Probing the photon polarization in B→ K ∗ γ with conversion
FFERMILAB-PUB-15-142-T
Probing the photon polarization in B → K ∗ γ with conversion Fady Bishara
1, 2, ∗ and Dean J. Robinson
3, 4, † Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, IL 60510, USA Department of Physics, University of California, Berkeley, CA 94720, USA Ernest Orlando Lawrence Berkeley National Laboratory,University of California, Berkeley, CA 94720, USA (Dated: October 10, 2018)
Abstract
We re-examine the possibility to measure the photon polarization in B → K ∗ γ decays, viadecays in which the photon subsequently undergoes nuclear conversion to a lepton pair. We obtaincompact expressions for the full decay-plus-conversion amplitude. With these results we show thatinterference between the B → ( K ∗ → Kπ ) γ decay and the γN → (cid:96) + (cid:96) − N conversion permitsboth the ratio and relative weak phase between the left- and right-handed photon amplitudes tobe probed by an angular observable, constructed from the final state dilepton, kaon and pionkinematic configuration. Exploiting this technique will be experimentally challenging. However,we present special kinematic cuts that enhance the statistical power of this technique by an O (1)factor. We verify this effect and extract pertinent angular kinematic distributions with dedicatednumerical simulations. ∗ E-mail:[email protected] † E-mail:[email protected] a r X i v : . [ h e p - ph ] J u l . INTRODUCTION Measurement of the photon polarization in b → sγ radiative decays has been of long-standing interest. Within the Standard Model (SM), flavor-changing electroweak interac-tions maximally violate parity, so that one expects the fraction of left-handed photons in b → sγ processes to be order unity, up to small corrections arising from either the non-zero strange quark mass or from higher order QCD contributions. In contrast, certain NewPhysics (NP) scenarios may generate b → sγ operators of comparable size to the SM terms,but with exotic parity structure, significantly modifying the expected ratio of left- versusright-handed photons – the photon polarization ratio. Measurement of this ratio thereforehas the potential to test the parity structure of b → sγ operators against SM expectations,as well as either constrain or detect the signatures of such NP scenarios.The b → sγ photon polarization ratio may be measured via various different approaches.Dominantly right-handed photon production in resonant B → ( K (1400) → Kππ ) γ gener-ates an up-down asymmetry of the photon momentum with respect to the K → Kππ decayplane [1, 2]. This up-down asymmetry was recently measured by the LHCb collaboration [3].However, a theoretical prediction for the asymmetry is not yet available, so that the photonpolarization ratio cannot yet be extracted from these results or compared to SM expecta-tions. Along similar lines, the photon polarization ratio may also be probed by measuringthe spin fraction of Λ’s in unpolarized Λ b → Λ γ decays [4], or by measuring an angularasymmetry between the Λ b spin and the outgoing photon momentum for polarized Λ b → Λ γ [5]. Other methods look for time dependent CP violation induced by mixing of B → K ∗ γ and ¯ B → K ∗ γ , which is proportional to the photon polarization ratio [6, 7]. Additionallyone may probe the polarization ratio by looking for asymmetries in angular observables inresonant B → Kπ(cid:96) + (cid:96) − or B → ππ(cid:96) + (cid:96) − [7, 8], or look for transverse asymmetries in thedilepton invariant mass for non-resonant B → Kπ(cid:96) + (cid:96) − [9].In this work we focus on the B → ( K ∗ → K + π − ) γ process, in which the on-shell pho-ton subsequently undergoes Bethe-Heitler (BH) nuclear conversion inside the detector toa lepton-antilepton pair. The cross-section for BH pair conversion of ∼ GeV photons isapproximately two (eight) orders of magnitude larger than the Compton (Rayleigh) scatter-ing cross-section ([10]; see chapter 32), so that to an excellent approximation the emittedphoton does not decohere before conversion. The B → K ∗ γ photon polarization ratio, r , is2recisely defined via the amplitude ratio re i ( φ + δ ) ≡ A ( B → K ∗ γ L ) A ( B → K ∗ γ R ) , (1)in which φ ( δ ) is a relative weak (strong) phase. In the SM, r is expected to be at most ∼ Λ qcd /m b [11], while the weak phase φ is suppressed by Λ qcd /m b | V ub V ∗ us /V tb V ∗ ts | (cid:28) r , as discussed above, but also φ may test SM expectations:Measuring either r or φ ∼ O (1) would be highly suggestive of NP effects.Measurement of r via BH conversion was first considered in Ref. [12]. In that analysis,the B meson was assumed to be at rest relative to the conversion nucleus. Further, theconversion itself was assumed be to a perfect linear polarizer of the photon, so that theconversion leptons and photon are constrained to be coplanar. However, in practice the B meson is typically at least semi-relativistic, and typically an O (1) fraction of conversionevents have non-negligible acoplanarity (see e.g. Ref. [13]). This leads to a richer phase spacestructure for the outgoing conversion leptons, kaon and pion. Moreover, interference effectsbetween the B → K ∗ γ decay and the BH conversion amplitudes were not included. The keymotivation to reconsider the above analysis, then, is to include these B -boost, acoplanarityand