Bs,d0 +- Decays in the [10pt] Aligned Two-Higgs-Doublet Model

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B

⁰_s,d

→ ℓ

⁺

ℓ

⁻

Decays in the Aligned Two-Higgs-Doublet Model

Jie Lu

IFIC, Valencia University, Spain

in collaboration with Antonio Pich and Xin-Qiang Li, 1404.5865

ICHEP 2014, Valencia Spain

July 4th, 2014

Jie Lu B0

s,d→ℓ+ℓ−Decays in the [10pt] Aligned Two-Higgs-Doublet Model 1 / 20

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1 Introduction

2 Aligned 2HDM

3 Calculation of Wilson Coefficients

4 Numerical Analysis

Jie Lu B0 →ℓ+ℓ−Decays in the [10pt] Aligned Two-Higgs-Doublet Model 2 / 20

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. . . . . Introduction

Motivation

Three important features of purely leptonic B-meson decays: B⁰_s,d→ℓ⁺ℓ⁻ Very small branching ratio;

loop induced only process in SM helicity-suppressed by the factor: mℓ/mb

GIM suppression

Theoretical clean: the only hadronic uncertainty factors are: fBs orfB_d; Sensitive to new physics: non-SM scalar and pseudoscalar interactions.

Golden Channel!

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. . . . . Introduction

B

⁰_s,d

→ ℓ

⁺

ℓ

⁻

: The Experimental status V.S. SM prediction

Current measurements by CMS and LHCb: weighted world average B(B⁰_s →µ⁺µ⁻)exp. = (2.9±0.7)×10⁻⁹ B(B⁰_d→µ⁺µ⁻)exp. = (

3.6^+1.6₋_1.4)

×10⁻¹⁰ [CMS-PAS-BPH-13-007]

SM predictions:

The latest predictions: NLO EW + NNLO QCD:

BBs→e⁺e⁻ = (8.54±0.55)×10⁻¹⁴, BBd→µ⁺µ⁻= (1.06±0.09)×10⁻¹⁰, BBs→µ⁺µ−= (3.65±0.23)×10⁻⁹, BB_d→e⁺e−=(2.48±0.21)×10−15, BBs→τ⁺τ⁻= (7.73±0.49)×10⁻⁷, BBd→τ⁺τ⁻= (2.22±0.19)×10⁻⁸, [Bobeth, Gorbahn, Hermann, Misiak, Stamou, Steinhauser, arXiv:1311.0903]

The error budgets forB⁰_s →µ⁺µ⁻andB⁰_d→µ⁺µ⁻

f_B_q CKM τ_H^q Mt αs other non- ∑

param. param.

BB⁰_s→µ⁺µ⁻ 4.0% 4.3% 1.3% 1.6% 0.1% <0.1% 1.5% 6.4%

BB⁰_d→µ⁺µ⁻ 4.5% 6.9% 0.5% 1.6% 0.1% <0.1% 1.5% 8.5%

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. . . . . Aligned 2HDM

General 2HDM and FCNC

Two Higgs doublet model: the simplest non-trivial extension on the SM Φ1 = 1

√2

( √

2(

G⁺cosβ−H⁺sinβ) vcosβ−hsinα+Hcosα+i(

G⁰cosβ−Asinβ) ) , Φ2 = 1

√2

( √

2(

G⁺sinβ+H⁺cosβ) vsinβ+hcosα+Hsinα+i(

G⁰sinβ+Acosβ) ) .

Two Yukawa coupling matrices⇒non-diagonal Yukawa matrices elements

⇒tree level ﬂavour changing neutral current(FCNC)

¯b

d

d¯

b h/H/A

¯b d

d¯

b

W W

u/c/t

u/c/t ξ^Dbd ξbd^D

(a) (b)

Possible solutions

make the non-diagonal Yukawa coupling vanish: impose aZ2 symmetryon the Lagrangian

[Glashow and Weinberg (1977)];

make the scalar couplings small enough: Type-III 2HDMmodel[Cheng and Sher (1987)]

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Aligned 2HDM: Yukawa coupling

It’s convenient to study Yukawa coupling inHiggs basis Φ1=

[ G⁺

√1

2(v+S1+iG⁰) ]

, Φ2=

[ H⁺

√1

2(S2+iS3) ]

,

Yukawa couplingin Higgs basis LY=−

√2 v

[Q¯^′_L(M^′_dΦ1+Y^′_dΦ2)d^′_R+ ¯Q^′_L(M^′_uΦ˜1+Y^′_uΦ˜2)u^′_R+ ¯L^′_L(M^′_ℓΦ1+Y^′_ℓΦ2)ℓ^′_R] +h.c. , Assuming the two diﬀerent Yukawa coupling proportional to each other, in

mass-eigenstate basis

Y_d,ℓ = ς_d,ℓM_d,ℓ, Yu = ς_u^∗Mu, The alignment parametersςf arearbitrary complex numbers.

