Power correction that scale with 1/mc2 [62] have also
been considered. They can be calculated quite analogously to those in the decay
Such analysis goes beyond the local OPE. An additional uncertainty of ±5% has been
In the high-q2 region, one encounters the breakdown of the heavy-mass expansion
(HME) at the end point of the dilepton mass spectrum: whereas the partonic
contribution vanishes, the 1/mb2 and 1/mb3 corrections tend towards a finite, non-zero value.
decay, no partial all-order resummation into a shape-function is possible. However,
for the integrated high-q2 spectrum an effective expansion is found in inverse
powless rapidly, and the convergence behaviour depends on the lower dilepton-mass cut
The large theoretical uncertainties could be significantly reduced by normalizing the
R(s0) =
For example, the uncertainty due to the dominating 1/mb3 term could be reduced
from 19% to 9% [66].
independent kinematical quantities. A hadronic invariant-mass cut is imposed in the
experiments. The high-dilepton-mass region is not affected by this cut, but in the
the relevance of the shape function. A recent analysis in soft-collinear effective theory
tion in the dilepton-mass spectrum can be accurately computed. Nevertheless, the
effects of subleading shape functions lead to an additional uncertainty of 5% [67, 68].
A more recent analysis [69] estimates the uncertainties due to subleading shape
functions more conservatively. By scanning over a range of models of these functions,
one finds corrections in the rates relative to the leading-order result to be between
decrease such uncertainties significantly by constraining both the leading and
subperforming the matching from QCD onto SCET at NNLO, and a prediction of the
zero of the forward-backward asymmetry in this semi-inclusive channel was provided.
• As already discussed, the cc¯ resonances can be removed by making appropriate
kinematic cuts in the invariant mass spectrum. However, nonperturbative contributions
away from the resonances within the perturbative windows are also important. In
the KS approach [72, 73] one absorbs factorizable long-distance charm rescattering
effects (in which the B¯ → Xscc¯r transition can be factorized into the product of s¯b
and cc¯ color-singlet currents) into the matrix element of the leading semileptonic
operator O9. Following the inclusion of nonperturbative corrections scaling with 1/mc2,
the KS approach avoids double-counting. For the integrated branching fractions one
due to the 1/mb corrections.
• The integrated branching fraction is dominated by this resonance background which
exceeds the nonresonant charm-loop contribution by two orders of magnitude. This
feature should not be misinterpreted as a striking failure of global parton-hadron
duality [74], which postulates that the sum over the hadronic final states, including
resonances, should be well approximated by a quark-level calculation [75]. Crucially,
of a phase-space integral over the absolute square of a correlator. For such a quantity
global quark-hadron duality is not expected to hold. Nevertheless, local quark-hadron
duality (which, of course, also implies global duality) may be reestablished by
resumming Coulomb-like interactions [74].
• Also electromagnetic perturbative corrections were calculated: NLL quantum
electrodynamics (QED) two-loop corrections to the Wilson coefficients are of O(2%) [55].
In the QED one-loop corrections to matrix elements, large collinear logarithms of the
form log(mb2/m`2) survive integration over phase space if only a restricted part of
the dilepton mass spectrum is considered. These collinear logarithms add another
contribution of order +2% in the low-q2 region of the dilepton mass spectrum in
Based on all these scientific efforts of various groups, the latest theoretical predictions have
been presented in ref. [66].
In the present manuscript, we make the effort to provide all missing relevant
perturwell-known, the angular decomposition of this inclusive decay rate provides three
independent observables, HT , HA, HL from which one can extract the short-distance electroweak
Wilson coefficients that test for possible new physics [77]:
(1 + z2)HT (q2) + 2(1 − z2)HL(q2) + 2zHA(q2) .
di-lepton rest frame, HA is equivalent to the forward-backward asymmetry [78], and the
q2 spectrum is given by HT + HL. The observables dominantly depend on the effective
Wilson coefficients corresponding to the operators O7, O9, and O10.
The paper is organized as follows. In section 2 we define the observables which we
consider in the present analysis. In section 3 the derivation of the log-enhanced terms is
presented. Master formulae for our observables are given in section 4, our phenomenological
results in section 5. We briefly discuss the new physics sensitivity of our observables in
section 6. Finally we explore the precise connection between experimental and theoretical
quantities using Monte Carlo techniques in section 7. The latter analysis updates, and
in parts supersedes, our previous statements in ref. [79]. We conclude in section 8. In
the appendices we collect various functions that arise in the computation of QED and
QCD corrections to the observables (appendix A), as well as formulas that parametrise the
observables in terms of ratios of high-scale Wilson coefficients (appendix B).
Definition of the observables
The z dependence of the double differential decay distribution presented in eq. (1.4) is
exact to all orders in QCD because it is controlled by the square of the leptonic current. The
inclusion of QED bremsstrahlung modifies the simple second order polynomial structure
and replaces it with a complicated analytical z dependence (see eqs. (3.28)–(3.33)). In
particular this implies that, as long as QED effects are observably large, a simple fit to a
quadratic polynomial will introduce non-negligible distorsions in the comparison between
theory and experiment. In this section we explain the procedure that we adopt to
construct various q2 differential distributions and suggest that experimental analyses follow
the same prescriptions.