interference effects. We exploit recent compact results for BH conversion spin-helicityamplitudes [13] to construct the full B → ( K ∗ → K + π − )( γ BH → e + e − ) amplitude, and showthat interference between its decay and conversion components permits both r and φ to beprobed by the kinematic configuration of the final state conversion leptons, kaon and pion.In particular, we develop an angular observable that probes both of these parameters. Thisnew analysis admits arbitrary boosts of the parent B meson relative to the BH conversionnucleus, and includes lepton-photon acoplanarity, which turns out to play an important rolein enhancing the r - and φ -sensitive interference effects.Performing an experiment to measure r and φ with this technique will be challenging.In the first instance, precise reconstruction of the leptonic momenta is required, which canonly be achieved, even in principle, if the leptonic opening angles are larger than the angularresolution of the detector. Typically, the leptonic opening angle after BH conversion is θ (cid:96)(cid:96) ∼ m e /E γ ∼ − , for typical photon energies in a B → K ∗ γ decay with a semirelativistic B . Specifically, for a photon of energy (cid:46) θ (cid:96)(cid:96) > − (10 − ) is ∼
98% (43%). Hence exquisite angular resolutions will be required. A further complicatingfactor is the multiple rescattering of the leptons in the detector material after conversion.3he rms rescattering angle in matter is (cid:39) (13 . /E (cid:96) ) (cid:112) x/x [10], where x/x is thepath length inside the detector in units of radiation length. For x/x ∼ few % – a typicalvalue – the rescattering angle is comparable to the typical opening angle, θ (cid:96)(cid:96) . Finally, theprobability of photon conversion itself is typically low at current and planned B factories,being at most of order a few percent. This probability depends mildly on the detector design.For example, it is approximately 3%, 2-3%, and 6% at BaBar [14], LHCb [15], and BelleII [16] respectively. For all these reasons, this technique will likely only be feasible with adedicated detector element that has a large scattering length, e.g. a gaseous TPC [17].For these reasons, in this work we shall restrict ourselves to a thought experiment-typeapproach. That is, we develop explicit analytic expressions for amplitudes and observableswith respect to the underlying B → ( K ∗ → K + π − )( γ BH → e + e − ) process alone, but do notinclude smearing from leptonic rescattering and limited kinematic resolution, or realistic de-tector simulations. Based on the results of this work, future studies may perhaps incorporatethese latter effects. II. AMPLITUDESA. Amplitude factorization
Keeping operators up to dimension five, the effective theory of interest for B → K ∗ γ and¯ B → ¯ K ∗ γ decays may be written in the general form L eff = g (cid:107) B † K ∗ µν F µν + g ⊥ B † K ∗ µν ˜ F µν + ¯ g (cid:107) ¯ B † ¯ K ∗ µν F µν + ¯ g ⊥ ¯ B † ¯ K ∗ µν ˜ F µν , (2)where ˜ X µν ≡ (cid:15) µνρσ X ρσ /
2, and the dimensionful couplings generically contain relative strongphases. We consider only K ∗ decays to charged pseudoscalars, i.e. K ∗ → K + π − or ¯ K ∗ → K − π + . The sign of the pion or kaon charge therefore tags the K ∗ versus the ¯ K ∗ , and hencetags the parent meson as either a B or ¯ B , up to electroweak loop suppressed corrections.Hence we neglect interference effects from B - ¯ B mixing.We assume the conversion nucleus is spin-0, e.g. a Si nucleus, which is the dominantsilicon isotope. The external quantum numbers for the full B → ( K ∗ → K + π − )( γ BH → e + e − )helicity amplitudes are then just the spins of the electron and positron, denoted r and4 = 1 , M rs = B K ∗ p , κ γ k , λ p K p π p − , rp + , sqP P + lepton exchanges= i g µν − p µ p ν /m K ∗ p − m K ∗ + im K ∗ Γ K ∗ (cid:88) λ = ± [ A B → K ∗ γ ] λµ [ A K ∗ → K + π − ] ν [ A BH ] λrs . (3)nearby to the K ∗ Breit-Wigner peak. Here λ = ± ( κ = ± ,
0) is the helicity of the photon( K ∗ ), and k µ ( p µ ) is the photon ( K ∗ ) momentum; P ( P (cid:48) ) denotes the incoming (outgoing)nuclear momentum, with nuclear mass P = P (cid:48) = M ; p ± ( p K,π ) denote the momenta ofthe leptons (kaon and pion); and finally q denotes the momentum exchange with the nucleus.Momentum and angular momentum conservation in the B → K ∗ γ process ensure that A B → K ∗ γ must annihilate the longitudinal component of the K ∗ propagator. Applying thepolarization completeness relation for p (cid:54) = 0, (cid:88) κ = ± , (cid:15) κµ ( p ) (cid:15) κ ∗ ν ( p ) = − g µν + p µ p ν /p , (4)we may factorize the full helicity amplitude into three helicity amplitude factors, M rs = ip − m K ∗ + im K ∗ Γ K ∗ (cid:88) λ,κ [ A B → K ∗ γ ] λκ [ A K ∗ → K + π − ] κ [ A BH ] λrs . (5a)Similarly for the ¯ B → ( ¯ K ∗ → K − π + )( γ BH → e + e − ) process M rs = ip − m K ∗ + im K ∗ Γ K ∗ (cid:88) λ,κ [ A ¯ B → ¯ K ∗ γ ] λκ [ A K ∗ → K − π + ] κ [ A BH ] λrs . (5b)Here and hereafter we neglect the mass splittings of the B - ¯ B and K ∗ - ¯ K ∗ systems, and denotethe masses (momenta) of both CP conjugate states by m B and m K ∗ ( p B and p ) respectively. B. Kinematics
