Is quark-hadron duality valid
in charm decays and mixing?

梅枝 宏之(中央研究院, 台湾)

Flavor physics workshop 2021
@オンライン開催, 2021年9月29日(paperunder 066
preparation)
Based on JHEP
09 (2021)+ 遂行中の研究

✔︎

D → K π → D̄
0

− +

0

theory: difficult{

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Exclusive?
Inclusive = sum of exclusive

concrete hadronic final states(quark-hadron duality)

Examples:◉ Heavy quark expansion (HQE).
1/mQ : expansion parameter
LO

c3
c2
Γ∝ 1+ 2 + 3 +⋯ ,
mQ mQ
(
)

— 1 / 14 —Q = b, chigher order corrections

theory: easy(?)(1) Is the above expansion convergent,expecially for charm quark?

(2) Is quark-hadron duality valid?

✔︎

✔︎

Inclusive processes(1) Semi-leptonic decays of B meson?
Γ[B̄ → Xclν] = Γ[B̄ → Dlν] + Γ[B̄ → D*lν] + ⋯◉ Decay rate with kinematical cut is considered in practice.
◉ This is applied for the determination of | Vcb | . also for c → u(2) Total width of B and D mesons
0

0

0

0

(3) Width differences in B − B̄ and D − D̄ systems◉ 1/mQ expansion & duality can be testable.

— 2 / 14 —Outline(A) Inclusive analysis— Decays: B and D mesons
0

0

0

0

— Mixings: B − B̄ and D − D̄(B) Quark hadron duality in the ’t Hooft model— Mixing: D 0 − D̄0— Decays: D mesons (preliminary)(C) Summary

Outline(A) Inclusive analysis— Decays: B and D mesons
0

0

0

0

— Mixings: B − B̄ and D − D̄(B) Quark hadron duality in the ’t Hooft model— Mixing: D 0 − D̄0— Decays: D mesons (preliminary)(C) Summary

𝒪𝒪

GF2 mQ522μμc6 < HQ | (Q̄q)(q̄Q) | HQ >π
G2
| VCKM | c3 − c3 2 + cG 2 + 3
Γ=
+⋯
3
192π
2mQ
2mQ mQ
MHQ[]

1/mQ2 corrections

B̄
b

{
W
W
b

The spectator does not join interaction.

q̄sp
q̄sp

LO

B̄

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If not, the convergence is questioned.

— 3 / 14 —{<latexit sha1_base64="EddfP4uEa8rhPnDsXjfppxY64so=">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</latexit>{<latexit sha1_base64="EddfP4uEa8rhPnDsXjfppxY64so=">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</latexit>

1/mQ expansion

total width of B and D(Q = b, c)

LO(1/mQ3 ) correction

vanish for mQ → ∞¯ Ds+(cs̄)
B +(b̄u), Bd(b̄d), Bs(b̄s) and D 0(cū), D +(cd),Lifetimes are common forif (1/mQ2 ) corrections are negligible.

lifetime ratio:

If the lifetime ratio ≈ 1, the 1/mQ expansion is convergent.
(provided that no cancellation occurs.)

What happens for realistic b, c?
Next slide✔︎✔︎Lifetime ratios for B and D mesons
King et al. [2109.13219]

Kirk, Lenz and Rauh [1711.02100v4]

including the Darwin operator◉ τ(Bs)/τ(Bd) and τ(B +)/τ(Bd) are close to 1.

◉ τ(D +)/τ(D 0) is much larger than 1.

1/mb expansion works.
1/mc expansion may be non-convergent series?

Nevertheless, a good agreement is obtained within theoretical errors.
— 4 / 14 —

2

24πMHQ(VCKM) [C(z) < HQ | (q̄Q)V−A(q̄Q)V−A | H̄Q >Γ12 = −

Γ12 : on − shell contribution< HQ | ℋe | H̄Q >i
M12 − Γ12 =2
2MHQ◉ D 0 − D̄0 mixing is strongly suppressed by GIM cancellation.

A tiny error may be enlarged.

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{

ff

Q

q
Box diagram

1/mQ corrections

1/mQ expansion
GF2 mQ2

W
W

Q̄

q̄

— 5 / 14 —ff

Mixings: D − D̄ and B − B̄

0
0
0
0

✔︎

Comparison between theory and experiment
D mesonHagelin 1981,

Box diagram

Bd mesonLenz and Tetlalmatzi-Xolocotzi 2019

Lenz and Tetlalmatzi-Xolocotzi 2019

SM ΔΓs = (0.091 ± 0.013) ps−1

SM ΔΓd = (2.6 ± 0.4) × 10−3 ps−1

Cheng 1982

Buras, Slominski and Steger 1984

Golowich and Petrov 2005
Bobrowski et al. 2010

NLO QCDy ' 6 · 10 7<latexit sha1_base64="y/aF49qvCb7anDjUlChhYutvXLY=">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</latexit>SM

Bs mesonsuppressed by GIM mechanismy = Γ12 /ΓD
HFLAV 2019Exp.

3
y = (6.51+0.63
)
⇥
10
0.69
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HFLAV 2019Exp.

ΔΓs = (0.090 ± 0.005) ps−1

HFLAV 2019Exp. ΔΓd = (0.002 ± 0.020) ps−1

● For Bs meson, the HQE gives good agreement with data.

● For Bd meson, the HQE is consistent with data within the error in data.

● For D meson, the order of magnitude is not reproduced within the leading-HQE.
— 6 / 14 —

✔︎

✔︎

Comparison between theory and experiment
D mesonHagelin 1981,

Box diagram

Cheng 1982

Violation of quark-hadron duality?

