( ) and feasibility of b X c b X s @ N 2 LO @ N 3 LO Miko - - PowerPoint PPT Presentation

SLIDE 1

b → Xsγ @ N2LO

(∗) and feasibility of b → Xcℓ¯

ν @ N3LO

Miko laj Misiak

University of Warsaw

(∗) In collaboration with Abdur Rehman and Matthias Steinhauser [arXiv:2002.01548],

as well as Mateusz Czaja, Tobias Huber and Go Mishima

• 1. Introduction
• 2. The radiative decay

(i) O(α2

s) contributions to ˆ

G17 and ˆ G27 (ii) Non-perturbative effects in ¯ B → Xsγ (iii) Updated SM predictions for Bsγ and Rγ

• 3. The semileptonic decay

(i) Motivation for O(α3

s)

(ii) Challenges

• 4. Summary

SLIDE 2

R(D) and R(D∗) “anomalies” [https://hflav.web.cern.ch] (3.1σ)

¯ ν τ W b c ¯ ν µ W b c

R(D(∗)) = B(B → D(∗)τ ¯ ν)/B(B → D(∗)µ¯ ν)

b → sℓ+ℓ− “anomalies” (> 5σ)

[see, e.g., J. Aebischer et al., arXiv:1903.10434]

Qℓ

9 =

bL sL l l

γα

Qℓ

10 =

bL sL l l

γαγ5 ℓ = e or µ

C7, the Wilson coefficient of Q7 =

b s

R L

γ

is an important input in the fits. 2

SLIDE 3

Sample Leading-Order (LO) contributions to C7 in the SM and beyond:

γ γ γ γ γ u, c, t u, c, t W ± W ± t t ˜ t ˜ t µ µ b W ± s b u, c, t s b H± s b χ± s b LQ s

• ⇒ MH± >

∼ 800 GeV in the 2HDM-II

3

SLIDE 4

Sample Leading-Order (LO) contributions to C7 in the SM and beyond:

γ γ γ γ γ u, c, t u, c, t W ± W ± t t ˜ t ˜ t µ µ b W ± s b u, c, t s b H± s b χ± s b LQ s

• ⇒ MH± >

∼ 800 GeV in the 2HDM-II

The strongest experimental constraint on C7 comes from Bsγ — — the CP- and isospin-averaged BR of ¯ B → Xsγ and B → X¯

sγ.

( ¯ B0, B−) (B0, B+)

3

SLIDE 5

Sample Leading-Order (LO) contributions to C7 in the SM and beyond:

γ γ γ γ γ u, c, t u, c, t W ± W ± t t ˜ t ˜ t µ µ b W ± s b u, c, t s b H± s b χ± s b LQ s

• ⇒ MH± >

∼ 800 GeV in the 2HDM-II

The strongest experimental constraint on C7 comes from Bsγ — — the CP- and isospin-averaged BR of ¯ B → Xsγ and B → X¯

sγ.

( ¯ B0, B−) (B0, B+)

HFLAV, arXiv:1909.12524: Bexp sγ = (3.32 ± 0.15) × 10−4 for Eγ > E0 = 1.6 GeV ≃ mb

3 ,

(±4.5%)

averaging CLEO, BELLE and BABAR with E0 ∈ [1.7, 2.0] GeV, and then extrapolating to E0 = 1.6 GeV. TH requirement: E0 should be large

• ∼ mb

2

• but not too close to the endpoint (mb − 2E0 ≫ ΛQCD).

3

SLIDE 6

Sample Leading-Order (LO) contributions to C7 in the SM and beyond:

γ γ γ γ γ u, c, t u, c, t W ± W ± t t ˜ t ˜ t µ µ b W ± s b u, c, t s b H± s b χ± s b LQ s

• ⇒ MH± >

∼ 800 GeV in the 2HDM-II

The strongest experimental constraint on C7 comes from Bsγ — — the CP- and isospin-averaged BR of ¯ B → Xsγ and B → X¯

sγ.

( ¯ B0, B−) (B0, B+)

HFLAV, arXiv:1909.12524: Bexp sγ = (3.32 ± 0.15) × 10−4 for Eγ > E0 = 1.6 GeV ≃ mb

3 ,

(±4.5%)

averaging CLEO, BELLE and BABAR with E0 ∈ [1.7, 2.0] GeV, and then extrapolating to E0 = 1.6 GeV. TH requirement: E0 should be large

• ∼ mb

2

• but not too close to the endpoint (mb − 2E0 ≫ ΛQCD).

With the full BELLE-II dataset, a ±2.6% uncertainty in the world average for Bexp

sγ

is expected.

SM calculations must be improved to reach a similar precision.

3

SLIDE 7

Determination of B( ¯ B → Xsγ) in the SM: B( ¯ B → Xsγ)Eγ>E0 = B( ¯ B → Xce¯ ν)exp

• V ∗

tsVtb

Vcb

• 2 6αem

π C [P (E0) + N(E0)] pert. non-pert. ∼ 96% ∼ 4% Γ[b→Xp

s γ]Eγ>E0

|Vcb/Vub|2 Γ[b→Xp

ue¯

ν] =

• V ∗

tsVtb

Vcb

• 2 6αem

π

P (E0) C =

• Vub

Vcb

• 2 Γ[ ¯

B→Xce¯ ν] Γ[ ¯ B→Xue¯ ν]

semileptonic phase-space factor

4

SLIDE 8

Determination of B( ¯ B → Xsγ) in the SM: B( ¯ B → Xsγ)Eγ>E0 = B( ¯ B → Xce¯ ν)exp

• V ∗

tsVtb

Vcb

• 2 6αem

π C [P (E0) + N(E0)] pert. non-pert. ∼ 96% ∼ 4% Γ[b→Xp

s γ]Eγ>E0

|Vcb/Vub|2 Γ[b→Xp

ue¯

ν] =

• V ∗

tsVtb

Vcb

• 2 6αem

π

P (E0) C =

• Vub

Vcb

• 2 Γ[ ¯

B→Xce¯ ν] Γ[ ¯ B→Xue¯ ν]

semileptonic phase-space factor

The effective Lagrangian: Lweak ∼

i Ci Qi Eight operators Qi matter for BSM sγ when the NLO EW and/or CKM-suppressed effects are neglected:

bL sL cL cL b s

R L

γ b s

R L

g bL sL q q

Q1,2 Q7 Q8 Q3,4,5,6 current-current photonic dipole gluonic dipole penguin 4

SLIDE 9

Determination of B( ¯ B → Xsγ) in the SM: B( ¯ B → Xsγ)Eγ>E0 = B( ¯ B → Xce¯ ν)exp

