Outline Motivation s + in the SM B 0 QED corrections in QCD - - PowerPoint PPT Presentation

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Enhanced QED correction to Bs → µ+µ−

(M. Beneke, C. Bobeth, R. Szafron, Phys. Rev. Lett. 120, 011801)

Robert Szafron

Technische Universit¨ at M¨ unchen

9 January 2018 XXIV Cracow Epiphany Conference

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Outline

◮ Motivation ◮ B0 s → µ+µ− in the SM ◮ QED corrections in QCD bound states ◮ Results ◮ Conclusions

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The Quest for New Physics

How can we discover BSM physics?

We need a measurement that shows significant discrepancy with the SM prediction

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The Quest for New Physics

How can we discover BSM physics?

We need a measurement that shows significant discrepancy with the SM prediction

Apart from the direct searches at the LHC, the good candidates are

◮ Precise low-energy measurements (e.g. g − 2, Lamb shift) ◮ Rare process suppressed/forbidden in the SM (e.g. CLFV,

FCNC)

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The Quest for New Physics

How can we discover BSM physics?

We need a measurement that shows significant discrepancy with the SM prediction

Apart from the direct searches at the LHC, the good candidates are

◮ Precise low-energy measurements (e.g. g − 2, Lamb shift) ◮ Rare process suppressed/forbidden in the SM (e.g. CLFV,

FCNC)

The key point is

We need a precise SM theoretical prediction!

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Strange B-meson

A good candidate for NP searches

◮ A meson composed of b and s

quarks

◮ B0 s and B s oscillate ◮ Among many its decay modes,

B0

s → µ+µ− is particularly

important for NP searches

b ¯ s Bs

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Bs → µ+µ−

In the SM the process is

◮ loop suppressed (FCNC)

SM helicity and loop suppression make it very sensitive to BSM scalar interactions.

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Bs → µ+µ−

In the SM the process is

◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson

decaying into energetic muons) SM helicity and loop suppression make it very sensitive to BSM scalar interactions.

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Bs → µ+µ−

In the SM the process is

◮ loop suppressed (FCNC) ◮ helicity suppressed (scalar meson

decaying into energetic muons)

◮ purely leptonic final state allows for

a precise SM prediction, QCD contained in the meson decay constant fBs SM helicity and loop suppression make it very sensitive to BSM scalar interactions.

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Experimental status

Bs → µ+µ− has been observed by LHCb and CMS in 2013

[R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 111, 101805] [S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett. 111, 101804]

Joint publication:

[CMS Collaboration and LHCb Collaboration, Nature 522, 68–72]

B(Bs → µ+µ−)LHCb+CMS = (2.8+0.7

−0.6) · 10−9

Most recent update B(Bs → µ+µ−)LHCb = (3.0+0.7

−0.6) · 10−9

[R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 118, 191801]

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Bs → µ+µ− in the SM

◮ Scales above mb can be integrated-out ◮ Electroweak physics is hidden in the matching coefficients of

weak interaction EFT

◮ Systematic expansion in mb mW ◮ Large logs are summed using usual RG evolution

[C. Bobeth, P. Gambino, M. Gorbahn, and U. Haisch, 2004; T. Huber, E. Lunghi, M. Misiak, and D. Wyler,2006]

L∆B=1 = 4GF √ 2

10

• i=1

CiQi + h.c.

b s W l + u,c,t l − Z u,c,t

b s l + l −

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Theory Status

Matching coefficients are known up to

◮ NLO EW [C. Bobeth, M. Gorbahn, E. Stamou, 2014 ] ◮ NNLO QCD [T. Hermann, M. Misiak, M. Steinhauser, 2013]

B(Bs → µ+µ−)TH = (3.65 ± 0.23) · 10−9

[C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, M. Steinhauser, Phys.Rev.Lett. 112 (2014) 101801]

◮ QED corrections below the mb scale not included ◮ Estimated size of the correction O(αem) ∼ 0.3%

Real radiation - only ultra-soft photons are important

◮ ISR is small (Low Soft-Photon Theorem) [Y. Aditya, K.Healey,

• A. Petrov, Phys.Rev. D87 (2013) 074028 ]

◮ FSR - included in the experimental analysis [A. J. Buras, J.