Buras, Slominski and Steger 1984

Golowich and Petrov 2005
Bobrowski et al. 2010

NLO QCDy ' 6 · 10 7<latexit sha1_base64="y/aF49qvCb7anDjUlChhYutvXLY=">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</latexit>SM

Bs meson discussed in B
d mesonPossibilities
theliterature
Lenz and Tetlalmatzi-Xolocotzi 2019

— 20% violation explains the data. (based on a simple model)
Jubb, Kirk, Lenz and Tetlalmatzi-Xolocotzi, 2017
−1 higher dim. operators?
(0.091 ± 0.013) psof
SM ΔΓs =Contribution
SM ΔΓd = (2.6 ± 0.4) × 10−3 ps−1

suggested by Georgi, 1992, prior to the experimental measurement

suppressed by GIM mechanism— it may give a source of SU(3) breaking linear in ms,
avoiding severe GIM cancellation?

y = Γ12 /ΓDx, y2019⇠ O(10HFLAV

HFLAV 2019Exp.

3
y = (6.51+0.63
)
⇥
10
0.69
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3) Bigi and Uraltsev, 2001 HFLAV 2019

— With some assumption about hadronic matrix elements,Exp.

ΔΓs = (0.090
ΔΓd and= (0.002
0.020) ps−1
Exp. Ligeti
x ⇠±y 0.005). 10 3ps−1
Falk, Grossman,
Petrov,±
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Beyond the standard model?

Golowich, Pakvasa and Petrov, 2007
● For Bs meson, the HQE gives good agreementwith data.
e.g.,

Golowich, Hewett, Pakvasa and Petrov, 2009

● For Bd meson, the HQE is consistent with data within the error in data.

● For D meson, the order of magnitude is not reproduced within the leading-HQE.
— 6 / 14 —Outline(A) Inclusive analysis— Decays: B and D mesons
0

0

0

0

— Mixings: B − B̄ and D − D̄(B) Quark hadron duality in the ’t Hooft model— Mixing: D 0 − D̄0— Decays: D mesons (preliminary result)(C) Summary✔︎✔︎✔︎✔︎Method to investigate duality violation?
Γinc ≠∑

Γexc

Comparison between d = 4 and d = 2
inclusive (HQE)

d=4
d=2

exclusive (Exp.) exclusive (Theo.)❌
❌
QCD is solvable in the large-Nc limit’t Hooft, 1974

The ’t Hooft model (QCD2 in the large-Nc limit)◎ Asymptotic free theory.

◎ Gluon is not dynamical.

◎ Confinement is built-in.

◎ Phase space offen has singularity in d = 2.

Duality violation is qualitatively testable.
— 7 / 14 —

Bound state equation in the ’t Hooft model
Bethe-Salpeter equation (in the light-cone gauge):
S
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the ’t Hooft equation:

Mk2 k (x) =

=
✓

m21

S

S
S
2

x

+

S

m22
1

2

x

◆

k (x)

2

Pr

Z 1
0

k (y)dy(x y)2

notation of QCD coupling:

Masses and wavefunctions for mesons can be determinedwithin the formalism.
— 8 / 14 —

00

Exclusive processes for D − D̄ mixing
Width difference in CP conserving limit:(k,m) (m,k)*
T
T
(D )
k+m
Γ12 =
(−1)
2 |p |∑
4M
D 0 kmk,m0TDphase space in 2D(k,m): color-allowed tree diagram

0

W+

M

−

M

+

M = π, K, + (excited states)

calculated by Grinstein and Lebed, 1998,,

: overlap integrals of meson wave functions— 9 / 14 —

0Inclusive result for D − D̄0

Γ12 = CA < D 0 | (ūαγ μγ5c α)(ūβγμγ5c β) | D 0 > +CP < D 0 | (ūαiγ5c α)(ūβiγ5c β) | D 0 >2GF2 2
(th)
(th)
(th)
(th)
2
2
2
2
CA = +(cV − cA2)V*
V
[(c
−
c
)(F
+
2G
)
−
(c
+
c
)(H
+
H
)]
uj
V
V
A
A
ci
ij
ij
ij
ji
MD0
2GF2 2
(th)
(th)
(th)
(th)
2
2
2
2
CP = −(cV − cA2)V*
V
[(c
−
c
)(G
+
2H
)
+
(c
+
c
)(H
+
H
)]
uj
V
V
A
A
ci
ij
ij
ij
ji
MD0

{

the standard model

generalized weak vertex:

cV =

1
,
2

cA = −

1
2

Matrix elements are calculated in the large-Nc factorization.

: 4D-like phase space: 2D-specific phase space: 2D-specific phase space
We present the two cases:{

zd = md2 /mc2{

zs = ms2 /mc2
expansion parameter(1) only 4D-like term(2) pure 2D result (all of the 4D-like and 2D-specific terms
are included)— 10 / 14 —00

Numerical result: D − D̄ mixing<latexit sha1_base64="1oydfaetzyx+Mf5Z/eSDf/8hj+g=">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</latexit>