• V ∗

tsVtb

Vcb

• 2 6αem

π C [P (E0) + N(E0)] pert. non-pert. ∼ 96% ∼ 4% Γ[b→Xp

s γ]Eγ>E0

|Vcb/Vub|2 Γ[b→Xp

ue¯

ν] =

• V ∗

tsVtb

Vcb

• 2 6αem

π

P (E0) C =

• Vub

Vcb

• 2 Γ[ ¯

B→Xce¯ ν] Γ[ ¯ B→Xue¯ ν]

semileptonic phase-space factor

The effective Lagrangian: Lweak ∼

i Ci Qi Eight operators Qi matter for BSM sγ when the NLO EW and/or CKM-suppressed effects are neglected:

bL sL cL cL b s

R L

γ b s

R L

g bL sL q q

Q1,2 Q7 Q8 Q3,4,5,6 current-current photonic dipole gluonic dipole penguin

Γ(b → Xp

sγ) =

G2

F m5 b, pole αem

32π4

• V ∗

tsVtb

• 2

8

• i,j=1

Ci(µb)Cj(µb) ˆ Gij,

( ˆ Gij= ˆ Gji)

4

SLIDE 10

Determination of B( ¯ B → Xsγ) in the SM: B( ¯ B → Xsγ)Eγ>E0 = B( ¯ B → Xce¯ ν)exp

• V ∗

tsVtb

Vcb

• 2 6αem

π C [P (E0) + N(E0)] pert. non-pert. ∼ 96% ∼ 4% Γ[b→Xp

s γ]Eγ>E0

|Vcb/Vub|2 Γ[b→Xp

ue¯

ν] =

• V ∗

tsVtb

Vcb

• 2 6αem

π

P (E0) C =

• Vub

Vcb

• 2 Γ[ ¯

B→Xce¯ ν] Γ[ ¯ B→Xue¯ ν]

semileptonic phase-space factor

The effective Lagrangian: Lweak ∼

i Ci Qi Eight operators Qi matter for BSM sγ when the NLO EW and/or CKM-suppressed effects are neglected:

bL sL cL cL b s

R L

γ b s

R L

g bL sL q q

Q1,2 Q7 Q8 Q3,4,5,6 current-current photonic dipole gluonic dipole penguin

Γ(b → Xp

sγ) =

G2

F m5 b, pole αem

32π4

• V ∗

tsVtb

• 2

8

• i,j=1

Ci(µb)Cj(µb) ˆ Gij,

( ˆ Gij= ˆ Gji)

NLO (O(αs)) – last missing pieces being evaluated by Tobias Huber and Lars-Thorben Moos

[arXiv:1912.07916]

Most important @ NNLO (O(α2

s)): ˆ

G77, ˆ G17, ˆ G27

known interpolated between the mc ≫ mb and mc = 0 limits [arXiv:1503.01791]

⇒ ± 3% uncertainty in BSM

sγ

4

SLIDE 11

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

5

SLIDE 12

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families). 5

SLIDE 13

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families).

• 2. Reduction to master integrals with the help of Integration By Parts (IBP) [KIRA, FIRE, LiteRed].

O(1 TB) RAM and weeks of CPU needed for the most complicated families. 5

SLIDE 14

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families).

• 2. Reduction to master integrals with the help of Integration By Parts (IBP) [KIRA, FIRE, LiteRed].

O(1 TB) RAM and weeks of CPU needed for the most complicated families.

• 3. Extending the set of master integrals Mk so that it closes under differentiation

with respect to z = m2 c/m2 b. This way one obtains a system of differential equations

d dz Mk(z, ǫ) =

• l

Rkl(z, ǫ) Ml(z, ǫ), (∗)

where Rnk are rational functions of their arguments. 5

SLIDE 15

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families).

• 2. Reduction to master integrals with the help of Integration By Parts (IBP) [KIRA, FIRE, LiteRed].

O(1 TB) RAM and weeks of CPU needed for the most complicated families.

• 3. Extending the set of master integrals Mk so that it closes under differentiation

with respect to z = m2 c/m2 b. This way one obtains a system of differential equations

d dz Mk(z, ǫ) =

• l

Rkl(z, ǫ) Ml(z, ǫ), (∗)

where Rnk are rational functions of their arguments.

• 4. Calculating boundary conditions for (∗) using automatized asymptotic expansions at mc ≫ mb.

5

SLIDE 16

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families).

• 2. Reduction to master integrals with the help of Integration By Parts (IBP) [KIRA, FIRE, LiteRed].

O(1 TB) RAM and weeks of CPU needed for the most complicated families.

• 3. Extending the set of master integrals Mk so that it closes under differentiation

with respect to z = m2 c/m2 b. This way one obtains a system of differential equations

d dz Mk(z, ǫ) =

• l

Rkl(z, ǫ) Ml(z, ǫ), (∗)

where Rnk are rational functions of their arguments.

• 4. Calculating boundary conditions for (∗) using automatized asymptotic expansions at mc ≫ mb.
• 5. Calculating three-loop single-scale master integrals for the boundary conditions.

5

SLIDE 17

Sample diagrams contributing to ˆ G27 @ NNLO:

c q b s b

• 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of

(a few) ×105 four-loop two-scale scalar integrals with unitarity cuts (O(500) families).

• 2. Reduction to master integrals with the help of Integration By Parts (IBP) [KIRA, FIRE, LiteRed].

O(1 TB) RAM and weeks of CPU needed for the most complicated families.

• 3. Extending the set of master integrals Mk so that it closes under differentiation

with respect to z = m2 c/m2 b. This way one obtains a system of differential equations

d dz Mk(z, ǫ) =

• l

Rkl(z, ǫ) Ml(z, ǫ), (∗)

where Rnk are rational functions of their arguments.

