( ∗ ) and feasibility of b → X c ℓ ¯ b → X s γ @ N 2 LO ν @ N 3 LO 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 s ) contributions to ˆ G 17 and ˆ (i) O ( α 2 G 27 (ii) Non-perturbative effects in ¯ B → X s γ (iii) Updated SM predictions for B sγ and R γ 3. The semileptonic decay (i) Motivation for O ( α 3 s ) (ii) Challenges 4. Summary
R ( D ) and R ( D ∗ ) “anomalies” [https://hflav.web.cern.ch] (3 . 1 σ ) 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” ( > 5 σ ) [see, e.g., J. Aebischer et al. , arXiv:1903.10434] γ α l l b L s L Q ℓ 9 = γ α γ 5 l l b L s L Q ℓ 10 = ℓ = e or µ γ b s C 7, the Wilson coefficient of Q 7 = R L 2 is an important input in the fits.
Sample Leading-Order (LO) contributions to C 7 in the SM and beyond: γ γ γ γ γ ˜ ˜ u, c, t u, c, t W ± W ± t t t t µ µ W ± H ± χ ± b s b u, c, t s b s b s b LQ s � �� � ⇒ M H ± > ∼ 800 GeV in the 2HDM-II 3
Sample Leading-Order (LO) contributions to C 7 in the SM and beyond: γ γ γ γ γ ˜ ˜ u, c, t u, c, t W ± W ± t t t t µ µ W ± H ± χ ± b s b u, c, t s b s b s b LQ s � �� � ⇒ M H ± > ∼ 800 GeV in the 2HDM-II The strongest experimental constraint on C 7 comes from B sγ — ¯ — the CP- and isospin-averaged BR of B → X s γ and B → X ¯ s γ . ( ¯ B 0 , B − ) ( B 0 , B + ) 3
Sample Leading-Order (LO) contributions to C 7 in the SM and beyond: γ γ γ γ γ ˜ ˜ u, c, t u, c, t W ± W ± t t t t µ µ W ± H ± χ ± b s b u, c, t s b s b s b LQ s � �� � ⇒ M H ± > ∼ 800 GeV in the 2HDM-II The strongest experimental constraint on C 7 comes from B sγ — ¯ — the CP- and isospin-averaged BR of B → X s γ and B → X ¯ s γ . ( ¯ B 0 , B − ) ( B 0 , B + ) HFLAV, arXiv:1909.12524: B exp sγ = (3 . 32 ± 0 . 15) × 10 − 4 for E γ > E 0 = 1 . 6 GeV ≃ m b 3 , ( ± 4 . 5%) averaging CLEO, BELLE and BABAR with E 0 ∈ [1 . 7 , 2 . 0] GeV, and then extrapolating to E 0 = 1 . 6 GeV. � � ∼ m b TH requirement: E 0 should be large but not too close to the endpoint ( m b − 2 E 0 ≫ Λ QCD ). 2 3
Sample Leading-Order (LO) contributions to C 7 in the SM and beyond: γ γ γ γ γ ˜ ˜ u, c, t u, c, t W ± W ± t t t t µ µ W ± H ± χ ± b s b u, c, t s b s b s b LQ s � �� � ⇒ M H ± > ∼ 800 GeV in the 2HDM-II The strongest experimental constraint on C 7 comes from B sγ — ¯ — the CP- and isospin-averaged BR of B → X s γ and B → X ¯ s γ . ( ¯ B 0 , B − ) ( B 0 , B + ) HFLAV, arXiv:1909.12524: B exp sγ = (3 . 32 ± 0 . 15) × 10 − 4 for E γ > E 0 = 1 . 6 GeV ≃ m b 3 , ( ± 4 . 5%) averaging CLEO, BELLE and BABAR with E 0 ∈ [1 . 7 , 2 . 0] GeV, and then extrapolating to E 0 = 1 . 6 GeV. � � ∼ m b TH requirement: E 0 should be large but not too close to the endpoint ( m b − 2 E 0 ≫ Λ QCD ). 2 With the full BELLE-II dataset, a ± 2 . 6% uncertainty in the world average for B exp is expected. sγ SM calculations must be improved to reach a similar precision. 3
Determination of B ( ¯ B → X s γ ) in the SM: � � 2 6 α em V ∗ ts V tb B ( ¯ B → X s γ ) Eγ>E 0 = B ( ¯ � � B → X c e ¯ ν ) exp π C [ P ( E 0 ) + N ( E 0 )] � � V cb pert. non-pert. ∼ 96% ∼ 4% � � � � Γ[ b → X p 2 6 α em 2 Γ[ ¯ V ∗ s γ ] Eγ>E 0 ts V tb � V ub B → X c e ¯ ν ] � � � � ν ] = P ( E 0 ) C = | V cb /V ub | 2 Γ[ b → X p Γ[ ¯ � � � V cb π V cb B → X u e ¯ ν ] u e ¯ semileptonic phase-space factor 4
Determination of B ( ¯ B → X s γ ) in the SM: � � 2 6 α em V ∗ ts V tb B ( ¯ B → X s γ ) Eγ>E 0 = B ( ¯ � � B → X c e ¯ ν ) exp π C [ P ( E 0 ) + N ( E 0 )] � � V cb pert. non-pert. ∼ 96% ∼ 4% � � � � Γ[ b → X p 2 6 α em 2 Γ[ ¯ V ∗ s γ ] Eγ>E 0 ts V tb � V ub B → X c e ¯ ν ] � � � � ν ] = P ( E 0 ) C = | V cb /V ub | 2 Γ[ b → X p Γ[ ¯ � � � V cb π V cb B → X u e ¯ ν ] u e ¯ semileptonic phase-space factor L weak ∼ � The effective Lagrangian: i C i Q i Eight operators Q i matter for B SM when the NLO EW and/or CKM-suppressed effects are neglected: sγ γ g c L c L q q b L s L b s b s b L s L R L R L Q 1 , 2 Q 7 Q 8 Q 3 , 4 , 5 , 6 current-current photonic dipole gluonic dipole penguin 4