(exc)
D
(inc)
D

MS,4D
mc = mc

larger than 103

ms /β

{

points: based only on the 4D-like phase spacecrosses: based on the 4D-like phase space + 2D-speci c one◎ The exclusive rate is enhanced by more than 103, confirmed for0.14 < ms /β < 0.25, when only the 4D-like phase space function is used.
fi— 11 / 14 —Lifetimes of D mesonsΓ = Γdec + ΓPI

u

W+

D q̄c

d¯ or s̄

1/mc correction

c
W+
c

D
q̄

sp

D
d¯ or s̄

sp

decay

c

D
d¯ or s̄Pauli interference (PI)neutral current
charged current

Fiertz rearranged

Weak interaction parameters are taken by the corresponding 4D values
CKM: from PDG
a1 ≈ 1.2, a2 ≈ − 0.38
exclusive{

inclusive → ongoing— 12 / 14 —T: color-allowed tree diagram

Numerical result: lifetimes for D mesons
Preliminary

MS (4D){τ[D +]/τ[D 0] = 1.4 − 1.5, τ[Ds]/τ[D 0] ≈ 1.03
τ[Ds]/τ[D 0] ≈ 1.02
τ[D +]/τ[D 0] ≈ 1.3,

pole (4D)(for mc = mcMS)(for mc = mcpole)◉ The convergence is better in 2D than in 4D,
because of no enhancement for the phase space factor in PI.
◉ Quark-hadron duality will be checked.
— 13 / 14 —Summary◎ We have studied quark-duality for D 0 − D̄0 mixing on the basis of
one certain dynamical mechanism.
— For the D 0 − D̄0 mixing, the order of magnitude for ΔΓD is enhancedby more than 103, confirmed for 0.14 < ms /β < 0.25, if the phase spacefunction is given by 4D-like one.

The result indicates that large duality violation in D 0 − D̄0 mixing
may be still possible.

◎ The lifetime of charmed mesons in the ’t Hooft model arecalculated by varying charm quark mass.

{

τ[D ]/τ[D ] = 1.4 − 1.5, τ[Ds]/τ[D 0] ≈ 1.03
τ[Ds]/τ[D 0] ≈ 1.02
τ[D +]/τ[D 0] ≈ 1.3,+

0(for mc = mcMS)

pole(for mc = mc )— Numerical check of quark-hadron duality including PI is ongoing.
— 14 / 14 —

Backup✔︎✔︎How duality violation appears?
?
Γinc ≠∑

Γexc◉ Accuracy of the HQE is limited up to non-perturbative effects in perturbation theory.
divergences◉ Duality violation is modeled by(1) Proliferation of Feynman diagrams(2) Renormalons(3) 1/mQ series?

{{(a) Instanton-based approach— takes accout of the effect of (fixed sized) background instanton— may represent the trancated effects for (3).

— considered more or less as an orientation.

(b) Resonance-based approach— large-Nc + linear Regge tragectory.
— factorial divergence (sign-flipping) for (3) is captured.

Weak annihilation in the ’t Hooft model

Inclusive:

Exclusive:∝ Nc ?

The quark-hadron duality may not be captured in the large-Nc limit.
finite-Nc + smearing with the Gausian weight

Grinstein, Lebed [9805404]

00Inclusive analysis for D − D̄ mixing(D,inc)2 (D,inc)Γ12 = λd2Γ(D,inc)
+
2λ
λ
Γ
+
λ
.
s d dss Γssdd

λk = VckV*uki, j = d or s

inc)
(th)
(th)
(th)
Γ(D,
∝
F
,
G
,
H
.
ij
ij
ij
ij

three types of phase space functions{: 4D-like phase space: 2D-specific phase space: 2D-specific phase space{zs = ms2 /mc2 ≪ 1
expansion parameter(D,inc)(D, inc)(D, inc)(D, inc)Γ12 ≃ λs2Γ(D,inc)
,
Γ
=
Γ
+
Γ
−
2Γ
ss
(GIM,1)(GIM,1)dd
sd

CKM unitarity + λb → 0 limit

inclusive observable:

zd = md2 /mc2 = 0{

2|
Γ(D,inc)∝zs(GIM,1) 4D−like|Γ(D,inc)∝ zs(GIM,1) 4D+2D

suppressed more strongly
for large mc

The order of magnitude for Γ(exc) /Γ(inc) strongly depends on the two cases.

Both cases are presented in what follows.

charged-current

neutral-current(Fiertz-transformed)

inclusive

exclusive{

to be improved byincluding mass dependences{

Exclusive formula is written by color-allowed tree diagram,
𝒟

𝒞

Formulas of D and B decays: overap integrals

Inclusive and exclusive processes
Quarkhadron<latexit sha1_base64="WY88TbmvOLK+1scWOK/sR77rY1Y=">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</latexit>e+<latexit sha1_base64="1wfkp7A3YTDXbnFdGt3U2bItji0=">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</latexit>

σ (inc)(e +e − → qq̄)q<latexit sha1_base64="+Und/47U9eMV8HQXuIy+k0hOrxk=">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</latexit>+e e ! hadrons + −e+<latexit sha1_base64="1wfkp7A3YTDXbnFdGt3U2bItji0=">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</latexit>

σ(e e → 2π) + σ(e +e − → 3π) + ⋯⇡⇡<latexit sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>e<latexit 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sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>

Poggio, Quinn and Weinberg, 1976 for the case with smearing

Quark-hadron duality: inclusive = sum of exclusive(one definition of)

Duality violation: inclusive ≠ sum of exclusive<latexit sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>✔︎Inclusive and exclusive processes
Quarkhadron<latexit sha1_base64="WY88TbmvOLK+1scWOK/sR77rY1Y=">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</latexit>e+<latexit sha1_base64="1wfkp7A3YTDXbnFdGt3U2bItji0=">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</latexit>