The amplitude factors in eqs. (5) are Lorentz invariants, and are naturally expressed withrespect to kinematic coordinates that are defined in different frames. That is, the full 2 → BN → K + π − (cid:96) + (cid:96) − N , for nucleus N – is factoredinto two 1 → B → K ∗ γ and K ∗ → K + π − , and one 2 → → K ∗ invariant masses. The former isfixed to k = 0 for an on-shell internal photon. With regard to the K ∗ invariant mass, it isconvenient to define hereafter the dimensionless quantityˆ s ≡ p /m K ∗ . (6)In the narrow width limit, the Breit-Wigner factor1 | (ˆ s − m K ∗ + im K ∗ Γ K ∗ | → π m K ∗ Γ K ∗ δ [(ˆ s − m K ∗ ] . (7)That is, the narrow width limit corresponds to an on-shell K ∗ . Since, however, the K ∗ hasa finite width – Γ K ∗ /m K ∗ ∼
5% – and need not be precisely on-shell, we shall treat ˆ s as aphase space variable: The Breit-Wigner ensures ˆ s is typically nearby to the K ∗ mass shellup to the K ∗ width, i.e. ˆ s (cid:39) ∼ Γ K ∗ /m K ∗ .In the case that the photon conversion material is cold, the lab frame coincides with theframe in which the BH conversion nucleus is at rest. The following choices, shown in Fig. 1,for the remaining nine coordinates then prove convenient for the construction of compactand intuitive results: the lepton polar and azimuthal angles θ ± and φ ± and the energies E ± ,defined in the nuclear rest frame – the lab frame – with respect to the photon momentumand the K ∗ - γ decay plane, defined by p and k ; the photon polar angle, θ γ , defined withrespect to the nuclear momentum, P , in the B rest frame; the K polar and azimuthal angles θ K and φ K , defined in the K ∗ rest frame with respect to the photon momentum, k , and theplane defined by P and k in that frame. Note that the K ∗ - γ decay plane is invariant underboosts between the lab, K ∗ and B rest frames, and therefore equivalent to the plane definedby P and k in either the K ∗ or B rest frames, as shown in Fig. 1. C. Helicity amplitude factors
With these choices, we now proceed to explicitly compute the amplitude factors [ M B → K ∗ γ ] λκ ,[ M K ∗ → K + π − ] κ , and [ M BH ] λrs . Applying a light-cone decomposition to p , we define its associ-6 − p − θ − p B p kφ + p + θ + N θ ‘‘ θ K p K k , p B P K ∗ φ K p π P kp θ γ B FIG. 1. Kinematic configuration and coordinate choices. B momentum is denoted by p B , andazimuthal angles are defined with respect to the K ∗ - γ decay plane (blue). This plane contains p and k ( P and k ) in the lab frame ( K ∗ or B rest frames); momenta lying in this plane in eachframe are shown in gray. Left: Lepton polar angles θ ± and azimuthal angles φ ± in the lab frame.Middle: θ K and φ K polar angles in the K ∗ rest frame. Right: The photon polar angle, θ γ , in the B rest frame. ated null momentum with respect to the photon, i.e.˜ p µ ≡ p µ − p k µ p · k , (8)and make the polarization gauge choices (cid:15) ± K ∗ µ ( p ) = ± (cid:104) k ∓ | σ µ | ˜ p ∓ (cid:105)√ (cid:104) k ∓ | ˜ p ± (cid:105) , (cid:15) ± γ µ ( k ) = ± (cid:104) ¯ k ∓ | σ µ | k ∓ (cid:105)√ (cid:104) ¯ k ∓ | k ± (cid:105) , (9)for ¯ k an arbitrary null reference momentum. From the effective theory (2) one may thenshow that the B → K ∗ γ and ¯ B → ¯ K ∗ γ helicity amplitudes are[ A B → K ∗ γ ] ±± = ( g (cid:107) ± ig ⊥ )( m B − m K ∗ ˆ s ) , [ A ¯ B → ¯ K ∗ γ ] ±± = (¯ g (cid:107) ± i ¯ g ⊥ )( m B − m K ∗ ˆ s ) , (10)and [ M B → K ∗ γ ] ∓± = [ M B → K ∗ γ ] ± = 0. Note that, by definition, A ( B → K ∗ γ R,L ) ≡ [ M B → K ∗ γ ] ±± . It follows from eq. (1) and its CP conjugate that g (cid:107) − ig ⊥ g (cid:107) + ig ⊥ = re i ( δ + φ ) and ¯ g (cid:107) + i ¯ g ⊥ ¯ g (cid:107) − i ¯ g ⊥ = re i ( δ − φ ) . (11)7ence [ A B → K ∗ γ ] ++ = ( g (cid:107) + ig ⊥ )( m B − m K ∗ ˆ s ) , [ A B → K ∗ γ ] −− = re i ( δ + φ ) ( g (cid:107) + ig ⊥ )( m B − m K ∗ ˆ s ) , [ A ¯ B → ¯ K ∗ γ ] ++ = re i ( δ − φ ) (¯ g (cid:107) − i ¯ g ⊥ )( m B − m K ∗ ˆ s ) , [ A ¯ B → ¯ K ∗ γ ] −− = (¯ g (cid:107) − i ¯ g ⊥ )( m B − m K ∗ ˆ s ) . (12)The reference gauge momentum ¯ k in eqs. (9) is so far arbitrary. However, a particularlyconvenient choice is ¯ k µ ≡ P · kM P µ − k µ , (13)where P = M is the nuclear mass. In the lab frame – the nuclear rest frame – thischoice (13) ensures that for k µ = E γ (1 , ˆ k ) then simply ¯ k µ = E γ (1 , − ˆ k ). We assume thenuclear scattering is coherent and quasi-elastic, i.e. that P (cid:48) = M – equivalently q ≡ P (cid:48) − P = q / (2 M ) in the lab frame – and that the outgoing nucleus is non-relativistic, sothat the momentum exchange with the nucleus | q | (cid:28) M (see Refs [18–20] for a review of BHconversion). With these assumptions and the choice of ¯ k in eq. (13), the BH spin-helicityamplitudes collapse to a simple form in the limit that the polar angles θ ± (cid:28) γ ± ≡ E ± /m (cid:29)
1, where m is the lepton mass [13]. At leading order in these limits,[ A BH ] λrs (cid:39) e α λrs , (14)with α − = − ( α +22 ) ∗ = 2 (cid:112) γ + γ − (cid:112) G ( q ) q (cid:18)
11 + γ θ −
11 + γ − θ − (cid:19) ,α − = +( α + ) ∗ = ± (cid:112) γ + γ − (cid:112) G ( q ) q γ ∓ γ + + γ − (cid:18) γ + θ + e − iφ + γ θ + γ − θ − e − iφ − γ − θ − (cid:19) ,α − = − ( α +11 ) ∗ = 0 , (15)in which − q (cid:39) m (cid:16) γ θ + γ − θ − + 2 γ − γ + θ − θ + cos( φ − − φ + ) (cid:17) + m (cid:20) γ + + 1 γ − (cid:21) . (16)Here G ( q ) is the BH quasi-elastic form factor for the photo-nuclear vertex [19, 20], G ( q ) = M a q / (1 − a q ) , (17)8n which a = 184 . . − / Z − / /m and Z is the atomic number of the nucleus. Thisform factor encodes electronic screening of the nucleus, and regulates the 1 /q pole in theamplitudes. From eqs. (15), one sees that the BH amplitudes are maximal at γ ± θ ± ∼
