-
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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mailto:[email protected]:[email protected]
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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 –
-
√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 –
-
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 –
-
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)
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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