σ (inc)(e +e − → qq̄)q<latexit sha1_base64="+Und/47U9eMV8HQXuIy+k0hOrxk=">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</latexit>+e e ! hadrons + −σ(e e → 2π) + σ(e +e − → 3π) + ⋯e+<latexit sha1_base64="1wfkp7A3YTDXbnFdGt3U2bItji0=">AAACjnichVG7SgNRED2urxhfqzaCTTAoghBmRVQEUbRJGROjgi9212tc3Be7m0AM/oBgq4WVgoX4CZY2/oBFPkEsI9hYOLtZEBV1LvfemXPnzNzDaK5p+AFRvUVqbWvv6Ex0Jbt7evv65YHBdd8pe7oo6o7peJua6gvTsEUxMAJTbLqeUC3NFBva0Ur4vlERnm849lpQdcWOpZZs48DQ1YChgtid3JPTlKHIUj8dJXbSiC3nyPfYxj4c6CjDgoCNgH0TKnxeW1BAcBnbQY0xjz0jehc4QZK5Zc4SnKEyesRniaOtGLU5Dmv6EVvnLiZvj5kpjNET3VKDHumOnun911q1qEb4lyrfWpMr3L3+0+HC278si+8Ah5+sPxga/0zlWDDjb20BDjAXaTJYoxshoVq92adyfNEozOfHauN0TS+s84rq9MBK7cqrfrMq8pdI8qCU72P56axPZZSZDK1Op5eW45ElMIJRTPBcZrGELHIoct8SznCOC0mWZqQFabGZKrXEnCF8MSn7AcPeklQ=</latexit>

⇡
⇡

⇡
⇡
⇡ ⇡
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D0<latexit sha1_base64="RHbWxkRrLTwGmd4mdwqv3+bcXWU=">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</latexit><latexit sha1_base64="2VH1Y1CLNfjGFrEKEpmrLodzoyc=">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</latexit>cq<latexit sha1_base64="+Und/47U9eMV8HQXuIy+k0hOrxk=">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</latexit>

(inc)
12

D̄0

u

<latexit sha1_base64="I5EPAtNSISnY40a8796E4VSsAT4=">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</latexit>

12 (⇡⇡)⇡zd<latexit sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>

D̄ 0

D0
ū

q̄
<latexit sha1_base64="p6Csxl0rKlfg7/Na7zvle/uatBg=">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</latexit>+<latexit sha1_base64="rhfL1Yf7n6HHY2jPDlXAqYyZ3qM=">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</latexit>

12 (⇢⇡) + · · ·ρ zd<latexit sha1_base64="6+eg/HzmKLT7izNQqVF3bJYNTkY=">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</latexit>

+

c̄

zd⇡
<latexit sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>+ (higher dimensional operators)⇡<latexit sha1_base64="0l+fhS1MtPcfj8vV9W0woJ5DPyA=">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</latexit>+ (all the other exclusive states)

Non-trivial point

Does duality violation possibly give a large correction to the box diagram? This talk

● Charm quark mass is a unique scale.
mc ⇡ 1.3 1.7 GeV<latexit sha1_base64="ctBuRPy8hsDptgkmFawRQTeNxLI=">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</latexit>— too heavy for ChPT

— too light for ⇤QCD /mc expansion?
Theoretically challenging● High statistics data are provided.
— in a good stage to test theories
00● D − D̄ mixing.
— ⇤QCD /mc expansion is not successful

LHCb [1810.06874]— experimental data are not quantitatively reproduced yet
perhaps, quark-hadron duality is violated?

Previous works for heavy meson decays◉ non-leptonic decays (color-allowed tree)[1] B. Grinstein and R. F. Lebed, Phys. Rev. D57, 1366-1378 (1998)[arXiv:hep-ph/9708396 [hep-ph]].

◉ semi-leptonic decay and also non-leptonic decay[2] I. I. Y. Bigi, M. A. Shifman, N. Uraltsev and A. I. Vainshtein,Phys. Rev. D59, 054011 (1999) [arXiv:hep-ph/9805241 [hep-ph]].

◉ non-leptonic decay (weak annhilation)[3] B. Grinstein and R. F. Lebed, Phys. Rev. D59, 054022 (1999)[arXiv:hep-ph/9805404 [hep-ph]].

◉ non-leptonic decays (weak annhilation, Pauli interference)[4] I. I. Y. Bigi and N. Uraltsev, Phys. Rev. D60, 114034 (1999)[arXiv:hep-ph/9902315 [hep-ph]]; Phys. Lett. B457, 163-169 (1999)[arXiv:hep-ph/9903258 [hep-ph]].

◉ semi-leptonic decay[5] R.~F. Lebed and N. G. Uraltsev, Phys. Rev. D62, 094011 (2000)[arXiv:hep-ph/0006346 [hep-ph]].

This work → heavy meson mixings
・Mixing is suppressed by the GIM cancellation.
Tiny duality violation is possibly enlarged after cancellation.

00 mixing

D − D̄

time evolution
0

D

D̄

0

Time evolution Eq.

CP-conserving limit
Mass eigenstate |D1,2 i = |D 0 i ± |D̄ 0 i<latexit sha1_base64="zd/sKJRg36WhTlYoP7Ibg7KY1JQ=">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</latexit>

observables{= 2M12 /mass difference<latexit sha1_base64="KGCPAh0pnlYPpEa3tFmzlb9Ow6Q=">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</latexit><latexit sha1_base64="yaxZMzUiZBdkkxCSbX5evQDlkRc=">AAACoHichVHLSsNAFD3Gd3202o3gprQoruqkiIogiC50p23tA6uUSZzWYF4kaaEWf8AfcOFCFATFT3Dpxh9w0U8QlwpuXHibBkRFnZCZc8/cc+8crmLrmusx1uqSunt6+/oHBkNDwyOj4cjYeN61ao4qcqqlW05R4a7QNVPkPM3TRdF2BDcUXRSUw7X2faEuHFezzG2vYYs9g1dNraKp3COqHIku765zw+Dlppw6ng1wJMGSzF+xn0AOQALB2rIid9jFPiyoqMGAgAmPsA4Ol74SZDDYxO2hSZxDSPPvBY4RIm2NsgRlcGIPaa9SVApYk+J2TddXq9RFp98hZQxT7JHdsBf2wG7ZE3v/tVbTr9F+S4NOpaMVdjl8MpF9+1dl0Onh4FP1h0Khl3GKBSn+9uahgkXfk0YebZ9pu1U7fepHpy/ZpcxUc5pdsmfyecFa7J6cmvVX9SotMmcI0aDk72P5CfKppDyfTKXnEiurwcgGMIk4ZmguC1jBBraQo74NnOMaN1Jc2pA2pXQnVeoKNFF8WdLOBzUKmN0=</latexit>