The extraction of multiple differential distributions from eq. (1.4) is
phenomenologically important because the various observables have different functional dependence on the
Wilson coefficients. For instance, at next-to-leading order in QCD and without including
any QED effect, the three HI defined in eq. (1.4) are given by [77]:
HT (q2) =
HL(q2) =
HA(q2) =
2
G2F mb5 |Vt∗sVtb| (1 − sˆ)2h C9 + 2 C7
2
G2F mb5 |Vt∗sVtb| (−4sˆ) (1 − sˆ)2 Re C10 C9 +
and of the forward-backward asymmetry dAFB/dq2:
dq2 ≡
dq2 ≡
−1 dq2dz
−1 dq2dz
with the understanding that AFB does not coincide with the coefficient of the linear term
We extract other single-differential distributions by projecting the double-differential
rate onto various Legendre polynomials, Pn(z). These polynomials are orthogonal in the
connection with the existing literature we choose the first two projections in such a way
to reproduce HT and HL in the limit of no QED radiation. For the higher order terms we
simply adopt the corresponding Legendre polynomials. The observables are defined as
and the weights we use are:
HI (q2) =
−1 dq2dz
WT =
WL =
WA =
P0(z) − 3
W3 = P3(z) ,
W4 = P4(z) ,
The unnormalized (defined in eq. (2.5)) forward-backward asymmetry receives
contributions form all odd powers of z in the Taylor expansion of the double differential rate and
is given by
In the literature the normalized differential and integrated forward-backward asymmetries
are often considered (see for instance ref. [66]):
dq2 HT (q2) + HL(q2)
dq2 ≡
AFB[qm2, qM2 ] ≡
−1
−1
−1
−1
dAFB =
4 HT (q2) + HL(q2) ,
ℓ−
p− ≡ p2
p+ ≡ x¯p1
ℓ−
from the effective weak Hamiltonian. The arrows indicate momentum rather than fermion flow. x
denotes the momentum fraction of the collinear photon.
The new observables H3 and H4 (obtained by employing the weights W3 and W4)
vanish exactly in the limit of no QED radiation but are still potentially important for
phenomenology because of their non trivial dependence on the Wilson coefficients.
find that projections with even higher Legendre polynomials are suppressed and will not
be considered further.
Note that the expected statistical experimental uncertainties (at a given luminosity) are
well understood in the total width (HT + HL) and forward-backward asymmetry (3/4 HA)
cases. On the other hand, HT , HL, H3 and H4 are obtained by projecting the double
differential rate with weights that (especially for W3 and W4) are essentially arbitrary. As
a consequence a simple rescaling of these weights implies a corresponding rescaling of the
central values we find. In section 6 we show how to use the squared weights (WI2) to assess
the expected Belle II reach for each of these observables.
The experimental procedure that we recommend is to use the weights WI to extract
single-differential distributions and to refrain from attempting polynomial fits to the data.
Log-enhanced QED corrections to the double differential decay rate
In this section we work out the formulas for the logarithmically enhanced electromagnetic
coefficients of the effective weak Hamiltonian are the same as in [66, 76]. The kinematics
can be inferred from figure 1.
Let us first consider the case without photon radiation. The momenta of the quarks
Moreover, we define
sij ≡
i ∈ {1, 2, s, b} .
y1 ≡
y2 ≡
decaying b-quark. From momentum conservation and by treating all final-state particles
s1s = 1 − y2 ,
s2s = 1 − y1 ,
s1b = y1 ,
s2b = y2 .
ssb = 1 − s12 ,
the b-quark and the positively charged lepton in the centre-of-mass system (c.m.s.) of the
final-state lepton pair. Hence
z =
y2 − y1
1 − s12
where all primed momenta are taken in the c.m.s. of the final-state lepton pair. It turns
out that z is simply given by [57]
At this point we stress that the l.h.s. of this equation is evaluated in the lepton c.m.s.,
whereas its r.h.s. is evaluated in the rest-frame of the decaying b-quark. The connection
We now switch on QED and consider the radiation of a collinear photon off a lepton
leg as shown in figure 1. The momentum of the positively (negatively) charged lepton is
lepton radiates the photon (left panel of figure 1), its momentum p+ after radiation is
and hence we have p
panel of figure 1), we obviously have p
− = x¯p2 and p+ = p1. In analogy to eq. (3.2),
± is the zero-component of p±, again evaluated in the rest-frame of the decaying
b-quark. We will also need the definition
As already discussed in refs. [66, 76], the logarithmically enhanced contributions
stem± ≡
s+− ≡ x¯ s12 .
ming from collinear photon radiation are evaluated by
invariant s12 = (p+ + p
the triple invariant, where the formulae look exactly the same as in the case without QED,
since we can lump the lepton and the collinear photon. We therefore arrive at
P F =
G2F mb|VtbVt∗s|2 ,
[1 + (1 − x)2]
The squared matrix elements |A|2 for the different operators read
f2(s12) =
|A|727 (s12, y1, y2) =
8mb4 [(1 − y2) y1 + (1 − y1) y2] ,
|A|729 (s12, y1, y2) = 4mb4 (1 − s12) ,
2
|A|710 (s12, y1, y2) = 4mb4 (y1 − y2) ,
2
|A|910 (s12, y1, y2) = 2mb4 s12 (y1 − y2) ,
|A|229 (s12, y1, y2) = αee f2(s12) |A|929 (s12, y1, y2) ,
|A|227 (s12, y1, y2) = αee f2(s12) |A|729 (s12, y1, y2) ,
|A|222 (s12, y1, y2) = αee2 |f2(s12)|2 |A|929 (s12, y1, y2) ,
|A|210 (s12, y1, y2) = αee f2(s12) |A|910 (s12, y1, y2) .
The function f2(s12) denotes the one-loop matrix element of P2 and is given by
− 9 (2 + yc)p|1 − yc|
1 + √
1 −
2 arctan √
1 − yc
1 − yc
yc − 1
when yc ≥ 1 ,
are complex. However, after taking into account the Wilson coefficients and adding the
appropriate complex conjugate expression, the double differential rate turns out to be real,
see eq. (3.28).
and changing variables according to eq. (3.5) we arrive at
1 − s12 z,
1 − s12 z
1 − s12 θ(1 − z) θ(1 + z) θ(s12) θ(1 − s12) .
The factor of two stems from the fact that both diagrams in figure 1 are relevant. Note
once all expressions on its r.h.s. are plugged in.