1. Thatis, the typical lepton polar angle θ ± ∼ m/E ± . It also follows that the typical momentumexchange − q ∼ m , i.e. | q | (cid:28) M in concordance with our assumption of non-relativisticscattering.It now remains to compute the K ∗ → K + π − and K ∗ → K − π + helicity amplitudes. Onlytransverse K ∗ modes are generated by the B → K ∗ γ amplitude. Since (cid:15) ± K ∗ · p = 0 and p = p K + p π , these amplitudes must therefore take the form[ A K ∗ → K + π − ] κ = g K ∗ (cid:15) κK ∗ µ ( p ) (cid:0) p µK − p µπ (cid:1) , (18)where g K ∗ is a dimensionless coupling, containing a strong phase. There are no other weakphases as K ∗ decays strongly to Kπ . Under the polarization conventions (9), and computingin the K ∗ rest frame defined by Fig. 1, the helicity amplitudes are just spherical harmonics[ A K ∗ → K + π − ] ± ( θ K , φ K ) = e ± iφ K √ g Kπ p Kπ sin θ K , (19)with the momentum p Kπ ≡ s / m K ∗ (cid:2) m K ∗ ˆ s − ( m K + m π ) (cid:3) / (cid:2) m K ∗ ˆ s − ( m K − m π ) (cid:3) / . (20)Under CP, note that the amplitude transforms as[CP A K ∗ → K + π − ] ± ( θ K , φ K ) = [ A K ∗ → K − π + ] ∓ ( θ K , φ K ) = [ A K ∗ → K + π − ] ± ( θ K , − φ K ) . (21)That is, defining φ K and θ K with respect to ¯ K ∗ rest frame just as for the K ∗ in Fig. 1, then[ A K ∗ → K − π + ] ± ( θ K , φ K ) = e ± iφ K √ g Kπ p Kπ sin θ K . (22) D. Full Amplitude
Applying all the results (12), (15), (19) and (22) to eqs. (5), and defining | α | ≡ (cid:80) λ,r,s | α λrs | , the unpolarized square amplitudes |M| = A ( r ) sin θ K (cid:110) | α | + 8 r r Re (cid:104) α − α − e i ( φ + δ − φ K ) (cid:105)(cid:111) , |M| = ¯ A ( r ) sin θ K (cid:110) | α | + 8 r r Re (cid:104) α − α − e i ( φ − δ − φ K ) (cid:105)(cid:111) , (23)9n which A ( r ) ≡ e r ) | g (cid:107) + ig ⊥ | | g Kπ | | m K ∗ (ˆ s −
1) + im K ∗ Γ K ∗ | p Kπ ( m B − m K ∗ ˆ s ) , ¯ A ( r ) ≡ e r ) | ¯ g (cid:107) − i ¯ g ⊥ | | ¯ g Kπ | | m K ∗ (ˆ s −
1) + im K ∗ Γ K ∗ | p Kπ ( m B − m K ∗ ˆ s ) . (24)Eqs. (23) compactly express the unpolarized square amplitude for the full B → ( K ∗ → K + π − )( γ BH → e + e − ) process in terms of just the BH conversion helicity amplitudes (15) andtrigonometric (exponential) functions of the kinematic observables θ K ( φ K ). The dependenceon the parameters r , φ and δ is explicit and elementary.As a cross-check of these results, we provide an alternative and more traditional deriva-tion of the square amplitude in Appendix A, via construction of linearly polarized pho-ton BH amplitudes. The consequent result (A4) and the square amplitude (23) are inexcellent numerical agreement in the γ ± (cid:29) θ ± (cid:28) B → ( K ∗ → K + π − )( γ BH → e + e − ) process. Note that the result (A4) does not incorporatethese approximations, so that the compact and explicit eqs. (23) are strictly an approxima-tion to eq. (A4). III. OBSERVABLESA. Differential rate
Making use of the explicit r and φ dependence in the square amplitude results (23), wemay now proceed to extract r and φ sensitive observables. First, however, we construct thefull differential rate. The factorization (5) ensures that the phase space with an on-shellinternal photon may be partitioned into a B → ( K ∗ → Kπ ) γ cascade decay and a γN → (cid:96) + (cid:96) − N conversion. That is, the differential rate for the full B → ( K ∗ → K + π − )( γ BH → e + e − )process can be written as d R = (cid:12)(cid:12) M (cid:12)(cid:12) d P B d P BH (25)where d P B ( d P BH ) is phase space factor of the decay (conversion). Note that d R here hasthe dimensions of a cross-section times a partial width.Each phase space factor is Lorentz invariant, and are naturally computed in differentframes, as shown in Fig. 1. Computing in the B rest frame followed by the K ∗ rest frame,10he phase space factor for the cascade decay is d P B = 12 m B π ) d p π E π d p K E K d k E γ δ ( p B − k − p K − p π ) , → m K ∗ p Kπ ˆ s / m B − m K ∗ ˆ s π ) m B d Ω K d cos θ γ d ˆ s , (26)performing all trivial integrals, including over the overall azimuthal orientation of the K ∗ - γ - N plane. Similarly, for the BH conversion, computing in the lab frame, d P BH = 12 M E γ π ) d p + E + d p − E − d P (cid:48) E (cid:48) δ ( P + k − P (cid:48) − p + − p − ) , → (cid:20) E − ∆ π ) M E (cid:21) d Ω + d Ω − d ∆ . (27)Here the lepton momenta been approximated in the measure via (cid:112) E ± − m (cid:39) E ± . Wehave further defined E ≡ E + + E − , ∆ ≡ E + − E − , (28)and enforced non-relativistic nuclear scattering, which implies E (cid:48) (cid:39) M or equivalently E γ (cid:39) E + + E − , up to q = q / M (cid:28) m corrections. Hence to an excellent approximation E ishalf the photon energy in the lab frame. Moreover, note that m ≤ E ± ≤ E γ implies that∆ ∈ ( m − E , E − m ) . (29)At an e + e − B -factory, such as Belle or BaBar, the e + e − → Υ → B ¯ B production factorizesfrom the subsequent B decays, because the B is a pseudoscalar. In this type of collider,the rapidity of parent B meson in the lab frame has a known prior probability distribution, f B ( η ) dη , determined by the collider configuration, and enters as an independent phase spacefactor in d R . In Appendix B we include a derivation of the B rapidity pdf (B6), for an e + e − machine. We shall