=

12 /

width difference: total width<latexit sha1_base64="pL8JRHGHXjHASNMhzmF2k5yHqvs=">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</latexit><latexit sha1_base64="F8wb4eXjO23vf9fdw6Gh2t+n0ig=">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</latexit>

✔︎

0

0

D − D̄ mixing: theory
Two methods◉Exclusive◉Inclusive

Hadronic-level analysis

Quark-level analysis

Hard to calculate{[D ! ⇡⇡]<latexit sha1_base64="R81tn1uCzIrm437BKQ8WACH3AOQ=">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</latexit><latexit sha1_base64="pL8JRHGHXjHASNMhzmF2k5yHqvs=">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</latexit><latexit sha1_base64="tpfrZCq/wTSEdZTTvdErBtTcKcw=">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</latexit>

Data are used

[D ! KK]

without data
purely theoretical method(quark-hadron duality is assumed)

00

D − D̄ mixing: experiment(x, y) = (0, 0) is excluded by >>11.5σ<latexit sha1_base64="U724Axsfk5bq3NMc2pZFUWSQ/NQ=">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</latexit>(|q/p|<latexit sha1_base64="fcS6lgzL0PwVzrl2+9ciXZD+wLk=">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</latexit>

1, Arg(q/p)) = (0, 0) is allowed

No signal for CP violation

<latexit sha1_base64="mAwsXSr8w5ohCsyqiBOkmpbfXOo=">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</latexit>⇤
i = Vci Vuid, s<latexit sha1_base64="06qpawDeKN9VigNTch1+rsDxn4o=">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</latexit>

D̄0D<latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>

0

D0

D̄0

<latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">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</latexit><latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">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</latexit><latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>

D̄0<latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>D0+ bottom loop

<latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">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</latexit>s̄, d¯d¯<latexit sha1_base64="fdGaS57QhSF+8KRzu3Rb7eltTgI=">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</latexit>

neglect<latexit sha1_base64="c8gVaEeowcNXdNrcxc2PHM/DgEI=">AAACdXichVHLSsNAFD2N7/po1I0gQrEqLqROVFBcFd24tNWq0JaSxFGHpklI0oCG/oA/4EI3CiriZ7jxB1z0E8SlghsX3qYB0aLeYWbOnLnnzpkZzTaE6zHWiEkdnV3dPb198f6BwaGEPDyy41o1R+d53TIsZ09TXW4Ik+c94Rl8z3a4WtUMvqtV1pv7uz53XGGZ296xzUtV9dAUB0JXPaLKslzUVCdw63PJEOzXy3KKpVkYyXagRCCFKDYt+RZF7MOCjhqq4DDhETagwqVWgAIGm7gSAuIcQiLc56gjTtoaZXHKUImt0HhIq0LEmrRu1nRDtU6nGNQdUiYxzZ7YHXtlj+yePbOPX2sFYY2ml2OatZaW2+XE6djW+7+qKs0ejr5Uf3r2cICV0Ksg73bING+ht/T+ydnr1mpuOphhV+yF/F+yBnugG5j+m36d5blzxOkDlJ/P3Q52FtLKYlrJLqUya9FX9GIck5il915GBhvYRJ7O9XGBa9zE3qUJaUqaaaVKsUgzim8hzX8CWCqQfw==</latexit>

For ms = md

CKM unitarity<latexit sha1_base64="T0uEWssUQ5rgGPvQycvCI1VSjhM=">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</latexit>

d+

s+

b =0

neglect

summation ∝<latexit sha1_base64="e+1Pdqdq7UuTQd6JfGklpeLjpCI=">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</latexit>

2d+2 d s+

2
s =0

⇤
i = Vci Vui<latexit sha1_base64="mAwsXSr8w5ohCsyqiBOkmpbfXOo=">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</latexit>d, s<latexit sha1_base64="06qpawDeKN9VigNTch1+rsDxn4o=">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</latexit>

D̄0D<latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>

0

D0

D̄0

<latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">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</latexit><latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">AAACZnichVHLSsNAFD2Nr1pfVREFN8WiuCq3VVBcFXXhsg/bCj5KEqc1mCYhSQu1+AOCW7twpSAifoYbf8CFf6C4VHDjwts0IFrUO8zMmTP33Dkzo1i65rhEjwGpq7unty/YHxoYHBoeCY+O5R2zaqsip5q6aW8psiN0zRA5V3N1sWXZQq4ouigoh2ut/UJN2I5mGptu3RK7FblsaCVNlV2msut7VAxHKUZeRDpB3AdR+JEyw9fYwT5MqKiiAgEDLmMdMhxu24iDYDG3iwZzNiPN2xc4Roi1Vc4SnCEze8hjmVfbPmvwulXT8dQqn6Jzt1kZwSw90A290j3d0jN9/Fqr4dVoeanzrLS1wiqOnExl3/9VVXh2cfCl+tOzixKWPa8ae7c8pnULta2vHTVfsyuZ2cYcXdIL+7+gR7rjGxi1N/UqLTLnCPEHxH8+dyfIJ2LxhVgivRhNrvpfEcQ0ZjDP772EJDaQQo7PLeMUZ2gGnqRhaUKabKdKAV8zjm8hRT4B91GKag==</latexit><latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>

