Study of the CKM angle γ sensitivity using flavor
untagged B
s
0
→ ˜
D
(∗)0
φ decays
D. Ao1, D. Decamp2, W. B. Qian1, S. Ricciardi3, H. Sazak4, S. T’Jampens2, V. Tisserand4, Z. R. Wang5,
Z. W. Yang5, S. N. Zhang6, X. K. Zhou1
1: University of Chinese Academy of Sciences, Beijing, China,
2: Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France, 3: STFC Rutherford Appleton Laboratory, Didcot, United Kingdom,
4: Universit´e Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France,
5: Center for High Energy Physics, Tsinghua University, Beijing, China,
6: School of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China, Abstract A sensitivity study for the measurement of the CKM angle γ from Bs0→ ˜D(∗)0φ decays is performed 1
using D meson reconstructed in the quasi flavour-specific modes Kπ, K3π, Kππ0, and CP -eigenstate modes
2
KK and ππ, where the notation ˜D0corresponds to a D0 or a D0meson. The LHCb experiment is taken as a
3
use case. A statistical uncertainty of about 8 − 19◦can be achieved with the pp collision data collected by the
4
LHCb experiment from year 2011 to 2018. The sensitivity to γ should be of the order 3−8◦after accumulating
5
23 fb−1 of pp collision data by 2025, while it is expected to further improve with 300 fb−1 by the second half
6
of the 2030 decade. The accuracy depends on the strong parameters r(∗)B and δB(∗), describing, together with γ,
7
the interference between the leading amplitudes of the B0s→ ˜D(∗)0φ decays. 8
Key words sensitivity study, CKM angle γ, B0s→ ˜D(∗)0φ decays 9
PACS 13.66.Bc, 14.20.Lq, 13.30.Eg
10
1
Introduction
11
Precision measurements of the CKM [1] angle γ (defined as arg[−VudVub∗/VcdVcb∗]) in a variety of B-meson
12
decay modes is one of the main goals of flavour physics. Such measurements can be achieved by exploiting the 13
interference of decays that proceed via the b → c¯us and b → u¯cs tree-level amplitudes, where the determination
14
of the relative weak phase γ is not affected by theoretical uncertainties. 15
Several methods have been proposed to extract γ [2–7]. At LHCb, the best precision is obtained by
16
combining the measurements of many decay modes, which gives γ = (74.0+5.0−5.8)◦ [8]. This precision dominates
17
the world average on γ from tree-level decays. LHCb has presented a new measurement based on the BPGGSZ 18
method [5] using the full Run 1 and Run 2 data. The result is γ = (68.7+5.2−5.1)◦[9] and constitutes the single best
19
world measurement on γ. A 2σ difference between B+ and B0s results was observed since summer 2018. The
20
Bs0 measurement is based on a single decay mode only with Run 1 data, i.e. B0s→ D∓
sK
±,∗ and has a large
21
uncertainty [10]. In October 2020, LHCb came with a similar analysis with the decay Bs0→ D∓
sK
±π+π−[11]
22
based on Run 1 and Run 2 data for which γ − 2βs= (42 ± 10 ± 4 ± 5)◦. The most recent LHCb combination
23
is then γ = (67 ± 4)◦ [12] and the new B+ (B0
s) result is (64+4−5)◦ (82+17−20)◦. Additional Bs0 decay modes will 24
help improve the level of measurement precision of Bs0 modes and the understanding of a possible discrepancy
25
with respect to the B+ modes. The two analyses based on Bs0→ D∓
sK
±(π+π−) use time-dependent methods
26
and therefore strongly rely on the B-tagging capabilities of the LHCb experiment. It should be noticed that 27
measurements exist at LHCb for B0mesons [13, 14], they offer also quite good prospects as their present average
28
is (82+8−9)◦. They are based on the decay B0→ D0K∗0, where the B0is self-tagged from the K∗0→ K+π−decay.
29
Received xxxx June xxxx
∗The inclusion of charge-conjugated processes is implied throughout the paper, unless otherwise stated.
©2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute
(a)
(b)
Figure 1. Feynman diagrams for (a, left) Bs0→ D(∗)0φ and (b, right) B0s→ D(∗)0φ decays.
As opposed to measurements with Bs0, those with B0 and B+ are also accessible at Belle II [15]. Prospects on
30
those measurements at LHCb are given in Ref. [16]. For the mode B0s→ D∓
sK
±one may anticipate a precision
31
on γ of the order of 4◦ after the end of LHC Run 3 in 2025, and 1◦ by 2035-2038. When combining with B0
32
and B+ modes the expected sensitivities are 1.5◦ and 0.35◦. The anticipated precision provided by Belle II is
33 1.5◦.
34
In this work, Bs0→ D(∗)0φ decays, whose observations were published by the LHCb experiment in 2013 [17]
35
and 2018 [18], are used to determine γ. A novel method presented in Ref. [18] showed also the feasibility 36
of measuring Bs0 → D∗0φ decays with a high purity. It uses a partial reconstruction method for the D∗0
37
meson [19]. A time-integrated method [20] is investigated where it was shown that information about CP 38
violation is preserved in the untagged rate of B0s→ ˜D(∗)0φ (or of B0→ ˜D0KS0), and that, if a sufficient number
39
of different D-meson final states are included in the analysis, this decay alone can, in principle, be used to 40
measure γ. The sensitivity to γ is expected to be much better with the case of Bs0→ ˜D(∗)0φ decay than for
41
B0→ ˜D0KS0, as it is proportional to the decay width difference y = ∆Γ/2Γ defined in Section 2 and equals to
42
(6.3 ± 0.3)%, for Bs0 mesons, and (0.05 ± 0.50)%, for B0 [21]. Sensitivity to γ from Bs0→ ˜D(∗)0φ modes comes
43
from the interference between two colour-suppressed diagrams shown in Fig. 1. The relatively large expected 44
value of the ratio of the ¯b → ¯uc¯s and ¯b → ¯cu¯s tree-level amplitudes (20 − 40 %, see Sect. 4.1) is an additional
45
motivation for measuring γ in Bs0 → ˜D(∗)0φ decays. In this study, five neutral D-meson decay modes Kπ,
46
K3π, Kππ0, KK and ππ are included, whose event yields are estimated using realistic assumptions based on 47
measurements from LHCb [18, 22, 23]. We justify the choice of those decays and also discuss the case of the 48
two decay modes ˜D0→ KS0π+π
−and K0
SK+K
−in Section 3.
