Polarized b baryon decay to pˇ · 2020. 11. 17. · p s= 7 and 8 TeV [17,18]. In future e+e...
Transcript of Polarized b baryon decay to pˇ · 2020. 11. 17. · p s= 7 and 8 TeV [17,18]. In future e+e...
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Prepared for submission to JHEP
Polarized Λb baryon decay to pπ and a dilepton pair
Diganta Dasa and Ria Sainb
aDepartment of Physics and Astrophysics, University of Delhi, Delhi 110007, IndiabThe Institute of Mathematical Sciences, Taramani, Chennai 600113, IndiabHomi Bhabha National Institute Training School Complex, Anushakti Nagar, Mumbai 400085, India
E-mail: [email protected], [email protected]
Abstract: The recent anomalies in b→ s`+`− transitions could originate from some NewPhysics beyond the Standard Model. Either to confirm or to rule out this assumption,
more tests of b → s`+`− transition are needed. Polarized Λb decay to a Λ and a dileptonpair offers a plethora of observables that are suitable to discriminate New Physics from
the Standard Model. In this paper, we present a full angular analysis of a polarized Λbdecay to a Λ(→ pπ)`+`− final state. The study is performed in a set of the operator wherethe Standard Model operator basis is supplemented with its chirality flipped counterparts,
and new scalar and pseudo-scalar operators. The full angular distribution is calculated
by retaining the mass of the final state leptons. At the low hadronic recoil, we use the
Heavy Quark Effective Theory framework to relate the hadronic form factors which lead
to simplified expression of the angular observables where short- and long-distance physics
factorize. Using the factorized expressions of the observables, we construct a number of
test of short- and long-distance physics including null tests of the Standard Model and its
chirality flipped counterparts that can be carried out using experimental data.
Keywords: Rare Decays, Polarised Baryon Decays, New Physics
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Contents
1 Introduction 1
2 Effective Hamiltonian 2
3 Angular distributions 4
4 Low-recoil Amplitudes in HQET 6
5 Low-recoil factorization 7
6 Numerical Analysis 11
7 Summary 13
A Nucleon spinor in Λ rest frame 14
B Leptonic helicity amplitudes 15
C Angular coefficients 16
1 Introduction
Several experimental results in the rare b → s`+`− processes have shown deviations fromthe Standard Model (SM) predictions. In the B → K(∗)`+`− decay the deviations in theRK(∗) observables hints to a possible violation of lepton flavor universality [1, 2]. Moreover,
the branching ratios of B → Kµ+µ− [3], B → K∗µ+µ− [4, 5], Bs → φµ+µ− [6], and theoptimized observables in B → K∗µ+µ− decay [7] show systematic deviation from the SMpredictions. Though inconclusive till now, New Physics (NP) beyond the SM could be the
origin of these deviations.
LHCb is now capable to study b→ s`+`− transitions in baryonic decays. Interestingly,the LHCb measurement of Λb → Λ`+`− branching ratio [8] shows deviations from the SMexpectations with the same trend as its mesonic counterparts. The Λb → Λ(→ pπ)`+`−
decay with unpolarized Λb, has been studied in the SM and beyond in Refs. [9–15] and
is described in terms of ten angular observables. Phenomenologically, the Λb → Λ(→pπ)µ+µ− decay could be richer than its mesonic counterpart as the Λb can be produced
in polarized state. If the Λb is polarized then the full angular distribution of Λb → Λ(→pπ)`+`− in the SM gives access to 34 angular observables [12] which were recently measured
by the LHCb [16]. In the LHCb, the polarization of Λb was measured as PΛb = 0.06±0.07±0.02 at
√s = 7 TeV [17]. At the CMS detector, PΛb = 0.00± 0.06± 0.02 was measured at
– 1 –
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√s = 7 and 8 TeV [17, 18]. In future e+e− collider, Λb can be produced in longitudinal
polarization also.
In this paper we calculate the full angular distributions of Λb → Λ(→ pπ)`+`− decayfor a polarized Λb. The calculation is done for a set of operators that include the SM
operator basis, the chirality-flipped counterpart of the SM operators (henceforth SM′),
and new scalar and pseudo-scalar operators (henceforth SP operators). We also retain the
masses of the final state leptons. In the presence of the SP operators, we find two additional
angular observables that are helicity suppressed. All the 36 angular observables depend
on ten form factors that have been calculated at large recoil in the framework of light-
cone-sum-rules [19–21]. The form factors also have been calculated in the lattice QCD [22]
which is valid at low recoil. At low recoil, or large dilepton invariant mass squared q2, the
decay can be described by a Heavy Quark Effective Theory (HQET) framework [23–25],
which ensures relations between the form factors known as Isgur-Wise relations [26–28].
The Isgur-Wise relations lead to the factorization of short- and long-distance physics in
the angular observables. In the SM, the factorized expressions can be used to construct
clean tests of form factors and short-distance physics. In this paper, we discuss how these
tests are affected by the presence of SM′ and SP operators. Additionally, we construct
combinations of observables that are sensitive to scalar NP only and therefore serve as null
tests of the SM+SM′. We also present a numerical analysis of several observables in the
SM and NP scenarios.
The organization of the paper is as follows: in Sec. 2 we discuss the effective operators
for b→ s`+`− transition and discuss the derivations of the decay amplitudes. In Sec. 3 wederived the full angular distribution of Λb → Λ(→ pπ)`+`− for a polarized Λb and discusshow to extract angular observables from the angular distribution. In section 4 we discuss
the relations between form factors in HQET. The low-recoil simplifications of the angular
observables due to the HQET and its consequences on the tests of short- and long-distance
physics are discussed in section 5. A numerical analysis is presented in Sec. 6 and our
results are summarized in Sec. 7. We also give several appendixes where details of the
derivations can be found.
2 Effective Hamiltonian
In the SM the rare b → s`+`− transition proceeds through loop diagrams which are de-scribed by radiative operator O7, semi-leptonic operators O9,10, and hadronic operatorsO1−6,8. The operators O7,9,10 are the dominant ones in the SM and read
O7 =mbe
[s̄σµνPRb
]Fµν , O9 =
[s̄γµPLb
][`γµ`
], O10 =
[s̄γµPLb
][`γµγ5`
]. (2.1)
The corresponding Wilson coefficients are C7, C9 and C10. We assume only the factorizablequark loop corrections to the hadronic operators O1−6,8 which are absorbed into the Wilsoncoefficients C7,9 (often written as Ceff7 , Ceff9 in the literature). For simplicity, we ignore thenon-factorizable corrections which are expected to play a significant role, particularly at
large recoil or low dilepton invariant mass squared q2 [29, 30].
– 2 –
-
Beyond the SM, effects of NP operators with the same Dirac structure as O9,10 can betrivially included by the modifications C9,10 → C9,10 + δC9,10. We additionally include thechirality flipped counterparts of O9,10 and new scalar and pseudo-scalar operators
O9′ =[s̄γµPRb
][`γµ`
], O10′ =
[s̄γµPRb
][`γµγ5`
],
OS(′) =[s̄PR(L)b
][``], OP (′) =
[s̄PR(L)b
][`γ5`
].
(2.2)
The Wilson coefficients corresponding to O9′,10′ , OS(′),P (′) are C9′,10′ and CS(′),P (′) , respec-tively. NP operators of the tensorial structure have been ignored for simplicity as these
operators lead to a large number of terms in the angular distributions that will be discussed
elsewhere. The Λb → Λ hadronic matrix elements for the set of operators (2.1) and (2.2)are conveniently defined in terms of ten q2 dependent helicity form factors fV,A0,⊥,t, f
T,T50,⊥
[31].