D0

D̄0

+ bottom loop

<latexit sha1_base64="Y+dZG3ca6fzI6yplPFy8feS9Ydg=">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</latexit><latexit sha1_base64="/CbBfx5oSoKYHqSBWpSIUr6yZ64=">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</latexit>s̄, d¯d¯<latexit sha1_base64="fdGaS57QhSF+8KRzu3Rb7eltTgI=">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</latexit>

neglect<latexit sha1_base64="c8gVaEeowcNXdNrcxc2PHM/DgEI=">AAACdXichVHLSsNAFD2N7/po1I0gQrEqLqROVFBcFd24tNWq0JaSxFGHpklI0oCG/oA/4EI3CiriZ7jxB1z0E8SlghsX3qYB0aLeYWbOnLnnzpkZzTaE6zHWiEkdnV3dPb198f6BwaGEPDyy41o1R+d53TIsZ09TXW4Ik+c94Rl8z3a4WtUMvqtV1pv7uz53XGGZ296xzUtV9dAUB0JXPaLKslzUVCdw63PJEOzXy3KKpVkYyXagRCCFKDYt+RZF7MOCjhqq4DDhETagwqVWgAIGm7gSAuIcQiLc56gjTtoaZXHKUImt0HhIq0LEmrRu1nRDtU6nGNQdUiYxzZ7YHXtlj+yePbOPX2sFYY2ml2OatZaW2+XE6djW+7+qKs0ejr5Uf3r2cICV0Ksg73bING+ht/T+ydnr1mpuOphhV+yF/F+yBnugG5j+m36d5blzxOkDlJ/P3Q52FtLKYlrJLqUya9FX9GIck5il915GBhvYRJ7O9XGBa9zE3qUJaUqaaaVKsUgzim8hzX8CWCqQfw==</latexit>

For ms = md

CKM unitarity<latexit sha1_base64="T0uEWssUQ5rgGPvQycvCI1VSjhM=">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</latexit>

d+

s+

b =0

neglect

2
2summation ∝ d + 2 d s + s = 0
Suppressed by the GIM mechanism.
<latexit sha1_base64="j1v43a5maMYUSgPUNMC4Cjkjx8M=">AAACZXichVHLSsNAFD2Nr1ofrQ9EcKFYFFfl1oovEIpuXGq1KqhIEqc6mCYhSQu1+APiVl24UhARP8ONP+DCLxBxqeDGhTdpUETUO8zMmTP33Dkzo9mGdD2ih4hSV9/Q2BRtjrW0trXHEx2dK65VcnSR1y3DctY01RWGNEXek54h1mxHqEXNEKva3py/v1oWjistc9mr2GKzqO6YsiB11WMqN0NbiSSlKIiBnyAdgiTCWLASV9jANizoKKEIARMeYwMqXG7rSINgM7eJKnMOIxnsCxwgxtoSZwnOUJnd43GHV+sha/Lar+kGap1PMbg7rBzAEN3TNb3QHd3QE73/Wqsa1PC9VHjWalphb8UPe5fe/lUVefaw+6X607OHAiYDr5K92wHj30Kv6cv7py9L07mh6jBd0DP7P6cHuuUbmOVX/XJR5M4QCz5gyo/xz+f+CVZGU+lMKrM4lszOhl8RRR8GMcLvPYEs5rGAPJ9bwBGOcRJ5VNqUbqWnlqpEQk0XvoXS/wEewIou</latexit><latexit sha1_base64="e+1Pdqdq7UuTQd6JfGklpeLjpCI=">AAACl3ichVHLSsNAFD3Gd31V3ShugkURhHJTBUUQRUFc2mqr4KMk6ajBNAlJWtDiD/gDCq4URMR/cOPC/oALP0FcVnDjwts0KirqDZk598w9d87MaI5peD7RQ51U39DY1NzSGmlr7+jsinb3ZDy74Ooirdum7a5pqidMwxJp3/BNsea4Qs1rpljV9uar66tF4XqGba34+47YzKs7lrFt6KrPVDaa2DC5OKdmc1sJeVT+TN+BFxn9gFwyLVM2GqM4BSH/BEoIYghjyY5eYgM52NBRQB4CFnzGJlR4/K1DAcFhbhMl5lxGRrAucIgIawtcJbhCZXaPxx3O1kPW4rza0wvUOu9i8u+yUsYQ3dMVVahM1/RIr7/2KgU9ql72edZqWuFku476ll/+VeV59rH7qfrTs49tTAZeDfbuBEz1FHpNXzw4rixPpYZKw3ROT+z/jB7olk9gFZ/1i6RInSLCD6B8v+6fIJOIK2NxJTkem50Ln6IFAxjECN/3BGaxiCWked8T3OAOZalfmpEWpMVaqVQXanrxJaTkG8Bdm1w=</latexit>

non-zero contributions
00
to D − D̄ mixing∝ SU(3) breaking:✓<latexit sha1_base64="D+COqY8OdeGzR2fXOxYoEsB+zC8=">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</latexit>

m2s

m2d
m2c

◆n

0

0

(1) D − D̄ mixing for individual flavors
D 0 → K −π + → D̄ 0{

solid line: inclusive
squares: sum of exclusive

D 0 → K −K + → D̄ 0{

dashed line: inclusive
points: sum of exclusive

Vertical dotted lines:
reference values corresponding to masses in 4D

β is fixed by 340 MeV.

(ansatz for fitting the string tension in QCD4)

ms /β is fixed by 0.32.

(MS mass at the scale of charm quark mass)
The vertical axis is normalized by 4GF2(cV2 − cA2 )2 βNc /π.

◎ For large mc, inclusive/exclusive agrees with each other.

