Table Of ContentDetermination of ∆Γ from Analysis
s
of Untagged Decays B0 J/ψ φ
s
3 →
0 by Using the Method of Angular Moments
0
2
n
a A. Bel’kov1 , S. Shulga2
∗ †
J
5
1 Particle Physics Laboratory, Joint Institute for Nuclear Research,
1
141980 Dubna, Moscow region, Russia
1
v
2 Francisk Skarina Gomel State University, Belarus
5
0
1
1
0
3 Abstract
0
/
h The performance of the method of angular moments on the ∆Γ determination
s
p from analysis of untagged decays B0(t),B0(t) J/ψ( l+l−)φ( K+K−) is
- s s → → →
p
examined. The results of Monte Carlo studies with evaluation of measurement
e
h errors are presented. The method of angular moments gives stable results for the
:
v estimate of ∆Γs and is found to bean efficient and flexible tool for the quantitative
Xi investigation of the Bs0 → J/ψφ decay. The statistical error of the ratio ∆Γs/Γs
r for values of this ratio in the interval [0.03, 0.3] was found to be independent on
a
this value, being 0.015 for 105 events.
1 Introduction
The study of decays B0(t),B0(t) J/ψ( l+l−)φ( K+K−), which is one of the gold
s s → → →
plated channels for B-physics studies at the LHC, looks very interesting from the physics
point of view. It presents several advantages related to the dynamics of these decays,
characterized by proper-time-dependent angular distributions, which can be described
in terms of bilinear combinations of transversity amplitudes. Their time evolution in-
volves, besides the values of two transversity amplitudes at the proper time t = 0 and
their relative strong phases, the following fundamental parameters: the difference and
average value of decay rates of heavy and light mass eigenstates of B0 meson, ∆Γ and
s s
∗E-mail: [email protected]
†E-mail: [email protected]
1
Γ , respectively, their mass difference ∆M , and the CP-violating weak phase φ(s). The
s s c
angular analysis of the decays B0(t),B0(t) J/ψ( l+l−)φ( K+K−) provides com-
s s → → →
plete determination of the transversity amplitudes and, in principle, gives the access to
all these parameters.
In the present paper we examine the performance of the angular-moments method
[1] applied to the angular analysis of untagged decays B0(t),B0(t) J/ψ( l+l−)φ(
s s → → →
K+K−)forthedeterminationof∆Γ . AftergivingthephysicsmotivationinSection2,we
s
describe in the next section the method of angular moments based on weighting functions
introducedinRef.[1]. Forthecaseof∆Γ determinationthismethodisproperlymodified
s
in Section 4. The SIMUB-package [2] for physics simulation of B-meson production and
decays has been used for Monte Carlo studies with two sets of weighting functions. In
Section 5 we present the results of these studies and concentrate on the evaluation of
measurement errors and their dependence on statistics.
2 Phenomenological description of the decays
B0(t),B0(t) J/ψ( l+l )φ( K+K )
s s − −
→ → →
The angular distributions for decays B0(t),B0(t) J/ψ( l+l−)φ( K+K−) are gov-
s s → → →
erned by spin-angular correlations (see [3]-[6]) and involve three physically determined
angles. In case of the so-called helicity frame [5], which is used in the present paper,
these angles are defined as follows (see Fig. 1):
The z-axis is defined to be the direction of φ-particle in the rest frame of the B0.
• s
The x-axis is defined as any arbitrary fixed direction in the plane normal to the
z-axis. The y-axis is then fixed uniquely via y = z x (right-handed coordinate
×
system).
The angles (Θ , χ ) specify the direction of the l+ in the J/ψ rest frame while
l+ l+
•
(Θ , χ ) give the direction of K+ in the φ rest frame. Since the orientation of
K+ K+
the x-axis is a matter of convention, only the difference χ = χ χ of the two
l+ K+
−
azimuthal angles is physically meaningful.