We now turn our attention to the more complicated case of the double invariant
p~1 + x¯p~2 −→ 0.
z =
x¯y2 − y1
(y1 + x¯y2)2 − 4x¯s12
Again, the primed momenta are evaluated in the lepton c.m.s., whereas the r.h.s. of the
equation is evaluated in the rest-frame of the b-quark. The differential decay width reads
favour of z according to eq. (3.16). This transformation reads
y2(±)(z) =
x2(1 − z2) + 4x¯
It turns out that this in an injective mapping only for s12 < x¯. For s12 > x¯ we have
to subdivide the y2-interval into two pieces, so that we get a total of three contributions.
ds+− dz
= P F
ds+− dz
= ±P F
Z 1−√s+−
∂∂z y2(+)(z) h
|A|2 (s12, 1 + s12 − y2, y2)i
×θ(1 − z) θ(1 + z) θ(s+−) θ(1 − s+−) ,
Z x−
1−√s+−
∂∂z y2(±)(z) h
|A|2 (s12, 1+s12 −y2, y2)i
y2 = y2(+)(z)
s12 = s+−/x¯
y2 = y2(±)(z)
s12 = s+−/x¯
Once the photon is radiated off `+, we apply very similar steps. As can be seen from
is determined by
x± =
1 − s+−
1 ∓ p(1 − z2)s+−
z =
y2 − x¯y1
(x¯y1 + y2)2 − 4x¯s12
favour of z according to eq. (3.23). This transformation reads
y1(±)(z) =
x2(1 − z2) + 4x¯
As mentioned before, this is an injective mapping only for s12 < x¯. For s12 > x¯ we have
to subdivide the y1-interval into two pieces, so that in this case we also get a total of three
ds+− dz
= −P F
ds+− dz
= ∓P F
Z 1−√s+−
∂∂z y1(−)(z) h
|A|2 (s12, y1, 1 + s12 − y1)i
×θ(1 − z) θ(1 + z) θ(s+−) θ(1 − s+−) ,
Z x−
1−√s+−
∂∂z y1(∓)(z) h
|A|2 (s12, y1, 1 + s12 − y1)i
The total contribution in case of the double invariant is now obtained by
ds+− dz
ds+− dz
ds+− dz
into eq. (3.8). This leads us to the following expression for the logarithmically enhanced
collinear decay width
+Re [C9C1∗0] ξ9(e1m0)(s, z) + αee2 Re h(C2 + CF C1) C7eff ∗ ξ2(e7m)(s, z)
where we assumed that the Wilson coefficients C1 and C2 are real, and we neglected
64 p1(s, z) √s ln q s
1−z2 −
1−z2 − 1
(z2 − 1)3 √s + z2 − 1
s(1−z2) −
s(z2 − 1)3 (s (z2 − 1) + 1)3/2
64 z p2(s, z) ln z1−+z1
s(z2 − 1)3
16 p4(s, z) ln(s)
s(z2 − 1)3
3s (z2 − 1)2 (s (z2 − 1) + 1) −
16(s − 1)2 p6(s, z) ln √2(1−s)
1−z2
y1 = y1(−)(z)
s12 = s+−/x¯
y1 = y1(∓)(z)
s12 = s+−/x¯
(z2 − 1)4
8 s3/2 p9(s, z) ln
1−z2 −
(z2 − 1)4
1−z2 − 1
(z2 − 1)4 (s + z2 − 1)5/2
+4(s − 1)2 sz2 + s − z2 + 1 ln
2(1 − s)
1 − z2
32 p12(s, z) ln(s)
(z2 − 1)3
64√s p14(s, z) ln
3(z2 − 1)3 (s + z2 − 1)2
1−z2 −
1−z2 − 1
(z2 − 1)3 (s + z2 − 1)3/2
64 z p11(s, z) ln z1−+z1
(z2 − 1)3
8 p13(s, z)
− (z2 − 1)2 (s + z2 − 1)
+32 (s − 1)2 ln
64 p15(s, z) sign(z) ln
32 s z p19(s, z) ln(s)
(z2 − 1)4
4s (√s−1)2 z p21(s, z)
(z2 − 1)3 (s+z2 −1)2 −
−16(s − 1)2 s z ln 1 −
(z2 − 1)2 ps (z2 − 1) + 1
(√s+1)√1−z2
32 z p17(s, z) ln 12 (√s + 1) √1 − z2
(z2 − 1)3 (s + z2 − 1)3/2
64 s z 9sz2 + 7s + 4z2 − 4 ln(s)
(z2 − 1)3
(z2 − 1)3
2(1 − s)
1 − z2
64 p15(s, z) ln
s(1−z2) −
(z2 − 1)2 ps (z2 − 1) + 1
−√s(z2−1)−√z2√s(z2−1)+1+1
(√s+1)√1−z2
8 (√s − 1)2 z p18(s, z)
(z2 − 1)2 (s + z2 − 1)
− 32(s − 1)2 z ln 1 −
16 s z p20(s, z) ln 12 (√s + 1) √1 − z2
(z2 − 1)4
(√s+1)√1−z2
(z2 − 1)4 (s + z2 − 1)5/2
The pi(s, z) are polynomials in s and z and are given in appendix A. In case of negative
or complex arguments, the logarithms and square-roots are defined as
z = p|z| ei/2 arg(z) ,
ln(z) = ln|z| + i arg(z) ,
squared matrix elements (see eq. (3.12)) are complicated functions of s12. We therefore
refrain from presenting their explicit expressions. They can easily be computed numerically
by applying the steps outlined above.
subsequently integrating over z. After proper normalisation one obtains the functions
to introduce the variable z we performed the calculation entirely in terms of the rescaled
energies yi. Moreover, there was more freedom in choosing the order of integrations since we
These two simplifications led to significantly simpler variable substitutions and shorter
expressions. With the ability to reproduce them by the more complicated calculation can
therefore be regarded as a non-trivial cross-check.