Assuming factorization between the hadronic and leptonic parts, and neglecting con-
tributions proportional to VubV∗us, the amplitude of the decay process Λ(p, sp) → Λ(k, sk)
`+(q+)`−(q−) can be written as
Mλ1,λ2(sp, sk) = −GF√
2VtbV
∗ts
αe4π
∑i=L,R
[∑λ
ηλHi,sp,skVA,λ L
λ1,λ2i,λ +H
i,sp,skSP L
λ1,λ2i
], (2.3)
where p, k, q+, q− are the momentum of Λb, Λ, `+ and `−, respectively, and sp, sk are
the projection of Λb and Λ spins on to the z-axis in their respective rest frames. The
polarization of the two leptons are denoted by λ1,2, λ = 0,±1, t are the polarization statesof virtual gauge boson that decays to the di-lepton pair, and η±1,0 = −1, ηt = +1. Theexpressions of the hadronic matrix elements H
L(R)VA,λ(sΛb , sΛ) and H
L(R)SP (sΛb , sΛ) can be
found in Ref. [11]. In literature, the angular observables are usually expressed in terms of
transversity amplitudes. For SM+SM′ set of operators the transversity amplitudes are [14]
AL,(R)⊥1 = −
√2N
(fV⊥√
2s−CL,(R)VA+ +2mbq2
fT⊥(mΛb +mΛ)√
2s−Ceff7), (2.4)
AL,(R)‖1 =
√2N
(fA⊥√
2s+CL,(R)VA− +2mbq2
fT5⊥ (mΛb −mΛ)√
2s+Ceff7), (2.5)
AL,(R)⊥0 =
√2N
(fV0 (mΛb +mΛ)
√s−q2CL,(R)VA+ +
2mbq2
fT0√q2s−Ceff7
), (2.6)
AL,(R)‖0 = −
√2N
(fA0 (mΛb −mΛ)
√s+q2CL,(R)VA− +
2mbq2
fT50√q2s+Ceff7
), (2.7)
A⊥t = −2√
2N(C10 + C10′)(mΛb −mΛ)√s+q2fVt , (2.8)
A‖t = 2√
2N(C10 − C10′)(mΛb +mΛ)√s−q2fAt , (2.9)
where s± = (mΛb ±mΛ)2 − q2, and the normalization constant is given by
N(q2) = GFVtbV∗tsαe
√√√√τΛb
q2√λ(m2Λb ,m
2Λ, q
2)
3.211m3Λbπ5
β` , β` =
√1−
4m2`q2
. (2.10)
– 3 –
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The combination of Wilson coefficients CL,(R)VA± are
CL(R)VA+ = (C9 + C9′)∓ (C10 + C10′) , (2.11)
CL(R)VA− = (C9 − C9′)∓ (C10 − C10′) . (2.12)
For the SP operators we follow the definition of transversity amplitudes from [35] and using
the scalar helicity amplitudes given in [11] we get
AL(R)⊥S =
√2NfVt
mΛb −mΛmb −ms
CL(R)SP+ , (2.13)
AL(R)‖S = −
√2NfAt
mΛb +mΛmb +ms
CL(R)SP− , (2.14)
where the Wilson coefficients are
CL(R)SP+ = (CS + CS′)∓ (CP + CP ′) , (2.15)
CL(R)SP− = (CS − CS′)∓ (CP − CP ′) . (2.16)
The subsequent parity violating weak decay amplitudes of Λ(k, sk) → p(k1)π(k2) are cal-culated following Ref. [10] and using the baryon spinor given in appendix A. In the angular
observables however, it is the parity violating parameter αΛ = 0.642 ± 0.013 [32] that isrelevant.
The leptonic helicity amplitudes are defined as
Lλ1λ2L(R) = 〈¯̀(λ1)`(λ2)|¯̀(1∓ γ5)`|0〉 , (2.17)
Lλ1λ2L(R),λ = �̄µ(λ)〈¯̀(λ1)`(λ2)|¯̀γµ(1∓ γ5)`|0〉 , (2.18)
where �µ is the polarization of the virtual gauge boson that decays to the dilepton pair.
The detailed expressions of the amplitudes are given in appendix B.
3 Angular distributions
The set of angles that forms the angular basis are θ, θb,` and φb,` [16]. Since transverse
polarization is considered, it is appropriate to define a normal vector n̂ = p̂{lab}beam × p̂
{lab}Λb
where p̂{lab}beam, p̂
{lab}Λb
are unit vectors in the lab frame. The angle θ between the n̂ and
the direction of Λ, in the Λb rest frame is defined as cos θ = n̂.p̂{Λb}Λ . To describe the
decay of Λ→ pπ and the dilepton system, we construct coordinate system {ẑb, ŷb, x̂b} and{ẑ`, ŷ`, x̂`}, respectively. The z-axes are as follows: ẑb = p̂
{Λb}Λ , ẑ` = p̂
{Λb}`+`− . The other two
axes are defined as: ŷb,` = n̂ × ẑb,` and x̂b,` = n̂ × ŷb,`. The angles θ` and φ` are madeby the `+ in the dilepton rest frame and the angles θb and φb are made by the proton in
the Λ rest frame as shown in figure figure 1. With this angular basis, the six fold angular
– 4 –
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Figure 1. Definition of the angular basis for decay of polarized Λb → Λ(→ pπ)`+`−
distribution of Λb → Λ(→ pπ)`+`− for the SM + SM′ + SP set of operators is
d6Bdq2 d~Ω(θ`, φ`, θb, φb, θ)
=3
32π2
( (K1 sin
2 θ` +K2 cos2 θ` +K3 cos θ`
)+(
K4 sin2 θ` +K5 cos
2 θ` +K6 cos θ`)
cos θb+
(K7 sin θ` cos θ` +K8 sin θ`) sin θb cos (φb + φ`) +
(K9 sin θ` cos θ` +K10 sin θ`) sin θb sin (φb + φ`) +(K11 sin
2 θ` +K12 cos2 θ` +K13 cos θ`
)cos θ+(
K14 sin2 θ` +K15 cos
2 θ` +K16 cos θ`)
cos θb cos θ+
(K17 sin θ` cos θ` +K18 sin θ`) sin θb cos (φb + φ`) cos θ+
(K19 sin θ` cos θ` +K20 sin θ`) sin θb sin (φb + φ`) cos θ+
(K21 cos θ` sin θ` +K22 sin θ`) sinφ` sin θ+
(K23 cos θ` sin θ` +K24 sin θ`) cosφ` sin θ+
(K25 cos θ` sin θ` +K26 sin θ`) sinφ` cos θb sin θ+
(K27 cos θ` sin θ` +K28 sin θ`) cosφ` cos θb sin θ+(K29 cos
2 θ` +K30 sin2 θ` +K35 cos θ`
)sin θb sinφb sin θ+(
K31 cos2 θ` +K32 sin
2 θ` +K36 cos θ`)
sin θb cosφb sin θ+(K33 sin
2 θ`)
sin θb cos (2φ` + φb) sin θ+(K34 sin
2 θ`)
sin θb sin (2φ` + φb) sin θ). (3.1)
– 5 –
-
We identify the angular observables with those given [10] as: K1ss = K1, K1cc = K2,
K1c = K3, K2ss = K4, K2cc = K5, K2c = K6, K4sc = K7, K4s = K8, K3sc = K9, and
K3s = K10. We obtain two new angular coefficients K35 and K36 that are absent in the
SM+SM′ set of operators [12]. These observables depend on the interference of scalar and
(axial-)vector amplitudes only and are helicity suppressed by m`/√q2. In appendix C we
express the observables in terms of transversity amplitudes. Since we have retained the
masses of the final state leptons, we write each of the Ki’s as
K{··· } = K{··· } +m`√q2K′{··· } +
m2`q2K′′{··· } . (3.2)
If the final states leptons are of two lightest flavors, then K′,′′ can be neglect for large recoilanalysis, but will be useful for di-tau final states.
Integrations (3.1) over the angles give differential decay distribution
dBdq2
= 2K1 +K2 . (3.3)
This is used to define normalized observables as
Mi =Ki
dB/dq2, (3.4)
The Mi’s can be extracted from (3.1) by convolution with different weight functions [12].
4 Low-recoil Amplitudes in HQET
At low recoil, lepton masses can be neglected for di-muon or di-electron final states. In
that case, the time-like amplitudes A⊥,‖t and hence the form factor fV,At do not contribute.
The HQET spin symmetry at leading order in 1/mb expansion and up to O(αs) correctionsimplies following relations between the rest of the form factors
ξ1 − ξ2 = fV⊥ = fV0 = fT⊥ = fT0 ,ξ1 + ξ2 = f
A⊥ = f
A0 = f
T5⊥ = f
T50 , (4.1)
where ξ1,2 are the leading Isgur-Wise form factors [31]. To exploit these relations we
assume the vector form factors fV,A⊥,0 as independent and use the relations (4.1) for tensor
form factors. Including one-loop corrections to the Isgur-Wise relations, the transversity
amplitudes read [10]
AL(R)⊥1 ' −2NC
L(R)+√s−f
V⊥ , A
L(R)‖1 = 2NC
L(R)−√s+f
A⊥ , (4.2)
AL(R)⊥0 '
√2NCL(R)+
mΛb +mΛ√q2
√s−f
V0 , (4.3)
AL(R)‖0 ' −
√2NCL(R)+
mΛb −mΛ√q2
√s+f
A0 , (4.4)
– 6 –
-
where the Wilson coefficients are given by
CL(R)+ =(
(C9 + C9′)∓ (C10 + C10′) +2κmbmΛb
q2C7),
CL(R)− =(
(C9 − C9′)∓ (C10 − C10′) +2κmbmΛb
q2C7).
(4.5)
The parameter κ ≡ κ(µ) = 1 − (αsCF /2π) ln(µ/mb) accounts for the radiative QCDcorrections to the form factors relations. The parameter κ is such that together with the
Wilson coefficients and the MS b-quark mass mb in (4.5), the amplitudes are free of the
renormalization scale µ.
The simplifications of the transversity amplitudes yield factorizations between short-
and long-distance physics in the angular observables. In the factorized expressions, the
vector and axial-vector Wilson coefficient contributes through the following short-distance
coefficients
ρ±1 =1
2
(|CR± |2 + |CL±|2
)= |C79 ± C9′ |2 + |C10 ± C10′ |2 ,
ρ2 =1
4
(CR+CR∗− − CL−CL∗+
)= Re(C79C∗10 − C9′C∗10′)− iIm(C79C∗9′ + C10C∗10′) .