• 4. Calculating boundary conditions for (∗) using automatized asymptotic expansions at mc ≫ mb.
• 5. Calculating three-loop single-scale master integrals for the boundary conditions.
• 6. Solving the system (∗) numerically [A.C. Hindmarsch, http://www.netlib.org/odepack]

along an ellipse in the complex z plane. Doing so along several different ellipses allows us to estimate the numerical error. 5

SLIDE 18

Sample three-loop propagator-type integrals that parameterize large-z expansions of the master integrals: 6

SLIDE 19

Sample three-loop propagator-type integrals that parameterize large-z expansions of the master integrals: Contributions to ˆ G27(E0 = 0) from diagrams with closed loops of massless fermions

massless, 2body

z 105 0.001 0.1 10 10 20 30 40 massless, 4body, BLM

z 105 0.001 0.1 10 1.0 0.5 0.0 0.5 1.0 1.5 2.0

massless, 4body , nonBLM

z 105 0.001 0.1 10 1.0 0.5 0.0 0.5 1.0 1.5 2.0 agrees with hep-ph/0302051 agrees with hep-ph/9903305 new arXiv:0707.3090 arXiv:1009.5685

∆ ˆ G27 ≃ 3[0.164 + 0.13z

1 2 − 21.51z

+68.10z

3 2 − 46.12z2

−(3.23z − 18.23z2) ln z]

And from diagrams with closed loops of massive fermions

⇑

massive mb, 2body

z 105 0.001 0.1 10 20 10 10 20 30 40

massive mc, 2body

z 105 0.001 0.1 10 20 20 40 60 agrees with arXiv:0707.3090 agrees with arXiv:0707.3090

UV renormalization has been carried out using the results from arXiv:1702.07674. 6

SLIDE 20

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). 7

SLIDE 21

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs 7

SLIDE 22

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs Isospin-averaged decay rate: Γ ≃ A + 1

2(B + C)(Qu + Qd) + DQs

≡ A + δΓc 7

SLIDE 23

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs Isospin-averaged decay rate: Γ ≃ A + 1

2(B + C)(Qu + Qd) + DQs

≡ A + δΓc Isospin asymmetry: ∆0− ≃

C−B 2Γ (Qu − Qd)

7

SLIDE 24

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs Isospin-averaged decay rate: Γ ≃ A + 1

2(B + C)(Qu + Qd) + DQs

≡ A + δΓc Isospin asymmetry: ∆0− ≃

C−B 2Γ (Qu − Qd)

⇒

δΓc/Γ ∆0−

≃

(B+C)(Qu+Qd)+2DQs (C−B)(Qu−Qd)

=

Qu+Qd Qd−Qu

• 1 + 2 D−C

C−B

• 7

SLIDE 25

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs Isospin-averaged decay rate: Γ ≃ A + 1

2(B + C)(Qu + Qd) + DQs

≡ A + δΓc Isospin asymmetry: ∆0− ≃

C−B 2Γ (Qu − Qd)

⇒

δΓc/Γ ∆0−

≃

(B+C)(Qu+Qd)+2DQs (C−B)(Qu−Qd)

=

Qu+Qd Qd−Qu

• 1 + 2 D−C

C−B

• ւ
• Qu + Qd + Qs = 0

SU(3)F violation

MM, arXiv:0911.1651 7

SLIDE 26

Non-perturbative contribution from gluon-to-photon conversion in the QCD medium.

It was first considered by Lee, Neubert & Paz in hep-ph/0609224. It originates from hard gluon scattering on the valence quark or a “sea” quark that produces an energetic photon. The quark that undergoes this Compton-like scattering is assumed to remain soft in the ¯ B-meson rest frame to ensure effective interference with the leading “hard” amplitude. Without interference the contribution would be negligible (O(α2

sΛ2/m2 b)).

Suppression by Λ/mb can be understood as originating from dilution of the target (size of the ¯ B-meson ∼ Λ−1). Dominant in ∆0−: Γ[B− → Xsγ] ≃ A + BQu + CQd + DQs, Γ[ ¯ B0 → Xsγ] ≃ A + BQd + CQu + DQs Isospin-averaged decay rate: Γ ≃ A + 1

2(B + C)(Qu + Qd) + DQs

≡ A + δΓc Isospin asymmetry: ∆0− ≃

C−B 2Γ (Qu − Qd)

⇒

δΓc/Γ ∆0−

≃

(B+C)(Qu+Qd)+2DQs (C−B)(Qu−Qd)

=

Qu+Qd Qd−Qu

• 1 + 2 D−C

C−B

• ւ
• Qu + Qd + Qs = 0

SU(3)F violation

MM, arXiv:0911.1651 δΓc Γ ≃ −1

3∆0−

• 1 + 2 D−C

C−B

• = −1

3(−0.48 ± 1.49 ± 0.97 ± 1.15)% × (1 ± 0.3) = (0.16 ± 0.74)%

• Belle, arXiv:1807.04236, E0 = 1.9 GeV

Recall: (x ± σx)(y ± σy) = xy ±

• (xσy)2 + (yσx)2 + (σxσy)2

7

SLIDE 27

The resolved photon contribution to the Q7-Q1,2 interference.

M.B. Voloshin, hep-ph/9612483; A. Khodjamirian, R. R¨ uckl, G. Stoll and D. Wyler, hep-ph/9702318;

• Z. Ligeti, L. Randall and M.B. Wise, hep-ph/9702322; G. Buchalla, G. Isidori, G. Rey, hep-ph/9705253;
• M. Benzke, S.J. Lee, M. Neubert, G. Paz, arXiv:1003.5012; A. Gunawardana, G. Paz, arXiv:1908.02812.

¯ B| | ¯ B

2 7

c

δN(E0) = (C2 − 1

6C1)C7

• −

µ2

G

27m2

c + Λ17

mb

• −

κV µ2

G

27m2

c

8

SLIDE 28

The resolved photon contribution to the Q7-Q1,2 interference.

M.B. Voloshin, hep-ph/9612483; A. Khodjamirian, R. R¨ uckl, G. Stoll and D. Wyler, hep-ph/9702318;

• Z. Ligeti, L. Randall and M.B. Wise, hep-ph/9702322; G. Buchalla, G. Isidori, G. Rey, hep-ph/9705253;
• M. Benzke, S.J. Lee, M. Neubert, G. Paz, arXiv:1003.5012; A. Gunawardana, G. Paz, arXiv:1908.02812.

¯ B| | ¯ B

2 7

c

δN(E0) = (C2 − 1

6C1)C7

• −

µ2

G

27m2

c + Λ17

mb

• −

κV µ2

G

27m2

c

Λ17 = 2

3Re

∞

−∞ dω1 ω1

• 1 − F
• m2

c−iε

mbω1

• + mbω1

12m2

c

• h17(ω1, µ)

ω1 ↔ gluon momentum, F (x) = 4x arctan2 1/√4x − 1

• 8

SLIDE 29

The resolved photon contribution to the Q7-Q1,2 interference.