49
In Section 2, the notations and the choice of D-meson decay final states are introduced. In Section 3, the 50
expected signal yields and their uncertainties are presented. In Section 4, the sensitivity which can be achieved 51
using solely these decays is shown, and further improvements are briefly discussed. In Section 5 the future 52
expected precision on γ with Bs0→ D(∗)0φ at LHCb are discussed for dataset available after LHC Run 3 by
53
2025, and after a possible second upgrade of LHCb, by 2038. Finally, conclusions are made in Section 6. 54
2
Formalism
55
Following the formalism introduced in Ref. [20], we define the amplitudes 56
A(Bs0→ D(∗)0φ) = A(∗)B , (1)
A(Bs0→ D(∗)0φ) = A(∗)B r(∗)B ei(δ(∗)
B +γ), (2)
where A(∗)B and r(∗)B are the magnitude of the Bs0 decay amplitude and the amplitude magnitude ratio between
57
the suppressed over the favoured Bs0decay modes, respectively, while δB(∗)and γ are the strong and weak phases,
58
respectively. Neglecting mixing and CP violation in D decays (see for example Ref. [24, 25]), the amplitudes 59
into the final state f (denoted below as [f ]D) and its CP conjugate ¯f are defined as
60 A(D0→ f ) = A(D0→ ¯f ) = Af, (3) A(D0→ f ) = A(D0→ ¯f ) = Afr f De iδf D, (4) 010201-2
where δDf and r f
D are the strong phase difference and relative magnitude, respectively, between the D0→ f and
61
the D0→ f decay amplitudes.
62
The amplitudes of the full decay chains are given by 63 ABf ≡ A(Bs0→ [f ]D(∗)φ) = A(∗)B A(∗)f h1 + r(∗)B rfDei(δ(∗) B +δ f D+γ) i , (5) AB ¯f ≡ A(Bs0→ [ ¯f ]D(∗)φ) = A(∗)B A (∗) f h rB(∗)e i(δ(∗) B +γ)+ rf De iδDfi. (6)
The amplitudes for the CP -conjugate decays are given by changing the sign of the weak phase γ 64 ¯ ABf ≡ A(B0s→ [f ]D(∗)φ) = A(∗)B A (∗) f h r(∗)B e i(δ(∗) B −γ)+ rf De iδDfi, (7) ¯ AB ¯f ≡ A(B0s→ [ ¯f ]D(∗)φ) = A(∗)B A (∗) f h 1 + rB(∗)r f De i(δ(∗)B +δfD−γ)i . (8)
Using the standard notations 65 τ = Γst, Γs= ΓL+ ΓH 2 , ∆Γs= ΓL− ΓH, y =∆Γs 2Γs , λf= q p. ¯ ABf ABf ,
and assuming |q/p| = 1 (|q/p| = 1.0003 ± 0.0014 [26]), the untagged decay rate for the decay Bs0/B0s→ [f ]D(∗)φ
is given by (Eq. (10) of Ref. [27])
dΓ(Bs0(τ ) → [f ]D(∗)φ)
dτ +
dΓ(B0s(τ ) → [f ]D(∗)φ)
dτ ∝
e−τ|ABf|2×(1 + |λf|2) cosh(yτ ) − 2Re(λf) sinh(yτ ) . (9)
2.1 Time acceptance
66
Experimentally, due to trigger and selection requirements and to inefficiencies in the reconstruction, the decay time distribution is affected by acceptance effects. The acceptance correction has been estimated from
pseudoexperiments based on a related publication by the LHCb collaboration [28]. It is described by an
empirical acceptance function
εta(τ ) =
(ατ )β
1 + (ατ )β(1 − ξτ ), (10)
with α = 1.5, β = 2.5 and ξ = 0.01. 67
Taking into account this effect, the time-integrated untagged decay rate is 68 Γ( ˜Bs0→ [f ]D(∗)φ) = Z ∞ 0 dΓ(Bs0(τ ) → [f ]D(∗)φ) dτ +dΓ(B 0 s(τ ) → [f ]D(∗)φ) dτ εta(τ )dτ. (11)
Defining the function
g(x) =
Z ∞
0
e−xτ(1 + ξτ (ατ )β)
1 + (ατ )β dτ, (12)
and using Eq. (9), one gets
where A = 1 − [g(1 − y) + g(1 + y)]/2 and B = 1 − [g(1 − y) − g(1 + y)]/2y. With y = (0.128 ± 0.009)/2 for the B0s 69
meson [21], one gets A = 0.488 ± 0.005 and B = 0.773 ± 0.008. Examples of decay-time acceptance distributions 70
are displayed in Fig. 2. 71
(ps)
τ
0
2
4
6
8
10
arbitrary unit
0
0.2
0.4
0.6
0.8
= 0.01
ξ
= 2.5 &
β
=1.0 &
α
= 0.01
ξ
= 2.5 &
β
=1.5 &
α
= 0.01
ξ
= 2.5 &
β
=2.0 &
α
Figure 2. Examples of decay-time acceptance distributions for three different sets of parameters α, β, and ξ (nominal in green).
72
2.2 Observables for D0 decays
73
The D-meson decays are reconstructed in quasi flavour-specific modes: f−(≡ f ) = K−π+, K−3π, K−π+π0,
74
and their CP -conjugate modes: f+(≡ ¯f ) = K+π−, K+3π, K+π−π0 as well as CP -eigenstate modes: f
CP = 75
K+K−, π+π−.
76
In the following, we introduce the weak phase βs is defined as βs= arg
−VtsVtb∗
VcsVcb∗
. From Eqs. (5), (7), (13)
and with λf= e2iβs
¯
ABf
ABf, for a given number of untagged B
0
s mesons produced in the pp collisions at the LHCb
interaction point, N (Bs0), we can compute the number of Bs0→ D0φ decays with the D meson decaying into
the final state f−. For the reference decay mode f−≡ K−π+ we obtain
NBs0→ [K−π+] D[K +K− ]φ= CKπ h − 2ByrBcos (δB+ 2βs− γ) + A 1 + rB2+ 4rBr Kπ D cos δBcos δ Kπ D + γ i , (14)
where, the terms proportional to (rKπ
D )2 1 and yr
Kπ
D 1 have been neglected (r
Kπ
D = 5.90+0.34−0.25 % [21]). The
best approximation for the scale factor CKπ is
CKπ= N (Bs0) × ε(Bs0→ [K − π+]D[K+K−]φ) × Br(Bs0→ [K − π+]D[K+K−]φ), (15) where, ε(Bs0 → [K−π+] D[K+K −]
φ) is the global detection efficiency of this decay mode, and Br(Bs0 →
77
[K−π+]
D[K+K
−]
φ) its branching fraction. The value of the scale factor CKπ is estimated from the LHCb
78
Run 1 data [18], the average fs/fd of the b-hadron production fraction ratio measured by LHCb [29] and the 79
different branching fractions [26]. 80
For a better numerical behaviour, we use the Cartesian coordinates parametrisation
x(∗)± = rB(∗)cos(δ (∗) B ± γ) and y (∗) ± = rB(∗)sin(δ (∗) B ± γ). (16) Then, Eq (14) becomes NBs0→ [K − π+]D[K+K−]φ= CKπ h
− 2By [x−cos(2βs) − y−sin(2βs)] +
A1 + x2−+ y2−+
2rKπ
D [(x++ x−) cos δKπD − (y+− y−) sin δKπD ]
. (17)
For three and four body final states K3π and Kππ0, there are multiple interfering amplitudes, therefore
their amplitudes and phases δDf vary across the decay phase space. However, an analysis which integrates over
the phase space can be performed in a very similar way to two body decays with the inclusion of an additional
parameter, the so-called coherence factor RfD which has been measured in previous experiments [36]. The
strong phase difference δf
D is then treated as an effective phase averaged over all amplitudes. For these modes,
we have an expression similar to (17)
NBs0→ [f−
]D[K+K−]φ=
CKπFf
h
− 2By [x−cos (2βs) − y−sin (2βs)] +
A1 + x2−+ y−2 + 2rfDR f D(x++ x−) cos δDf − (y+− y−) sin δDf i , (18)
where Ff is the scale factor of the f decay relative to the Kπ decay and depends on the ratios of detection
efficiencies and branching fractions of the corresponding modes
Ff= Cf CKπ = ε(D → f ) ε(D → Kπ)× [Br(D0→ f ) + Br(D0→ f )] [Br(D0→ K−π+) + Br(D0→ K−π+)]. (19)
The value of Fffor the different modes used in this study is determined from LHCb measurements in B
±→ DK±
81
and B±→ Dπ± modes, with two or four-body D decays [22, 23].