◎ For small mc, certain difference between inclusive/exclusive appears.

◎ The agreement beween inclusive/exclusive is better for K −π + than K −K +.

◎ Obvious spikes are observed for K −K + whereas it is not seen for K −π +.

(2) Bq0 − B̄0q (q = d, s) mixing for individual flavors
Bd0 → D −π + → B̄0d
Bd0 → D −D + → B̄0d{

solid line: inclusive
squares: sum of exclusive

Bs0 → Ds−K + → B̄0s
Bs0 → Ds−Ds+ → B̄0s{

dashed line: inclusive
points: sum of exclusive◎ The patters are similar for the Bd0 − B̄0d and Bs0 − B̄0s mixings.

◎ The disagreement between inclusive/exclusive is larger for Bq0 → DD → B̄0q.

Bs0 − B̄0s mixing{

If the domain of
mc < mcpole,4D
is considered:

points: based only on the 4D-like phase spacecrosses: based on the 4D-like phase space + 2D-speci c one

mb /β = 13.7mb /β = 14.1

mb /β = 15.5mb /β = 17.0

◎ The corrections are up to (20 % , 18 % , 11 % , 8%) for mb /β = (13.7, 14.1, 15.5, 17.0).
The result is consistent with 4D within 1σ for the latter two.
(th)
(The ratio of the HFLAV data to the HQE gives ΔΓ(ex)/ΔΓ
= 0.99 ± 0.15.)
B
B
s

fi

A. Lenz and G. Tetlalmatzi-Xolocotzi, 2020s

Analytical check of local duality
the standard model

generalized weak vertex:

cV =

1
,
2

cA = −

1
2◉ Inclusive width differenceΓ12 = CA < D 0 | (ūαγ μγ5c α)(ūβγμγ5c β) | D 0 > +CP < D 0 | (ūαiγ5c α)(ūβiγ5c β) | D 0 >2GF2 2
(th)
(th)
(th)
2
2
2
2 (th)CA = +(cV − cA2)V*
V
[(c
−
c
)(F
+
2G
)
−
(c
+
c
)(I
+
I
)]
ud
V
V
A
A dd
cd
dd
dd
dd
MD0
2GF2 2
(th)
(th)
(th)
2
2
2
2 (th)
CP = −(cV − cA2)V*
V
[(c
−
c
)(G
+
2H
)
+
(c
+
c
)(I
+
I
)]
ud
V
V
A
A dd
cd
dd
dd
dd
MD0

{

(th)
(th)
(th) (th)
Fdd
, Gdd
, Hdd
, Idd : phase space functions

c

D

d

u

0

d¯

c̄

1 − 4md2 /mc2

large-Nc factorization:

D̄
ū

(th)
Fdd
=

down quark massless limit:

0

{

2
2 2
2 2
Γ12 → 4(cV − cA) V*
V G f M
cd ud F D 0 D 0Analytical check of local duality
the standard model

generalized weak vertex:

cV =

1
,
2

cA = −

1
2◉ Sum of exclusive width difference¯
massless limits for u and d quark for π(ud)(k,m) (m,k)*
T
T
(D )
k+m
Γ12 =
(−1)
∑
4MD2 0 | pkm |
k,m
0

(m)

(k)

k = 0, m = 0 : ground states∝ f π(k) pμ{f π(k) =Nc 1
ϕk(x)dxπ ∫0(a) exact solution, ϕ0(x) = 1.
∞∑

ϕk(x)ϕ*(y) = δ(x − y)k{0Nc /π k = 0(b) completeness:

k=0

f π(k) =k≠0

Analytical check of local duality
the standard model

generalized weak vertex:

cV =

1
,
2

cA = −

1
2◉ Sum of exclusive width difference¯
massless limits for u and d quark for π(ud)analog of the Pauli interference (Bigi and Uraltsev, 1999)(k,m) (m,k)*
T
T
0
(D )
k+m
Γ12 =
(−1)
∑
4MD2 0 | pkm |k,m(0,0) (0,0)*

1

T T
Nc
2
2
2
2
→−
= 4(cV − cA) GFV*Vud MD0
ϕD0(x)ϕπ(x)2cd4MD0 | p00 |π ( ∫0)2

2= 4(cV2 − cA2 )2GF2V*Vfud
D 0 MD 0
cd

agrees with the inclusive result

(m)

(k)

k = 0, m = 0 : ground states∝ f π(k) pμ{f π(k) =Nc 1
ϕk(x)dxπ ∫0(a) exact solution, ϕ0(x) = 1.
∞∑

ϕk(x)ϕ*(y) = δ(x − y)k{0Nc /π k = 0(b) completeness:

k=0

f π(k) =k≠0

Numerical evalutation of local duality
Motivations(1) Check whether local duality exists for massive final states.
(2) Check the net size of observables in the presence of
the GIM cancellation. (main motivation)— We calculate D → ππ, D → Kπ, D → KK and implement the cancellation.

