4.2.1 Matrix Element and Differential Decay Distribution
At tree level in the effective theory, the matrix element of the effective Hamiltonian (4.1.1) for the decay ¯B →K¯∗(→Kπ)`+`− can be written, in naive factorization1 as [31,83,84]
4π is the fine-structure constant and all contributions fromH(u)eff have been ignored for the moment.
The matrix element above is expressed in terms of ¯B → Kπ matrix elements, however form factors in Eqs. (2.1.9), (2.1.10), (2.1.13) and (2.1.15) are defined for B → V (V being a vector meson) transitions. Thus, we need to rewrite the matrix elementshKπ| Oi|B¯i in terms ofB →V form factors. In order to do so, one possibility is to assume the K∗ to be produced resonantly, which warrants the use of the narrow width approximation for describing theK∗ →Kπtransition.
In this approximation, the full K∗ propagator simplifies to 1 This allows us to disentangle the form factors from theK∗Kπ coupling gK∗Kπ [99, 100], because it cancels between the vertex factor and the width
ΓK∗= gK2∗Kπ The matrix elements in equations (2.1.9) and (2.1.10), except for theK∗ polarization vector, can be written in the following compact form
hK¯∗(pK∗)|Jµ|B(p)¯ i=ε∗νAνµ. (4.2.5)
1Here using naive factorisation for decomposing the matrix elements of the electromagnetic operator O7 and the semileptonic operatorsO9,10 into products of matrix elements of currents is entirely justified. Because for all these operators one current is hadronic and the other leptonic or electromagnetic, no ”non-factorisable” (in the naive-factorisation sense, see Section3.1) interactions between final states can occur.
4.2. Differential Decay Distribution and Spin Amplitudes 51 The Aνµ element in the previous equation is a tensor encapsulating all the form factors. Then, the correspondingB →Kπ matrix element reads
hK(p¯ K)π(pπ)|Jµ|B¯(p)i=−DK∗(p2K∗)WνAνµ, (4.2.6) where [100]
|DK∗(p2K∗)|2=g2K∗Kπ π mK∗ΓK∗
δ(p2K∗−m2K∗) = 48π2 β3m2K∗
δ(p2K∗−m2K∗), (4.2.7)
Wν ≡
gνµ− pνK∗pµK∗
m2K∗
(pπ−pK)µ=QνπK −m2K−m2π p2K∗
pµK∗, QµπK =pµK−pµπ. (4.2.8) Assuming that K∗ is produced on-shell, only four independent kinematical variables are needed to fully characterize all the quantities in the decay: the dilepton mass squared q2 and the angles θK∗, θ` and φ (all of them defined in Appendix A). Squaring the matrix element, summing over spins of the final states and taking advantage of several kinematics identities (see Appendix A), one obtains the full angular distribution of the ¯B0 →K¯∗0(→K+π−)`+`−,
d4Γ
dq2dcosθ`dcosθK∗dφ = 9
32πJ(q2, θ`, θK∗, φ), (4.2.9) where
J(q2, θ`, θK∗, φ) =J1ssin2θK∗+J1ccos2θK∗+ (J2ssin2θK∗+J2ccos2θK∗) cos 2θ` +J3sin2θK∗sin2θ`cos 2φ+J4sin 2θK∗sin 2θ`cosφ
+J5sin 2θK∗sinθ`cosφ
+ (J6ssin2θK∗+J6ccos2θK∗) cosθ`+J7sin 2θK∗sinθ`sinφ
+J8sin 2θK∗sin 2θ`sinφ+J9sin2θK∗sin2θ`sin 2φ. (4.2.10) In a similar way, the differential angular distribution of the CP-conjugate mode Bd0 → K∗0(→ K−π+)`+`− is obtained
d4Γ¯
dq2dcosθ`dcosθK∗dφ = 9
32πJ(q¯ 2, θ`, θK∗, φ), (4.2.11) with ¯J(q2, θ`, θK∗) having the same structure asJ(q2, θ`, θK∗) but replacing [100]
J1s,1c,2s,2c,3,4,7 −→J¯1s,1c,2s,2c,3,4,7, J5,6s,6c,8,9 −→ −J¯5,6s,6c,8,9, (4.2.12) where ¯Ji equals Ji but with all weak phases conjugated. The extra minus sign in (4.2.12) is due to the kinematical convention used in the definition of the angles.
The angular coefficientsJi, which are functions ofq2 only, are usually expressed in term of the K∗ transversity amplitudes. In the remaining of this section, we are going to define these so-called transversity amplitudes and show their relations with the angular coefficients.
4.2.2 Transversity Amplitudes
A particular context that allows for a natural introduction of the transversity amplitudes is the study of the decay ¯B → K¯∗V∗, where the ¯K∗0 is assumed to be on-shell while V∗ is a virtual gauge boson (either aγ∗ or aZ0). Then, as in Eq. (4.2.5), the amplitude for the process takes the form [31]
M(m,n)(B →K∗V∗) =ε∗µK∗(m)Mµνε∗νV∗(n), (4.2.13) where Mµν is the tensor associated to the hadronic current, εµV∗ is the polarization vector of the virtual gauge boson andεµK∗ is the K∗ polarization vector.