restrict ourselves hereafter to the case that f B ( η ) is known. In this case,note that the energy E is fully specified by η , ˆ s and θ γ , viz. E ( η, θ γ , ˆ s ) ≡ m B − m K ∗ ˆ s m B (cid:0) cosh η + cos θ γ sinh η (cid:1) , (30)so that the lepton energies can be expressed in terms of η , ˆ s , θ γ and ∆, via eqs. (28). Inour discussion of the kinematics above, it was convenient to express the amplitudes in termsof the ten phase space coordinates ˆ s , θ γ , E ± , Ω K and Ω ± . We see here, however, that for11he differential rate, it is more natural to choose η and ∆ as phase space coordinates ratherthan E ± . Combining the above results together, the full differential rate d R = (cid:12)(cid:12) M (cid:12)(cid:12) f B ( η ) m K ∗ p Kπ ˆ s / (cid:20) m B − m K ∗ ˆ s π ) m B (cid:21)(cid:20) E ( η, θ γ , ˆ s ) − ∆ M E ( η, θ γ , ˆ s ) (cid:21) d Ω K d Ω + d Ω − d ∆ d cos θ γ d ˆ sdη . (31) B. Polarization and weak phase observables
Let us now define a further change of azimuthal angular coordinates, modulo 2 πψ ≡ φ + + φ − + 2 φ K , ¯ ψ ≡ φ + + φ − − φ K ,ϕ ≡ φ + − φ − . (32)The angle ϕ encodes the acoplanarity of the leptons with respect to the photon, with copla-narity corresponding to ϕ = π . Note that φ K and φ ± are defined with opposite orientationsaround the photon momentum direction ˆ k (see Fig. 1). For coplanar conversion leptonsand a stationary B in the lab frame, ψ then corresponds to the relative twist between thepositron-electron conversion plane and the K - π decay plane. Similarly, ¯ ψ would then cor-respond to the averaged orientation of the positron-electron conversion plane and the K - π decay plane with respect to the K ∗ - γ decay plane.From eq. (16), the momentum exchange has the form q ∝ ζ cos ϕ , with ζ <
1. Itfollows from eqs. (15), (23) and (31) that the differential rate can be written in the form d R = (cid:16) (cid:88) k a k cos k ( ϕ ) (cid:17)(cid:104) A + A cos( ψ + ϕ − φ − δ )+ A cos( ψ − ϕ − φ − δ ) + A cos( ψ − φ − δ ) (cid:105) , (33)where a k and A i are purely functions of the phase space orthogonal to ψ , ¯ ψ and ϕ . Thatis, a k and A i are functions of ˆ s , η , θ γ,K, ± and ∆. Integrating over all phase space except dψ , we see that the marginal differential rates for B → ( K ∗ → K + π − )( γ BH → e + e − ) and¯ B → ( ¯ K ∗ → K − π + )( γ BH → e + e − ) must respectively have the form d R dψ = R π (cid:104) − r r C cos( ψ − φ − δ ) (cid:105) ,d R dψ = R π (cid:104) − r r C cos( ψ − φ + δ ) (cid:105) . (34)12qs. (23) tell us that the cosine coefficient C arises from a ratio of BH interference termsto the BH squared amplitude, and is therefore independent of r or φ . In other words, thiscoefficient is the same for both of the CP conjugate processes, and is B → K ∗ γ operatorindependent. We have chosen the relative sign of C in eqs (34) to anticipate the choicethat ensures C >
0. Further, we have chosen the normalization of C in eqs (34) to ensure,via positive semi-definiteness of d R /dψ , that |C| ≤
1, noting that the r dependent factor2 r/ (1+ r ) ≤ r . The coefficient C may then be interpreted as the maximum possibleratio of the amplitude of d R /dψ oscillations to their average value, R / π . Hereafter we callthis ratio the relative oscillation amplitude.Eqs (34) are the main results of this paper. Once the coefficient C is computed, thenmeasurement of the relative oscillation amplitude in d R /dψ permits extraction of r up tothe two-fold ambiguity r ↔ /r . Further, measurement of the average phase shift (phaseshift difference) between d R /dψ and d R /dψ permits extraction of the weak (strong) phase φ ( δ ). Equivalently, one may construct two forward-backward type asymmetries. Defining thefour quadrants I : ψ ∈ [0 , π/ ψ ∈ [ π/ , π ], III : ψ ∈ [ π, π/
2] and IV : ψ ∈ [3 π/ , π ]then Ψ ψ ≡ R − (cid:90) − I − II+III+IV d R dψ dψ = 2 π r r C sin( φ + δ )Ω ψ ≡ R − (cid:90) − I+II+III − IV d R dψ dψ = 2 π r r C cos( φ + δ ) , (35a)and moreover Ψ ψ ≡ R − (cid:90) − I − II+III+IV d R dψ dψ = 2 π r r C sin( φ − δ )Ω ψ ≡ R − (cid:90) − I+II+III − IV d R dψ dψ = 2 π r r C cos( φ − δ ) . (35b)Note that all four symmetries have an upper bound 2 /π . For known C , one may extract r and φ ± δ from these two sets of asymmetries. C. Statistics and Sensitivity Enhancements
Before proceeding to numerical computation of C , let us pause to consider the statisticalconfidence in the extraction of r and φ . We focus on their extraction from the asymmetries(35). These asymmetries are expectation values of a random variable defined to take the13alues ± N (cid:29) σ X = (cid:112) (1 − X ) /N (cid:39) / √ N , for X = Ψ ψ , Ω ψ , Ψ ψ , andΩ ψ . The statistical confidence at which one rejects the SM values X SM – thus measuringNP effects – is then characterized by the chi-square statistic ( X − X SM ) /σ X .As shown in Ref. [13] and below, special ‘sensitivity parameter’ kinematic cuts may en-hance C on the resulting remaining phase space. The construction of these cuts is motivatedby the observation, from eqs. (23), that C is enhanced on those areas of phase space inwhich the BH interference term, ∼ α − α − , is comparable or larger than terms in the to-tal BH square amplitude | α | = (cid:80) λ,r,s | α λrs | . For example, one may define the sensitivityparameter S ≡ | α − α − | / | α − | (cid:39) − cos[ | φ + − φ − | − π ]) (cid:20) γ + γ − ( γ + + γ − ) (cid:21)(cid:20) γ + θ + γ − θ − ( γ θ − γ − θ − ) (cid:21) (1 + γ θ )(1 + γ − θ − ) . (36)Requiring S (cid:38) O (1) en-hancements of C on the remaining phase space, as will be verified below. Note that the(1 − cos[ | φ + − φ − | − π ]) factor in eq. (36) implies that S (cid:39) S typically favors events with higher acoplanarity. One mayalso consider other sensitivity parameters, such as T ≡ | α − α − | / | α | , (37)which is normalized such that T ∈ [0 , C ( N ) to be the relative oscillation amplitude (number of events) in theabsence of S or T cuts, and write C c ≡ C [ S > S c , T > T c ] , N c ≡ N [ S > S c , T > T c ] . (38)Compared to the S c = T c = 0 case, the application of sensitivity parameter cuts scales theNP statistical confidence by the factorΣ ≡ (cid:18) X − X SM σ X (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) S > S c , T > T c (cid:18) σ X X − X SM (cid:19) (cid:39) (cid:18) C c C (cid:19) N c N , (39)for X = Ψ ψ , Ω ψ , Ψ ψ , and Ω ψ . That is, the enhancement C c / C achieved by the sensitivityparameter cuts competes with the corresponding increase in the statistical error, (cid:112) N /N c ,14ince necessarily N c < N . We shall see in the next section that there are choices of S c and T c for which Σ >
1. For the purpose of measuring NP effects, this is equivalent to aneffective increase in the sample size N (cid:55)→ Σ N – an increase in the effective statistics. IV. SIMULATIONSA. Relative Oscillation Amplitude
Extraction of the relative oscillation amplitude coefficient C is achieved numerically viaMonte-Carlo (MC) generation of B → ( K ∗ → K + π − )( γ BH → e + e − ) events according tothe differential rate (31). We use the matrix element in eq. (A4), generated from linearlypolarized photon amplitudes. Though it does not provide the same analytical insight asthe matrix element generated from spinor-helicity methods (23), this matrix element isas numerically stable as the latter, and moreover, acts as a convenient cross-check of theanalytic results in eqs. (23) and (34).For the numerical results shown in this paper, we use a private MC code written in C/Python . For simplicity, we apply the narrow K ∗ limit (7), which fixes the K ∗ to beon-shell. We assume the nominal Belle II parameters [16] (see also App. B) in order todetermine the B rapidity distribution f B ( η ), which is peaked at βγ (cid:39) .
29. More details ofthe operation of this MC generator are included in Appendix C. We have further checkedthe numerical results with a second private MC, written in
C++/Java , that makes use ofthe matrix element (23). In both codes, we discard the overall normalization of the matrixelement – e.g. A ( r ) in eqs. (23) – since we are concerned only with the relative oscillationamplitude, ∼ C .In order to account for limited angular resolution, we include hereafter cuts on the leptonpolar angles, θ ± , and opening angle, θ (cid:96)(cid:96) , defined in Fig. 1. We will consider a uniform polarcut θ (cid:96)(cid:96), ± > θ c , (40)for various values of θ c . In particular, we consider two benchmark cases θ c = 10 − and 5 × − . The former captures almost all conversion leptons in the B → ( K ∗ → K + π − )( γ BH → e + e − ) process for semirelativistic B ’s, while the latter might be plausi-bly achievable in the near- to mid-term future. To extract C , we fit eq. (34) to the15 − − − − θ c . . . . . . C ( θ ‘‘ , ± > θ c ) . . . . . ψ/π . . . . . N o r m a li ze d e v e n t s (0 . , . , π/ . , π/ . , π/ FIG. 2. Left: The fit value for C with the ± σ error band as a function of the polar angle cuts θ (cid:96)(cid:96), ± > θ c (see Fig. 1). The peak value of C approximately coincides with the peak of the θ ± marginal distribution (see the left panel of Fig. 6). Right: Normalized differential distribution d R /dψ for four different ( r, φ + δ ) couplets and θ c = 10 − . Also shown are theory predictions(gray) for the input values of ( r, φ + δ ) and the extracted value C [ θ c = 10 − ] in eq. (41). d R /dψ histograms for various choices of r and φ + δ , including the couplets { r, φ + δ } = { (0 . , , (0 . , π/ , (0 . , π/ , (1 . , π/ } . In Fig. 2 we show the extracted C as a functionof the θ c cuts. The maximal relative oscillation amplitude one can expect is of O (20%), andthe benchmark extracted C values are C [ θ c = 10 − ] = 0 . ± . , and C [ θ c = 5 × − ] = 0 . ± . , (41)where the errors are purely statistical in origin. We also show in Fig. 2 typical d R /dψ histograms for various choices of r and φ + δ . The expected shifted and amplitude-modulatedcosine can be clearly seen. Applying the extracted value for C , these histograms are inexcellent agreement with the theory predictions (34). B. Statistics Enhancements
Incorporating the S and T kinematic cuts, we show in Fig. 3 the absolute enhancementof C as a function of the net cut efficiency, (cid:15) , for the two benchmark polar angle cuts. Thenet cut efficiency is defined hereafter to be the fraction of events kept after application ofboth kinematic and polar cuts. The pure S cuts – that is, C ( S > S c , T >
0) – provide thelarger enhancement at high cut efficiencies. At lower efficiencies the pure T cut provides thelarger enhancement. The C and (cid:15) dependence on S c may be read off from Fig. 4.16 . . . C θ ‘‘, ± > − ST Σ = 1 θ ‘‘, ± > × − − − Σ − − (cid:15) FIG. 3.