Master formulas for the observables
We start again from the double differential decay width
dz dq2 =
(1 + z2)HT (q2) + 2zHA(q2) + 2(1 − z2)HL(q2) ,
di-lepton rest frame. This formula is modified once QED corrections are taken into account
(see sections 2 and 3) due to the appearance of higher powers of z. As stated in section 2,
lepton forward-backward asymmetry; the q2-spectrum is given by HT + HL,
C =
dq2 =
dAFB =
−1
−1
dz dq2 = HT (q2) + HL(q2) ,
dz dq2 sign(z) =
HI (q2) =
G2F mb5,pole |Vt∗sVtb|2 Φ`I`(sˆ),
Moreover, we normalise the observables to the inclusive semi-leptonic
corrections), and also use the ratio [55, 80]
Consequently, our expression of the normalised angular observables HI reads
ϕ(1) =
3 − 3
ϕ(2) = nh −
+ O(αes3, κ2, αsκ, αesΛ2/mb2, Λ3/mb3) ,
e
2mb2 − 2mb2
− 27
Φ`I`(sˆ) = X Re hCieff (μb) Cjeff∗(μb) HiIj(μb, sˆ)i ,
i≤j
− 27
As explained in detail in [76], a consistent perturbative expansion in inclusive
¯
B → Xs`+`− in the presence of QED corrections is done in αes = αs(μb)/(4π) and
also exist QED corrections at O(αsκ) which could be computed in principle. However,
e
infrared safe observable with respect to collinear photon radiation. We therefore neglect
this contribution, but include it lateron in the quantity R(s0), where QED logs will be
present in the normalisation.
of heavy and light quark flavours, respectively, and β(5) = 23/3 is the one-loop QCD
0
terms represent the matrix element of the kinetic energy and magnetic moment operator,
respectively, and are defined as
λ2 = −hB|h¯iσμνGμνh|Bi/(12MB) ≈ 4 (MB2 ∗ − MB2 ) .
of products of the low-scale Wilson coefficients and various functions arising from the
matrix elements,
and (16) in [76]. Their low-scale Wilson coefficients are also given explicitly (analytically
HiIj =
N=7,9,10
N=7,9,10
|MiN |2 SNIN + Re(Mi7Mi9∗) S7I9 + ΔHiIi ,
for i = j ,
2MiN MjN∗ SNIN +
Mi7Mj9∗ + Mi9Mj7∗
For I = A the formula is simpler,
HiAj =
N=7,9
MiN Mj10∗ + Mi10MjN∗
The coefficients MiA are listed in table 6 of [76]. The building blocks SNIM have the following
SNM = σNM (sˆ) n1 + 8 αes ωN(1M),I (sˆ) + 16 αes2 ω(2)
I I
From (4.11) and (4.12) we see that the possible combinations of indices are N M
NM (sˆ) read
for i = j ,
to appendix A.
NM,I (sˆ) can be extracted from [50] and have already
NM,I (sˆ) have so far only been available for the
q2-spectrum [84–87], but not for the double differential rate. Due to a recent calculation
in QCD [88], they can be extracted for N M
= 99, 1010 and I = T, L as well as for
the authors of [88, 89] and we can therefore present them here for the first time. All
NM,I (sˆ) are rather lengthy and we therefore relegate their explicit expressions
corrections can be obtained from [57] (see also [56, 59]) and were previously computed
in [77]. We confirm their expressions,
(1 − sˆ)(5sˆ + 3) ,
(sˆ − 1)(3sˆ + 13) ,
3sˆ2 + 2sˆ + 3 ,
3sˆ2 + 2sˆ − 9 ,
χ2,99(sˆ) = sˆ 15sˆ2 − 14sˆ − 5 ,
T
2 −17sˆ2 + 10sˆ + 3 ,
χ2,710(sˆ) = −4 9sˆ2 − 10sˆ − 7 ,
A
χ2,910(sˆ) = −2sˆ 15sˆ2 − 14sˆ − 9 .
A
Here the contributions biIj represent finite bremsstrahlung corrections that appear at NNLO.
only include them for these two cases, but not for HT and HL separately. This is still an
excellent approximation since the effect of finite bremsstrahlung corrections is very small
anyway. The explicit formulas can be found in [48, 51] and will therefore not be repeated.
differential rate can be inferred from that paper. One obtains
c2Tj = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F (r)
c1Tj = − 61 c2Tj ,
for j 6= 1, 2 ,
c2T2 = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F (r)
c1T1 = +αsκ 247λm2c2 (1 − sˆ)2(1 + 3sˆ) F (r)
e
c1T2 = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F ∗(r)
M27∗ +
M29∗ ,
M17∗ +
M19∗ ,
Mj9∗ ,
c2Lj = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F (r) Mj7∗ +
c1Lj = − 61 c2Lj ,
for j 6= 1, 2 ,
Mj7∗ +
Mj9∗ ,
for j 6= 1, 2 ,
− 6
M27∗ +
M29∗
for j 6= 1, 2 ,
c2L2 = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F (r) M27∗ +
c1L1 = +αsκ 247λm2c2 (1 − sˆ)2(3 − sˆ) F (r) M17∗ +
e
c1L2 = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F ∗(r)
M29∗ ,
M19∗ ,
− 6
M27∗ +
M29∗
we also include factorisable non-perturbative charm contributions which we implement by
means of the Kru¨ger-Sehgal approach [72, 73]. We elaborated extensively on this approach
and also the formulas by means of which these corrections are taken into account in ref. [66].
Given their length we do not repeat these formulas here but refer the inclined reader to
refs. [66, 72, 73] for all necessary details.
Finally, the coefficients eiIj collect the ln(mb2/m`2)-enhanced electromagnetic corrections
which we calculated in section 3 for the double differential rate. Their contribution to the
HI can be derived from (3.28) by applying the projections given in section 2. One finds
eI11 =
eI12 =
eI1j =
for j = 7, 9 ,
for I = T, L, while for I = A one gets
e910 = 8 αsκ σ9A10(sˆ) ω9(e1m0,)A(sˆ) ,
A
We consider the observables HI (or equivalently HI ) in the low-q2 region only, because
their sensitivity to New Physics is highest in this region [77]. Besides, there are two more
of the forward-backward asymmetry, which we extract numerically from HA by means of
the formulas given above. Moreover, there is the branching ratio. In principle, it can be
obtained by taking the sum of HT and HL. Its master formula has already been given
in [76]. We therefore only highlight two small pieces which are available for the branching
ratio only, but not for HT and HL individually. These are only the finite bremsstrahlung
In the high-q2 region we consider two observables. The first one is the branching ratio,
where we include the same terms as in the low-q2 region. As far as QED corrections are
obtained from a numerical fit. To take into account our most recent input parameters (see
table 1), we re-did the fits and collected the results in appendix A. In addition, the
twoloop QCD matrix element functions F17,2(sˆ) and F19,2(sˆ), which were originally computed
in [53], were given explicitly only in [54]. We implement these formulas in our numerical
code. Moreover, non-perturbative 1/mb3 corrections become sizable in the high-sˆ region.