ρ±3 =1
2
(|CR± |2 − |CL±|2
)= 2 Re(C79 ± C9′)(C10 ± C10′)∗
ρ4 =1
4
(CR+CR∗− + CL−CL∗+
)=(|C79|2 − |C9′ |2 + |C10|2 − |C10′ |2
)− iImC79 C∗10′ − C9′ C∗10 ,
(4.6)
where we have abbreviated
C79 ≡ C9 +2κmbmΛb
q2C7 . (4.7)
The ρ±3 and ρ4 contributes due to the secondary parity violating decay. These coefficients
appear in other b → s`+`− decays also. For example, ρ±1 and ρ2 appear in B → K∗(→Kπ)`+`− [33], B → Kπ`+`− [34], and Λb → Λ∗(→ NK̄)`+`− decay [35], ρ4 appears inΛb → Λ∗(→ NK̄)`+`− decay, and ρ−3 and ρ4 appear in B → Kπ`+`− decay. In the SMfollowing simplifications take place
ρ+1 = ρ−1 = ρ1 = 2Re(ρ4) , ρ
+3 = ρ
−3 = ρ3 , (4.8)
Im(ρ2) = 0 , Im(ρ4) = 0 , (4.9)
so that the independent coefficients are ρ1, ρ3 and Re(ρ2). The scalar short-distance coef-
ficients that appear in angular observables are
ρ±S = |CLSP±|2 + |CRSP±|2 ,
ρS1 = 2(CLSP+CL∗SP− + CRSP+CR∗SP−) .(4.10)
5 Low-recoil factorization
Using the HQET simplifications of the previous section, we obtain factorization between
short- and long-distance physics in the angular observables. We reiterate that the lep-
ton mass is neglected to obtain these expressions. The factorized expression for the 10
– 7 –
-
observables that also appear in the unpolarized Λb decay are
K1 = 2s+
(|fA⊥ |2 +
(mΛb −mΛ)2
q2|fA0 |2
)ρ−1 + 2s−
(|fV⊥ |2 +
(mΛb +mΛ)2
q2|fV0 |2
)ρ+1
+1
m2b[|fVt |2s+ρS+(mΛb −mΛ)
2 + |fAt |2s−ρS−(mΛb +mΛ)2] , (5.1)
K2 = 4s+|fA⊥ |2ρ−1 + 4s−|fV⊥ |2ρ+1 +
1
m2b[|fVt |2s+ρS+(mΛb −mΛ)
2
+ |fAt |2s−ρS−(mΛb +mΛ)2] , (5.2)
K3 = 16√s+s−f
A⊥f
V⊥Re(ρ2) , (5.3)
K4 = −8αΛ√s+s−
(fA⊥f
V⊥ +
(m2Λb −m2Λ)
q2fV0 f
A0
)Re(ρ4)
− αΛ√s+s−f
Vt f
At (mΛb
2 −mΛ2)Re(ρS1) , (5.4)K5 = −16αΛ
√s+s−f
A⊥f
V⊥Re(ρ4)− αΛ
√s+s−f
Vt f
At (mΛb
2 −mΛ2)Re(ρS1) , (5.5)K6 = −4αΛs+|fA⊥ |2ρ−3 − 4αΛs−|f
V⊥ |2ρ+3 , (5.6)
K7 = −8αΛ√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ −
(mΛb −mΛ)√q2
fA0 fV⊥
)Re(ρ4) (5.7)
K8 = 4s+αΛ(mΛb −mΛ)√
q2fA0 f
A⊥ρ−3 − 4s−αΛ
(mΛb +mΛ)√q2
fV0 fV⊥ ρ
+3 , (5.8)
K10 = 8αΛ√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ +
(mΛb −mΛ)√q2
fA0 fV⊥
)Im(ρ4) , (5.9)
For the the additional observables that appear due to polarization, the factorization looks
like
K11 = −8PΛb√s+s−
(fA0 f
V0
(m2Λb −m2Λ)
q2− fA⊥fV⊥
)Re(ρ4)
− PΛbN2αΛf
Vt f
At Re[ρS1]
√s+s−(mΛb
2 −mΛ2) , (5.10)K12 = 16PΛb
√s+s−f
A⊥f
V⊥Re(ρ4)− PΛbN
2αΛfVt f
At Re[ρS1]
√s+s−(mΛb
2 −mΛ2) , (5.11)K13 = 4PΛbs+|f
V⊥ |2ρ−3 + 4PΛbs−|f
A⊥ |2ρ+3 , (5.12)
K14 = −2αΛPΛbs−(|fV⊥ |2 − |fV0 |2
(mΛb +mΛ)2
q2
)ρ+1
− 2αΛPΛbs+(|fA⊥ |2 − |fA0 |2
(mΛb −mΛ)2
q2
)ρ−1 (5.13)
+ PΛbαΛN2 1
m2b[|fVt |2s+ρS+(mΛb −mΛ)
2 + |fAt |2s−ρS−(mΛb +mΛ)2] , (5.14)
K15 = −4αΛPΛbs−|fV⊥ |2ρ+1 − 4αΛPΛbs+|f
A⊥ |2ρ−1
+ PΛbαΛN2 1
m2b[|fVt |2s+ρS+(mΛb −mΛ)
2 + |fAt |2s−ρS−(mΛb +mΛ)2] , (5.15)
K16 = −16αΛPΛb√s+s−f
A⊥f
V⊥Re(ρ2) , (5.16)
– 8 –
-
K17 = −4αΛPΛbs−(mΛb +mΛ)√
q2fV0 f
V⊥ ρ
+1 + 4αΛPΛbs+
(mΛb −mΛ)√q2
fA0 fA⊥ρ−1 , (5.17)
K18 = −16αΛPΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ −
(mΛb −mΛ)√q2
fA0 fV⊥
)Re(ρ2) , (5.18)
K19 = 8αΛPΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ +
(mΛb −mΛ)√q2
fA0 fV⊥
)Im(ρ2) , (5.19)
K22 = −8PΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ −
(mΛb −mΛ)√q2
fA0 fV⊥
)Im(ρ4) , (5.20)
K23 = 8PΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ +
(mΛb −mΛ)√q2
fA0 fV⊥
)Re(ρ4) , (5.21)
K24 = 4PΛbs−(mΛb +mΛ)√
q2fV0 f
V⊥ ρ
+3 + 4PΛbs+
(mΛb −mΛ)√q2
fA0 fA⊥ρ−3 , (5.22)
K25 = 8αΛPΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ −
(mΛb −mΛ)√q2
fA0 fV⊥
)Im(ρ2) , (5.23)
K27 = −4αΛPΛbs−(mΛb +mΛ)√
q2fV0 f
V⊥ ρ
+1 − 4αΛPΛbs+
(mΛb −mΛ)√q2
fA0 fA⊥ρ−1 , (5.24)
K28 = −8αΛPΛb√s+s−
((mΛb +mΛ)√
q2fV0 f
A⊥ +
(mΛb −mΛ)√q2
fA0 fV⊥
)Re(ρ2) , (5.25)
K29 = −PΛbαΛfVt f
At Im(ρS1)
√s+s−(mΛb
2 −mΛ2) (5.26)
K30 = +8αΛPΛb√s+s−
(m2Λb −m2Λ)
q2fA0 f
V0 Im(ρ2)
+ PΛbαΛfVt f
At Im(ρS1)
√s+s−(mΛb
2 −mΛ2) (5.27)
K31 = PΛbαΛ1
m2b[−|fVt |2s+ ρS+ (mΛb −mΛ)
2 + |fAt |2s− ρS− (mΛb +mΛ)2] (5.28)
K32 = −2αΛPΛbs−(mΛb +mΛ)
2
q2|fV0 |2ρ+1 + 2αΛPΛbs+
(mΛb −mΛ)2
q2|fA0 |2ρ−1 (5.29)
+ αΛPΛb1
m2b[−|fVt |2s+ρS+(mΛb −mΛ)
2 + |fAt |2s−ρS−(mΛb +mΛ)2] , (5.30)
K33 = −2αΛPΛbs−|fV⊥ |2ρ+1 + 2αΛPΛbs+|f
A⊥ |2ρ−1 , (5.31)
K34 = 8αΛPΛb√s+s−f
A⊥f
V⊥ Im(ρ2) . (5.32)
Note that the observables K20,21 vanish in the HQET. Moreover, K35 and K36 are helicity
suppressed by m`/√q2 and therefore are not given above.
Using the factorized expressions, we derive tests of operator product expansion through
form factors and the short-distance physics. In the SM+SM′+SP set of operators, the
combinations of K2 and K33 are
K2 −2
αΛPΛbK33 = |N |2(8|fA⊥ |2s+ρ−1 + |f
Vt |2s+
(mΛb −mΛ)2
m2bρ+S + |f
At |2s−
(mΛb +mΛ)2
m2bρ−S ) ,
(5.33)
– 9 –
-
K2 +2
αΛPΛbK33 = |N |2(8|fV⊥ |2s−ρ+1 + |f
Vt |2s+
(mΛb −mΛ)2
m2bρ+S + |f
At |2s−
(mΛb +mΛ)2
m2bρ−S ) .