Upper panels:
The coefficient C as a function of the cut efficiency for the S (blue) and T (gold) kinematic cuts, with polar cuts θ c = 10 − (left) and θ c = 5 × − (right). The coloredregions depict the ± σ statistical error bands. The equivalent effective statistics curve Σ = 1, i.e. C = C / (cid:112) N/N , is also shown (gray). Lower panels : Statistics enhancement Σ as a function ofthe cut efficiency. The maxima correspond to the optimum cuts S > S optc and T > T optc (coloreddots in all panels). Comparing to the equivalent effective statistics curve Σ = 1 – i.e. C = C / (cid:112) N/N –we see in Fig. 3 that the C dependence initially rises faster than C / (cid:112) N/N . This meansthat for low values of S c and T c , the statistics on the S and T cut phase space is enhancedcompared to the full data set. The explicit Σ dependence is also shown in Fig. 3. Inparticular, one sees that for θ c = 5 × − , the statistical enhancement Σ is optimized at S optc = 1 . T optc = 0 .
62) corresponding to Σ = 3 . . r and φ + δ , along with their statistical errors, with and withoutthe optimal S optc and T optc cuts. We use an MC sample generated from an input couplet( r, φ + δ ) = (0 . , π/ N = 10 events. This roughly correspondsto 50 ab − of data – a benchmark luminosity at Belle II [16] – and a percent level photon17 S c . . . C ( S > S c ) (cid:15) ; θ c = 10 − (cid:15) ; θ c = 5 × − C ; θ c = 10 − C ; θ c = 5 × − − − N e t c u t e ffi c i e n c y (cid:15) FIG. 4. The enhancement in C as a function of S . The secondary y -axis shows the correspondingcut efficiency. The colored regions depict the ± σ statistical error bands. conversion rate with ideal acceptance efficiency. The polar cut is θ c = 5 × − . Theextracted values are obtained by two different methods, first from a fit to the differentialrate eq. (34) and second from the quadrant asymmetries (35). Method S , T > S > S optc T > T optc r ( φ + δ ) /π r ( φ + δ ) /π r ( φ + δ ) /πd R /dψ . ± .
017 0 . ± .
026 0 . ± .
010 0 . ± .
015 0 . ± .
013 0 . ± . ψ , Ω ψ . ± .
005 0 . ± .
157 0 . ± .
003 0 . ± .
088 0 . ± .
003 0 . ± . TABLE I. Extracted values of r and ( φ + δ ) from an MC sample with input values (0 . , π/ The statistical errors for the optimized kinematic S and T cuts are a O (1) factor smallerthan for the full data set, as expected from the above numerical analysis: Application ofthese sensitivity parameter cuts improves the statistical power of the r and φ extractionfrom B → ( K ∗ → K + π − )( γ BH → e + e − ) events. Moreover, even without these enhancements,both r and φ are extracted with sufficient numerical precision to probe NP effects at the r ∼ φ ∼ V. CONCLUSIONS
In this paper we have presented the helicity amplitudes and differential rate for the B → K ∗ γ → K + π − γ process, in which the photon undergoes subsequent nuclear conversion18o a lepton pair. Interference between the intermediate, on-shell photon polarizations inthe coherent B → ( K ∗ → K + π − )( γ BH → e + e − ) process produces oscillations in the angularkinematic observable ψ . Measuring the amplitude and phase of these oscillations – or equiv-alently two quadrant-type asymmetries Ψ ψ and Ω ψ – permits extraction in principle of thepolarization ratio, r , and the relative weak phase, φ , of the right- and left-handed B → K ∗ γ amplitudes. In this manner, SM expectations for both r and φ may be tested.We have employed private Monte Carlo simulations to compute the ψ distribution andasymmetries as a function of r , φ and kinematic cuts. In particular, kinematic cuts withrespect to the sensitivity parameters S and T may sufficiently amplify these oscillations,such that the overall statistical power of the r and φ measurement is increased by an O (1)factor.Implementing this approach using converted photons will be experimentally challenging,not least because of the high angular resolution required to reconstruct the conversion leptonkinematics. Moreover, a detector whose thickness is on the order of one radiation lengthor less is required to avoid multiple leptonic rescatterings, that otherwise smear the leptonkinematics. Nonetheless, the theoretically clean nature of the r - and φ -sensitive observablespresented in this work may perhaps encourage the use of this technique in a future dedicateddetector element. ACKNOWLEDGMENTS
The authors thank Yuval Grossman, Roni Harnik, Jacques Lefrancois, Zoltan Ligeti,Marie-H´el`ene Shune, and Jure Zupan for helpful discussions. The work of FB is supportedin part by the Fermilab Fellowship in Theoretical Physics and by the University of Cincinnatiphysics department Mary J. Hanna fellowship. Fermilab is operated by Fermi Research Al-liance, LLC under Contract No. DE-AC02-07CH11359 with the United States Departmentof Energy. The work of DR is supported by the NSF under grant No. PHY-1002399.19 ppendix A: The B → Kπe + e − squared matrix element by polarization decomposi-tion The effective Lagrangian in eq. (2) gives the following Feynman rule for the BK ∗ γ vertex: γ ( k ) λ , µK ∗ ( p ) κ , ν = i (cid:2) g (cid:107) (cid:0) ∆ m BK ∗ g µν − k ν p µ (cid:1) + 2 g ⊥ (cid:15) µνρσ k ρ p σ (cid:3) (A1)where ∆ m BK ∗ ≡ m B − m K ∗ . The amplitude for B → γ λ Kπ with λ being the photon helicityis then given by N λ = g K ∗ (cid:110) g (cid:107) (cid:2) ∆ m BK ∗ ε ∗ λ · ( p K − p π ) − ε ∗ λ · p [ k · ( p K − p π )] (cid:3) − g ⊥ (cid:15) µνρσ ( p K − p π ) µ ε λ ∗ ν k ρ p σ (cid:111) (A2)The BH squared amplitude in the nuclear rest frame for a linearly polarized photon in the+ˆ z direction with polarizations λ = { , } isBH