In the most general form the angular distribution for the decay B0(t) J/ψ(
s → →
l+l−)φ( K+K−) in case of a tagged B0 sample can be expressed as
→ s
d4Ntag(B0) 9 6
s = (t)g (Θ ,Θ ,χ). (1)
i i l+ K+
dcosΘ dcosΘ dχdt 32π O
l+ K+ i=1
X
Here (i = 1,...,6) are time-dependent bilinear combinations of the transversity ampli-
i
O
tudes A (t), A (t) and A (t) for the weak transition B0(t) J/ψφ [7] (we treat these
0 || ⊥ s →
combinations as observables):
= A (t) 2, = A (t) 2, = A (t) 2,
1 0 2 || 3 ⊥
O | | O | | O | |
= Im A∗(t)A (t) , = Re A∗(t)A (t) , = Im A∗(t)A (t) , (2)
O4 || ⊥ O5 0 || O6 0 ⊥
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
2
n
x
χ + p*
K K+
χ J/ψ ϕ Θ +
l+ K
χ Θ B0 n
l+ s z
p*
n l+ p*-
y K
Figure 1: Definition of physical angles for description of decays B0(t),B0(t) J/ψ(
s s → →
l+l−)φ( K+K−) in the helicity frame
→
and the g are functions of the angles Θ , Θ , χ only [5]:
i l+ K+
g = 2cos2Θ sin2Θ ,
1 K+ l+
g = sin2Θ (1 sin2Θ cos2χ),
2 K+ l+
−
g = sin2Θ (1 sin2Θ sin2χ),
3 K+ l+
−
g = sin2Θ sin2Θ sin2χ,
4 K+ l+
−
1
g = sin2Θ sin2Θ cosχ,
5 √2 l+ K+
1
g = sin2Θ sin2Θ sinχ. (3)
6 √2 l+ K+
For the decay B0(t) J/ψ( l+l−)φ( K+K−) in case of a tagged B0 sample the
s → → → s
angular distribution is given by
d4Ntag(B0) 9 6
s = (t)g (Θ ,Θ ,χ) (4)
i i l+ K+
dcosΘ dcosΘ dχdt 32π O
l+ K+ i=1
X
with the same angular functions g and
i
= A¯(t) 2, = A¯ (t) 2, = A¯ (t) 2,
1 2 || 3 ⊥
O | | O | | O | |
= Im A¯∗(t)A¯ (t) , = Re A¯∗(t)A¯ (t)), = Im A¯∗(t)A¯ (t) , (5)
O4 || ⊥ O5 0 || O6 0 ⊥
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)
where A¯ (t), A¯ (t) and A¯ (t) are the transversity amplitudes for the transition B0(t)
0 || ⊥ s →
J/ψφ.
3
Thetimedependence ofthetransversity amplitudesforthetransitionsB0(t),B0(t)
s s →
J/ψφ is not of purely exponential form due to the presence of B0 B0 mixing. This
s − s
mixing arises due to either a mass difference or a decay-width difference between the
mass eigenstates of the (B0 B0) system. The time evolution of the state B0(t) of an
s − s | s i
initially, i.e. at time t = 0, present B0 meson can be described in general form as follows:
s
B0(t) = g (t) B0 +g (t) B0 , g (t = 0) = 1, g (t = 0) = 0,
| s i + | si − | si + −
i.e., the state B0(t) at time t is a mixture of the flavor states B0 and B0 with prob-
| s i | si | si
abilities defined by the functions g (t) and g (t). In analogous way, the time evolution
+ −
0 0
of the state B (t) of an initially present B meson is described by the relation
| s i s
B0(t) = g¯ (t) B0 +g¯ (t) B0 , g¯ (t = 0) = 0, g¯ (t = 0) = 1.
| s i + | si − | si + −
Diagonalization of the full Hamiltonian (see [8] for more details) gives
1 α
g (t) = e−iµLt +e−iµHt , g (t) = e−iµLt e−iµHt ,
+ −
2 2 −
(cid:16) (cid:17) (cid:16) (cid:17)
g¯ (t) = g (t)/α2, g¯ (t) = g (t). (6)
+ − − +
Here µ M (i/2)Γ are eigenvalues of the full Hamiltonian corresponding
L/H L/H L/H
≡ −
to the masses and total widths of “light” and “heavy” eigenstates B , and α is
L/H
| i
a phase factor defining the CP transformation of flavor eigenstates of the neutral B -
s
meson system: CP B0 = α B0 . In the case α = 1 the probability for B0 to oscillate
| si | si | | 6 s
to a B0 is not equal to the probability of a B0 to oscillate to a B0. Such an asymmetry
s s s
in mixing is often referred to as indirect CP violation, which is negligibly small in case
of the neutral B-meson system.