(5.34)
If the SP operators are absent then these combinations depend only on ρ− and ρ+ respec-
tively. The K8 and K24 can be combined to form two relations that can be used to extract
ρ+3 and ρ−3 , and K17 and K27 can be combined to form two relations that can be used to
extract ρ±1 even in the presence of SP operators
PΛbK8 + αΛK24 = −8|N |2PΛbαΛf
V0 f
V⊥mΛb +mΛ√
q2s−ρ
+3 , (5.35)
PΛbK8 − αΛK24 = 8|N |2PΛbαΛf
A0 f
A⊥mΛb −mΛ√
q2s+ρ
−3 , (5.36)
K27 +K17 = 8|N |2PΛbαΛfA0 f
A⊥mΛb −mΛ√
q2s+ρ
−1 , (5.37)
K27 −K17 = −8|N |2PΛbαΛfV0 f
V⊥mΛb +mΛ√
q2s−ρ
+1 . (5.38)
In the SM+SM′+SP set of operators, several ratios of short-distance coefficients can
be determined without any hadronic effects
PΛbK8 + αΛK24K27 −K17
= −ρ−3
ρ−1,
PΛbK8 − αΛK24K27 +K17
=ρ+3ρ+1
, (5.39)
K16K34
= −2Re(ρ2)Im(ρ2)
,K25K22
= −αΛIm(ρ2)Im(ρ4)
,K23K10
= −PΛbRe(ρ4)αΛIm(ρ4)
. (5.40)
In the SM+SM′ the ratio of K3 and K5 is proportional to Re(ρ2)/Re(ρ4) but is modified
as following in the presence of SP operators
K3K5
= −16m2bf
A⊥f
V⊥Re(ρ2)
16αΛm2bfA⊥f
V⊥Re(ρ4) + αΛ(m
2Λb−m2Λ)fAt fVt Re(ρS1)
. (5.41)
If the SP operators are absent, then this ratio is equal to Re(ρ2)/Re(ρ4). On the other
hand, in SM+SM′, the ratios K5/K7 and K5/K23 are independent of any short distance
physics but are modified in the presence of SP operators as
K5K7
=
√q2[32fA⊥f
V⊥Re(ρ4) + (m
2Λb−m2Λ)fAt fVt Re(ρS1)
]16m2bRe(ρ4)
[(mΛb +mΛ)f
A⊥f
V0 − (mΛb −mΛ)fA0 fV⊥
] , (5.42)
K5K23
=
√q2[32fA⊥f
V⊥Re(ρ4) + (m
2Λb−m2Λ)fAt fVt Re(ρS1)
]16m2bPΛbRe(ρ4)
[(mΛb +mΛb)f
A⊥f
V0 + (mΛb −mΛ)fA0 fV⊥
] . (5.43)These two relations could be regarded as null test of SM+SM′.
– 10 –
-
We find a ratios involving K18, K28, and K3 that are independent of any short distance
physics in the SM+SM′+SP set of operators
K18 +K28K3
= −PΛbαΛmΛb +mΛ√
q2fV0fV⊥
, (5.44)
K18 −K28K3
= PΛbαΛmΛb −mΛ√
q2fA0fA⊥
. (5.45)
The ratios can be used to extract fA0 /fA⊥ and f
V0 /f
V⊥ .
The angular observable K29 vanishes in the SM+SM′ in the limit of zero lepton mass.
If SP operators are present then it is proportional to the imaginary part of ρS1 as given in
equation (5.26). The real part of ρS1 can be extracted from the combination
PΛb(K4 −K5)− αΛK11 = PΛbαΛfAt f
Vt
(m2Λb −m2Λ)
m2b
√s+s−Re(ρS1) , (5.46)
so that the ρS1 can be determined completely.
K31 given in equation (5.28) also vanishes in the SM if m` is neglected but depends on
ρS± if scalar operators are present. This observable therefore serves as a null test of SM.
We find another null test of ρS± from the following combination
K2 +K15PΛbαΛ
= 2
(|fVt |2s+
(mΛb −mΛ)2
m2bρ+S + |f
At |2s−
(mΛb +mΛ)2
m2bρ−S
). (5.47)
This equation in combination with (5.28) can be used to extract ρ+S and ρ−S . We find that
the following relation holds in the presence of SM+SM′+SP set of operators
PΛbαΛK2 −K15 = 2(PΛbαΛK1 −K14
). (5.48)
Since we work in the m` → 0 limit, the relations K13αΛ = −PΛbK6 and K16 =−PΛbαΛK3 remain unchanged when SM+SM′ is supplemented by the SP operators [12] inlow-recoil HQET.
6 Numerical Analysis
In this section we perform a numerical analysis of polarized Λb decay to Λ(→ pπ)µ+µ− atthe low recoil region 15 GeV2 ≤ q2 ≤ 20 GeV2. At this region the masses of the leptonsare neglected. Due to low-recoil HQET, only four form factors contribute through the SM
amplitudes (4.2)-(4.4) and the scalar amplitudes (2.13) and (2.14). The form factors, as
well as their uncertainties are taken from lattice QCD calculations [22]. In addition to
uncertainties coming from the form factors, there are additional sources of uncertainties in
our analysis. We consider a 10% corrections between the amplitudes (4.2)-(4.4) to account
for the neglected terms of the O(ΛQCD/mb) and O(mΛ/mΛb) to derive the leading Isgur-Wise relations (4.1). These corrections are implemented by scaling the amplitudes A⊥,‖0,
A⊥,‖0 by uncorrelated real scale factors.
– 11 –
-
There are theoretical uncertainties emanating from a virtual photon connected to the
operators O1−6 and O8 which are of non-local in nature. At low recoil, the HQET combinedwith low recoil OPE in 1/Q, where Q ∼ (mb,
√q2), the non-local effects are calculated in
terms of local matrix elements suppressed by the powers of Q and absorbed in the Wilson
coefficients C7,9 [27]. The uncertainties due to the neglected terms in the OPE of the orderO(αsΛQCD/mb,m4c/Q4) are included in our analysis by a uncorrelated 5% corrections tothe amplitudes A
L,(R)⊥,‖0 , A
L(R)⊥,‖0 . Other sources of uncertainties are parametric uncertainties,
and scale dependence of the SM Wilson coefficients Ci(µ) for varying the scale µ in therange mb/2 < µ < 2mb. At large-q
2, uncertainties due to quark-hadron duality violation
in the integrated observables are expected to be small and are neglected in our analysis
[39].
SM Set1 Set2 Scalar
15 16 17 18 19 200.00
0.05
0.10
0.15
0.20
0.25
0.30
q2(GeV2)
M12
SM Set1 Set2 Scalar
15 16 17 18 19 20
0.00
0.05
0.10
0.15
0.20
q2(GeV2)
M14
SM Set1 Set2 Scalar
15 16 17 18 19 20-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
q2(GeV2)
M15
SM Set1 Set2 SM +Scalar
15 16 17 18 19 20-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
q2(GeV2)
M32
Figure 2. The angular observables M12,14,15,32 for polarized Λb → Λ(→ pπ)µ+µ− decay in differentq2 bins in the SM and NP scenarios. For (axial-)vector operators we choose two sets from global
fits to b → sµ+µ− data. Set1 corresponds to set δC9 = −C′9 = −1.16, δC10 = C′10 = −0.38 andSet2 corresponds to δC9 = −C′9 = −1.15, δC10 = C′10 = −0.01. The set of scalar couplings areCS = 1.5 + i0.03, C′S = −2 + i0.2, CP = 1.8 + i0.1, C′P = −1.1− i0.04.
Though there are a total of 36 observables, for simplicity we study the ones that
otherwise do not appear in unpolarized Λb decay, and where both SP and (axial-)vector
operators contribute. We also show in our plots only the interesting variants of observables
for a representative set of NP couplings. A detailed NP analysis for all the observables
will be presented elsewhere. To estimate the effects of NP operators we summarize the
– 12 –
-
constraints on the NP Wilson coefficients that we use for our analysis. For simplicity
we assume that the vector and axial vector, and scalar and pseudo-scalar NP contributes
separately. From the global fits to b→ sµ+µ− data we we choose two sets for illustrations[36–38], Set1: δC9 = −C′9 = −1.16 , δC10 = +C′10 = −0.38 and Set2: δC9 = −C′9 =−1.15 , δC10 = −C′10 = −0.01. For scalar Wilson coefficients we follow the fit presentedin [40].
All our determination are for PΛb = 1. In figure 2 we show the SM predictions and
NP sensitivities of M11, M12, M14, M32 different high-q2 bins. Here the bands correspond
to the uncertainties coming from form factor and other sources. These plots indicate that
the observables are sensitive to NP effects.
SM Scalar
15 16 17 18 19 20-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
q2(GeV2)
K5/K
23
SM Scalar
15 16 17 18 19 20-0.1
0.0
0.1
0.2
0.3
q2(GeV2)
K2+
K15
PΛ
bαΛ
Figure 3. The angular observables K5/K23 (left) and K2 + K15/(PΛbαΛ) (right) for polarized
Λb → Λ(→ pπ)µ+µ− decay in different q2 bins in the SM and in the presence of SP operators. Theset of scalar couplings are CS = −1.5 + i0.03, C′S = 2.2 + i0.2, CP = 2.5 + i0.1, C′P = −3− i0.04.
The relations (5.42) and (5.43) are quite interesting. In the SM+SM′ they are indepen-
dent of any short-distance physics but are modified by SP operators. These are null test of
the SM+SM′. In figure 3 we show the ratio K5/K23 in the SM and for set of representative
scalar couplings in different q2 bins. The relations (5.46) and (5.47) are the ones where
the combinations of angular observables depend on the SP operators only. In 3 figure we
plot K2 + K15/(PΛbαΛ) for a set of scalar couplings. This combination is also a null test
of SM+SM′.
7 Summary
In this paper we derive a full angular distribution of polarized Λb decay Λb → Λ(→ pπ)`+`−.The distribution is obtained for a set of operators where the Standard Model operator basis
is supplemented with its chirality flipped counterparts and additional scalar and pseudo
scalar operators. We retain the mass of the final state leptons. We apply a Heavy Quark
Effective Theory framework valid at low hadronic recoil and obtain factorization of long
and short-distance physics in the angular observables. Using the factorized expressions we
construct several tests of form factors and Wilson coefficients, including some null test of
– 13 –
-
the Standard Model and its chirality flipped counterparts. Our analysis shows that new
insight to b→ s`+`− transition can be obtained from this mode.
Acknowledgements
DD acknowledges the DST, Govt. of India for the INSPIRE Faculty Fellowship (grant
number IFA16-PH170).
A Nucleon spinor in Λ rest frame
The angles θb and φb made by the p in the pπ rest frame (denoted as pπ-RF) are shown in
figure 1. In this frame, characterized by k2 = m2Λ, the four-momentum kµ1,2 read
kµ1∣∣pπ−RF = (Ep, |kpπ| sin θb cosφb, |kpπ| sin θb sinφb, |kpπ| cos θb) ,
kµ2∣∣pπ−RF = (Eπ,−|kpπ| sin θb cosφb,−|kpπ| sin θb sinφb,−|kpπ| cos θb) ,
(A.1)
where
Ep =k2 +m2p −m2π
2√k2
, Eπ =k2 +m2π −m2p
2√k2
, (A.2)
and
|kpπ| =
√λ(k2,m2p,m
2π)
2√k2
. (A.3)
The Dirac spinors for the Λ are
u(k,+1/2) =
√2mΛ
0
0
0
, u(k,−1/2) =
0
√2mΛ
0
0
, (A.4)
an the spinor for p are [41]
u(k1,+1/2) =1√
2mΛ
√r+ cos
θb2
√r+ sin
θb2 e
iφb
√r− cos
θb2
√r− sin
θb2 e
iφb
, u(k1,−1/2) =
1√2mΛ
−√r+ sin θb2 e−iφb
√r+ cos
θb2
√r− sin
θb2 e−iφb
−√r− cos θb2
.
(A.5)
The secondary weak decay is governed by the Hamiltonian
Heff∆S=1 =4GF√
2V ∗udVus[d̄γµPLu][ūγ
µPLs] . (A.6)
– 14 –
-
The Λ(k, sΛ)→ p(k1, sp)π(k2) matrix elements are parametrized as
H2(sΛ, sΛ) ≡ 〈p(k1, sp)π−(k2)|[d̄γµPLu
][ūγµPLs] |Λ(k, sΛ)〉
=[ū(k1, sp)
(ξ γ5 + ω
)u(k, sΛ)
]. (A.7)
In terms of the kinematics variables the secondary decay amplitudes are
H2(+1/2,+1/2) = (√r+ ω −
√r− ξ) cos
θΛ2,
H2(+1/2,−1/2) = − (√r+ ω +
√r− ξ) sin
θΛ2eiφ ,
H2(−1/2,+1/2) = − (−√r+ ω +
√r− ξ) sin
θΛ2e−iφ ,
H2(−1/2,−1/2) = (√r+ ω +
√r− ξ) cos
θΛ2. (A.8)
In the final distribution only the parity violating parameter is relevant
α =−2Re(ωξ)√
r−r+|ξ|2 +
√r+r−|ω|2
≡ αexp . (A.9)
B Leptonic helicity amplitudes
To calculate the lepton helicity amplitudes for Λ(p, sp) → Λ(k, sk) `+(q+)`−(q−) in thedilepton rest-frame we have to calculate the following two currents
ū`1(1∓ γ5)v`2 , and �̄µ(λ)ū`1γµ(1∓ γ5)v`2 . (B.1)
In the dilepton rest-frame (2`RF), the four momentum of the two leptons are
qµ−
∣∣∣2`RF
= (E`,−|q2`| sin θ` cosφ`,−|q2`| sin θ` sinφ`,−|q2`| cos θ`) , (B.2)
qµ+
∣∣∣2`RF
= (E`, |q2`| sin θ` cosφ`, |q2`| sin θ` sinφ`, |q2`| cos θ`) , (B.3)
with
|q2`| =β`2
√q2 , E` =
√q2
2, β` =
√1−
4m2`q2
. (B.4)
Following [41] we give the explicit expressions of the spinors in this frame are
u`−(λ) =
√E` +m`χuλ2λ√E` −m`χuλ
, χu+ 12
=
cos θ`2−e−iφl sin θ`2
, χu− 12
=
eiφl sin θ`2cos θ`2
, (B.5)
v`+(λ) =
√E` −m`χv−λ−2λ√E` +m`χ
v−λ
, χv+ 12
=
sin θ`2e−iφl cos θ`2
, χv− 12
=
−eiφl cos θ`2sin θ`2
.(B.6)
– 15 –
-
Using these spinors we obtain the following non-zero expressions of the leptonic helicity
amplitudes for different combinations of λ1 and λ2
L+ 1
2+ 1
2L = −
√q2(1 + β`)e
iφl , L− 1
2− 1
2R =
√q2(1 + β`)e
−iφl , (B.7)
L− 1
2− 1
2L = −
√q2(1− β`)e−iφl , L
+ 12
+ 12
R =√q2(1− β`)eiφl , (B.8)
L+ 1
2+ 1
2L,+1 =
√2m` sin θ`e
2iφl , L+ 1
2+ 1
2R,+1 =
√2m` sin θ`e
2iφl , (B.9)
L− 1
2− 1
2L,−1 =
√2m` sin θ`e
−2iφl , L− 1
2− 1
2R,−1 =
√2m` sin θ`e
−2iφl , (B.10)
L− 1
2− 1
2L,+1 = −
√2m` sin θ` , L
− 12− 1
2R,+1 = −
√2m` sin θ` , (B.11)
L+ 1
2+ 1
2L,−1 = −
√2m` sin θ` , L
+ 12
+ 12
R,−1 = −√
2m` sin θ` , (B.12)
L+ 1
2− 1
2L,+1 =
√q2
2(1− β`)(1− cos θ`)eiφl , L
− 12
+ 12
R,−1 = −√q2
2(1− β`)(1− cos θ`)e−iφl ,
(B.13)
L− 1
2+ 1
2L,+1 = −
√q2
2(1 + β`)(1 + cos θ`)e
iφl ,−L+12− 1
2R,−1 = −
√q2
2(1 + β`)(1 + cos θ`)e
−iφl ,
(B.14)
L+ 1
2− 1
2R,+1 =
√q2
2(1 + β`)(1− cos θ`)eiφl , L
− 12
+ 12
L,−1 = −√q2
2(1 + β`)(1− cos θ`)e−iφl , (B.15)
L− 1
2+ 1
2R,+1 = −
√q2
2(1− β`)(1 + cos θ`)eiφl ,−L
+ 12− 1
2L,−1 = −
√q2
2(1− β`)(1 + cos θ`)e−iφl ,
(B.16)
L+ 1
2+ 1
2L,0 = 2m` cos θ`e
iφl , L− 1
2− 1
2L,0 = −2m` cos θ`e
−iφl , (B.17)
L+ 1
2+ 1
2R,0 = 2m` cos θ`e
iφl , L− 1
2− 1
2R,0 = −2m` cos θ`e
−iφl , (B.18)
L+ 1
2− 1
2L,0 =
√q2(1− β`) sin θ` , L
− 12
+ 12
R,0 =√q2(1− β`) sin θ` , (B.19)
L− 1
2+ 1
2L,0 =
√q2(1 + β`) sin θ` , L
+ 12− 1
2R,0 =
√q2(1 + β`) sin θ` , (B.20)
L+ 1
2+ 1
2L,t = −2m`e
iφlL− 1
2− 1
2L,t = −2m`e
−iφl , (B.21)
L+ 1
2+ 1
2R,t = 2m`e
iφlL− 1
2− 1
2R,t = 2m`e
−iφl . (B.22)
The combinations of λ1 and λ2 for which the amplitudes vanish are not shown.