Girrbach, D. Guadagnoli, G. Isidori, Eur.Phys.J. C72 (2012) 2172 ]

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Scales in the problem

Leptonic decay of Bs is a multi-scale problem

◮ Electroweak scale mW ◮ Hard scale mb ◮ Hard-collinear scale

• mbΛQCD

◮ Soft scale ΛQCD ◮ Collinear scale mµ

We take ΛQCD ∼ mµ so the soft scale of HQEFT is also a soft scale of SCETI

SM Weak EFT SCETI ⊗ HQEFT SCETII ⊕ HQEFT

m2

W → ∞

m2

b → ∞

mbΛQCD → ∞

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QED corrections in QCD bound-states

The final state has no strong interaction – QCD is contained in the decay constant 0|¯ qγµγ5b| ¯ Bq(p) = ifBqpµ This is no longer true when QED effects are included – non-local time ordered products have to be evaluated 0|

• d4x T{jQED(x), L∆B=1(0)}| ¯

Bq This can be done for QED bound-states but QCD is non-perturbative at low scales

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Helicity suppression

Can the helicity suppression be relaxed?

b ¯ q ℓ ¯ ℓ Bs ¯ ℓγµγ5ℓ → mℓ

mb

¯ ℓcγ5ℓ¯

c

For mℓ → 0 the amplitude has to vanish Annihilation and helicity flip take place at the same point r

1 mb

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Helicity suppression

Can the helicity suppression be relaxed?

b ¯ q ℓ ¯ ℓ Bs ¯ ℓγµγνℓ →

mℓ ΛQCD ¯

ℓcγ5ℓ¯

c

Annihilation and helicity flip can be separated by r ∼

1

√

mbΛQCD

It is still a short distance effect since the size of the meson is r ∼

1 ΛQCD

For mℓ → 0 the amplitude still vanishes

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SCET approach

b ¯ q ℓ ¯ ℓ γ h.c.

EFT approach allows to perform systematic expan- sion in

ΛQCD mb .

Two step matching is required: Ef- fective weak interaction → SCETI → SCETII Modes

◮ Hard collinear, p2 ∼ ΛQCDmb ◮ Collinear, p2 ∼ Λ2 QCD ∼ m2 µ ◮ Soft p2 ∼ Λ2 QCD

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SCET approach

b ¯ q ℓ ¯ ℓ γ h.c.

The quark is only non-local along the direction of the lepton Soft quarks do not contribute to the power enhanced correction The quark has a hard- collinear virtuality – soft glu-

• ns are power suppressed

0| q(x−)βhv(0)α

• Bq(p)
• = −ifBqmb

4 × ∞ dωe−iωt 1 + / v 2

• φ+(ω)/

n+ + φ−(ω)

• /

n− − n−lγν

⊥

∂ ∂lν

⊥

• γ5
• αβ

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Operators

Q9 = αem 4π (¯ qγµPLb)(¯ ℓγµℓ) Q10 = αem 4π

• ¯

qγµPLb ¯ ℓγµγ5ℓ

• Q7

= e 16π2 mb

• ¯

qσµνPRb

• Fµν

C eff

7

C eff

9

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

◮ Tree level amplitude

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

◮ Helicity suppression × power enhancement factor

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

◮ Convolution from the hard-scale matching

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

◮ Convolution with the light-cone distribution function

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The correction at amplitude level

b ¯ q γ C9,10 ¯ ℓ ℓ ¯ q ℓ

b ¯ q γ C7 ¯ ℓ ℓ ¯ q ℓ γ b ¯ q γ Ci ¯ ℓ ℓ q′ γ ℓ ¯ q

iA = mℓfBqN C10 ¯ ℓγ5ℓ + αem 4π QℓQq mℓmB fBqN ¯ ℓ(1 + γ5)ℓ ×

• 1

0 du (1 − u) C eff 9 (um2 b)

∞

dω ω φB+(ω)