The Yukawa coupling after alignment LY=−

√2 v H⁺{

¯u[

ς_dVM_dP_R−ςuM^†_uVP_L]

d+ς_ℓνM¯ _ℓP_Rℓ}

−1 v

∑

φ⁰_i,f

y^φ_f⁰ⁱ φ⁰_i [¯fMfPRf] +h.c. ,

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Aligned 2HDM and Z

₂

symmetry

three new universal parameters introducedςf: an arbitrary complex number;

⇒new sources of CP violation;

all theneutral-currentinteractions are diagonal in ﬂavour;

allleptonic couplingsare diagonal in ﬂavour due to the absence ofνR; The usualZ2 symmetric models recovered by the following assignment ofςf

Model ςd ςu ςl

Type I cotβ cotβ cotβ

Type II −tanβ cotβ −tanβ

Type X cotβ cotβ −tanβ

Type Y −tanβ cotβ cotβ

Inert 0 0 0

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. . . . . Calculation of Wilson Coefficients

The effective weak Hamiltonian

The eﬀective weak Hamiltonian ofB⁰_s,d→ℓ⁺ℓ⁻ Heff = − GFα

√2πs²_W [

V_tbV^∗_tq

10,S,P∑

i

(CiOi+C^′_iOi^′

)+h.c.

] ,

O10 = (¯qγµPLb) (¯ℓγ^µγ5ℓ), O10^′ = (¯qγµPRb) (¯ℓγ^µγ5ℓ), OS = mℓmb

M²_W (¯qPRb) (¯ℓℓ), OS^′ = mℓmb

M²_W (¯qPLb) (¯ℓℓ), OP = mℓmb

M²_W (¯qPRb) (¯ℓγ5ℓ), OP^′ = mℓmb

M²_W (¯qPLb) (¯ℓγ5ℓ),

Operators withℓγ¯ ^µℓ: vanish when contracted with the B-meson momentumpµ; Oi^′: proportional to the light-quark massmq≪mband can be neglected;

Tensor operators: have no contributions due to⟨0|¯qσµνb|B¯⁰_q(p)⟩= 0

Anomalous dimensionofOi⁽^′⁾ is zero: no contribution from renormalization due to QCD correction.

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Automatic calculation of Feynman Diagrams in Mathematica

Write model ﬁles forFeynRules, include the Lagrangian and constraints;

generate all the Feynman rules and a model ﬁle forFeynArtsautomatically;

generate the Feynman diagrams and original amplitudes for the physical process by FeynArts;

evaluate the Feynman amplitudes:

compute the analytic results inFeynArtsdirectly;

pass the Feynman amplitudes toFeynCalcfor further evaluation;

pass the Feynman amplitudes toFormCalcfor further evaluation (only in Feynman gauge);

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The Wilson Coefficient

The Wilson CoeﬃcientsCi: require equality of 1PI Green functions calculated in the fulland in theeﬀectivetheory.

In the full theory: need to evaluate variousbox,penguinandself-energydiagrams . The sum of Wilson coeﬃcients in Feynman gauge

C10 = C^SM₁₀ + C^{Z penguin,}₁₀ ^A2HDM,

CS = C^box,_S ^SM +C^box,_S ^A2HDM+C^φ_S⁰ⁱ^,^A2HDM, C_P = C^box,_P ^SM +C^{Z penguin,}_P ^SM +CGB penguin,SM

P +C^box,_P ^A2HDM +C^{Z penguin,}_P ^A2HDM+CGB penguin,A2HDM

P +C^φ_P⁰ⁱ^,^A2HDM. All calculations are performed in both theFeynman (ξ= 1)and theunitary (ξ=∞)gauges, to check the gauge independence.

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The Wilson Coefficient in SM: C

₁₀

C10in the SM: generated from the W-box and Z-penguin diagrams;

[Inami and Lim, ’81; Buchalla and Buras, ’99; Bobeth, Gorbahn, Hermann, Misiak, Stamou and Steinhauser, 1311.0903]

W^±

W^± (1.1)

t t t t

W^± (1.3)

W^± G^±

G^± (1.2)

G^±

G^± (1.4) s

b

s

ℓ

ℓ νℓ

b

s

ℓ

ℓ νℓ

b

s

ℓ

ℓ νℓ

b

s

ℓ

νℓ C^SM₁₀ =−η^EWY η^QCD_Y Y0(xt)

Z Z

t

t ℓ

ℓ ℓ

ℓ b

s W^±

b

s G^±

W^±

(2.1) (2.2) (2.3)

(2.5) (2.6)

t b

s

W^±

G^± G^±

W^±

Z ℓ

ℓ b

s b t

G^±

Z ℓ

ℓ b

s b t

W^±

(2.9) (2.10)