They were originally computed in [60] and we implement the formulas of refs. [60, 61]. The
second observable is the ratio R(s0) which we have already mentioned in the introduction.
uncertainties that stem from poorly known parameters in the 1/mb2 and 1/mb3
powercorrections can be significantly reduced, as we will see in our numerical analysis in section 5.
In terms of our perturbative quantities, it reads
R(s0) =
= 4
Vt∗sVtb
2 Rsˆ10 dsˆ Φ``(sˆ)
We would rather
and which was absent in [66]. Once the integration over sˆ is restricted to the high-q2
Φu(sˆ) αesκ = 8 αsκ (1 − sˆ)2 (1 + 2sˆ) ω9(e9m)(sˆ)
e
Let us conclude this section by a few remarks on the renormalisation schemes for the
that are present in the definition of sˆ and in several loop functions suffer from renormalon
ambiguities [90, 91]. We therefore convert them analytically to short-distance schemes (1S
and MS, respectively) before any numerical evaluation of the observables is carried out. In
the mass of the top quark is concerned we take the pole mass as input and convert it to
1Note that we use a different pre-factor here.
q2 ∈ [1, 6] GeV2
q2 ∈ [1, 3.5] GeV2
the integrated observable and its QED correction normalized to the total low-q2 branching ratio
From inspection of the left plot in figure 12 we see that, in the low-q2 region HT is
much smaller than HL. We can understand the origin of this effect by looking at the ratio
HT /HL at leading order:
C120 + (C9 + 2C7)2
The suppression comes from the small 2sˆ . 1 factor and from the accidental strong
cancellation between C9 and 2C7/sˆ at low sˆ (in fact, the combination C9 + 2C7/sˆ vanishes for
C9 + 2C7/sˆ > C9 + 2C7 and the integrated HT and HL observables at low-q2 would assume
very similar values.
In table 4 we present the results we obtain by integrating the Monte Carlo
generated b → s`` histograms. For each bin ([s1, s2]) and for each observable O (HT + HL,
HT and HL) we show the total integrated observable (Rss12 O/ R16(HT + HL)), the total
amongst the three observables (with the effect on HT being only slightly larger) and that
the suppression of HT with respect to HL is responsible for very large relative effects in
the 30–50% range.
Finally we must point out that the numerical estimates presented in table 4 are affected
by sizable uncertainties that are hard to quantify and that only the analytical results
presented in table 2 should be utilized. The Monte Carlo study was nevertheless extremely
valuable to build confidence in our study.
search for new physics via quark flavour observables. It is theoretically clean, while the
exclusive mode is affected by unknown power corrections. Thus, besides allowing for a
nontrivial check of the recent LHCb data on the exclusive mode, it contains complementary
information both in Standard Model predictions and in pinning down new physics. It is
therefore a precious channel to be measured at Belle II, and might be accessible even
In the present article we perform a complete angular analysis of the inclusive decay
able to date. We confirm the findings of ref. [77] that a separation of the double differential
decay width into three observables HT,A,L(q2), as well as subdivision of the low-q2 region
into two bins (see also [66]), provides significantly more information than the branching
ratio or forward-backward asymmetry in the entire low-q2 region alone.
We compute logarithmically enhanced QED corrections to these observables and find
double differential decay width in the absence of QED corrections. We therefore propose
to project out HT,A,L(q2) using weight functions, and argue that the Legendre polynomials
Pn(z) are the optimal choice for the latter. Besides reproducing HT (q2) and HL(q2) in
the absence of QED radiation, they allow to construct observables H3,4(q2) (eq. (2.6))
that vanish if only QCD corrections are taken into account, and are therefore particular
sensitive to QED effects. In view of the benefits of the Legendre weight functions we
urgently recommend the experiments to use the weights (2.6) to extract single-differential
distributions, and to refrain from attempting polynomial fits to the data.
The absolute values of the QED effects that we compute are natural in size. However,
due to the phase-space and Wilson coefficient suppression of HT (q2) the relative size of the
QED corrections is large in this observable. We argue carefully that this does clearly not
indicate a breakdown of perturbation theory. On the contrary, we can benefit from the
fact that QED corrections lift the smallness of HT (q2) to a certain extent, which makes it
an observable that is particular sensitive to QED radiation.
To supplement our calculation we carry out a dedicated Monte Carlo study, whose
main purpose is three-fold. First, we investigate how the electromagnetic logarithms are
treated correctly in the presence of angular and energy cuts. We find that our analytical
predictions can be directly applied, with the exception of the electron channel at BaBar,
where our numbers have to be modified according to eqs. (7.1) and (7.2). Second, the size
of the QED corrections, in particular their large relative size in HT (q2), are confirmed by
the Monte Carlo (cf. tables 2 and 4). Last but not least, it consitutes also a validation
of PHOTOS, which is used by experiments to estimate QED effects in the calculation
of efficiencies.