The time evolution of the transversity amplitudes A (t) (f = 0, , ) is given by the
f
|| ⊥
equations
1 1
A (t) = A (0) g (t)+g (t) ξ(s) , A¯ (t) = A (0) g¯ (t)+g¯ (t) ξ(s) . (7)
f f + − ηf α f f f + − ηf α f
(cid:20) CP (cid:21) (cid:20) CP (cid:21)
Here ηf are eigenvalues of CP-operator acting on the transversity components of the
CP
final state which are eigenstates of CP-operator:
CP J/ψφ = ηf J/ψφ , (f = 0, , ),
| if CP| if || ⊥
η0 = 1, η|| = 1, η⊥ = 1,
CP CP CP −
(s)
and ξ is the CP-violating weak phase [9]:
f
ξ(s) = e−iφc(s) , φ(s) = 2[arg(V∗V ) arg(V∗V )] = 2δγ,
f c ts tb − cq cb −
where δ is the complex phase in the standard parameterization of the CKM matrix
elements V (i u,c,t , j d,s,b ), and γ is the third angle of the unitarity triangle.
ij
∈ { } ∈ { }
4
The phase φ(s) is very small and vanishes at leading order in the Wolfenstein expan-
c
sion. Taking into account higher-order terms in the Wolfenstein parameter λ = sinθ =
C
0.22 gives a non-vanishing result [10]:
φ(s) = 2λ2η = 2λ2R sinγ.
c − − b
Here
1 V
ub
R | |
b
≡ λ V
cb
| |
is constrained by present experimental data as R = 0.36 0.08 [11]. Using the experi-
b
±
mental estimate γ = (59 13)o [12], the following constrain can be obtained for the phase
±
φ(s):
c
φ(s) = 0.03 0.01. (8)
c − ±
According to Eq. (7) at time t = 0, the transversity amplitudes of B0,B0 J/ψφ
s s →
decays depend on the same observables A (0) , A (0) , A (0) and on the two CP-
0 || ⊥
| | | | | |
conserving strong phases, δ arg[A∗(0)A (0)] and δ arg[A∗(0)A (0)]. Time-
1 ≡ || ⊥ 2 ≡ 0 ⊥
reversal invariance of strong interactions forces the form factors parameterizing quark
currents to be all relatively real and, consequently, naive factorization leads to the fol-
lowing common properties of the observables:
Im[A∗(0)A (0)] = 0, Im[A∗(0)A (0)] = 0, Re[A∗(0)A (0)] = A (0)A (0) .
0 ⊥ || ⊥ 0 || ±| 0 || |
Moreover, in the absence of strong final-state interactions, δ = π and δ = 0.
1 2
In the framework of the effective Hamiltonian approach the two body decays, both
B0 J/ψφ and B0 J/ψK⋆, correspond to the transitions ¯b s¯c¯c with topologies
s → d → →
of color-suppressed spectator diagrams shown in Fig. 2. Factorizing the hadronic matrix
elements of the four-quark operators of the effective Hamiltonian into hadronic matrix
elements of quark currents, the transversity amplitudes A (0) , A (0) , A (0) of decays
0 || ⊥
| | | | | |
B0,B0 J/ψV((q,V) (s,φ),(d,K⋆) ) can be expressed in terms of effective Wilson
q q → ∈ { }
coefficient functions, constants of J/ψ decay, and form factors of transitions B V
q
→
induced by quark currents [1]. In Table 1 we collect the predictions of Ref. [1] for the
transversity amplitudes of B0 J/ψφ (B0 J/ψK⋆) calculated with B K⋆ form
s → d → →
factors given by different models [13, 14, 15]. The B K⋆ form factors can be related
→
to the B φ case by using SU(3) flavor symmetry. The most precise polarization
→
measurements performed recently in decays B J/ψK⋆:
→
A (0) 2 = 0.60 0.04, A (0) 2 = 0.16 0.03 (BaBar [16]),
0 ⊥
| | ± | | ±
A (0) 2 = 0.62 0.04, A (0) 2 = 0.19 0.04 (Belle [17]),
0 ⊥
| | ± | | ±
confirm the predictions based on the model [15].