Determination of B ( ¯ B → X s γ ) in the SM: � � 2 6 α em V ∗ ts V tb B ( ¯ B → X s γ ) Eγ>E 0 = B ( ¯ � � B → X c e ¯ ν ) exp π C [ P ( E 0 ) + N ( E 0 )] � � V cb pert. non-pert. ∼ 96% ∼ 4% � � � � Γ[ b → X p 2 6 α em 2 Γ[ ¯ V ∗ s γ ] Eγ>E 0 ts V tb � V ub B → X c e ¯ ν ] � � � � ν ] = P ( E 0 ) C = | V cb /V ub | 2 Γ[ b → X p Γ[ ¯ � � � V cb π V cb B → X u e ¯ ν ] u e ¯ semileptonic phase-space factor L weak ∼ � The effective Lagrangian: i C i Q i Eight operators Q i matter for B SM when the NLO EW and/or CKM-suppressed effects are neglected: sγ γ g c L c L q q b L s L b s b s b L s L R L R L Q 1 , 2 Q 7 Q 8 Q 3 , 4 , 5 , 6 current-current photonic dipole gluonic dipole penguin 8 G 2 F m 5 b, pole α e m � � � � 2 C i ( µ b ) C j ( µ b ) ˆ Γ( b → X p � V ∗ s γ ) = ts V tb G ij , ( ˆ Gij = ˆ Gji ) 32 π 4 i,j =1 4
Determination of B ( ¯ B → X s γ ) in the SM: � � 2 6 α em V ∗ ts V tb B ( ¯ B → X s γ ) Eγ>E 0 = B ( ¯ � � B → X c e ¯ ν ) exp π C [ P ( E 0 ) + N ( E 0 )] � � V cb pert. non-pert. ∼ 96% ∼ 4% � � � � Γ[ b → X p 2 6 α em 2 Γ[ ¯ V ∗ s γ ] Eγ>E 0 ts V tb � V ub B → X c e ¯ ν ] � � � � ν ] = P ( E 0 ) C = | V cb /V ub | 2 Γ[ b → X p Γ[ ¯ � � � V cb π V cb B → X u e ¯ ν ] u e ¯ semileptonic phase-space factor L weak ∼ � The effective Lagrangian: i C i Q i Eight operators Q i matter for B SM when the NLO EW and/or CKM-suppressed effects are neglected: sγ γ g c L c L q q b L s L b s b s b L s L R L R L Q 1 , 2 Q 7 Q 8 Q 3 , 4 , 5 , 6 current-current photonic dipole gluonic dipole penguin 8 G 2 F m 5 b, pole α e m � � � � 2 C i ( µ b ) C j ( µ b ) ˆ Γ( b → X p � V ∗ s γ ) = ts V tb G ij , ( ˆ Gij = ˆ Gji ) 32 π 4 i,j =1 NLO ( O ( α s )) – last missing pieces being evaluated by Tobias Huber and Lars-Thorben Moos s )): ˆ G 77 , ˆ G 17 , ˆ [arXiv:1912.07916] G 27 Most important @ NNLO ( O ( α 2 known interpolated between the m c ≫ m b and m c = 0 limits [arXiv:1503.01791] 4 ± 3% uncertainty in B SM ⇒ sγ
Sample diagrams contributing to ˆ G 27 @ NNLO: c q b s b 5
Sample diagrams contributing to ˆ G 27 @ NNLO: c q b s b 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of (a few) × 10 5 four-loop two-scale scalar integrals with unitarity cuts ( O (500) families). 5
Sample diagrams contributing to ˆ G 27 @ NNLO: c q b s b 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of (a few) × 10 5 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
Sample diagrams contributing to ˆ G 27 @ NNLO: c q b s b 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of (a few) × 10 5 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 M k so that it closes under differentiation with respect to z = m 2 c /m 2 b . This way one obtains a system of differential equations d � dz M k ( z, ǫ ) = R kl ( z, ǫ ) M l ( z, ǫ ) , ( ∗ ) l where R nk are rational functions of their arguments. 5
Sample diagrams contributing to ˆ G 27 @ NNLO: c q b s b 1. Generation of diagrams and performing the Dirac algebra to express everything in terms of (a few) × 10 5 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 M k so that it closes under differentiation with respect to z = m 2 c /m 2 b . This way one obtains a system of differential equations d � dz M k ( z, ǫ ) = R kl ( z, ǫ ) M l ( z, ǫ ) , ( ∗ ) l where R nk are rational functions of their arguments. 4. Calculating boundary conditions for ( ∗ ) using automatized asymptotic expansions at m c ≫ m b . 5
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