λλ (cid:48) (cid:39) e G q (cid:40) g λλ (cid:48) (cid:2) E γ q + ( k · p − + k · p + ) (cid:3) ( k · p − )( k · p + ) − (cid:18) E p + p λ − k · p − + E p − p λ + k · p + (cid:19) (cid:32) E p + p λ (cid:48) − k · p − + E p − p λ (cid:48) + k · p + (cid:33)(cid:41) , (A3)where terms of O ( q /E ± ) were dropped (see Appendix C in [13] for details). The squaredamplitude is then given by |M| = (cid:88) λ,λ (cid:48) ∈{ , } N λ N ∗ λ (cid:48) BH λλ (cid:48) . (A4)A numerical comparison between the above expression and eq. (23) shows excellent agree-ment over the entire phase space (sampled uniformly). Appendix B: B rapidity distribution Consider an e + e − → Υ → B ¯ B factory, and let Θ denote the polar angle of the B ’s withrespect to the electron beamline in the center of mass frame. The amplitude for production M prod ∼ sin Θ, and so the probability distribution p Ω (cos Θ) = 34 (cid:2) − cos Θ (cid:3) . (B1)20ere and in the following we neglect effects of lab frame angular acceptance cuts, which maynon-trivially restrict the domain of both η and Θ.In the center of mass frame – the rest frame of the Υ – each B has energy E ∗ = m Υ / ∗ superscript denotes center-of-mass frame quantities. The corresponding ra-pidity, which we choose to be positive by convention on the branch Θ ∈ [0 , π ], η ∗ = cosh − ( m Υ / m B ) , (B2)and the B speed in this frame β ∗ = tanh( η ∗ ).At B -factories the lab frame electron and position beam energies, E ± , are asymmetric,but are chosen such that the Υ is on shell, i.e. 4 E + E − = m . For example, at Belle II thebeams are planned to be E + = 7 GeV and E − = 4 GeV. The boost rapidity of the center ofmass frame with respect to the lab frame is correspondingly η Υ = cosh − [( E + + E − ) /m Υ ] . (B3)The rapidity of the B in the lab frame may now be written as a function of cos Θ, viz. η (cos Θ) = cosh − (cid:110) cosh η ∗ (cid:2) cosh η Υ + β ∗ cos Θ sinh η Υ (cid:3)(cid:111) , (B4)and its pdf, by definition f B ( η ) = (cid:90) − d cos Θ p Ω (cos Θ) δ [ η − η (cos Θ)] . (B5)Under a change of variables ζ = η (cos Θ), one finds f B ( η ) = 34 β ∗ cosh η ∗ (cid:90) η Υ + η ∗ | η Υ − η ∗ | dζ δ [ η − ζ ] sinh ζ sinh η Υ (cid:20) − (cid:18) cosh ζ − cosh η ∗ cosh η Υ sinh η ∗ sinh η Υ (cid:19) (cid:21) = 3 sinh η η ∗ sinh η Υ (cid:20) − (cid:18) cosh η − cosh η ∗ cosh η Υ sinh η ∗ sinh η Υ (cid:19) (cid:21) , | η Υ − η ∗ | < η < η ∗ + η Υ , (B6)and zero otherwise. Note that f B itself has zeroes at each end of its non-trivial domain, i.e.at η = η ∗ + η Υ and | η Υ − η ∗ | . The boost at the pdf peak is γ peak = ( E + + E − ) / (2 m ). E.g.for the Belle II parameters, the peak βγ peak = sinh cosh − ( γ peak ) = 0 .
29. This is the boostof B ’s emitted at Θ = π/
2, and matches the quoted B design boost at Belle II [16].21 − − − − − θ . . . . . . P ( θ ‘‘ > θ ) . . . . . E γ /m B . . . . . . N o r m a li ze d e v e n t s θ ‘‘ > − θ ‘‘ > − FIG. 5. Left: The cumulative distribution function (CDF) of the opening angle between theleptons. Right: The normalized distribution of the photon energy in units of the B mass for twodifferent θ (cid:96)(cid:96) cuts. Appendix C: The Monte Carlo event generator
This appendix describes in more detail the MC event generator written in C and Python .The B → ( K ∗ → Kπ ) γ phase space is generated as follows. The B rapidity is sampled fromthe PDF given in eq. (B6) while the photon polar angle θ γ and the K polar and azimuthalangles are generated uniformly in the appropriate frame. On the other hand, since in BHphoton conversion the leptons are produced with preferentially small angles with respect tothe photon direction, the lepton polar angles are generated uniformly on a log scale. This isimplemented via the transformed variables t ± = log θ ± where t ± are uniformly distributedand with t ± ∈ [ − , − δφ )is peaked around π and so, to improve the efficiency of the generator, δφ is sampled from aCauchy distribution. All other BH variables are generated uniformly.The weight associated with each event is proportional to the matrix element (A4). Theevents are unweighted using the standard procedure. That is, the weights are normalizedto the largest weight and the event is kept if its normalized weight is larger than a randomnumber on [0 , r, φ + δ ) coupletswith 500k events per sample. Some representative distributions from the (0 . ,
0) sampleare shown in Figs 5 and 6. In particular, the left panel in Fig. 5 shows the cumulative22 − − − − θ + N o r m a li ze d e v e n t s θ ‘‘ > − θ ‘‘ > − . . . . . . E + /E γ . . . . . . . . . . N o r m a li ze d e v e n t s θ ‘‘ > − θ ‘‘ > − FIG. 6. Left: The normalized polar angle distribution of the positron for two different values ofthe opening angle cut θ (cid:96)(cid:96) . Right: the positron energy as a fraction of the photon energy for twovalues of θ (cid:96)(cid:96) . The distribution exhibits the expected behavior for BH conversion. It is symmetricabout 1 / distribution function for the opening angle between the leptons θ (cid:96)(cid:96) while the right panelshows the distribution of photons energies. Figure 6 shows the polar angle and fractionalenergy distribution of the positron. [1] M. Gronau, Y. Grossman, D. Pirjol, and A. Ryd, Phys.Rev.Lett. , 051802 (2002),arXiv:hep-ph/0107254 [hep-ph].[2] M. Gronau and D. Pirjol, Phys.Rev. D66 , 054008 (2002), arXiv:hep-ph/0205065 [hep-ph].[3] R. Aaij et al. (LHCb Collaboration), Phys.Rev.Lett. , 161801 (2014), arXiv:1402.6852[hep-ex].[4] T. Mannel and S. Recksiegel, Acta Phys.Polon.
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