M.B. Voloshin, hep-ph/9612483; A. Khodjamirian, R. R¨ uckl, G. Stoll and D. Wyler, hep-ph/9702318;

• Z. Ligeti, L. Randall and M.B. Wise, hep-ph/9702322; G. Buchalla, G. Isidori, G. Rey, hep-ph/9705253;
• M. Benzke, S.J. Lee, M. Neubert, G. Paz, arXiv:1003.5012; A. Gunawardana, G. Paz, arXiv:1908.02812.

¯ B| | ¯ B

2 7

c

δN(E0) = (C2 − 1

6C1)C7

• −

µ2

G

27m2

c + Λ17

mb

• −

κV µ2

G

27m2

c

Λ17 = 2

3Re

∞

−∞ dω1 ω1

• 1 − F
• m2

c−iε

mbω1

• + mbω1

12m2

c

• h17(ω1, µ)

ω1 ↔ gluon momentum, F (x) = 4x arctan2 1/√4x − 1

• The soft function h17:

h17(ω1, µ) =

• dr

4πMBe−iω1r ¯

B|(¯ hS¯

n)(0)¯

n

• iγ⊥

α ¯

nβ(S†

¯ ngGαβ s S¯ n)(r¯

n)(S†

¯ nh)(0)| ¯

B (mb−2E0 ≫ ΛQCD) A class of models for h17:

h17(ω1, µ) = e−

ω2 1 2σ2

n a2nH2n

• ω1

σ √ 2

• ,

σ < 1 GeV

Hermite polynomials

Constraints on moments (e.g.):

• dω1h17 = 2

3µ2 G,

• dω1ω2

1h17 = 2 15(5m5 + 3m6 − 2m9).

8

SLIDE 30

The resolved photon contribution to the Q7-Q1,2 interference.

M.B. Voloshin, hep-ph/9612483; A. Khodjamirian, R. R¨ uckl, G. Stoll and D. Wyler, hep-ph/9702318;

• Z. Ligeti, L. Randall and M.B. Wise, hep-ph/9702322; G. Buchalla, G. Isidori, G. Rey, hep-ph/9705253;
• M. Benzke, S.J. Lee, M. Neubert, G. Paz, arXiv:1003.5012; A. Gunawardana, G. Paz, arXiv:1908.02812.

¯ B| | ¯ B

2 7

c

δN(E0) = (C2 − 1

6C1)C7

• −

µ2

G

27m2

c + Λ17

mb

• −

κV µ2

G

27m2

c

Λ17 = 2

3Re

∞

−∞ dω1 ω1

• 1 − F
• m2

c−iε

mbω1

• + mbω1

12m2

c

• h17(ω1, µ)

ω1 ↔ gluon momentum, F (x) = 4x arctan2 1/√4x − 1

• The soft function h17:

h17(ω1, µ) =

• dr

4πMBe−iω1r ¯

B|(¯ hS¯

n)(0)¯

n

• iγ⊥

α ¯

nβ(S†

¯ ngGαβ s S¯ n)(r¯

n)(S†

¯ nh)(0)| ¯

B (mb−2E0 ≫ ΛQCD) A class of models for h17:

h17(ω1, µ) = e−

ω2 1 2σ2

n a2nH2n

• ω1

σ √ 2

• ,

σ < 1 GeV

Hermite polynomials

Constraints on moments (e.g.):

• dω1h17 = 2

3µ2 G,

• dω1ω2

1h17 = 2 15(5m5 + 3m6 − 2m9).

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1(GeV)

h17(GeV)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1(GeV)

h17(GeV)

• 8

SLIDE 31

The resolved photon contribution to the Q7-Q1,2 interference.

M.B. Voloshin, hep-ph/9612483; A. Khodjamirian, R. R¨ uckl, G. Stoll and D. Wyler, hep-ph/9702318;

• Z. Ligeti, L. Randall and M.B. Wise, hep-ph/9702322; G. Buchalla, G. Isidori, G. Rey, hep-ph/9705253;
• M. Benzke, S.J. Lee, M. Neubert, G. Paz, arXiv:1003.5012; A. Gunawardana, G. Paz, arXiv:1908.02812.

¯ B| | ¯ B

2 7

c

δN(E0) = (C2 − 1

6C1)C7

• −

µ2

G

27m2

c + Λ17

mb

• −

κV µ2

G

27m2

c

Λ17 = 2

3Re

∞

−∞ dω1 ω1

• 1 − F
• m2

c−iε

mbω1

• + mbω1

12m2

c

• h17(ω1, µ)

ω1 ↔ gluon momentum, F (x) = 4x arctan2 1/√4x − 1

• The soft function h17:

h17(ω1, µ) =

• dr

4πMBe−iω1r ¯

B|(¯ hS¯

n)(0)¯

n

• iγ⊥

α ¯

nβ(S†

¯ ngGαβ s S¯ n)(r¯

n)(S†

¯ nh)(0)| ¯

B (mb−2E0 ≫ ΛQCD) A class of models for h17:

h17(ω1, µ) = e−

ω2 1 2σ2

n a2nH2n

• ω1

σ √ 2

• ,

σ < 1 GeV

Hermite polynomials

Constraints on moments (e.g.):

• dω1h17 = 2

3µ2 G,

• dω1ω2

1h17 = 2 15(5m5 + 3m6 − 2m9).

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1(GeV)

h17(GeV)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

1(GeV)

h17(GeV)

• G+P numerically:

Λ17 ∈ [−24, 5] MeV for mc = 1.17 GeV. Factor-of-3 improvement w.r.t. BLNP. In our code: κV = 1.2 ± 0.3. Warning: scheme for mc! 8

SLIDE 32

Non-perturbative contribution proportional to |C8|2

• A. Kapustin, Z. Ligeti & H. D. Politzer [hep-ph/9507248],
• A. Ferroglia & U. Haisch [arXiv:1009.2144],

focused on the collinear logs ln mb

ms in the corresponding contribution to P (E0).

⇒ fragmentation functions ⇒ effects below 1% in Bsγ. 9

SLIDE 33

Non-perturbative contribution proportional to |C8|2

• A. Kapustin, Z. Ligeti & H. D. Politzer [hep-ph/9507248],
• A. Ferroglia & U. Haisch [arXiv:1009.2144],

focused on the collinear logs ln mb

ms in the corresponding contribution to P (E0).