C Angular coefficients
K1 =1
4
(2|AR‖0 |
2 + |AR‖1 |2 + 2|AR⊥0 |
2 + |AR⊥1 |2 + {R↔ L}
)+
1
2
(|ARS⊥|2 + |ARS‖|
2 + {R↔ L}), (C.1)
K′1 = Re(ARS⊥A
∗⊥t +A
RS‖A
∗‖t − {R↔ L}
), (C.2)
– 16 –
-
K′′1 = −(|AR‖0 |
2 + |AR⊥0 |2 + {R↔ L}
)+
(|A⊥t|2 + {⊥↔ ‖}
)+ 2Re
(AR⊥0A
∗L⊥0 +A
R⊥1A
∗L⊥1 + {⊥↔ ‖}
)−(|ARS⊥|2 + |ARS‖|
2 + {R↔ L})− Re
(ARS‖A
∗LS‖ +A
RS⊥A
∗LS⊥
), (C.3)
K2 =1
2
(|AR‖1 |
2 + |AR⊥1 |2 + {R↔ L}
)+
1
2
(|ARS⊥|2 + |ARS‖|
2 + {R↔ L}), (C.4)
K′2 = Re(A⊥tA
∗RS⊥ +A‖tA
∗RS‖ − {R↔ L}
), (C.5)
K′′2 =(|AR‖0 |
2 − |AR‖1 |2 + |AR⊥0 |
2 − |AR⊥1 |2 + {R↔ L}
)+
(|A⊥t|2 + {⊥↔ ‖}
)−(|ARS⊥|2 + |ARS‖|
2 + {R↔ L})
+ 2Re
(AR⊥0A
∗L⊥0 +A
R⊥1A
∗L⊥1 −A
RS⊥A
∗LS⊥ + {⊥↔ ‖}
), (C.6)
K3 = −β`(AR⊥1A
∗R‖1 − {R↔ L}
), (C.7)
K′3 = β`Re(ARS‖A
∗R‖0 +A
RS⊥A
∗R⊥0 + {R↔ L}
+ ALS‖A∗R‖0 +A
RS‖A
∗L‖0 + {‖ ↔⊥}
), (C.8)
K′′3 = 0 , (C.9)
K4 =αΛ2
Re
(2AR⊥0A
∗R‖0 +A
R⊥1A
∗R‖1 + 2A
RS⊥A
∗RS‖ + {R↔ L}
), (C.10)
K′4 = αΛRe(ARS‖A
∗⊥t +A
RS⊥A
∗‖t − {R↔ L}
), (C.11)
K′′4 = −2αΛRe(AR⊥0A
∗R‖0 +A
RS⊥A
∗RS‖ + {R↔ L}
− AR⊥0A∗L‖0 −A
R⊥1A
∗L‖1 + {‖ ↔⊥}
− A⊥tA∗‖t +ALS⊥A
∗RS‖ +A
RS⊥A
∗LS‖
), (C.12)
K5 = αΛRe(AR⊥1A
∗R‖1 +A
RS⊥A
∗RS‖ + {R↔ L}
), (C.13)
K′5 = αΛRe(ARS‖A
∗⊥t +A
RS⊥A
∗‖t + {R↔ L}
), (C.14)
K′′5 = 2αΛRe(AR⊥0A
∗R‖0 −A
R⊥1A
∗R‖1 + {R↔ L}
+ AR⊥0A∗L‖0 +A
R‖0A
∗L⊥0 +A
R⊥1A
∗L‖1 +A
R‖1A
∗L⊥1
+ A⊥tA∗‖t − (A
RS⊥A
∗RS‖ + {R↔ L}+A
LS⊥A
∗RS‖ +A
RS⊥A
∗LS‖ )
), (C.15)
– 17 –
-
K6 = −αΛβ`
2
(|AR⊥1 |
2 + |AR‖1 |2 − {R↔ L}
), (C.16)
K′6 = αΛβ`Re(ARS⊥A
∗R‖0 +A
RS‖A
∗R⊥0 + {R↔ L}
+ ARS‖A∗L⊥0 +A
LS‖A
∗R⊥0 + {‖ ↔⊥}
), (C.17)
K′′6 = 0 , (C.18)
K7 =αΛ√
2Re
(AR⊥1A
∗R‖0 −A
R‖1A
∗R⊥0 + {R↔ L}
), (C.19)
K′7 = 0 , (C.20)
K′′7 = 2√
2αΛRe
(AR‖1A
∗R⊥0 −A
R⊥1A
∗R‖0 + {R↔ L}
), (C.21)
K8 =αΛβ`√
2Re
(AR⊥1A
∗R⊥0 −A
R‖1A
∗R‖0 − {R↔ L}
), (C.22)
K′8 =αΛβ`√
2Re
(ARS⊥A
∗R‖1 −A
RS‖A
∗R⊥1 + {R↔ L}
+ ALS⊥A∗R‖1 +A
RS⊥A
∗L‖1 − {‖ ↔⊥}
), (C.23)
K′′8 = 0 , (C.24)
K9 =αΛ√
2Im
(AR⊥1A
∗R⊥0 −A
R‖1A
∗R‖0 + {R↔ L}
), (C.25)
K′9 = 0 , (C.26)
K′′9 = 2√
2αΛIm
(AR‖1A
∗R‖0 −A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.27)
K10 =αΛβ`√
2Im
(AR⊥1A
∗R‖0 −A
R‖1A
∗R⊥0 − {R↔ L}
), (C.28)
K′10 =αΛβ`√
2Im
(ARS⊥A
∗R⊥1 −A
RS‖A
∗R‖1 + {R↔ L}
+ ALS⊥A∗R⊥1 +A
RS⊥A
∗L⊥1 + {⊥↔ ‖}
), (C.29)
K′′10 = 0 , (C.30)
K11 =PΛb2
Re
(2AR‖0A
∗R⊥0 −A
R‖1A
∗R⊥1 + {R↔ L}
+ 2ARS⊥A∗RS‖ + {R↔ L}
), (C.31)
K′11 = PΛbRe(ARS⊥A
∗‖t +A
RS‖A
∗⊥t − {R↔ L}
), (C.32)
K′′11 = −2PΛbRe(AR‖0A
∗R⊥0 +A
RS⊥A
∗RS‖ + {R↔ L} −A⊥tA
∗‖t
+ (AR‖1A∗L⊥1 −A
R‖0A
∗L⊥0 +A
LS⊥A
∗RS‖ + {‖ ↔⊥})
), (C.33)
– 18 –
-
K12 = −PΛbRe(AR‖1A
∗R⊥1 + {R↔ L} −A
RS⊥A
∗RS‖ − {R↔ L}
), (C.34)
K′12 = PΛbRe(ARS⊥A
∗‖t +A
RS‖A⊥t − {R↔ L}
), (C.35)
K′′12 = 2PΛbRe(AR‖0A
∗R⊥0 +A
R‖1A
∗R⊥1 −A
RS⊥A
∗RS‖ + {R↔ L}
+ A⊥tA∗‖
+ AR‖0A∗L⊥0 −A
R‖1A
∗L⊥1 + {‖ ↔⊥}
− ALS⊥A∗RS‖ − {R↔ L}), (C.36)
K13 =PΛbβl
2
(|AR‖1 |
2 + |AR⊥1 |2 − {R↔ L}
), (C.37)
K′13 = PΛbβlRe(ARS‖A
∗R⊥0 +A
RS⊥A
∗R‖0 + {R↔ L}
+ ARS‖A∗L⊥0 +A
LS⊥A
R⊥0 + {‖ ↔⊥}
), (C.38)
K′′13 = 0 , (C.39)
K14 =PΛbαΛ
4Re
(2|AR‖0 |
2 + 2|AR⊥0 |2 − |AR‖1 |
2 − |AR⊥1 |2 + {R↔ L}
+ 2(|ARS‖|2 + |ARS⊥|2 + {R↔ L})
), (C.40)
K′14 = PΛbαΛRe(ARS⊥A
∗⊥t +A
RS‖A
∗‖t − {R↔ L}
), (C.41)
K′′14 = −PΛbαΛ(|AR‖0 |
2 + |AR⊥0 |2 + {R↔ L}
− (|A‖t|2 + |A⊥t|2)+ (|ARS⊥|2 + |ARS‖|
2 + {R↔ L})
− 2Re(AL‖0A∗R‖0 −A
L‖1A
∗R‖1 −A
RS‖A
∗LS‖ + {‖ ↔⊥})
), (C.42)
K15 =PΛbαΛ
2
(− (|AR⊥1 |
2 + |AR‖1 |2 + {R↔ L})
+ (|ARS‖|2 + |ARS⊥|2 + {R↔ L})
), (C.43)
K′15 = PΛbαΛRe(ARS‖A
∗‖t +A
RS⊥A
∗⊥t − {R↔ L}
), (C.44)
K′′15 = PΛbαΛ[|AR‖0 |
2 + |AR‖1 |2 + |AR⊥0 |
2 + |AR⊥1 |2 + {R↔ L}
+ (|A‖t|2 + |A⊥t|2)− (|ARS⊥|2 + |ARS‖|2 + {R↔ L})
+ 2Re(AL‖0A∗R‖0 −A
L‖1A
∗R‖1 −A
RS‖A
∗LS‖ + {‖ ↔⊥})
], (C.45)
– 19 –
-
K16 = PΛbαΛβlRe(AR‖1A
∗R⊥1 − {R↔ L}
), (C.46)
K′16 = PΛbαΛβlRe(ARS‖A
∗R‖0 +A
RS⊥A
∗R⊥0 + {R↔ L}
+ ARS‖A∗L‖0 +A
LS‖A
∗R‖0 + {‖ ↔⊥}
), (C.47)
K′′16 = 0 , (C.48)
K17 = −PΛbαΛ√
2Re
(AR‖1A
∗R‖0 −A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.49)
K′17 = 0 , (C.50)
K′′17 = 2√
2PΛbαΛRe
(AR‖1A