Numerical method to solve the ’t Hooft equation
for q1q2 bound statebased on the BSW-improved Multhopp technique Brower, Spence and Weis, 1979ϕ=N

a sin(kθ),∑ k
k=1

1 + cos θx=2eigenvalue problem: M 2ai = (H0 + V )ijaj

M 2: eigenvalue
ai: eigenvector

Definition of amplitude and overlap integrals
Grinstein, Lebed 1997, Bigi, Uraltsev 1999

Amplitude{<latexit sha1_base64="wCS2/yeq4C8nVDyXFm6AuUL5xTc=">AAACpnichVE9S8NQFD3G7/pVdVFcikXpVG5FVJyKLk7iV61gS03S1/owTUKSFmoRd/+Ag5OCgwj6Axxd/AMO/gRxrODi4G0aEC3WF/Leuefdc+87XM02pOsRvXQonV3dPb19/aGBwaHhkfDo2K5rlR1dpHTLsJw9TXWFIU2R8qRniD3bEWpJM0RaO1pt3KcrwnGlZe54VVtkS2rRlAWpqx5TufBkRhNFadZ0ruGehDLCzAc4F45SnPwVaQWJAEQRrA0r/IAM8rCgo4wSBEx4jA2ocPnbRwIEm7ksasw5jKR/L3CCEGvLnCU4Q2X2iPciR/sBa3LcqOn6ap27GPw7rIxghp7phur0RLf0Sp9/1qr5NRpvqfKpNbXCzo2cTWx//Ksq8enh8FvVRqHxy1SOBSvae/NQwJLvSbJH22cabvVmn8rxeX17eWumNktX9MY+L+mFHtmpWXnXrzfF1gVCPKjE77G0gt25eGIhTpvz0eRKMLI+TGEaMZ7LIpJYwwZS3PcU17jDvRJT1pWUkm6mKh2BZhw/lnLwBaxBnH8=</latexit>

Overlap int.

Kinematical val.

q 2 = Mk2 for on-shell amplitudes

Expressions in the SMi2

M12<latexit sha1_base64="aT5/OD6cygI6omb2PGD/x7TmPFQ=">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</latexit>

0

0

0

0

12 = B hD |O1 |D̄ i + C hD |O2 |D̄ i{ O = (c̄u) (c̄u)

O1 = (c̄u)V A (c̄u)V A

Two contributions<latexit sha1_base64="l1ON8B0wexEIXWrHA2kwFjSVecs=">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</latexit><latexit sha1_base64="Jv3A1RdBTJqbl3mYeFzPLug8dr8=">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</latexit><latexit sha1_base64="IYQ8a3stXPhWPCMYHqd3iB0XMdk=">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</latexit>

2

S P

S P

Flavor sum

Explicit formula{<latexit sha1_base64="mAwsXSr8w5ohCsyqiBOkmpbfXOo=">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</latexit>⇤
i = Vci Vui<latexit sha1_base64="Jv3A1RdBTJqbl3mYeFzPLug8dr8=">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</latexit>qiu<latexit sha1_base64="p3LK17nff5KyxrWM6AikBlpdxJ8=">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</latexit><latexit sha1_base64="G49kYEMAgxyNI/i9SSrZHr54J+U=">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</latexit>c̄

CKM unitarityc<latexit sha1_base64="yGcaqV7whz/FYeohB5oPGeh47+E=">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</latexit>

d+

s+

b =0

<latexit sha1_base64="XwUp89KrGyhv6wVA8vdiARaikoI=">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</latexit>

d =

s

b

<latexit sha1_base64="PVTWn0W3cwB2CMGPvhyRqO8aByE=">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</latexit>ū<latexit sha1_base64="kkhiKmqovyTCEhZ8MB4i6RQYyRg=">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</latexit>q̄j i, j = (d, s, b)<latexit sha1_base64="WJg+Z/4mA4dv63o8Wx9N8cdRvac=">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</latexit><latexit sha1_base64="eIp1vXZgTX/fsOf5GqN57mAiVKk=">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</latexit>{<latexit sha1_base64="Jv3A1RdBTJqbl3mYeFzPLug8dr8=">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</latexit>

U : loop function<latexit sha1_base64="nPXjUZdvwAMU4yidy3Dg0dVnPYg=">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</latexit>

Buras, Slominski and Steger,
Nucl. Phys. B245, 369 (1984)

Structure of higher-dimensional operators
Ohl, Ricciardi and Simmons [9301212]
D=6

Leading in

D=9Subleading in<latexit sha1_base64="fXt3Kg4ZMXyFjV/gGaz3n75qPpk=">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</latexit>i : color/Dirac structureHiger-dimensional operators with condensates
Bobrowski, Lenz and Riedl [1002.4794]

D=9

D=12

✔︎

Motivations to study duality violation(1) For the D 0 − D̄0 mixing, an inclusive calculation, based on theleading operator in HQE, disagrees with data.
HFLAV 2019[0506185]<latexit sha1_base64="y/aF49qvCb7anDjUlChhYutvXLY=">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</latexit>

y ' 6 · 107(2) For, e.g., B̄ → Xqlν̄ (q = c, u), one can get the systematic uncertainty
in the OPE.
(3) Crucial to explain the short lifetime of Λb?
before 2003

data updated.
after 2003
[1405.3601]

Topological amplitudes in the 1/Nc counting

naive countings:{<latexit sha1_base64="Jv3A1RdBTJqbl3mYeFzPLug8dr8=">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</latexit>(A does not contribute to the neutral meson mixings)◉ Even in the presence of intermediate resonances,the color-allowed tree diagram is still dominant. [9805404]

dominant topology in large-Nc limit:00
Numerical result: Bq − B̄q (q = d, s)

Bd0 − B̄0d00
Bs − B̄smb = mbpole,4DΔΓ(exc)qΔΓ(inc)
q

mc /β

mc /β

For mc < mcpole,4D◎ For the Bd0 − B̄0d mixing, the correction to the inclusive rate up to 40% is observed.
The correction of this size is not excluded yet.
(The experimental data for ΔΓBd /ΓBd is still consistent with zero.)◎ For the Bs0 − B̄0s mixing, the correction to the inclusive rate up to 18% is observed.
The result is consistent with what is currently indicated in 4D.
(th)± 0.15.)(The ratio of the HFLAV data to the HQE gives ΔΓ(ex)
/ΔΓ
=
0.99
B
B
s

A. Lenz and G. Tetlalmatzi-Xolocotzi, 2020s