⇒ fragmentation functions ⇒ effects below 1% in Bsγ. Such logs were varied in the range [ln 10, ln 50] ≃

• ln mB

mK , ln mB mπ

• in the phenomenological analyses, which roughly reproduced

the fragmentation function estimates. 9

SLIDE 34

Non-perturbative contribution proportional to |C8|2

• A. Kapustin, Z. Ligeti & H. D. Politzer [hep-ph/9507248],
• A. Ferroglia & U. Haisch [arXiv:1009.2144],

focused on the collinear logs ln mb

ms in the corresponding contribution to P (E0).

⇒ fragmentation functions ⇒ effects below 1% in Bsγ. Such logs were varied in the range [ln 10, ln 50] ≃

• ln mB

mK , ln mB mπ

• in the phenomenological analyses, which roughly reproduced

the fragmentation function estimates.

• M. Benzke, S.J. Lee, M. Neubert & G. Paz [arXiv:1003.5012]

pointed out non-perturbative effects that are unrelated to the collinear logs. Their estimated range is [−0.3, 1.9]% of Bsγ for the overall non-perturbative effect being proportional to |C8|2, w.r.t. the mb

ms = 50 case in P (E0), for µb = 1.5 GeV and E0 = 1.6 GeV.

9

SLIDE 35

Non-perturbative contribution proportional to |C8|2

• A. Kapustin, Z. Ligeti & H. D. Politzer [hep-ph/9507248],
• A. Ferroglia & U. Haisch [arXiv:1009.2144],

focused on the collinear logs ln mb

ms in the corresponding contribution to P (E0).

⇒ fragmentation functions ⇒ effects below 1% in Bsγ. Such logs were varied in the range [ln 10, ln 50] ≃

• ln mB

mK , ln mB mπ

• in the phenomenological analyses, which roughly reproduced

the fragmentation function estimates.

• M. Benzke, S.J. Lee, M. Neubert & G. Paz [arXiv:1003.5012]

pointed out non-perturbative effects that are unrelated to the collinear logs. Their estimated range is [−0.3, 1.9]% of Bsγ for the overall non-perturbative effect being proportional to |C8|2, w.r.t. the mb

ms = 50 case in P (E0), for µb = 1.5 GeV and E0 = 1.6 GeV.

Numerically, we can reproduce this range by performing a replacement

ln mb

ms → κ88 ln 50

with κ88 = 1.7 ± 1.1

in all the perturbative contributions proportional to |C8|2. 9

SLIDE 36

Non-perturbative contribution proportional to |C8|2

• A. Kapustin, Z. Ligeti & H. D. Politzer [hep-ph/9507248],
• A. Ferroglia & U. Haisch [arXiv:1009.2144],

focused on the collinear logs ln mb

ms in the corresponding contribution to P (E0).

⇒ fragmentation functions ⇒ effects below 1% in Bsγ. Such logs were varied in the range [ln 10, ln 50] ≃

• ln mB

mK , ln mB mπ

• in the phenomenological analyses, which roughly reproduced

the fragmentation function estimates.

• M. Benzke, S.J. Lee, M. Neubert & G. Paz [arXiv:1003.5012]

pointed out non-perturbative effects that are unrelated to the collinear logs. Their estimated range is [−0.3, 1.9]% of Bsγ for the overall non-perturbative effect being proportional to |C8|2, w.r.t. the mb

ms = 50 case in P (E0), for µb = 1.5 GeV and E0 = 1.6 GeV.

Numerically, we can reproduce this range by performing a replacement

ln mb

ms → κ88 ln 50

with κ88 = 1.7 ± 1.1

in all the perturbative contributions proportional to |C8|2. The [ln 10, ln 50] range remains used in other (small) terms where collinear logs arise. 9

SLIDE 37

Updated SM predictions for Bsγ and Rγ ≡ B(s+d)γ/Bcℓ¯

ν (with E0 = 1.6 GeV):

Bsγ = (3.40 ± 0.17) × 10−4

compare to (3.36 ± 0.23) × 10−4 in arXiv:1503.01789

(±5.0%)

(±6.9%)

Rγ = (3.35 ± 0.16) × 10−3

compare to (3.31 ± 0.22) × 10−3 in arXiv:1503.01789

(±4.8%)

(±6.7%)

10

SLIDE 38

Updated SM predictions for Bsγ and Rγ ≡ B(s+d)γ/Bcℓ¯

ν (with E0 = 1.6 GeV):

Bsγ = (3.40 ± 0.17) × 10−4

compare to (3.36 ± 0.23) × 10−4 in arXiv:1503.01789

(±5.0%)

(±6.9%)

Rγ = (3.35 ± 0.16) × 10−3

compare to (3.31 ± 0.22) × 10−3 in arXiv:1503.01789

(±4.8%)

(±6.7%)

Current uncertainty budget in Bsγ: ±3% higher-order, ±3% interpolation in mc, ±2.5% parametric (including δΓc

Γ , κV and κ88)

10

SLIDE 39

Updated SM predictions for Bsγ and Rγ ≡ B(s+d)γ/Bcℓ¯

ν (with E0 = 1.6 GeV):

Bsγ = (3.40 ± 0.17) × 10−4

compare to (3.36 ± 0.23) × 10−4 in arXiv:1503.01789

(±5.0%)

(±6.9%)

Rγ = (3.35 ± 0.16) × 10−3

compare to (3.31 ± 0.22) × 10−3 in arXiv:1503.01789

(±4.8%)

(±6.7%)

Current uncertainty budget in Bsγ: ±3% higher-order, ±3% interpolation in mc, ±2.5% parametric (including δΓc

Γ , κV and κ88)

When the interpolation gets removed but nothing else changes: √ 32 + 2.52% = 3.9% – still somewhat behind the expected experimental ±2.6%. 10

SLIDE 40

Updated SM predictions for Bsγ and Rγ ≡ B(s+d)γ/Bcℓ¯

ν (with E0 = 1.6 GeV):

Bsγ = (3.40 ± 0.17) × 10−4

compare to (3.36 ± 0.23) × 10−4 in arXiv:1503.01789

(±5.0%)

(±6.9%)

Rγ = (3.35 ± 0.16) × 10−3

compare to (3.31 ± 0.22) × 10−3 in arXiv:1503.01789

(±4.8%)

(±6.7%)

Current uncertainty budget in Bsγ: ±3% higher-order, ±3% interpolation in mc, ±2.5% parametric (including δΓc

Γ , κV and κ88)

When the interpolation gets removed but nothing else changes: √ 32 + 2.52% = 3.9% – still somewhat behind the expected experimental ±2.6%. Shifts in uncertainties related to δΓc

Γ , κV and κ88:

formerly: 1.25% + 2.85% + 1.10% = 5.20% (in quadrature: 3.30%) at present: 0.74%+0.88%+0.92% = 2.54% (in quadrature: 1.48%) √ 1.482 + 2.012% = 2.49% ≃ 2.5% 10

SLIDE 41

Summary for the radiative decay

• Perturbative NNLO calculations of Γ[b → Xp

s γ] that aim at removing

the mc-interpolation have been finalized for diagrams involving closed fermion loops on the gluon lines. We confirm several published results, and supplement them with a previously unknown (tiny) piece.