5
c J Ψ c
b s
W
0
B V
q
q q
Figure 2: Color suppressed diagrams for decays B0 J/ψV ((q,V) (s,φ),(d,K⋆) )
q → ∈ { }
Table 1: Predictions for B0 J/ψφ (in brackets – for B0 J/ψK⋆) observables
s → d →
obtained in Ref. [1] for various model estimates of the B K⋆ form factors [13, 14, 15]
→
(the normalization condition A (0) 2 + A (0) 2 + A (0) 2 = 1 is implied)
0 || ⊥
| | | | | |
Observable BSW [13] Soares [14] Cheng [15]
A (0) 2 0.55 (0.57) 0.41 (0.42) 0.54 (0.56)
0
| |
A (0) 2 0.09 (0.09) 0.32 (0.33) 0.16 (0.16)
⊥
| |
3 Angular-moments method
The angular distributions for decays B0(t),B0(t) J/ψ( l+l−)φ( K+K−) in case
s s → → →
of tagged B0 and B0(t) samples (see Eqs. (1) and (4), respectively) as well as in case of
s s
the untagged sample can be expressed in the most general form in terms of observables
b (t):
i
9 6
f(Θ ,Θ ,χ;t) = b (t)g (Θ ,Θ ,χ). (9)
l+ K+ i i l+ K+
32π
i=1
X
The explicit time dependence of observables is given by the following relations:
b (t) = A (0) 2G (t),
1 0 L
| |
b (t) = A (0) 2G (t),
2 || L
| |
b (t) = A (0) 2G (t),
3 ⊥ H
| |
b (t) = A (0) A (0) Z (t),
4 || ⊥ 1
| || |
b (t) = A (0) A (0) G (t)cos(δ δ ),
5 0 || L 2 1
| || | −
b (t) = A (0) A (0) Z (t), (10)
6 0 ⊥ 2
| || |
where we have used the general compact notations:
1
G (t) = (1 cosφ(s))e−ΓLt +(1 cosφ(s))e−ΓHt ,
L/H 2 ± c ∓ c
1 h i
Z (t) = e−ΓHt e−ΓLt cosδ sinφ(s)
1,2 2 − 1,2 c
(cid:16) (cid:17)
6
– for observables b ( + )/2 in case of the untagged sample with equal initial
i i i
≡ O O
numbers of B0 and B0, while
s s
G(Bs0)/(B0s)(t) = G (t) e−Γstsin(∆Mt)sinφ(s),
L/H L/H ± c
Z(Bs0)/(B0s)(t) = Z (t) e−Γst sinδ cos(∆Mt) cosδ sin(∆Mt)sinφ(s)
1,2 1,2 ± 1,2 − 1,2 c
h i
– for observables b(Bs0) and b(B0s) in case of tagged B0 and B0(t) samples,
i ≡ Oi i ≡ Oi s s
respectively, with Γ (Γ + Γ )/2. It is easy to see that both in the tagged and
s L H
≡
untagged case we have
G (t) = e−ΓL/Ht.
L/H |φ(cs)=0
According to Ref. [1], the observables b (t) can be extracted from distribution func-
i
tion (9) by means of weighting functions w (Θ , Θ , χ) for each i such that
i l+ K+
9
dcosΘ dcosΘ dχ w (Θ ,Θ ,χ) g (Θ ,Θ ,χ) = δ , (11)
l+ K+ i l+ K+ j l+ K+ ij
32π
Z
projecting out the desired observable alone:
b (t) = dcosΘ dcosΘ dχ f(Θ ,Θ ,χ;t) w (Θ ,Θ ,χ). (12)
i l+ K+ l+ K+ i l+ K+
Z
The angular-distribution function (9) obeys the condition
L(t) dcosΘ dcosΘ dχ f(Θ ,Θ ,χ;t) = b (t)+b (t)+b (t). (13)
l+ K+ l+ K+ 1 2 3
≡
Z
For decays B J/ψ( l+l−)φ( K+K−), the explicit expressions of weighting
→ → →
functions, given in Table 5 of Ref. [1] for physically meaningful angles in the transversity
frame, get the following form (Set A) after transformation into the helicity frame:
w(A) = 2 5cos2Θ ,
1 − l+
w(A) = 2 5sin2Θ cos2χ,
2 − l+
w(A) = 2 5sin2Θ sin2χ,
3 − l+
5
w(A) = sin2Θ sin2χ,
4 −2 K+
25
(A)
w = sin2Θ sin2Θ cosχ,
5 4√2 K+ l+
25
(A)
w = sin2Θ sin2Θ sinχ. (14)
6 4√2 K+ l+