82
The time-integrated untagged decay rate for Bs0→ [ ¯f ]Dφ is given by Eq. (13) by substituting ABf→ ¯AB ¯f
and λf→ ¯λf¯= λ
−1
f = e
−2iβs(A
B ¯f/ ¯AB ¯f) which is equivalent to the change βs→ −βs and γ → −γ (i.e. x±→ x∓
and y±→ y∓). Therefore, the observables are
NBs0→ [K+π−]D[K+K−]φ=
CKπ
h
− 2By [x+cos (2βs) + y+sin (2βs)] +
A1 + x2++ y2++
2rKπ
D (x++ x−) cos δDKπ+ (y+− y−) sin δDKπ
i
and for the modes f+≡ K+3π, K+π−π0 NBs0→ [f+]D[K+K−] φ = CKπFf h
− 2By [x+cos (2βs) + y+sin (2βs)] +
A1 + x2++ y2++ 2rDfRf(x++ x−) cos δ f D+ (y+− y−) sin δ f D i . (21) Obviously, any significant asymmetries on the yield of observable corresponding to Eq. 17 with respect to 83
Eq. 20, or Eq. 18 with respect to Eq. 21, is a clear signature for CP violation. 84
For the CP -eigenstate modes D → h+h−(h ≡ K, π), we have r
D = 1 and δD = 0. Following the same
approach than for quasi flavour-specific modes, the observables can be written as
NBs0→ [h+h − ]D[K+K−]φ= 4CKπFhh h A 1 + x2++ y+2+ x++ x− − By(1 + x++ x−+ x+x−+ y+y−) cos (2βs) + (y+− y−+ y+x−− x+y−) sin (2βs) i . (22)
In analogy with Ff, Fhh is defined as
Fhh= Chh CKπ = ε(D → hh) ε(D → Kπ)× Br(D0→ hh) [Br(D0→ K−π+) + Br(D0→ K−π+)] (23)
and their values are determined in the same way than Ff.
85
For the modes KS0π+π− and K0
SK +K− (i.e. K0 Shh) one obtains NBs0→ [KS0hh]D[K+K−] φ = 2CKπFK0 Shh× h − By [(x++ x−) cos(2βs) + (y+− y−) sin(2βs)] + A1 + x2−+ y−2+ 2(x++ x−) rKS0hh D (m2+, m2−)κ K0 Shh D (m2+, m2−) cos δ K0 Shh D (m2+, m2−) i , (24) where FK0
Shhis defined as for Eq. 23. The strong parameters r
K0 Shh D (m2+, m2−), κ K0 Shh D (m2+, m2−), and cos δ K0 Shh D (m2+, m2−) 86
vary over the Dalitz plot (m2+, m2−) ≡ (m2(KS0π
+), m2(K0
Sπ
−)) and are defined in Sec. 3.
87
2.3 Observables for D∗0 decays
88
For the D∗0 decays, we considered the two modes: D∗0→ D0π0 and D∗0→ D0γ, where the D0 mesons are
reconstructed, as in the above, in quasi flavour-specific modes: Kπ, K3π, Kππ0 and CP -eigenstate modes:
ππ and KK . As shown in Ref. [30], the formalism for the cascade B0s→ D∗0φ, D∗0→ D0π0 is similar to the
Bs0→ D0φ. Therefore, the relevant observables can be written similarly to Eqs. (17), (18), (20), (21) and (22),
by substituting CKπ→ CKπ,Dπ0, rB→ rB∗ and δB→ δ∗B (i.e. x±→ x∗±and y±→ y∗±)
NBs0→ [[K−π+] Dπ 0] D∗[K+K −] φ = CKπ,Dπ0 h − 2Byx∗ −cos (2βs) − y ∗ −sin (2βs) + A1 + x∗−2+ y ∗2 −+ 2rKπ D x∗++ x∗− cos δ Kπ D − y ∗ +− y−∗ sin δ Kπ D i , (25) 010201-6
NBs0→ [[K+π−]Dπ0]D∗[K+K − ]φ= CKπ,Dπ0 h − 2Byx∗ +cos (2βs) + y ∗ +sin (2βs) + A1 + x∗+2+ y+∗2+ 2rKπ D x∗ ++ x∗− cos δ Kπ D + y ∗ +− y−∗ sin δ Kπ D i , (26) NBs0→ [[f−] Dπ 0] D∗[K+K −] φ = CKπ,Dπ0Ff h − 2Byx∗ −cos (2βs) − y ∗ −sin (2βs) + A1 + x∗−2+ y ∗2 −+ 2rDfRf x∗++ x∗− cos δ f D− y ∗ +− y−∗ sin δ f D i , (27) NBs0→ [[f+]Dπ0]D∗[K+K − ]φ= CKπ,Dπ0Ff h − 2Byx∗ +cos (2βs) + y ∗ +sin (2βs) + A1 + x∗2 + + y+∗2+ 2rDfRf x∗ ++ x∗− cos δ f D+ y ∗ +− y−∗ sin δ f D i , (28) NBs0→ [[h+h−]Dπ0]D∗[K+K − ]φ= 4CKπ,Dπ0Fhh h A 1 + x∗2 + + y∗+2+ x∗++ x∗− − By 1 + x∗++ x∗−+ x ∗ +x∗−+ y ∗ +y∗− cos (2βs) + y∗+− y∗ −+ y ∗ +x∗−− x ∗ +y∗− sin (2βs) i . (29)
In the case D∗0→ D0γ, the formalism is very similar, except that there is an effective strong phase shift of
π with respect to the D∗0→ D0π0 [30]. The observables can be derived from the previous ones substituting
CKπ,Dπ0→ CKπ,Dγ and δB∗→ δB∗+ π (i.e. x∗±→ −x∗±and y±∗→ −y∗±)
NBs0→ [[K−π+] Dγ]D∗[K+K −] φ = CKπ,Dγ h 2Byx∗ −cos (2βs) − y ∗ −sin (2βs) + A1 + x∗2 −+ y ∗2 −+ 2rKπ D h − x∗ ++ x ∗ − cos δ Kπ D + y+∗− y∗ − sin δKπ D i i , (30) NBs0→ [[K+π−]Dγ]D∗[K+K − ]φ= CKπ,Dγ h 2Byx∗+cos (2βs) + y ∗ +sin (2βs) + A1 + x∗+2+ y+∗2+ 2rKπ D h − x∗ ++ x∗− cos δ Kπ D − y∗ +− y∗− sin δKπ D i i , (31)