11

SLIDE 42

Summary for the radiative decay

• Perturbative NNLO calculations of Γ[b → Xp

s γ] that aim at removing

the mc-interpolation have been finalized for diagrams involving closed fermion loops on the gluon lines. We confirm several published results, and supplement them with a previously unknown (tiny) piece.

• The isospin asymmetry ∆0− measured by Belle in 2018 helps to

suppress non-perturbative uncertainties in Bsγ, especially those arising in the Q7-Q8 interference.

11

SLIDE 43

Summary for the radiative decay

• Perturbative NNLO calculations of Γ[b → Xp

s γ] that aim at removing

the mc-interpolation have been finalized for diagrams involving closed fermion loops on the gluon lines. We confirm several published results, and supplement them with a previously unknown (tiny) piece.

• The isospin asymmetry ∆0− measured by Belle in 2018 helps to

suppress non-perturbative uncertainties in Bsγ, especially those arising in the Q7-Q8 interference.

• The 2019 reanalysis of non-perturbative effects in the Q1,2-Q7

interference by Gunawardana and Paz implies that the corresponding uncertainty gets reduced by a factor of three.

11

SLIDE 44

Summary for the radiative decay

• Perturbative NNLO calculations of Γ[b → Xp

s γ] that aim at removing

the mc-interpolation have been finalized for diagrams involving closed fermion loops on the gluon lines. We confirm several published results, and supplement them with a previously unknown (tiny) piece.

• The isospin asymmetry ∆0− measured by Belle in 2018 helps to

suppress non-perturbative uncertainties in Bsγ, especially those arising in the Q7-Q8 interference.

• The 2019 reanalysis of non-perturbative effects in the Q1,2-Q7

interference by Gunawardana and Paz implies that the corresponding uncertainty gets reduced by a factor of three.

• The updated SM predictions read Bsγ = (3.40 ± 0.17) × 10−4

and Rγ = (3.35 ± 0.16) × 10−3 for E0 = 1.6 GeV.

11

SLIDE 45

Determination of |Vcb| from the inclusive ¯ B → Xcℓν rate and spectra |Vcb| = (42.00 ± 0.64) × 10−3

[P. Gambino, K. J. Healey and S. Turczyk, arXiv:1606.06174] 1.5% roughly:

• (1.0%)2 + (1.1%)2 ≃ 1.5%

perturbative

• ther

O(α3

s)

12

SLIDE 46

Determination of |Vcb| from the inclusive ¯ B → Xcℓν rate and spectra |Vcb| = (42.00 ± 0.64) × 10−3

[P. Gambino, K. J. Healey and S. Turczyk, arXiv:1606.06174] 1.5% roughly:

• (1.0%)2 + (1.1%)2 ≃ 1.5%

perturbative

• ther

O(α3

s)

Impact on the uncertainty in the SM prediction for B(Bs → µ+µ−):

• (3.0%)2 + (2.3%)2 ≃ 3.8%

|Vcb|2

• ther

[ C. Bobeth, M. Gorbahn, T. Hermann, MM, E. Stamou and M. Steinhauser, arXiv:1311.0903], [ M. Beneke, C. Bobeth and R. Szafron, arXiv:1908.07011]. 12

SLIDE 47

Determination of |Vcb| from the inclusive ¯ B → Xcℓν rate and spectra |Vcb| = (42.00 ± 0.64) × 10−3

[P. Gambino, K. J. Healey and S. Turczyk, arXiv:1606.06174] 1.5% roughly:

• (1.0%)2 + (1.1%)2 ≃ 1.5%

perturbative

• ther

O(α3

s)

Impact on the uncertainty in the SM prediction for B(Bs → µ+µ−):

• (3.0%)2 + (2.3%)2 ≃ 3.8%

|Vcb|2

• ther

[ C. Bobeth, M. Gorbahn, T. Hermann, MM, E. Stamou and M. Steinhauser, arXiv:1311.0903], [ M. Beneke, C. Bobeth and R. Szafron, arXiv:1908.07011].

Impact on the uncertainty in the SM prediction for ǫK:

• (5.3%)2 + (6.4%)2 ≃ 8.3%

(roughly) |Vcb|4

• ther

using Eq. (17) of [ J. Brod, M. Gorbahn and E. Stamou, arXiv:1911.06822 ]. 12

SLIDE 48 ① ★⑩☛ ✁ é ➺ ➓ ✺➓ ⑩❘ ➮ ✁ ❢ ➫ ❵✂✄☎ ✆ ✝ ü Þ ✞ ✹✟ ✞ ✢ ❢ ➫ ✠ ✡ ➊ ♦ ✞ ✆ ☞ ✌ ä ➒ ✆ ✟ ❦ ✡➬ ❵✆ ✍ ✎ ✄✹✍ ➁ ♦ ä ✡➂ ✏ ❬ ✈ ❂ ✚ ✈ ✑ ✒ ✰ ✓ ➫ ✇ ✎ ✞ ✟ ☞ ✆ ✇✝ ✞ ✎ ☎ ✆ ✍ ✇ ✞ ➾ ✟ ✄✂✂ ✞ ✍ ✆ ✎ ✏ ❾✔ ✕ ❾✔ ý ✖ ❾✔ ✗ ✙ ✘ ÿ ✙ ✚ ÿ ✈ ✖ ❾✔ ✗ ❛ ✘ ÿ