The expressions of Eq. (14) are not unique and there are many legitimate choices of
weighting functions. A particular set can be derived by linear combination of angular
functions g (see [1] for more discussions):
i
6
w (Θ ,Θ ,χ) = λ g (Θ ,Θ ,χ), (15)
i l+ K+ ij j l+ K+
j=1
X
7
where the 36 unknown coefficients λ are solutions of 36 equations
ij
9 6
λ dcosΘ dcosΘ dχ g (Θ ,Θ ,χ)g (Θ ,Θ ,χ) = δ . (16)
ij l+ K+ j l+ K+ k l+ K+ ik
32π
j=1 Z
X
The weighting functions (set B) corresponding to the linear combination of the angular
functions (3) are given by
1
w(B) = [28cos2Θ sin2Θ 3sin2Θ (1+cos2Θ )],
1 12 K+ l+ − K+ l+
1
w(B) = [4cos2Θ sin2Θ 29sin2Θ (1 sin2Θ cos2χ)
2 − 8 K+ l+ − K+ − l+
+21sin2Θ (1 sin2Θ sin2χ)],
K+ l+
−
1
w(B) = [4cos2Θ sin2Θ +21sin2Θ (1 sin2Θ cos2χ)
3 − 8 K+ l+ K+ − l+
29sin2Θ (1 sin2Θ sin2χ)],
K+ l+
− −
25
w(B) = sin2Θ sin2Θ sin2χ,
4 − 8 K+ l+
(B) (A)
w = w ,
5 5
(B) (A)
w = w . (17)
6 6
For a limited number of experimental events N in the time bin around the fixed value
of the proper time t, distributed according to the angular function (9), it is convenient
to introduce the normalized observables
¯
b (t) b (t)/L(t) (18)
i i
≡
with normalization factor L(t) given by Eq. (13). Then, as it follows from the Eq. (12),
¯
the observables b (t) (18) are measured experimentally by
i
1 N
¯b(exp) = wj (19)
i N i
j=1
X
with summation over events in a time bin around t. Here wj w (Θj ,Θj ,χj), where
i ≡ i l+ K+
Θj , Θj and χj are angles measured in the j-th event. The statistical measurement
l+ K+
error of the observable (19) can be estimated as
1 N
δ¯b(exp) = (¯b(exp) wj)2,
i Nvu i − i
ujX=1
u
t
with summation over all events in the same time bin.
4 Time-integrated observables
For data analysis it is rather convenient to use the time-integrated observables defined as
˜ 1 T0
b (T ) = dt dcosΘ dcosΘ dχw (Θ ,Θ ,χ) f(Θ ,Θ ,χ;t) (20)
i 0 L˜(T) Z0 Z l+ K+ i l+ K+ l+ K+
8
with argument T T, where T is the maximal value of the B-meson proper time
0
≤
measured for the sample of events being used, and L˜(T) is a new normalization factor,
which has the form:
T T
L˜(T) L(t) = dt dcosΘ dcosΘ dχ f(cosΘ ,cosΘ ,χ;t) =
l+ K+ l+ K+
≡
Z0 Z0 Z
= ( A (0) 2 + A (0) 2)G˜ (T)+ A (0) 2G˜ (T), (21)
0 || L ⊥ H
| | | | | |
where, in the compact notations used in Eq. (10),
T
G˜ (T) dtG (t).
L/H L/H
≡
Z0
˜ ˜ ˜
The following normalization condition is valid for the observables (20): b +b +b = 1.
1 2 3
For a limited number of experimental events N(T), measured in the proper time region
t [0,T], Eq. (20) reduces to
∈
1 N(T0)
˜b(exp)(T ) = wj (22)
i 0 N(T) i
j=1
X
with summation over all events N(T ) in the time interval t [0,T ] for T T. In case
0 0 0
∈ ≤
of the untagged sample we have
1 e−ΓLT 1 e−ΓHT 1
G˜ (T) = (1 cosφ(s)) − +(1 cosφ(s)) − (23)
L/H −2 ± c Γ ∓ c Γ
(cid:20) L H (cid:21)
and
1 T
Z˜(T) dtZ (T)
≡ cosδ1,2sinφ(cs) Z0 1,2
1
= (e−ΓHT 1)/Γ (e−ΓLT 1)/Γ .