Table 1. Integrated luminosities and cross-sections of LHCb Run 1 and Run 2 data. The integrated lumi-nosities come from Ref. [31] and cross-sections from Refs. [32, 33]
Years/Run √s (TeV) int. lum.( fb−1) cross section equiv. 7 TeV data
2011 7 1.1 σ2011= 38.9 µb 1.1 2012 8 2.1 1.17 × σ2011 2.4 Run 1 – 3.2 – 3.5 2015-2018 (Run 2) 13 5.9 2.00 × σ2011 11.8 Total – 9.1 – 15.3 NBs0→ [[f − ]Dγ]D∗[K+K − ]φ= CKπ,DγFf h 2Byx∗−cos (2βs) − y ∗ −sin (2βs) + A1 + x∗2 −+ y ∗2 −+ 2rDfRf h − x∗ ++ x∗− cos δ f D+ y∗ +− y−∗ sin δDfi i, (32) NBs0→ [[f+]Dγ]D∗[K+K − ]φ= CKπ,DγFf h 2Byx∗ +cos (2βs) + y ∗ +sin (2βs) + A1 + x∗+2+ y+∗2+ 2rfDRf− x ∗ ++ x∗− cos δ f D− y ∗ +− y−∗ sin δ f D i , (33) NBs0→ [[h+h−] Dγ]D∗[K+K −] φ = 4CKπ,DγFhh h A 1 + x∗2 + + y∗+2− x∗+− x∗− − By 1 − x∗ +− x∗−+ x ∗ +x∗−+ y ∗ +y∗− cos (2βs) + −y∗ ++ y−∗+ y ∗ +x∗−− x ∗ +y∗− sin (2βs) i . (34)
CKπ,Dπ0 and CKπ,Dγ are determined in the same way CKπ, i.e. from the LHCb Run 1 data [18] and taking 89
into account the fraction of longitudinal polarization in the decay Bs0→ D∗0φ: f
L= (73±15±4)% [18] and the
90
branching fractions Br(D∗0→ D0π0) and Br(D∗0→ D0γ) [26].
91
3
Expected yields
92
The LHCb collaboration has measured the yields of Bs0 → ˜D(∗)0φ, ˜D0 → Kπ modes using Run 1 data,
93
corresponding to an integrated luminosity of 3 fb−1 (Ref. [18]). Taking into account cross-section differences
94
among different centre-of-mass energies, the equivalent integrated luminosities in different data taking years at 95
LHCb are summarized in Table 1. The corresponding expected yields of ˜D0 meson decaying into other modes
96
are also estimated according to Ref. [22], [23], and [34], the scaled results are listed in Table 2, where the 97
longitudinal polarisation fraction fL= (73±15±4) % [18] of Bs0→ ˜D
∗0φ is considered so that the CP eigenvalue
98
of the final state is well defined and similar to that of the Bs0→ ˜D0φ mode.
99
There are some extra parameters used in the sensitivity study, as shown in Table 3. Most of which come 100
from D decays, and the scale factors F are calculated by using the data from Ref. [22] and [23], and branching 101
fractions from PDG [26]. 102
Table 2. Expected yield of each mode for 9.1 fb−1 (Run 1 and Run 2 data). The expected yields for the Bs0→ ˜D
∗0
φ sub-modes are scaled by the longitudinal fraction of polarisation fL= (73 ± 15) %. To be scaled
by 6.3 (90) for prospects after 2025 (2038)(see Section 5).
Expect. yield (Run 1 only)
B0s→ ˜D0(Kπ)φ 577 (132 ± 13 [18]) B0s→ ˜D0(K3π)φ 218 B0s→ ˜D0(Kππ0)φ 58 B0s→ ˜D0(KK)φ 82 B0s→ ˜D0(ππ)φ 24 B0s→ ˜D0(KS0ππ)φ 54 B0s→ ˜D0(KS0KK)φ 8 B0s→ ˜D∗0φ mode D0π0 D0γ B0s→ ˜D∗0(Kπ)φ 337 184 (119 [18]) B0s→ ˜D ∗0(K3π)φ 127 69 B0s→ ˜D∗0(Kππ0)φ 34 18 B0s→ ˜D∗0(KK)φ 48 26 B0s→ ˜D ∗0(ππ)φ 14 8 103
The expected numbers of signal events are also calculated from the full expressions given in Sections 2.2 104
and 2.3, by using detailed branching fraction derivations explained in Ref. [18] and scaling by the LHCb Run 105
1 and Run 2 integrated luminosities as listed in Table 1. The obtained normalisation factors CKπ, CKπ,Dπ0,
106
and CKπ,Dγ are respectively 608±67, 347±56, and 189±31. To compute the uncertainty on the normalisation
107
factors, we made the assumption that it is possible to improve by a factor 2 the global uncertainty on the 108
measurement of the branching fraction of the decay modes Bs0→ D(∗)0φ, and of the polarisation of the mode
109
Bs0→ D∗0φ, when adding LHCb data from Run 2 [18]. The values of the three normalisation factors are in
110
good agreement with the yields listed in Table 2. 111
Table 3. Other external parameters used in the sensitivity study. The scale factors F are also listed.