The V∗ is an off-shell gauge boson, so it has 3 (spin 1) polarisations plus an extra time-like (spin 0) polarisation. The three spin 1 components (n=±,0) are orthogonal to the momentumqµ transferred toV∗during theB meson decay, i.e. qµεµV∗(n) = 0, while the spin 0 component (n=t) is proportional to qµ, i.e. εµV∗(t) = qµ/p
q2. The set {εµV∗(±), εµV∗(0), εµV∗(t)} of independent polarisation vectors constitutes a basis on the polarisation space ofV∗. Assuming that the gauge boson V∗ propagates along the z-axis,
qµ= (q0,0,0, qz), (4.2.14)
in the B meson rest frame, then the four polarisation vectors of this basis may be written as [99,101]
εµV∗(±) = 1
√2(0,1,∓i,0), (4.2.15)
εµV∗(0) = 1
pq2(−qz,0,0,−q0), (4.2.16) εµV∗(t) = 1
pq2(q0,0,0, qz). (4.2.17) It can be proved that the polarization vectors in Eqs. (4.2.15)-(4.2.17) satisfy the following or-thonormality and completeness relations
ε∗µV∗(n)εV∗µ(n0) =gnn0, (4.2.18) X
n,n0
ε∗µV∗(n)ενV∗(n0)gnn0 =gµν, (4.2.19) withn=±,0, t andgnn0 = diag(+,−,−,−,).
On a different note, theK∗ is on-shell, thus it has only 3 possible polarisations available and all of them are orthogonal with respect to the momentumpµK∗= (k0,0,0, kz) carried by the meson.
Again, in theB meson rest frame, these polarisation vectors read
εµV∗(±) = 1
√2(0,1,±i,0), (4.2.20)
εµV∗(0) = 1 mK∗
(kz,0,0, k0), (4.2.21)
(4.2.22)
4.2. Differential Decay Distribution and Spin Amplitudes 53 and satisfy the relations
ε∗µK∗(n)εK∗µ(n0) =−δmm0, (4.2.23) X
m,m0
ε∗µK∗(m)ενK∗(m0)δmm0 =−gµν+pµK∗pνK∗
m2K∗
. (4.2.24)
TheK∗ helicity amplitudes H0,H+1 andH−1 can now be projected out fromMµν by contracting it with the explicit polarization vectors in (4.2.33), obtaining [31,101]
H±1 =ε∗νK∗(±)ε∗µV∗(±)Mνµ, (4.2.25) H0 =ε∗νK∗(0)ε∗µV∗(0)Mνµ, (4.2.26) which are calledhelicity amplitudes as they are attached to a specific helicity value ofm. Using a more compact notation
Hm=M(m,m)(B →K∗V∗), m= 0,+1,−1 (4.2.27) Alternatively, one can work within a different framework provided by combinations of helicityHm amplitudes, which we calltransversity amplitudes [31,83]:
A⊥,k ≡ 1
√2(H+1∓H−1), A0 ≡H0. (4.2.28) However, not all the information available is contained in A±,0. In B → K∗V∗ with V∗ being virtual there is yet another possible contraction of the hadronic tensorMµν with the polarization vectors in (4.2.33)
At=ε∗νK∗(0)ε∗µV∗(t)Mνµ=M(0,t)(B →K∗V∗). (4.2.29) This transversity amplitude has no counterpart in theK∗helicity basis, however it may be regarded to aK∗ polarization vector which is longitudinal in theK∗ rest frame and simultaneously time-like in theV∗ rest frame.
Consider now that theV∗ decays into a lepton-antilepton pair, then the amplitude becomes
M(B→K∗V∗(→`+`−))(m)∝ε∗νK∗(m)MνµX
n,n0
ε∗µV∗(n)ερV∗(n0)gnn0 `γ¯ ρ, PL,R`
, (4.2.30) where theV →`+`− coupling has been omitted, explaining the proportionality sign.
This amplitude admits a decomposition in terms of six transversity amplitudes AL⊥,k,0 and AR⊥,k,0, the LandR superindices refer to the chirality of the leptonic current, and also the seventh transversity amplitude At. This last At amplitude cannot be split into left- and right-handed parts since the polarization vector needed to projectAtisεµ(t) =qµ/p
q2 and, because of current conservation, one has that
qµ `γ¯ µ`
= 0, (4.2.31)
qµ `γ¯ µγ5`
= 2im` `γ¯ 5`
. (4.2.32)
Thus, the time-like component ofV∗ only couples to axial vectors but not to vectors. At the same time, this proves that At must vanish in the limit of massless leptons.
As we have shown, this formalism incorporates all contributions to the amplitude of the decay B¯ → K¯∗V∗(→ ``) with vector and axial-vector currents. Therefore, contributions coming from effective operators O(0)7,9,10, including their chirally-flipped counterparts, are accounted for in the transversity amplitudes parametrising the decay.
Notice that pseudoscalar currents can be transformed into axial-vectors currents by means of the equation of motion for these currents in Eq. (4.2.32). Therefore, pseudoscalar contributions to the amplitude can be reabsorbed in the time-like transversity amplitude At. Unfortunately, this does not apply to the scalar contributions and need to be embedded in a new amplitude AS. Finally, accounting for tensor and pseudotensor contributions require the addition of more transversity amplitudes (up to six new amplitudes as argued in [102]).
In summary, we have shown that the amplitude of the decay B → K∗V∗(→ `+`−) can be unambiguously characterised by means of seven transversity amplitudes AL,R⊥,k,0 and At. If tensor and pseudotensor contributions can be neglected, then the most general framework also involves an eight amplitude accounting for scalar operators.
Given the matrix element in equation (4.2.1), the explicit form of the transversity amplitudes, up to corrections of orderO(αs) and O(ΛQCD/mb), reads [31,83]
4.3. Angular Coefficients 55