∗R‖0 −A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.51)
K18 =PΛbαΛβl√
2Re
(AR⊥1A
∗R‖0 −A
R‖1A
∗R⊥0 − {R↔ L}
), (C.52)
K′18 =PΛbαΛβl√
2Re
(ARS‖A
∗R‖1 −A
RS⊥A
∗R⊥1 + {R↔ L}
+ ALS‖A∗R‖1 +A
RS‖A
∗L‖1 − {‖ ↔⊥}
), (C.53)
K′′18 = 0 , (C.54)
K19 = −PΛbαΛ√
2Im
(AR‖1A
∗R⊥0 −A
R⊥1A
∗R‖0 − {R↔ L}
), (C.55)
K′19 = 0 , (C.56)
K′′19 = 2√
2PΛbαΛIm
(AR‖1A
∗R⊥0 −A
R⊥1A
∗R‖0 − {R↔ L}
), (C.57)
K20 =PΛbαΛβl√
2Im
(AR⊥1A
∗R⊥0 −A
R‖1A
∗R‖0 − {R↔ L}
), (C.58)
K′20 =PΛbαΛβl√
2Im
(ARS‖A
∗R⊥1 −A
RS⊥A
∗R‖1 + {R↔ L}
+ ALS‖A∗R⊥1 +A
RS‖A
∗L⊥1 − {‖ ↔⊥}
), (C.59)
K′′20 = 0 , (C.60)
K21 =PΛb√
2Im
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.61)
K′21 = 0 , (C.62)
K′′21 = −2√
2PΛbIm
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.63)
K22 = −PΛbβl√
2Im
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 − {R↔ L}
), (C.64)
K′22 =PΛbβl√
2Im
(ARS‖A
∗R‖1 +A
RS⊥A
∗R⊥1 + {R↔ L}
+ ALS‖A∗R‖1 +A
RS‖A
∗L‖1 + {‖ ↔⊥}
), (C.65)
– 20 –
-
K′′22 = 0 , (C.66)
K23 = −PΛb√
2Re
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 + {R↔ L}
), (C.67)
K′23 = 0 , (C.68)
K′′23 = 2√
2PΛbRe
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 + {R↔ L}
), (C.69)
K24 =PΛbβl√
2Re
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 − {R↔ L}
), (C.70)
K′24 =PΛbβl√
2Re
(ARS‖A
∗R⊥1 +A
LS⊥A
∗L‖1 + {R↔ L}
+ ALS‖A∗R⊥1 +A
RS‖A
∗L⊥1 + {‖ ↔⊥}
), (C.71)
K′′24 = 0 , (C.72)
K25 =PΛbαΛ√
2Im
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 + {R↔ L}
), (C.73)
K′25 = 0 , (C.74)
K′′25 = 2√
2PΛbαΛIm
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 + {R↔ L}
), (C.75)
K26 =PΛbαΛβl√
2Im
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 − {R↔ L}
), (C.76)
K′26 = −PΛbαΛβl√
2
(ARS‖A
∗R⊥1 +A
RS⊥A
∗R‖1 + {R↔ L}
+ ALS‖A∗R⊥1 +A
RS‖A
∗L⊥1 + {‖ ↔⊥}
), (C.77)
K′′26 = 0 , (C.78)
K27 = −PΛbαΛ√
2Re
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.79)
K′27 = 0 , (C.80)
K′′27 = 2√
2PΛbαΛRe
(AR‖1A
∗R‖0 +A
R⊥1A
∗R⊥0 + {R↔ L}
), (C.81)
K28 =PΛbαΛβl√
2Re
(AR‖1A
∗R⊥0 +A
R⊥1A
∗R‖0 − {R↔ L}
), (C.82)
K′28 =PΛbαΛβl√
2Re
(ARS‖A
∗R‖1 +A
RS⊥A
∗R⊥1 + {R↔ L}
+ ALS‖A∗R‖1 +A
RS‖A
∗L‖1 + {‖ ↔⊥}
), (C.83)
K′′28 = 0 , (C.84)
K29 = PΛbαΛIm(ARS⊥A
∗RS‖ + {R↔ L}
), (C.85)
K′29 = −PΛbαΛIm(ARS‖A
∗⊥t −ARS⊥A∗‖t − {R↔ L}
), (C.86)
– 21 –
-
K′′29 = −2PΛbαΛIm(AR‖0A
∗R⊥0 +A
RS⊥A
∗LS‖ + {R↔ L}
+ AR‖0A∗L⊥0 +A
R⊥0A
∗L‖0 + {‖ ↔⊥}
− A⊥tA∗‖t), (C.87)
K30 = −PΛbαΛIm(AR‖0A
∗R⊥0 −A
RS⊥A
∗RS‖ + {R↔ L}
), (C.88)
K′30 = −PΛbαΛIm(ARS‖A
∗⊥t −ARS⊥A∗‖t − {R↔ L}
), (C.89)
K′′30 = 2PΛbαΛIm(AR‖0A
∗R⊥0 −A
RS⊥A
∗RS‖ + {R↔ L}
− AR‖0A∗L⊥0 +A
R⊥0A
∗L‖0 +A⊥tA
∗‖t
− ALS⊥A∗RS‖ − {‖ ↔⊥}), (C.90)
K31 = −PΛb2
(|ARS⊥|2 − |ARS‖|
2 + {R↔ L}), (C.91)
K′31 = −PΛbαΛRe(ARS‖A
∗‖t −A
RS⊥A
∗⊥t − {R↔ L}
), (C.92)
K′′31 = −PΛbαΛ(|AR‖0 |
2 − |AR⊥0 |2 + {R↔ L}
+ |A‖t|2 − |A⊥t|2
+ |ARS⊥|2 − |ARS‖|2 + {R↔ L}
+ 2Re(AL‖0A∗R‖0 −A
L⊥0A
∗R⊥0 −A
RS‖A
∗LS‖ +A
RS⊥A
∗LS⊥)
), (C.93)
K32 = −PΛbαΛ
2
(|AR‖0 |
2 − |AR⊥0 |2 + {R↔ L}
+ |ARS‖|2 − |ARS⊥|2 + {R↔ L}
), (C.94)
K′32 = −PΛbαΛRe(ARS‖A
∗‖t −A
RS⊥A
∗⊥t − {R↔ L}
), (C.95)
K′′32 = −PΛbαΛ(|AR⊥0 |
2 − |AR‖0 |2 + {R↔ L}+ |A‖t|2 − |A⊥t|2
+ |ARS⊥|2 − |ARS‖|2 + {R↔ L}
+ 2Re(AL‖0A∗R‖0 −A
RS‖A
∗LS‖ − {‖ ↔⊥})
), (C.96)
K33 = −PΛbαΛ
4
(|AR‖1 |
2 − |AR⊥1 |2 + {R↔ L}
), (C.97)
K′33 = 0 , (C.98)
K′′33 = PΛbαΛ(|AR‖1 |
2 − |AR⊥1 |2 + {R↔ L}
), (C.99)
– 22 –
-
K34 = −PΛbαΛ
2Im
(AR‖1A
∗R⊥1 + {R↔ L}
), (C.100)
K′34 = 0 , (C.101)
K′′34 = 2PΛbαΛIm(AR‖1A
∗R⊥1 + {R↔ L}
), (C.102)
K35 = 0 , (C.103)
K′35 = PΛbαΛβlIm(ARS⊥A
∗R‖0 −A
RS‖A
∗R⊥0 + {R↔ L}
+ AL⊥0A∗RS‖ +A
R⊥0A
∗LS‖ +A
RS⊥A
∗L‖0 +A
LS⊥A
∗R‖0
), (C.104)
K′′35 = 0 , (C.105)K36 = 0 , (C.106)
K′36 = −PΛbαΛβlRe(ARS‖A
∗R‖0 −A
RS⊥A
∗R⊥0 + {R↔ L}
+ ALS‖A∗R‖0 +A
RS‖A
∗L‖0 − {‖ ↔⊥}
), (C.107)
K′′36 = 0 (C.108)
References
[1] R. Aaij et al. [LHCb Collaboration], Search for lepton-universality violation in
B+ → K+`+`− decays, Phys. Rev. Lett. 122, no. 19, 191801 (2019), [arXiv:1903.09252[hep-ex]].