Parameter Value -2βS[mrad] −36.86 ± 0.82 [35] y = ∆Γs/2Γs (%) 6.40 ± 0.45 [21] rKπ D (%) 5.90+0.34−0.25 [21] δKπ D [deg] 188.9+8.2−8.9 [21] rK3π D (%) 5.49 ± 0.06 [36] RK3π D (%) 43+17−13 [36] δK3π D [deg] 128+28−17 [36] rKππ0 D (%) 4.47 ± 0.12 [36] RKππ0 D (%) 81 ± 6 [36] δKππ0 D [deg] 198+14−15 [36]
Scale factor (wrt Kπ) (stat. uncertainty only)
FK3π (%) 37.8 ± 0.1 [22]
FKππ0 (%) 10.0 ± 0.1 [23]
FKK (%) 14.2 ± 0.1 [22]
Fππ (%) 4.2 ± 0.1 [22]
The number of expected event yields and the value of the coherence factor, RD, listed in Table 2 and 3 113
justify a posteriori our choice of performing the sensitivity study on γ with the D-meson decay modes Kπ, K3π, 114
Kππ0, KK and ππ. By definition the value of RD is one for two-body decays and rD= 1 for CP -eigenstates,
115
while for K3π RD is about 43% and larger, 81%, for Kππ0. The larger is RD the strongest is the sensitivity to
116
γ. As from Eqs. 20, 21, and 22 it is clear that the largest sensitivity to γ is expected to be originated from the
117
ordered D0 decay modes: KK, ππ, Kπ, Kππ0, and K3π, for the same number of selected events. Therefore,
118
even with lower yields the modes Kππ0 and ππ should be of interest; it is discussed in Section 4.9.
119
Coming back to the modes D0 → KS0ππ and KS0KK, the scale factors are FK0
Sππ = (9.3 ± 0.1)% and
FK0
SKK= (1.4±0.1)% [34]. The strong parameters r
KS0hh D (m2+, m2−), κ KS0hh D (m2+, m2−), and cos δ KS0hh D (m2+, m2−) can
be defined following the effective method presented in Ref. [37], while using quantum-correlated ˜D0 decays and
where the phase space (m2+, m2−) is split in N tailored regions or “bins” [38], such that in bin of index i
p Ki/K−i= r KS0hh D,i , ci= κ K0Shh D,i cos δ K0Shh D,i , and si= κ K0Shh D,i sin δ KS0hh D,i , where, δK0Shh
D,i is the strong phase difference and κ
KS0hh
D,i is the coherence factor. There is a recent publication by
120
the BES-III collaboration [39] that combines its data together with the results of CLEO-c [38], while applying 121
the same technique to obtain the value of the ci, si, and K±i parameters varying of the phase space. The
122
binning schemes are symmetric with respect to the diagonal in the Dalitz plot (m2+, m2−) (i.e. ±i). Those
123
results are also compared to an amplitude model from the B-factories BaBar and Belle [40]. When porting 124
result between the BES-III/CLEO-c combination, obtained with quantum correlated ˜D0decays, and LHCb for
125
Bs0→ ˜D(∗)0φ measurements, one needs to be careful about the bin conventions so there might be a minus sign in 126
phase (which only affects si). The expected yield listed in Table 2 for the mode D0→ KS0ππ is 54 events. It is
127
8 events for the decay D0→ K0
SKK. Though the binning scheme that latter case is only 2×2, it has definitely 128
a too small expected yield to be further considered. For D0 → KS0ππ, the binned method of Refs. [38, 39]
129
supposes to split the selected ˜D0 events over 2 × 8 bins such that with Run 1 and Run 2 only about 3 events
130
only may populate each bin. That is the reason why, though the related observable is presented in Eq. 24, 131
we decided to not include that mode in the sensitivity study. This choice could eventually be revisited after 132
Run 3, when about 340 Bs0→ D0(KS0ππ)φ events should be available, and then about 20 events may populate
133
each bin. 134
4
Sensitivity study for Run 1 & 2 LHCb dataset
135The sensitivity study consists in testing and measuring the value of the unfolded γ, r(∗)B , and δ
(∗)
B parameters
136
and their expected resolution, after having computed the values of the observables according to various initial 137
configurations and given external inputs for the other involved physics parameters or associated experimental 138
observables. To do this, a procedure involving global χ2 fit based on the CKMfitter package [41] has been
139
established to generate pseudoexperiments and fit samples of Bs0→ ˜D(∗)0φ events.
140
This Section is organized as follow. In Subsection 4.1 we explain the various configurations that we tested for 141
the nuisance strong parameters rB(∗), and δB(∗), as well as the value of γ. Then, in Subsection 4.2 we explain how
142
the pseudoexperiments have been generated. In Subsection 4.3 the first step of the method is illustrated with 143
one- and two-dimension p-value profiles for the γ, r(∗)B , and δ
(∗)
B parameters. Before showing how the γ, r
(∗)
B , and
144
δ(∗)B parameters are unfolded from the generated pseudoexperiments in Subsection 4.6, we discuss the stability of
145
the former one-dimension p-value profile for γ against changing the time acceptance parameters (Subsection 4.4) 146
and for a newly available binning scheme for the D → K3π decay (Subsection 4.5). Then, unfolded values for 147
γ and precisions, i.e. sensitivity for Run 1 & 2 LHCb dataset, for the various generated configurations of δ(∗)B 148
and r(∗)B are presented in Subsection 4.7. We finally conclude this Section with Subsections 4.8 and 4.9, in
149
which we study the intriguing case where γ = 74◦(see LHCb 2018 combination [8], recently superseded by [12])
150
and we test the effect of dropping or not the least abundant expected decays modes Bs0 → ˜D(∗)0(ππ)φ and
151
Bs0→ ˜D(∗)0(Kππ0)φ in Run 1 & 2 LHCb dataset.
152
4.1 The various configurations of the γ, rB(∗), and δ
(∗)
B parameters
153
The sensitivity study was performed with the CKM angle γ true value set to be (65.66+0.90−2.65)◦ (i.e. 1.146
154
rad) as obtained by the CKMfitter group, while excluding any measured values of γ in its global fit [35]. As 155
a reminder, the average of the LHCb measurements is γ = (74.0+5.0−5.8)◦ [8], therefore, the value γ = 74◦, is also
156
tested (see Sec. 4.8). 157
The value of the strong phases δB(∗) is a nuisance parameter that cannot be predicted or guessed by any
158
argument, and therefore, six different values are assigned to it: 0, 1, 2, 3, 4, 5 rad (0◦, 57.3◦, 114.6◦, 171.9◦,
159
229.2◦, 286.5◦). This corresponds to 36 tested configurations (i.e. 6 × 6).