• ✚

ÿ ✈ ✖ ❾✔ ✁ ✸ ✂ ✄ ☎ ✚ ✄ ✈ ✖ ❾✔ ✁ ➄ ✆ ✂ ✄ ✝ ✞ ✚ ✄ ✈ ✖ ✿ ✿ ✿ ❢ ❾ ✟ æ ✆ ✟ ✞ ☎ ✹ ✢❵ ➒ ✂ ✞ ➓ ✄✍ ❵ ✠ ❦ ✡☛ ❦ ✇ ❘ ✡ é➺ ✠ ä ☞ ✌ ❢ ü Þ ✞ ✍ ✹✍ ➍ ❵ ✞ ✟ ✂ ➒ ✟ ✇ ✆ ✂✄ ☞ ✞ ➓ ✌ ✍ ✆ ✢ ✄☎ ✎ ✄ ✎ ✞ ✍ ☎ ✝ ✹ ✎ ✞ ➓ ✄✍ ✂ ✹ ✂ Þ ✞ ♦ ä ✡ ❵✆ ✟ ✆ ✢✞ ✂ ✞ ✟ ✎ ✏ ✖ ✍ ❀ ✖ ✎ ❀ ✏ ✑ ❀ ✏ ✒✓ ✘ ❤ ★ ❥ ✔ ✕ ➫ ✗ ✛ ✜ ✢ ✢ ✢ ✗ ✛ ✣ ✟ ✜ ✤ ✤ ✤ ✣ ✕ ➫ ❥ ★ ✐ ❢ ♦ ä ✡ ❵✆ ✟ ✆ ✢✞ ✂ ✞ ✟ ✎ ✆ ✟ ✞ ✞ ➬ ✂ ✟ ✆ ☎ ✂ ✞ ➓ ➬ ✟ ✹ ✢ ➓ ✆ ✂ ✆ q ✥ ✦ ✧ ✩ ✦ ➾ ✪✫ ✬ ✭ ✮ ✯✰ ✮✱ ✬ ✲ ✳ ✲ ✱ ✴ ✵ ✮✮ ✭ ✶✱ ✷ ✹ ✵ ✶✺ ✯✭ ✻ ✱ ↔ ➒ ⑩ ✶ ✼ ß Þ➮ ✯✼ ✬ ✽✽✾ ➁ÿÿ ÿ ✁

• ✽

☎ ✈ ✺ ✲ ✮ ✳ ✃ ✭ ✶ ➀ ✭ ✹ ✱ ✴ ✵ ✮✮ ✭ ✶✱ æ ✭ ✻ ✼ ✴ ✰ ➀ ✼ ß Þ➮ ✯✼ ✆ ✆ ➁ÿÿ þ ✽ ✁ ÿ ✾ ÿ ÿ ✆ ✼ ✴ ✼ ❵ ✵ ✭ ✶ ✬ ➍➇ ➒ ✮⑩ Þ ❿ ✵ ✮ ✼ ÿ þ ✯ ✺ ÿÿ ÿ ÿ ✆

SLIDE 49

SLIDE 50 ❽ ➓ ➓ ☛ ✉ ☛ ÿ ✉ ☛

• ✉

✒ ✸ ✸ ✸ ♠ ❿ ➓ Ô❘ ✁ ➓ ✆ ❥ ➃ ➒ Þ★❥ ↔ ✂ ✄ ✔☎✝ ➁☎✞✟✞✠ ☎ ✡ ❝ ❘ ✢ ✁ ✇ ★☛ ➓ ❽ ❘ ☞ q ❥ ↔ ✂ ✄ ✘ ☎ ✞ ➁ ✌✍✍✎✠ ✘ ✘✡ ✏ ➓ ☞★ ✇ ✆ ☛ ➺ ❥ ✂ ß ✄ ✗✗✗ ➁ ✌✍✍ ✟✠ ✔✔ ✗ ✡ ✂ ❘ ✆ ❥ ✰ Ô❘ ❦ ★ ➓ ✑ ✆ ✇ ❥ ✂ ✓ ✕ ✘✟ ➁ ✌✍✍ ✟✠ ☎☎✝✍ ☎✎ q ✖ ✙ ✸ ✸ ♠ ✄ ➓ ✑ Þ ➓ ❦ ❥ ✄ ☛ ☛ ✹ ❥ ß ➒ ★ ✃ Þ ✇ ❥ ❿ ✚ ✛ ✂ ✍ ✘ ☎ ✌ ➁ ✌✍✍ ✘✠ ✍ ✗ ✌ q ✖ ❛ ✸ ✸ ♠ ✜☞✁ ➓ ❦ ❽ ✇ ❥ ❝ ❘ ✢ ✁ ✇ ★ ☛ ❥ ↔ ❘ ★ ➀✇ ❥ ❿ ✚ ✛ ✂ ☎✝✍☎ ➁ ✌✍☎✝ ✠ ☎✝✘✡ ✏ ❘ ★★ ➓ ☞❥ ✂ ✇ ➺ ☛ ➺ ❘ ❦ ☛ ➺ ❥ ✓ ☛ ✹ ➓ ★❽Þ ❘ ☞❥ ✂ ✓ ✕ ✞ ✌ ➁ ✌✍☎✎ ✠ ✍✎ ✝✍✌✎ q ✣ ✤ ✸ ✸ ✏ ❘ ★★ ➓ ☞❥ ✂ ✇ ➺ ☛ ➺ ❘ ❦ ☛ ➺ ❥ ✂ ✓ ✕ ☎ ✍✍ ➁ ✌✍☎ ✞✠ ✍ ✞ ✔✍✍ ☎ q ✣ ➄✥ ✸ ♠ þ❂ ✦ ✧ ✈ ✸ ✕ ❘ ✹✹✇ ★ ✃➓ ❦ ❥ ✏ ❘ ★★ ➓ ☞❥ ü ➒ ❦ ✑ Ô ✩ ✆ ❥ ❿ ✚ ✛ ✂ ✍ ✘ ✍✔ ➁ ✌✍✍✘✠ ✍ ✟✘ þ❂ ✦ ✪ ✈ ✸ ✏ ❘ ★★ ➓ ☞❥ ü ➒ ❦ ✑ Ô ✩ ✆ ❥ ✫ ❦ ❘ ☞❽ ✹➓ ➺ ❥ ❿ ✚ ✛ ✂ ☎✍☎☎ ➁ ✌✍☎✍ ✠ ☎✍ ✞ ✦ ❋ ✬ ✭ ✈ ✸ ✸ ✄ ✇ ✃✇ ❥ ð Þ ✇ ✮ ✢ ❘ ★ ❥ ✫ ❦ ❘ ☞❽ ✹ ➓ ➺ ❥ ✯ ❘ ✇ ★ ✹ Þ❽ ➓ ✇ ★❥ ✂ ✓ ✕ ✎ ✗ ➁ ☎ ✞✞✘✠ ✝✍☎ ✘ ✡ ✰ Ô❘ ❦ ★ ➓ ✑ ✆ ✇ ❥ ✏ ➓ ☞★ ✇ ✆ ☛ ➺ ❥ ✫ ❦ ❘ ☞❽ ✹➓ ➺ ❥ ✂ ✓ ß ✟ ✍ ➁ ☎ ✞ ✞✟✠ ✔☎ ✟✞q ✏ q ❵ ❘ ➓ ☞ ✄ ➍ ß ➒ ★ ✑ Þ ❿ ❘ ★ q ÿ þ ✹ ❽ ÿÿ ÿ ÿ þ