(2.7) s

Z ℓ

ℓ t

b

s

Z ℓ

s ℓ t

G^± b

Z ℓ

ℓ

Z ℓ

ℓ t

b

s

(2.4) G^±

G^±

Z ℓ

s ℓ t

W^± b

Z ℓ

ℓ t

b

s

(2.8) s

η_Y^EW = 0.977: the NLO EW matching corrections and QED RG running;

[Bobeth, Gorbahn and Stamou, 1311.1348]

η_Y^QCD = 1.010: the NLO and NNLO QCD

corrections;

[Hermann, Misiak and Steinhauser,1311.1347]

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The Wilson Coefficient in SM: C

_S

and C

_P

C_SandCPin SM: generated from the W-box, Z-penguin, Higgs-penguin and Goldstone-penguin diagrams.

hSM hSM

t

t ℓ

ℓ ℓ

ℓ b

s W^±

b

s G^±

W^±

(3.1) (3.2) (3.3)

(3.5) (3.6)

t b

s

W^±

G^± G^±

W^±

hSM

ℓ

ℓ b

s b t

G^±

hSM

ℓ

ℓ b

s b t

W^±

(3.9) (3.10)

(3.7) s

hSM

ℓ

ℓ t

b

s

hSM

ℓ

s ℓ t

G^± b

hSM

ℓ

hSM

ℓ

ℓ t

b

s (3.4)

G^±

hSM

ℓ

s ℓ t

W^± b

hSM

ℓ

ℓ t

b

s

(3.8) s

C^SM_S

= C^box,_S,_Unitary^SM +C^{h penguin,}_S,_Unitary^SM

= C^box,_S,_Feynman^SM +C^{h penguin,}_S,Feynman^SM

G⁰ G⁰

t

t ℓ

ℓ ℓ

ℓ b

s W^±

b

s G^±

(4.1) (4.2) (4.3) (4.4)

t b

s

W^±

G^± G^±

W^± G⁰

ℓ

ℓ t

b

s

G⁰ ℓ

ℓ b

s b t

G^±

G⁰ ℓ

ℓ b

s b t

W^±

(4.5) (4.6) (4.7)

G⁰ ℓ

ℓ s

t G^± b

G⁰ ℓ

ℓ s

t W^± b G⁰

ℓ

(4.8)

s s

C^SM_P

= C^box,_P,Unitary^SM +C^{Z penguin,}_P,_Unitary^SM

= C^box,_P,_Feynman^SM +C^{Z penguin,}_P,Feynman^SM +CGB penguin,SM

P,Feynman

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The Wilson Coefficient in A2HDM: I

Z penguin diagramsin A2HDM: contribute toC^A2HDM₁₀ andCP

Z t

t ℓ

ℓ ℓ b

s H^±

(5.2) (5.4)

s Z

ℓ

ℓ s

t H^± b

(5.1) H^±

H^±Z ℓ

ℓ t

b

s

Z ℓ

ℓ b

s b t

H^± (5.3)

C^A2HDM₁₀

=|ςu|²F1(xt,x_H±)

Box diagramsin A2HDM: contribute toCSandCP

H^±

W^± (6.1)

t t t t

H^± (6.3)

H^±/G^± W^±

H^± (6.2)

H^±

G^±/H^± (6.4) s

b

s

ℓ

ℓ νℓ

b

s

ℓ

ℓ νℓ

b

s

ℓ

ℓ νℓ

b

s

ℓ

ℓ νℓ

Goldstone-Boson penguin diagramsin A2HDM: contribute toCP

G⁰ t

t

ℓ

ℓ b

s H^±

(7.1) (7.3)

s

G⁰ ℓ

s ℓ t

H^± b

G⁰ ℓ

ℓ b

s b t

H^± (7.2)

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The Wilson Coefficient in A2HDM: II

one-loop penguin+tree-level FCNCcontribution fromLFCNC

ϕ⁰i t

t b

s H^±

(9.1)

ℓ ϕ⁰i t

t ℓ b

s W^±

(9.5)

(9.2) H^±

H^±

W^±

W^± (9.6)

ϕ⁰i ℓ

ℓ t

b

s ϕ⁰i

ℓ

ℓ t

b

s

ϕ⁰i b

s b t

H^± (9.3)

ϕ⁰i b

s b t

W^± (9.7)

ϕ⁰i ℓ

s ℓ t

W^± b

ϕ⁰i ℓ

ℓ t

b

s (9.9) W^±

H^±

(9.10) t b

s H^±

W^±ϕ⁰i ℓ

ℓ ℓ

ℓ

ℓ ℓ

ℓ _(9.8)

ϕ⁰i t

t b

s G^±

(9.11) (9.12)