160
Since both interfering diagrams displayed in Fig. 1 are colour-suppressed, the value of the ratio of the 161
¯b → ¯uc¯s and ¯b → ¯cu¯s tree-level amplitudes, r(∗)
B , is expected to be |VubVcs|/|VcbVus| ∼ 0.4. This assumption
162
is well supported by the study performed with Bs0→ D∓
sK
± decays by the LHCb collaboration , for which a
163
value rB= 0.37+0.10−0.09 has been measured [10]. But, as the decay B0s→ D
∓
sK
±is colour-favoured, it is important
164
to test other values originated from already measured colour-suppressed B-meson decays, as non factorizing 165
final state interactions can modify the decay dynamics [42]. Among them, the decay B0→ DK∗0 plays such
166
a role for which LHCb obtains rB = 0.22+0.17−0.27 [35], confirmed by a more recent and accurate computation:
167
rB = 0.265 ± 0.023 [14]. The value of rB is known to strongly impact the precision on γ measurements as
168
1/rB [43]. Therefore, the two extreme values 0.22 and 0.40 for rB(∗) have been tested for the sensitivity study,
169
while the values for rB and rB∗ are expected to be similar.
170
This leads to a total of 72 tested configurations for the rB(∗), δB, and δ
∗
B parameters (i.e. 2 × 6 × 6).
171
4.2 Generating pseudoexperiments for various configurations of parameters
172
At a first step, different configurations for observables corresponding to Sec. 2.2 and 2.3 are computed. 173
The observables are obtained with the value of the angle γ and of the four nuisance parameters r(∗)B and δ
(∗)
B 174
fixed to various sets of initial true values (see Sec. 4.1), while the external parameters listed in Table 3 and the 175
normalisation factors CKπ, CKπ,Dπ0, and CKπ,Dγ have been left free to vary within their uncertainties. In a
176
second step, for the obtained observables, including their uncertainties that we assume to be their square root, 177
and all the other parameters, except γ, r(∗)B , and δ
(∗)
B , a global χ2 fit is performed to compute the resulting
178
p-value distributions of the γ, rB(∗), and δB(∗) parameters. Then, at a third step, for the obtained observables, 179
including their uncertainties, and all the other parameters, except γ, r(∗)B , and δ(∗)B , 4000 pseudoexperiments
180
are generated according to Eqs. (14)-(34), for the various above tested configurations. And in a fourth step, 181
for each of the generated pseudoexperiment, all the quantities are varied within their uncertainties. Then a 182
global χ2 fit is performed to unfold the value of the parameters γ, r(∗)B0
s, and δ
(∗)
B0
s, for each of the 4000 generated
183
pseudoexperiments. In a fifth step, for the distribution of the 4000 values of fitted γ, r(∗)B , and δ(∗)B , an extended
184
unbinned maximum likelihood fit is performed to compute the most probable value for each of the former five 185
parameters, together with their dispersion. The resulting values are compared to their injected initial true 186
values. The sensitivity to γ, r(∗)B , and δ
(∗)
B is finally deduced and any bias correlation is eventually highlighted
187
and studied. 188
4.3 One- and two-dimension p-value profiles for the γ, r(∗)B , and δ
(∗)
B parameters
189
Figure 3 displays the one-dimension p-value profile of γ, at the step two of the procedure described in 190
Sec. 4.2. The Figure is obtained for an example set of initial parameters: γ = 65.66◦ (1.146 rad), r(∗)
B = 0.4,
191
δB= 3.0 rad, and δ∗B= 2.0 rad. The integrated luminosity assumed here is that of LHCb data collected in Run
192
1 & 2. The corresponding fitted value is γ = 65.7+6.3−33.8
◦
, thus in excellent agreement with the initial tested 193
true value. The Fig. 3 also shows the corresponding distribution obtained from a full frequentist treatment on 194
Monte-Carlo simulation basis [44], where γ = 65.7+6.9−34.9
◦
. This has to be considered as a demonstration that 195
the two estimates on γ are in quite fair agreement at least at the 68.3 % confidence level (CL), such that no 196
obvious under-coverage is experienced with the nominal method, based on the ROOT function TMath::Prob [45]. 197
On the upper part of the distribution the relative under-coverage of the “Prob” method is about 6.3/6.9 ' 91 %. 198
As opposed to the full frequentist treatment on Monte-Carlo simulation basis, the nominal retained method 199
allows performing computations of very large number of pseudoexperiments within a reasonable amount of 200
time and for non-prohibitive CPU resources. For the LHCb Run 1 & 2 dataset, 72 configurations of 4000 201
pseudoexperiments were generated (i.e. 288 000 pseudoexperiments in total). The whole study was repeated 202
another two times for prospective studies with future anticipated LHCb data, such that more than about 864 203
000 pseudoexperiments were generated for this publication (see Sec. 5). In the same Figure, one can also see 204
the effect of modifying the value of rB(∗)from 0.4 to 0.22, for which γ = 65.7+12.0−60.7
◦
, where the upper uncertainty 205
scales roughly as expected as 1/rB(∗) (i.e. 6.3 × 0.4/0.22 = 12.6). Compared to the full frequentist treatment
206
on Monte-Carlo simulation, where γ = 65.7+13.2−∞
◦
, the relative under-coverage of the “Prob” method is about 207
12.0/13.2 ' 91 %. Finally, the p-value profile of γ is also displayed when dropping the information provided 208
by the Bs0 → ˜D
∗0φ mode, and thus keeping only that of the B0
s → ˜D0φ mode. In that case, γ is equal to 209
65.7+6.3(12.0)−∞ ◦
, rB(∗)= 0.4 (0.22), such that the CL interval is noticeably enlarged on the lower side of the γ
210
angle distribution (more details can be found in Sec. 5.3). 211
For the same set of initial parameters (i.e. γ = 65.66◦(1.146 rad), r(∗)
B = 0.4, δB= 3.0 rad, and δB∗= 2.0 rad)
212
and the same projected integrated luminosity, Fig. 4 displays the one-dimension p-value profile of the nuisance 213
parameters rB(∗) and δB(∗). It can be seen that the p-value is maximum at the initial tested value, as expected.
214
Two-dimension p-value profile of the nuisance parameters r(∗)B and δ
(∗)
B as a function of γ are provided in
215
Fig. 5-8. Figures 5 and 6 correspond to two other example configurations γ = 1.146 rad, δB= 1.0 rad, and
216 δ∗ B= 5.0 rad, and r (∗) B = 0.4 and r (∗)
B = 0.22, respectively. Figures 7 and 8 stand for the configurations γ = 1.146
217
rad, δB= 3.0 rad, δ∗B= 2.0 rad, and r
(∗)
B = 0.4 and r
(∗)
B = 0.22, respectively. Those two-dimension views allows
218
to see the correlation between the different parameters. In general large correlations between δ(∗)B and γ are
219
observed. In case of configurations where r(∗)B = 0.4, large fraction of the δ
(∗)
B vs. γ plane can be excluded at 220
95 % CL, while the fraction is significantly reduced for the corresponding r(∗)B = 0.22 configurations. For the
221
δ∗
B vs. γ plane, one can easily see the advantage of our Cartesian coordinates approach (see Sec. 2.2 and 2.3) 222
together with the fact that in the case of the mode D∗0→ D0γ there is an effective strong phase shift of π with
223
respect to the D∗0→ D0π0 [30], such that additional constraints allow to remove fold-ambiguities with respect
224
to the associated δB vs. γ plane.