We update the Standard Model predictions for all angular observables integrated over
two bins in the low-q2 region. The branching ratio and the observable R(s0) are also
evaluated in the high-q2 region. Moreover, we provide our prediction for the zero crossing of
the forward-backward asymmetry (or, equivalently, HA). The parametric and perturbative
branching ratio, where the relative errors are much larger. In the former case the reason
is the zero crossing of HA which entails a cancellation between the central values of the
two bins in the low-q2 region. In the latter case we suffer from poorly known hadronic
same cut in q2 [61].
parameters in the 1/mb2,3 power-corrections, a drawback that is circumvented in the ratio
independent way. We give all observables in terms of ratios R7,8,9,10 of high-scale Wilson
coefficients, which we assume to be altered by the new interactions. We also study
cortightest constraints. On the other hand, if deviations from the Standard Model are seen,
all observables become crucial to pin down the structure of new physics.
ferent from unity, one might wonder whether this sign of lepton non-universality could
be traced back to logarithmically enhanced QED corrections. LHCb uses the PHOTOS
Monte Carlo to eliminate the impact of collinear photon emissions from the final state
electrons. Therefore, the corrections calculated in this paper do not seem to apply to the
ratio RK . Given that the agreement between PHOTOS and our analytical calculations is
not perfect (see e.g. tables 2 and 4), it would be advisable to correct for photon radiation
using data-driven methods that do not rely on PHOTOS.
Acknowledgments
We would like to thank Javier Virto for useful discussions, and Kevin Flood, Chris Schilling
and Owen Long for logistic and technical support that allowed the Monte Carlo study
presented in section 7. We are indebted to Mathias Brucherseifer, Fabrizio Caola, and Kirill
studies [88, 89]. T. Huber acknowledges support from Deutsche Forschungsgemeinschaft
within research unit FOR 1873 (QFET). T. Hurth thanks the CERN theory group for
its hospitality during his regular visits to CERN. All authors are grateful to the Mainz
Institute for Theoretical Physics (MITP) for its hospitality at the Capri-Institute in May
2014, where part of this work was done.
QED and QCD functions
QED functions for the double differential rate
QED corrections to the double differential rate in eq. (3.28).
p3(s, z) = s
3 z2 − 1
+ s2 z2 − 1
+s 6z6 + 37z4 − 36z2 − 7 + 5z4 + 24z2 + 3 ,
p5(s, z) = s4 13z8 − 56z6 + 210z4 − 112z2 − 55
−15z8 + 31z6 − 127z4 + 149z2 + 154
+3s2 5z8 − 9z6 + 55z4 − 31z2 − 84
+s −13z8 + 65z6 − 285z4 + 355z2 + 262
−13z6 + 37z4 − 299z2 − 109 ,
p6(s, z) = s z2 − 1 − z2 − 1 ,
p8(s, z) = s2
−z10 + 3z8 + 32z6 + 364z4 + 289z2 + 17
− 19z8 + 106z6 + 102z4 − 173z2 − 19
+2 −z8 + 7z6 − 9z4 + z2 + 2 ,
+s2 z2 − 1
+s z2 − 1
2 46z6 + 889z4 + 1030z2 + 87
3 256z4 + 483z2 + 51 + z2 − 1
−26z10 + 173z8 − 2504z6 − 2098z4 + 7690z2 + 989
−13z12 + 122z10
− 1190z8 + 830z6 + 8809z4 − 7288z2 − 1270
+s2 z2 − 1
−s z2 − 1
2 15z8 − 18z6 + 397z4 + 3716z2 + 706
3 15z6 + 19z4 − 403z2 − 143 + z2 − 1
4 13z4 − 22z2 + 1 ,
p11(s, z) = s2 5z2 + 3 + z2 − 1 ,
p12(s, z) = s2 z6 − 6z4 − 9z2 − 2 − s z2 − 1
+ z4 − 1 ,
−s 2z6 + 17z4 − 24z2 + 5 + z2 − 1
+s z2 − 1
2 7z2 − 1 − z2 − 1
p15(s, z) = s z2 − 1 + z2 + 1 ,
+s z2 − 1
2 19z2 + 5 + z2 − 1
−2 s z2 − 5 z2 − 1
+ z2 − 1
+ z2 − 1
2 z4 − 2z2 + 5 ,
2 z4 − 27z2 − 34
+s z2 − 1
+ z2 − 1
3 3z4 − 10z2 − 33 ,
2 42z6 + 717z4 + 742z2 + 63 + 5s z2 − 1
+s2 z2 − 1
+ z2 − 1