H L
−2 − − −
(cid:20) (cid:21)
For the untagged sample the explicit form of time-integrated normalized observables
˜ ˜
(20) in terms of the functions G (T) and Z(T) is given by
L/H
˜b (T ) = A (0) 2G˜ (T )/L˜(T),
1 0 0 L 0
| |
˜b (T ) = A (0) 2G˜ (T )/L˜(T),
2 0 || L 0
| |
˜b (T ) = A (0) 2G˜ (T )/L˜(T),
3 0 ⊥ H 0
| |
˜b (T ) = A (0) A (0) Z˜(T )cosδ sinφ(s)/L˜(T),
4 0 | || || ⊥ | 0 1 c
˜b (T ) = A (0) A (0) G˜ (T )cos(δ δ )/L˜(T),
5 0 0 || L 0 2 1
| || | −
˜b (T ) = A (0) A (0) Z˜(T )cosδ sinφ(s)/L˜(T). (24)
6 0 | 0 || ⊥ | 0 2 c
In the Standard Model (SM) sinφ(s) 0 and the observables ˜b (T ) are vanishing. In
c ≈ 4,5 0
case of a new physics signal the values of sinφ(s) and ˜b (T ) can be sizable, however.
c 4,5 0
9
The following relations are valid for the observables (24):
˜ ˜
b (T )b (T )
˜b (T ) = cosδ sinφ(s)Z˜(T ) 2 0 3 0 ,
4 0 1 c 0 vuG˜L(T0)G˜H(T0)
u
t
˜ ˜ ˜
b (T ) = cos(δ δ ) b (T )b (T ),
5 0 2 1 1 0 2 0
−
q
˜ ˜
b (T )b (T )
˜b (T ) = cosδ sinφ(s)Z˜(T ) 1 0 3 0 .
6 0 2 c 0 vuG˜L(T0)G˜H(T0)
u
t
If we introduce the function
γ˜(T) G˜ (T)/G˜ (T), (25)
H L
≡
then, the values of initial transversity amplitudes at t = 0 and the strong-phase difference
˜ ˜
(δ δ ) are determined from the observables b (T) b (T = T ) by
2 1 i i 0
− ≡
˜
b (T)
A (0) 2 = 1 ,
| 0 | ˜b (T)+˜b (T)+˜b (T)/γ˜(T)
1 2 3
˜
b (T)
A (0) 2 = 2 ,
| || | ˜b (T)+˜b (T)+˜b (T)/γ˜(T)
1 2 3
˜
b (T)/γ˜(T)
A (0) 2 = 3 ,
| ⊥ | ˜b (T)+˜b (T)+˜b (T)/γ˜(T)
1 2 3
˜
b (T)
5
cos(δ δ ) = , (26)
2 1
− ˜ ˜
b (T)b (T)
1 2
q
where we consider the initial amplitudes normalized as A (0) 2+ A (0) 2+ A (0) 2 = 1.
0 || ⊥
| | | | | |
We have also:
˜b (T) G˜ (T)G˜ (T)
sinφ(s)cosδ = 4,6 L H . (27)
c 1,2 ˜b (T)˜b (T) q Z˜(T)
2,1 3
q
For extraction of the B0-width difference ∆Γ Γ Γ from experimental data it
s s ≡ H − L
is convenient to use a special set of the time-integrated normalized observables:
ˆb (T ) = 1 T0 dt dcosΘ dcosΘ dχw (Θ ,Θ ,χ) eΓ′tf(Θ ,Θ ,χ;t),
i 0 ˜ l+ K+ i l+ K+ l+ K+
L(T) Z0 Z
(28)
where Γ′ is some arbitrary initial approximation of the B0-meson total decay width.
s
These observables can be extracted from the experimental events N(T), measured in the
proper time region t [0,T], by using the formula
∈
1 N(T0)
ˆb(exp)(T ) = Wj, (29)
i 0 N(T) i
j=1
X
where Wj eΓ′tj wj, and summation is performed over all events N(T ) in the time
i ≡ i 0
interval tj [0,T ].
0
∈
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