225
4.4 Effect of the time acceptance parameters
226
Figure 9 shows for a tested configuration γ = 1.146 rad, rB(∗)= 0.4, and δ
(∗)
B = 1.0 rad, that the impact of
227
the time acceptance parameters A and B can eventually be non negligible and has an impact on the profile 228
distribution of the p-value of the global χ2 fit to γ. For the given example the fitted value of γ is either
229
65.3+14.3−38.4 ◦
or 66.5+13.8−51.0 ◦
, when the time acceptance is either or not accounted for. The reason why the 230
precision improves when the time acceptance is taken into account may be not intuitive. This is because 231
for B/A ' 1.6, as opposed to the case B/A ' 1.0, the impact of the first term in Eq. 14, which is directly 232
proportional to cos (δB+ 2βs− γ), is amplified with respect to the second term, for which the sensitivity to γ is
233
more diluted. 234
Table 4. Expected value of γ, as a function of different time acceptance parameters. The second line
corresponds to the nominal values. The nominal set of parameters A and B is written in bold style.
α β ξ A B B/A fitted γ (◦) 1.0 2.5 0.01 0.367 0.671 1.828 66.5+13.8−40.1 1.5 2.5 0.01 0.488 0.773 1.584 65.3+14.3−38.4 2.0 2.5 0.01 0.570 0.851 1.493 65.3+13.2−37.8 1.5 2.0 0.01 0.484 0.751 1.552 65.9+13.2−39.0 1.5 3.0 0.01 0.491 0.789 1.607 66.5+13.2−38.4 1.5 2.5 0.02 0.480 0.755 1.573 66.5+13.8−39.5 1.5 2.5 0.005 0.492 0.783 1.591 65.3+13.8−36.7 235
Even if the parameters A and B are computed to a precision at the percent level (Sec. 2.1), we investigated 236
further the impact of changing their values. Note that for this study, the overall efficiency is kept constant, 237
while the shape of the acceptance function is varied. The values α, β and ξ were changed in Eq. 10, and the 238
results of those changes are listed in Table 4. When α increases, both A and B turn larger, but the value of the 239
ratio B/A decreases. When β or ξ decreases, the 3 values of A, B, and B/A increase. The effect of changing 240
β or ξ alone is small. A modification of α has a much larger impact on A and B. However, all these changes
241
have a weak impact on the precision of the fitted γ value. This is good news as this means that the relative 242
efficiency loss caused by time acceptance effects will not cause much change in the sensitivity to the CKM 243
γ angle. As a result, time acceptance requirements can be varied without much worry to improve the signal
244
purity and statistical significance, when analysing the Bs0→ ˜D(∗)0φ decays with LHCb data.
245
4.5 Effect of a new binning scheme for the D → K3π decay
246
According to Ref. [46], averaged values of the K3π input parameters over the phase space defined as 247 RK3π D e −iδK3π D = R A∗ D0→K3π(x)AD0→K3π(x)dx AD0→K3πAD0→K3π , (35)
(rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0
Full frequent. treat. on MC basis Prob (rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r
Full frequent. treat. on MC basis Prob (rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r only φ D → s B (rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r only φ D → s B
Figure 3. Profile of the p-value of the global χ2 fit to γ after the observables are computed (top left), for the set of true initial parameters: γ = 1.146 rad, r(∗)B = 0.4, δB= 3.0 rad, and δB∗ = 2.0 rad (the corresponding
distribution obtained from a full frequentist treatment on Monte-Carlo simulation basis [44] is superimposed to the same distribution). The related p-value profile for r(∗)B = 0.22 is also presented (top right). The integrated luminosity assumed here is that of LHCb data collected in Run 1 & 2. Profile of the p-value of the global χ2 fit to γ after the observables are computed, where only the decay mode B0s→ ˜D0φ is used
and for the set of true initial parameters: γ = 1.146 rad, r(∗)B = 0.4 (bottom left) and 0.22 (bottom right), δB= 3.0 rad, and δ∗B= 2.0 rad. On each figure the vertical dashed red line indicates the initial γ true value,
and the two horizontal dashed black lines, refer to 68.3 and 95.4 % CL.
B r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r Bs→ D(*)φ B r* 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r Bs→ D(*)φ (rad) B δ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r Bs→ D(*)φ (rad) B * δ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 p-value 0.0 0.2 0.4 0.6 0.8 1.0 LHCb Run 1 & 2 CKM f i t t e r Bs→ D(*)φ
Figure 4. Profile of the p-value distribution of the global χ2fit to rB(∗)(top left (right)) and δ(∗)B (bottom left (right)), after that the observables are computed, for the set of initial true parameters: γ = 65.66◦ (1.146 rad), rB(∗)= 0.4, δB= 3.0 rad, and δ∗B= 2.0 rad. On each figure the vertical dashed red line indicates the
initial r(∗)B and δ(∗)B true values, and the two horizontal dashed black lines, refer to 68.3 and 95.4 % CL.
(deg) γ 0 20 40 60 80 100 120 140 160 180 B r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 B r* 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg)B δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg) B * δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value
Figure 5. Two-dimension p-value profile distribution of the nuisance parameters rB(∗) and δ(∗)B as a function of γ. On each figure the dashed black lines indicate the initial true values: γ = 65.66◦(1.146 rad), δB= 57.3◦
(1.0 rad), and δ∗B= 286.5 ◦ (5.0 rad), and r(∗)B = 0.4. (deg) γ 0 20 40 60 80 100 120 140 160 180 B r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 B r* 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg)B δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg) B * δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value
Figure 6. Two-dimension p-value profile distribution of the nuisance parameters rB(∗) and δ
(∗)
B as a function
of γ. On each figure the dashed black lines indicate the initial true values: γ = 65.66◦(1.146 rad), δB= 57.3◦
(1.0 rad), and δ∗B= 286.5
◦
are used here and corresponds to relatively limited value for the coherence factor: RK3π
D = (43+17−13)% [36].
248
A more attractive approach could be to perform the analysis in disjoint bins of the phase space. In this case, 249
the parameters are re-defined within each bin. New values for RK3π
D and δ
K3π
D in each bins from Ref. [46]
250
have alternatively been employed. No noticeable change on γ and rB(∗) fitted p-value profiles were seen, but
251
it is possible that some fold-effects on δB(∗), as seen e.g. in Figs. 7-5, become less probable. The lack of
252
significant improvement is expected, as the ˜D0→ K3π mode is not the dominant decay and also because the
253
new measurements of RK3π
D and δ
K3π
D in each bin still have large uncertainties.