3(sˆ− 1)2
3(sˆ− 1)2
ω7(19),T(sˆ) = − 34 log μb
mb −
ω9(19),T(sˆ) = (√sˆ+ 1)2(8sˆ3/2
(5sˆ+ 1)log(1 − sˆ) + sˆ(3sˆ+ 1)log(sˆ) +
6(sˆ− 1)2
− 15sˆ+ 4 sˆ− 5)Li2(1 − sˆ)
6(sˆ− 1)2√sˆ
36(sˆ− 1)2√sˆ
2(sˆ2 − 12sˆ− 5)Li2(1 − √sˆ)
3(sˆ− 1)2√sˆ
3(sˆ− 1)2
3(sˆ− 1)2
6(sˆ− 1)2sˆ
A.2 Functions for the QCD corrections to the HI
The one-loop QCD functions [50, 77] can be computed analytically,
ω7(17),T(sˆ) = − 38 log mμbb −
(√sˆ+ 1)2(sˆ3/2 − 10sˆ+ 13√sˆ− 8)Li2(1 − sˆ)
6(sˆ− 1)2
+ 5sˆ3 − 54sˆ2 + 57sˆ− 8
18(sˆ− 1)2
3(sˆ− 1)2
36(sˆ− 1)2
sˆ(5sˆ+ 1)log(sˆ) +
3(sˆ− 1)2
− log(1 − sˆ) +
3(sˆ− 1)2
2√sˆ(sˆ+ 3)Li2(1 − √sˆ) π2(16sˆ+ 29√sˆ+ 19)(√sˆ− 1)2
36(sˆ− 1)2
sˆ2 + 18sˆ− 19
(2sˆ+ 1)log(1 − sˆ) + 2 log(1 − sˆ)log(sˆ),
3sˆ 3
√sˆ)log(sˆ)
3(sˆ− 1)2
3(sˆ− 1)2
3(sˆ− 1)2
3(sˆ − 1)2
3(sˆ − 1)2
(2sˆ3 − 11sˆ2 + 10sˆ − 1) log(1 − sˆ)
2sˆ(2sˆ − 5) log(1 −
3(sˆ − 1)2sˆ
( sˆ + 1)2(4sˆ3/2
− 7sˆ + 2 sˆ − 3)Li2(1 − sˆ)
3(sˆ − 1)2
√sˆ) log(sˆ)
− 3
9sˆ2 −38sˆ+29
6(sˆ − 1)2
7sˆ2 − 2sˆ − 5
6(sˆ − 1)2
(sˆ − 7)sˆ log(sˆ)
3(sˆ − 1)2
( sˆ + 1)2(sˆ3/2
4(sˆ2 − 6sˆ − 3)Li2(1 −
3(sˆ − 1)2√sˆ
3(sˆ − 1)2sˆ
√sˆ)
(sˆ3 − 3sˆ + 2) log(1 − sˆ)
2(sˆ2 − 3sˆ − 3) log(sˆ)
π2(8sˆ3/2 + 13sˆ + 2 sˆ + 9)(√sˆ − 1)2
√
18(sˆ − 1)2√sˆ
4√sˆ(sˆ + 3)Li2(1 −
√sˆ)
3(sˆ − 1)2
18(sˆ−1)2
log(1 − sˆ) log(sˆ) ,
− 8sˆ + 3 sˆ − 4)Li2(1 − sˆ)
3(sˆ − 1)2
log(1 − sˆ) log(sˆ) ,
( sˆ + 1)2(4sˆ−9 sˆ+3)Li2(1−sˆ)
3(sˆ−1)2
(2sˆ + 1) log(1 − sˆ)
4√sˆ(sˆ2 − 12sˆ − 5)Li2(1 −
√sˆ)
3(sˆ − 1)2
3(sˆ − 1)2√sˆ
3(sˆ − 1)2
for five active flavours.
sˆ + 8)(√sˆ − 1)2
18(sˆ − 1)2
4sˆ3 − 51sˆ2 + 42sˆ + 5
6(sˆ − 1)2
3(sˆ − 1)2
log(1 − sˆ) log(sˆ) .
− log(1 − sˆ)
The two-loop QCD functions [88, 89] are obtained from least-squares fits and are also valid
for all q2. The necessary data was kindly provided by the authors of [88, 89].
ω9(19),T (sˆ) + 54.919(1 − sˆ)4 − 136.374(1 − sˆ)3
+ 119.344(1 − sˆ)2 − 15.6175(1 − sˆ) − 31.1706 ,
ω9(11)0,A(sˆ) + 74.3717(1 − sˆ)4 − 183.885(1 − sˆ)3
+ 158.739(1 − sˆ)2 − 29.0124(1 − sˆ) − 30.8056 ,
ω9(19),L(sˆ) − 5.95974(1 − s)3 + 11.7493(1 − s)2
+ 12.2293(1 − s) − 38.6457 .
Functions for the QED corrections to the HI
The following functions are again obtained by least-squares fits. They are valid in the
low-q2 region (1 GeV2 < q2 < 6 GeV2) only.
−0.768521 − 80.8068 sˆ4 + 70.0821 sˆ3 − 21.2787 sˆ2 + 2.9335 sˆ − 0.0180809
sˆ
19.063 + 2158.03sˆ4 − 2062.92sˆ3 + 830.53sˆ2 − 186.12sˆ + 0.324236
sˆ
8(1 − sˆ)2
−6.03641 − 896.643sˆ4 + 807.349sˆ3 − 278.559sˆ2 + 47.6636sˆ − 0.190701
sˆ
21.5291 + 3044.94sˆ4 − 2563.05sˆ3 + 874.074sˆ2 − 175.874sˆ + 0.121398
sˆ
2.2596 + 157.984 sˆ4 − 141.281 sˆ3 + 52.8914 sˆ2 − 13.5377 sˆ + 0.0284049
sˆ
8(1 − sˆ)2
4(1 − sˆ)2
2sˆ(1 − sˆ)2
(1 − sˆ)2
4(1 − sˆ)2
8(1 − sˆ)2
8(1 − sˆ)2
4(1 − sˆ)2
−8.01684 − 1121.13sˆ4 + 882.711sˆ3 − 280.866sˆ2 + 54.1943sˆ − 0.128988
sˆ
4(1 − sˆ)2
+ i −2.14058 − 588.771sˆ4 + 483.997sˆ3 − 124.579sˆ2 + 12.3282sˆ + 0.0145059
sˆ
4.54727+330.182sˆ4 −258.194sˆ3 +79.8713sˆ2 −19.6855sˆ+ 0.0371348
sˆ
2sˆ(1 − sˆ)2
2sˆ(1 − sˆ)2
−2.27221 − 298.369sˆ4 + 224.662sˆ3 − 65.1375sˆ2 + 11.5686sˆ − 0.0233098
sˆ
(1 − sˆ)2
+ i −0.666157 − 120.303sˆ4 + 109.315sˆ3 − 28.2734sˆ2 + 2.44527sˆ + 0.00279781
sˆ
(1 − sˆ)2
1 − 5 sˆ
1 − sˆ
7 − 16 √sˆ + 9 sˆ
4 (1 − sˆ)
ln(1 −
√sˆ) −
+ ln(1 −
√sˆ) +
5 − 16 sˆ + 11 sˆ
4 (1 − sˆ)
(1 − 3 sˆ) ln sˆ
(1 − sˆ)
2sˆ(1 − sˆ)2
9sˆ(1 − sˆ)2
(1 − sˆ)2
9(1 − sˆ)2
−1.71832 − 234.11sˆ4 + 162.126sˆ3 − 37.2361sˆ2 + 6.29949sˆ − 0.00810233
sˆ
8(224.662sˆ3 −2.27221−298.369sˆ4 −65.1375sˆ2 +11.5686sˆ− 0.0233098 )
sˆ
1 − sˆ
− (1 − sˆ)
24sˆ(1 − sˆ)2
7.98625 + 238.507 (sˆ − a) − 766.869 (sˆ − a)2
24sˆ(1 − sˆ)2
with a = (4mc2/mb2)2.