[2] R. Aaij et al. [LHCb Collaboration], Test of lepton universality with B0 → K∗0`+`− decays,JHEP 1708, 055 (2017), [arXiv:1705.05802 [hep-ex]].
[3] R. Aaij et al. [LHCb], Differential branching fractions and isospin asymmetries of
B → K(∗)µ+µ− decays, JHEP 06, 133 (2014), [arXiv:1403.8044 [hep-ex]].
[4] R. Aaij et al. [LHCb], Differential branching fraction and angular analysis of the decay
B0 → K∗0µ+µ−, JHEP 08, 131 (2013) [arXiv:1304.6325 [hep-ex]].
[5] R. Aaij et al. [LHCb], Measurements of the S-wave fraction in B0 → K+π−µ+µ− decays andthe B0 → K∗(892)0µ+µ− differential branching fraction, JHEP 11, 047 (2016)[arXiv:1606.04731 [hep-ex]].
[6] R. Aaij et al. [LHCb], Angular analysis and differential branching fraction of the decay
B0s → φµ+µ−, JHEP 09, 179 (2015) [arXiv:1506.08777 [hep-ex]].
[7] R. Aaij et al. [LHCb Collaboration], Measurement of CP -averaged observables in the
B0 → K∗0µ+µ− decay, arXiv:2003.04831 [hep-ex].
[8] R. Aaij et al. [LHCb Collaboration], Differential branching fraction and angular analysis of
Λ0b → Λµ+µ− decays, JHEP 1506, 115 (2015), [arXiv:1503.07138 [hep-ex]].
[9] T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij and P. Santorelli, Rare baryon
decays Λb → Λl+l−(l = e, µ, τ) and Λb → Λγ : differential and total rates, lepton- andhadron-side forward-backward asymmetries, Phys. Rev. D 87, 074031 (2013),
[arXiv:1301.3737 [hep-ph]].
– 23 –
-
[10] P. Böer, T. Feldmann and D. van Dyk, Angular Analysis of the Decay Λb → Λ(→ Nπ)`+`−,JHEP 1501, 155 (2015), [arXiv:1410.2115 [hep-ph]].
[11] D. Das, Model independent New Physics analysis in Λb → Λµ+µ− decay, Eur. Phys. J. C 78,no. 3, 230 (2018), [arXiv:1802.09404 [hep-ph]].
[12] T. Blake and M. Kreps, Angular distribution of polarised Λb baryons decaying to Λ`+`−,
JHEP 1711, 138 (2017), [arXiv:1710.00746 [hep-ph]].
[13] S. Roy, R. Sain and R. Sinha, Lepton mass effects and angular observables in
Λb → Λ(→ pπ)`+`−, Phys. Rev. D 96, no. 11, 116005 (2017), [arXiv:1710.01335 [hep-ph]].
[14] D. Das, On the angular distribution of Λb → Λ(→ Nπ)τ+τ− decay, JHEP 07, 063 (2018),[arXiv:1804.08527 [hep-ph]].
[15] H. Yan, Angular distribution of the rare decay Λb → Λ(→ Nπ)`+`−, [arXiv:1911.11568[hep-ph]].
[16] R. Aaij et al. [LHCb], Angular moments of the decay Λ0b → Λµ+µ− at low hadronic recoil,JHEP 09, 146 (2018) [arXiv:1808.00264 [hep-ex]].
[17] LHCb collaboration, R. Aaij et al. Measurements of the Λ0b → J/ψΛ decay amplitudes andthe Λ0b polarisation in pp collisions at
√s = 7 TeV, Phys. Lett. B 724, 27-35 (2013),
[arXiv:1302.5578 [hep-ex]].
[18] CMS collaboration, Measurement of Lambda b polarization and the angular parameters of the
decay Lambdab → J/psiLambda, CMS-PAS-BPH-15-002.
[19] Y. m. Wang, Y. Li and C. D. Lu, Rare Decays of Λb → Λ + γ and Λb → Λ + `+`− in theLight-cone Sum Rules, Eur. Phys. J. C 59, 861-882 (2009) [arXiv:0804.0648 [hep-ph]].
[20] T. Mannel and Y. M. Wang, Heavy-to-light baryonic form factors at large recoil, JHEP 12,
067 (2011), [arXiv:1111.1849 [hep-ph]].
[21] Y. M. Wang and Y. L. Shen, Perturbative Corrections to Λb → Λ Form Factors from QCDLight-Cone Sum Rules, JHEP 02, 179 (2016), [arXiv:1511.09036 [hep-ph]].
[22] W. Detmold and S. Meinel, Λb → Λ`+`− form factors, differential branching fraction, andangular observables from lattice QCD with relativistic b quarks, Phys. Rev. D 93, no. 7,
074501 (2016), [arXiv:1602.01399 [hep-lat]].
[23] N. Isgur and M. B. Wise, WEAK TRANSITION FORM-FACTORS BETWEEN HEAVY
MESONS Phys. Lett. B 237, 527-530 (1990)
[24] N. Isgur and M. B. Wise, Heavy baryon weak form-factors, Nucl. Phys. B 348, 276-292 (1991)
[25] N. Isgur and M. B. Wise, Weak Decays of Heavy Mesons in the Static Quark Approximation,
Phys. Lett. B 232, 113-117 (1989)
[26] N. Isgur and M. B. Wise, Relationship Between Form-factors in Semileptonic B̄ and D
Decays and Exclusive Rare B̄ Meson Decays, Phys. Rev. D 42, 2388-2391 (1990)
[27] B. Grinstein and D. Pirjol, Exclusive rare B → K∗`+`− decays at low recoil: Controlling thelong-distance effects, Phys. Rev. D 70, 114005 (2004), [hep-ph/0404250].
[28] T. Mannel, W. Roberts and Z. Ryzak, Baryons in the heavy quark effective theory, Nucl.
Phys. B 355, 38-53 (1991)
[29] M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive B → V l+l−, V γdecays, Nucl. Phys. B 612, 25-58 (2001), [arXiv:hep-ph/0106067 [hep-ph]].
– 24 –
-
[30] M. Beneke, T. Feldmann and D. Seidel, Exclusive radiative and electroweak b→ d and b→ spenguin decays at NLO, Eur. Phys. J. C 41, 173 (2005), [hep-ph/0412400].
[31] T. Feldmann and M. W. Y. Yip, Form Factors for Λb → Λ Transitions in SCET, Phys. Rev.D 85, 014035 (2012), Erratum: [Phys. Rev. D 86, 079901 (2012)], [arXiv:1111.1844 [hep-ph]].
[32] C. Patrignani et al. [Particle Data Group], Review of Particle Physics, Chin. Phys. C 40, no.
10, 100001 (2016).
[33] C. Bobeth, G. Hiller and D. van Dyk, The Benefits of B̄ → K̄∗l+l− Decays at Low Recoil,JHEP 07, 098 (2010) [arXiv:1006.5013 [hep-ph]].
[34] D. Das, G. Hiller, M. Jung and A. Shires, The B → Kπ`` and Bs → KK`` distributions atlow hadronic recoil, JHEP 09, 109 (2014) [arXiv:1406.6681 [hep-ph]].
[35] D. Das and J. Das, The Λb → Λ∗(1520)(→ NK̄)`+`− decay at low-recoil in HQET, JHEP07, 002 (2020), [arXiv:2003.08366 [hep-ph]].
[36] B. Capdevila, U. Laa and G. Valencia, Anatomy of a six-parameter fit to the b→ s`+`−anomalies, Eur. Phys. J. C 79, no.6, 462 (2019), [arXiv:1811.10793 [hep-ph]].
[37] B. Capdevila, A. Crivellin, S. Descotes-Genon, J. Matias and J. Virto, Patterns of New
Physics in b→ s`+`− transitions in the light of recent data, JHEP 01, 093 (2018),[arXiv:1704.05340 [hep-ph]].
[38] T. Blake, S. Meinel and D. van Dyk, Bayesian Analysis of b→ sµ+µ− Wilson Coefficientsusing the Full Angular Distribution of Λb → Λ(→ p π−)µ+µ− Decays, Phys. Rev. D 101,no.3, 035023 (2020) [arXiv:1912.05811 [hep-ph]].
[39] M. Beylich, G. Buchalla and T. Feldmann, Theory of B → K(∗)`+`− decays at high q2: OPEand quark-hadron duality, Eur. Phys. J. C 71, 1635 (2011), [arXiv:1101.5118 [hep-ph]].
[40] F. Beaujean, C. Bobeth and S. Jahn, Eur. Phys. J. C 75, no.9, 456 (2015), [arXiv:1508.01526
[hep-ph]].
[41] H. E. Haber, Spin formalism and applications to new physics searches, In *Stanford 1993,
Spin structure in high energy processes* 231-272 [hep-ph/9405376].
– 25 –
1 Introduction2 Effective Hamiltonian 3 Angular distributions4 Low-recoil Amplitudes in HQET 5 Low-recoil factorization 6 Numerical Analysis 7 Summary A Nucleon spinor in rest frame B Leptonic helicity amplitudes C Angular coefficients