254 (deg) γ 0 20 40 60 80 100 120 140 160 180 B r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 B r* 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg)B δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg) B * δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value
Figure 7. Two-dimension p-value profile distribution of the nuisance parameters rB(∗)and δ(∗)B as a function of γ. On each figure the dashed black lines indicate the initial true values: γ = 65.66◦(1.146 rad), δB= 171.9◦
(3.0 rad), and δ∗B= 114.6 ◦ (2.0 rad), and r(∗)B = 0.4. (deg) γ 0 20 40 60 80 100 120 140 160 180 B r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 B r* 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg)B δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value (deg) γ 0 20 40 60 80 100 120 140 160 180 (deg) B * δ 0 50 100 150 200 250 300 350 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 p-value
Figure 8. Two-dimension p-value profile distribution of the nuisance parameters rB(∗) and δ(∗)B as a function of γ. On each figure the dashed black lines indicate the true values: γ = 65.66◦(1.146 rad), δB= 171.9◦(3.0
rad), and δB∗= 114.6
◦
(2.0 rad), and rB(∗)= 0.22.
4.6 Unfolding the γ, rB(∗), and δ
(∗)
B parameters from the generated pseudoexperiments
255
As explained in Sec. 4.2, for each of the tested γ, r(∗)B , and δ(∗)B configurations, 4000 pseudoexperiments
256
are generated, which values of γ, r(∗)B , and δB(∗) are unfolded from global χ2 fits (See Sec. 4.3 for illustrations).
257
(rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0 Run 1-2 No time acceptance (rad) γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p-value 0.0 0.2 0.4 0.6 0.8 1.0 Run 1-2
With time acceptance nominal A/B parameters
Figure 9. Profile of the p-value distribution of the global χ2 fit to γ, for the set of true initial parameters: γ = 1.146 rad, r(∗)B = 0.4, and δB(∗)= 1.0 rad. The integrated luminosity assumed here is that of LHCb data collected in Run 1 & 2, when the time acceptance A and B are set to 1 in Eqs. (14)-(34): no time acceptance (top left) or to their nominal values A = 0.488 ± 0.005 and B = 0.773 ± 0.008 (top right), as computed in Sec. 2.1. The dashed red line shows the initial γ true value γ = 65.66◦(1.146 rad).
Figure 10 displays the extended unbinned maximum likelihood fits to the nuisance parameters r(∗)B and δ
(∗)
B . 258
The initial configuration is γ = 65.66◦ (1.146 rad), r(∗)
B = 0.4, δB = 171.9
◦ (3 rad), and δ∗
B = 114.6
◦ (2 rad)
259
and an integrated luminosity equivalent of LHCb Run 1 & 2 data. It can be compared with Fig. 4. All the 260
distributions are fitted with the Novosibirsk empirical function, whose description contains a Gaussian core 261
part and a left or right tail, depending on the sign of the tail parameter [47]. The fitted values of r(∗)B are
262
centered at their initial tested values 0.4, with a resolution of 0.14, and no bias is observed. For δ(∗)B , the fitted
263
value is (176 ± 42)◦ ((104 ± 13)◦) for an initial true value equal to 171.9◦ (114.6◦). The fitted value for δ∗
B is 264
slightly shifted by about 2/3 of a standard deviation, but its measurement is much more precise than that of 265
δB, as it is measured both from the D∗0→ D0γ and the D∗0→ D0π0 observables.
266
Figure 11 shows the corresponding fit to the CKM angle γ, where the value rB(∗) = 0.22 is also tested.
267
This Figure can be compared to the initial p-value profiles shown in Fig. 3. As shown in Figs. 7 and 8, γ is 268
correlated with the nuisance parameters r(∗)B and δ
(∗)
B , such correlations may generate long tails in its distribution
269
as obtained from 4000 pseudoexperiments. To account for those tails, extended unbinned maximum likelihood 270
fits, constituted of two Novosibirsk functions, with opposite-side tails, are performed to the γ distributions. 271
With an initial value of 65.66◦, the fitted value for γ returns a central value equal to µ
γ= (65.9 ± 0.3)◦, with
272
a resolution of σγ = (8.8 ± 0.2)◦, when r(∗)B = 0.4 and respectively, µγ = (66.6 ± 0.7)◦, with a resolution of
273
σγ = (14.4 ± 0.5)
◦, when r(∗)
B = 0.22. The worse resolution obtained with r
(∗)
B = 0.22 follows the empirical
274
behaviour 1/r(∗)B (i.e. 8.8 × 0.4/0.22 ' 16.0). There again, no bias is observed.
275
Finally, Fig. 12 displays the two-dimension distributions of the nuisance parameters rB(∗) and δ
(∗)
B as a 276
function of γ obtained from 4000 pseudoexperiments. It can be compared with the corresponding p-value 277
profiles shown in Fig. 7. 278
4.7 Varying δB(∗) and r
(∗)
B 279
According to Sec. 4.1, 72 configurations of nuisance parameters δB(∗)and r
(∗)
B have been tested for γ = 65.66
◦
280
(1.146 rad) and 4000 pseudoexperiments have been generated for each set, according to the procedure described 281
in Sec. 4.2 and illustrated in Sec. 4.6. The integrated luminosity assumed in this Section is that of LHCb data 282
collected in Run 1 & 2. 283
The fitted mean value of γ (µγ), for r(∗)B = 0.4 and 0.22, as a function of δ
(∗)
B , for an initial true value of
284
65.66◦ (1.146 rad) are given in Table 5, while the corresponding resolutions (σ
γ) are listed in Table 6. The
285
fitted means are in general compatible with the true γ value within less than one standard deviation. For 286
r(∗)B = 0.4, the resolution varies from σγ= 8.3◦ to 12.9◦. For rB(∗)= 0.22, the resolution is worse, as expected,
287
it varies from σγ= 13.9◦to 18.7◦. For r(∗)B = 0.22, the distribution of γ of the 4000 pseudoexperiments has its
288
maximum above 90◦for δ(∗)
B = 286.5
◦ and is therefore not considered.
289
The obtained values for µγ and σγ are also displayed in Figs. 13 and 14. It is clear that the resolution on γ
290
depends to first order on r(∗)B , then to the second order on δ
(∗)
B . The best agreement with respect to the tested
291
initial true value of γ is obtained when δB(∗)= 0
◦ (0 rad) or 180◦ (π rad), and, there also, the best resolutions
292
are obtained (i.e. the lowest values of σγ). The largest CP violation effects and the best sensitivity to γ are
![Table 5. Fitted mean value of γ (µ γ ) (in [deg]), for r ( B ∗ ) = 0.4 (top) and 0.22 (bottom), as a function of δ B ( ∗ ) , for an initial true value of 65.66 ◦ .](https://thumb-eu.123doks.com/thumbv2/123doknet/14006957.456281/20.918.150.712.239.639/table-fitted-mean-value-function-initial-true-value.webp)