squares fit (for fixed values of mb and mc) read
The respective high-q2 functions for the branching ratio that are obtained by a
least8(1 − sˆ)2(1 + 2sˆ)
8(1 − sˆ)2(1 + 2sˆ)
96(1 − sˆ)2
9(1 − sˆ)2(1 + 2sˆ)
New physics formulas
+ 0.204994 |R7|2 + 0.00230146 |R8|2 + 0.244813 |R9|2
+ 1.74294 |R10|2 + 0.632156
× 10−7 ,
+ 0.0631028 |R7|2 + 0.000727107 |R8|2 + 0.273706 |R9|2
+ 1.96638 |R10|2 + 0.257773
× 10−7 ,
+ 0.268097 |R7|2 + 0.00302857 |R8|2 + 0.518519 |R9|2
+ 3.70932 |R10|2 + 0.889929
× 10−7 ,
+ 0.206371 |R7|2 + 0.00230943 |R8|2 + 0.179467 |R9|2
+ 1.28881 |R10|2 + 0.436438
× 10−7 ,
− 0.0027675 |R9|2 − 0.0192329 |R10|2 − 0.0115297
× 10−7 ,
+ 0.0650616 |R7|2 + 0.000738436 |R8|2 + 0.239011 |R9|2
+ 1.72527 |R10|2 + 0.123204
× 10−7 ,
+ 0.271433 |R7|2 + 0.00304786 |R8|2 + 0.418478 |R9|2
+ 3.01408 |R10|2 + 0.559642
× 10−7 ,
− 0.00174093 |R9|2 − 0.0120987 |R10|2 + 0.0121072
× 10−7 ,
− 0.00174093 |R9|2 − 0.0120987 |R10|2 + 0.0131242
× 10−7 ,
− 0.0027675 |R9|2 − 0.0192329 |R10|2 − 0.0113078
× 10−7 ,
− 0.00450843 |R9|2 − 0.0313316 |R10|2 + 0.00181642
× 10−7 ,
× 10−9 , (B.13)
× 10−9 , (B.14)
× 10−9 , (B.15)
× 10−9 , (B.16)
× 10−9 ,
× 10−9 , (B.18)
+ 0.482688 |R9|2 + 0.0892516 |R7|2 + 0.00051617 |R8|2
+ 3.35446 |R10|2 + 1.6742
× 10−9 ,
+ 0.662601 |R9|2 + 0.112159 |R7|2 + 0.00064865 |R8|2
+ 4.60478 |R10|2 + 2.20357
× 10−9 ,
+ 0.200654 |R9|2 + 0.0371021 |R7|2 + 0.000214573 |R8|2
+ 1.39446 |R10|2 + 0.697498
× 10−9 ,
+ 0.0747905 |R9|2 + 0.00952261 |R7|2 + 0.0000550722 |R8|2
+ 0.51976 |R10|2 + 0.22061
× 10−9 ,
+ 0.275445 |R9|2 + 0.0466247 |R7|2 + 0.000269645 |R8|2
+ 1.91422 |R10|2 + 0.918108
× 10−9 ,
+ 0.00589466 |R7|2 + 0.000128527 |R8|2 + 0.575967 |R9|2
+ 4.20578 |R10|2 + 0.806915
× 10−7 ,
+ 0.00631092 |R7|2 + 0.0000975709 |R8|2 + 0.439598 |R9|2
+ 3.20293 |R10|2 + 0.701014
× 10−7 ,
+ 0.0122056 |R7|2 + 0.000226098 |R8|2 + 1.01556 |R9|2
+ 7.40871 |R10|2 + 1.50793
× 10−7 ,
+ 0.00897313 |R7|2 + 0.000146331 |R8|2 + 0.609248 |R9|2
+ 4.43707 |R10|2 + 0.914888
× 10−7 ,
+ 0.00748476 |R7|2 + 0.00010436 |R8|2 + 0.466941 |R9|2
+ 3.39295 |R10|2 + 0.791074
× 10−7 ,
+ 0.0164579 |R7|2 + 0.000250691 |R8|2 + 1.07619 |R9|2
+ 7.83003 |R10|2 + 1.70596
× 10−7 ,
+ 0.210889 |R7|2 + 0.0028916 |R8|2 + 0.813297 |R9|2
+ 5.94874 |R10|2 + 1.46402
× 10−7 ,
+ 0.0694138 |R7|2 + 0.000881518 |R8|2 + 0.714084 |R9|2
+ 5.16931 |R10|2 + 0.985134
× 10−7 ,
+ 0.280302 |R7|2 + 0.00377311 |R8|2 + 1.52738 |R9|2
+ 11.1181 |R10|2 + 2.44915
× 10−7 ,
+ 0.215344 |R7|2 + 0.00291736 |R8|2 + 0.78123 |R9|2
+ 5.7259 |R10|2 + 1.3762
× 10−7 ,
+ 5.11822 |R10|2 + 0.940534
× 10−7 ,
+ 0.287891 |R7|2 + 0.003817 |R8|2 + 1.48796 |R9|2
+ 10.8441 |R10|2 + 2.31673
× 10−7 ,
+ 0.00287361 |R7|2 + 0.0000373632 |R8|2 + 0.211548 |R9|2
+ 1.50748 |R10|2 + 0.200589
× 10−7 ,
+ 0.00370104 |R7|2 + 0.0000421485 |R8|2
+ 0.234333 |R9|2 + 1.66583 |R10|2 + 0.292268
× 10−7 ,
+ 0.00293717 |R7|2 + 0.0000444449 |R8|2 + 0.228597 |R9|2
+ 1.6322 |R10|2 + 0.174573
× 10−3 ,
+ 0.00393316 |R7|2 + 0.000050205 |R8|2 + 0.256024 |R9|2
+ 1.82281 |R10|2 + 0.266662
× 10−3 .
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