G^±

G^±ϕ⁰i ℓ

ℓ t

b

s

(9.15) G^±

H^±ϕ⁰i ℓ

ℓ t

b

s _(9.16)

H^±

G^±ϕ⁰i ℓ

ℓ t

b

s

ϕ⁰i ℓ

ℓ t

b

s ϕ⁰i b

s b t

G^± (9.13)

(9.17) W^±

G^±

(9.18) t b

s G^±

W^±ϕ⁰i ℓ

ℓ ℓ

ℓ

(9.4) s

ϕ⁰i ℓ

s ℓ t

H^± b

(9.14) s

ϕ⁰i ℓ

s ℓ t

G^± b

In CP conserving case: C^φ_S⁰ⁱ^,^A2HDMfromφ⁰i = (h,H) C^φ_P⁰ⁱ^,^A2HDMfromφ⁰_i =A

FCNC

ϕ⁰i

ℓ

s ℓ b

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. . . . . Numerical Analysis

The Branching Ratio of B

⁰_s,d

→ ℓ

⁺

ℓ

⁻

The CP conserved averaged time-integrated branching ratio (include the eﬀect of Bq−B¯q mixing)

[De Bruyn, Fleischer, Knegjens etal, 1204.1735; 1204.1737; Buras, Fleischer, Girrbach, Knegjens, 1303.3820]

B(B⁰_q→ℓ⁺ℓ⁻) = G⁴_FM⁴_W 8π⁵Γ^q_H

V_tbV^∗_tqC^SM₁₀²f²_B_qM_B_qm²_ℓ vu ut1−4m²_ℓ

M²_B_q

× [

|P|²+ (

1−∆Γq

Γ^q_L )|S|²

] .

P≡ C10

C^SM₁₀ + M²_B_q 2M²_W

( mb

m_b+mq

) CP

C^SM₁₀ , S≡ vu ut1−4m²_ℓ

M²_B_q M²_B_q 2M²_W

( mb

m_b+mq

) C_S C^SM₁₀

the ratio of branching ratio

R_qℓ ≡ B(B⁰_q→ℓ⁺ℓ⁻) B(B⁰_q→ℓ⁺ℓ⁻)SM

= [

|P|²+( 1−∆Γq

Γ^q_L )|S|²

] ,

Rsµ= 0.79±0.20andRdµ= 3.38^+1.53_−1.35are used as constraints on the model parameters.

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Choice of the model parameters

10 free parameters in our results:

ςu,d,ℓ, (MH,MA,M_H±) α,˜ (λ3, λ7), CR(MW) Four parameters can be ﬁxed for simplicity:

the mixing angleα:˜ |cosα˜|>0.90(68%CL)[Celis, Ilisie and Pich, 1302.4022;

1310.7941]

|λ3,7|.8π;[Gunion, Haber,Kane and Dawson (2000); Branco, Ferreira, Lavoura, Rebelo, Sher and Silva(2011)]

C_R(M_W): no strong constraint yet.

They are less sensitive toRsµ, so

λ3=λ7= 1, cosα˜= 0.95, CR(MW) = 0 Six parameters to be analysed later:

neutral scalar masses:MH≥Mh≃126GeV,MH∈[130,500]GeV, M_A∈[80,500]GeV

charged Higgsmass andςu:M_H±∈[80,500]GeVwith requiring|ςu| ≤2. [Celis, Ilisie and Pich, 1302.4022; 1310.7941; Jung, Pich and Tuz on, 1006.0470; Jung, Li and Pich, 1208.1251]

ςdandςℓ:|ςd,ℓ| ≤50

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Numerical Analysis I: small ς

_d

, ς

_ℓ

|ςd,ℓ| ∼ |ςu| ≤2: CS,Pare negligible,C^A2HDM₁₀ only involveςuandM_H±

The ratioRsµputs strong upper bound on the parameter: |ςu| ≤0.49 (0.97)with M_H± = 80 (500)GeV, at95%CL.

For lagerM_H±, the bound become weaker: lim

x_H+→∞C^A2HDM₁₀ = 0.

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Numerical Analysis II: large ς

_d

, ς

_ℓ

ςd,ℓ∈[−50,50]: C_SandCPcan induce a signiﬁcant enhancement.

Mass1: M_H±=M_A= 80GeV

MH= 130GeV

Mass2:

M_H±=MA=MH= 200GeV

Mass3:

M_H±=MA=MH= 500GeV

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Numerical Analysis III: Z

2

symmetric models

The discreteZ2symmetric models: particular cases of the CP-conserving A2HDM

type-I,type-Xand type-Yare almost indistinguishable;

tanβ≥1.5at 95%

CL under constraint fromRsµ;

Fortype-IImodel, an enhancement ofRsµ

is still possible in the large tanβregion.