SLIDE 51

SLIDE 52

Feasibility of b → Xcℓ¯ ν @ N3LO

b c b b c b ℓ ν M

− →

contribution to Γ contribution to dΓ/dq2 for q2 = M 2 15

SLIDE 53

Feasibility of b → Xcℓ¯ ν @ N3LO

b c b b c b ℓ ν M

− →

contribution to Γ contribution to dΓ/dq2 for q2 = M 2

Let us consider q2 = m2

c:

Im    

from Real boundary condition for the differential equations at mc ≫ mb 15

SLIDE 54

Feasibility of b → Xcℓ¯ ν @ N3LO

b c b b c b ℓ ν M

− →

contribution to Γ contribution to dΓ/dq2 for q2 = M 2

Let us consider q2 = m2

c:

Im    

from Real boundary condition for the differential equations at mc ≫ mb Possible IBP outsourcing: Fraunhofer Institute for Industrial Mathematics [D. Bendle et al., arXiv:1908.04301] 15

SLIDE 55

BACKUP SLIDES

16

SLIDE 56

The “hard” contribution to ¯ B → Xsγ

• J. Chay, H. Georgi, B. Grinstein PLB 247 (1990) 399.

A.F. Falk, M. Luke, M. Savage, PRD 49 (1994) 3367.

Goal: calculate the inclusive sum ΣXs

• C7(µb)Xsγ|O7| ¯

B + C2(µb)Xsγ|O2| ¯ B + ...

• 2

γ γ q q ¯ B ¯ B

7 7

Im{ } ≡ Im A

The “77” term in this sum is “hard”. It is related via the

• ptical theorem to the imaginary part of the elastic forward

scattering amplitude ¯ B( p = 0)γ( q) → ¯ B( p = 0)γ( q): When the photons are soft enough, m2

Xs = |mB(mB − 2Eγ)| ≫ Λ2 ⇒ Short-distance dominance ⇒ OPE.

However, the ¯ B → Xsγ photon spectrum is dominated by hard photons Eγ ∼ mb/2. Once A(Eγ) is considered as a function of arbitrary complex Eγ, ImA turns out to be proportional to the discontinuity of A at the physical cut. Consequently, Im Eγ 1 Emax

γ

Re Eγ [GeV] ≃ 1

2mB

Emax

γ

1 GeV

dEγ Im A(Eγ) ∼

• circle

dEγ A(Eγ). Since the condition |mB(mB − 2Eγ)| ≫ Λ2 is fulfilled along the circle, the OPE coefficients can be calculated perturbatively, which gives A(Eγ)|

circle ≃

• j
• F (j)

polynomial(2Eγ/mb)

mnj

b (1 − 2Eγ/mb)kj + O (αs(µhard))

• ¯

B( p = 0)|Q(j)

local operator| ¯

B( p = 0). Thus, contributions from higher-dimensional operators are suppressed by powers of Λ/mb. At (Λ/mb)0: ¯ B( p)|¯ bγµb| ¯ B( p) = 2pµ ⇒ Γ( ¯ B → Xsγ) = Γ(b → Xparton

s

γ) + O(Λ/mb). At (Λ/mb)1: Nothing! All the possible operators vanish by the equations of motion. At (Λ/mb)2: ¯ B( p)|¯ bvDµDµbv| ¯ B( p) ∼ mB µ2

π,

¯ B( p)|¯ bvgsGµνσµνbv| ¯ B( p) ∼ mB µ2

G,

The HQET heavy-quark field: bv(x) = 1

2(1 + v

/)b(x) exp(imb v · x) with v = p/mB. 17

SLIDE 57

The same method has been applied to the 3-loop counterterm diagrams

[MM, A. Rehman, M. Steinhauser, PLB 770 (2017) 431]

Master integrals:

I1 I7 I13

x

I2 I8 I14

x

x

I3 I9 I15

x

I4 I10 I16

x

I5 I11 I17 I6 I12 I18

18

SLIDE 58

Results for the bare NLO contributions up to O(ǫ):

ˆ G(1)2P

27

= − 92

81ǫ + f0(z) + ǫf1(z) z→0

− → − 92

81ǫ − 1942 243 + ǫ

• −26231

729 + 259 243π2

10 7 10 5 0.001 0.1 10 5 5 10 10 7 10 5 0.001 0.1 10 40 30 20 10

f0(z) f1(z) z z

Dots: solutions to the differential equations and/or the exact z → 0 limit. Lines: large- and small-z asymptotic expansions Small-z expansions of ˆ G(1)2P

27

: f0 from C. Greub, T. Hurth, D. Wyler, hep-ph/9602281, hep-ph/9603404,

• A. J. Buras, A. Czarnecki, MM, J. Urban, hep-ph/0105160,

f1 from H.M. Asatrian, C. Greub, A. Hovhannisyan, T. Hurth and V. Poghosyan, hep-ph/0505068.

2 7

19

SLIDE 59

Analogous results for the 3-body final state contributions (δ = 1): ˆ G(1)3P

27

= g0(z) + ǫg1(z)

z→0

− → − 4

27 − 106 81 ǫ

2 7

10 7 10 5 0.001 0.1 10 0.20 0.15 0.10 0.05 0.00 0.05 0.10 10 7 10 5 0.001 0.1 10 1.5 1.0 0.5 0.0 0.5

g0(z) g1(z) z z

Dots: solutions to the differential equations and/or the exact z → 0 limit. Lines: exact result for g0, as well as large- and small-z asymptotic expansions for g1. g0(z) =    − 4

27 − 14 9 z + 8 3z2 + 8 3z(1 − 2z) s L + 16 9 z(6z2 − 4z + 1)

• π2

4 − L2

, for z ≤ 1

4,

− 4

27 − 14 9 z + 8 3z2 + 8 3z(1 − 2z) t A + 16 9 z(6z2 − 4z + 1) A2,

for z > 1

4,

where s = √1 − 4z, L = ln(1 + s) − 1

2 ln 4z, t = √4z − 1, and A = arctan(1/t).

20