• ln mbω

m2

ℓ

+ ln u 1 − u

• − QℓC eff

7

∞

dω ω φB+(ω)

• ln2 mbω

m2

ℓ

− 2 ln mbω m2

ℓ

+ 2π2 3

◮ Double logarithmic enhancement due to endpoint singularity

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Non-perturbative contribution

Non-perturbative physics is encoded in the moments of B-meson light-cone distribution function [M. Beneke, G. Buchalla, M. Neubert,

and C. T. Sachrajda, 1999]

1 λB(µ) ≡ ∞ dω ω φB+(ω, µ), σn(µ) λB(µ) ≡ ∞ dω ω lnn µ0 ω φB+(ω, µ) λB(1 GeV) = (275 ± 75) MeV σ1(1 GeV) = 1.5 ± 1 σ2(1 GeV) = 3 ± 2 Power-enhancement factor mB ∞ dω ω φB+(ω) lnk ω ∼ mB λB × σk

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Numerical prediction: the power-enhanced correction

Total power-enhanced correction changes the branching fraction by: −(0.3 − 1.1)% Cancellation between C7 and C9 part central value: −0.6% = 1.1% − 1.7% (C7, C9 parts ). Uncertainty comes from λB, σ1, σ2. C7 part is surprisingly large thanks to double logs enhancement. Previous estimate of QED uncertainty was 0.3%, obtained by scale variation method. This uncertainty is still present. Not enhanced QED corrections: work in progress

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Numerical prediction: the branching fraction

For the rate, the correction can be incorporated by C10 → C10 + αem 4π QℓQq∆QED . with ∆QED = (33 − 119) + i (9 − 23), New prediction for the branching ratio B(Bs → µ+µ−)SM = (3.57 ± 0.17) · 10−9 Uncertainty:

◮ parametric: ±0.167 (now dominates but it is expected to be

reduced in the future)

◮ non-parametric non-QED: ±0.043 ◮ QED +0.022 −0.030 (∼ 0.84%)

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QED effects in other observables

QED generates a scalar operator that can mimic New Physics, but the effect is small. For observables related to the time-dependent rate asymmetry

[K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, A. Pellegrino, N. Tuning Phys. Rev. Lett. 109, 041801] we find

Aλ

∆Γ

= 1 − r2|∆QED|2 ≈ 1 − 1.0 · 10−5 , Cλ = −ηλ 2r Re(∆QED) ≈ ηλ 0.6% , Sλ = 2r Im(∆QED) ≈ −0.1% , where r ≡ αem

4π QℓQq C10

and ηL/R = ±1.

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Conclusions

◮ Radiative corrections in bound states can have surprisingly

complex pattern

◮ QED correction to QCD bound states can exhibit power

enhancement that cannot be anticipated without detailed computation

◮ Systematic progress is possible thanks to EFT approach

(NRQED, HQEFT, SCET)

◮ In precision physics, scale variation is not the most reliable

way to estimate uncertainty

◮ Coming soon: Next-to-leading power QED corrections

(non-enhanced terms)

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Auxiliary slide: Input parameters

Parameter Value Reference α(5)

s

(mZ ) 0.1181(11) PDG 2016 1/Γs

H [ps]

1.609(10) PDG 2016 fBs [MeV] 228.4(3.7)

• S. Aoki et al., Eur. Phys. J. C77, 112 (2017)

|V ∗

tbVts/Vcb|

0.982(1)

• M. Bona (UTfit), PoS ICHEP2016, 554 (2016)

|Vcb| 0.04200(64)

• P. Gambino et al., Phys. Lett. B763, 60 (2016)

Remaining parameters are the same as in [C. Bobeth, M. Gorbahn, T.

Hermann, M. Misiak, E. Stamou, and M. Steinhauser, Phys. Rev. Lett. 112, 101801 (2014)]

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Auxiliary slide: Uncertainty breakdown

◮ Parametric: ±0.167;

4.7%

◮ Non-parametric non-QED: ±0.043;

1.2%

◮ QED: +0.022 −0.030; +0.6 −0.8%

added in quadrature The QED uncertainty is almost as large as the non-parametric non-QED uncertainty.