Angular distribution as an effective probe of new physics in semi-hadronic three-body meson decays
LLDU-18-004
Angular distribution as an e ff ective probe of new physicsin semi-hadronic three-body meson decays C. S. Kim, ∗ Seong Chan Park, † and Dibyakrupa Sahoo ‡ Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea (Dated: December 4, 2018)We analyze, in a fully model-independent manner, the e ff ects of new physics on a few semi-hadronic three-body meson decays of the type P i → P f f f , where P i , P f are well chosen pseudo-scalar mesons and f , denote fermions out of which at least one gets detected in experiments. We find that the angular distribution ofevents of these decays can probe many interesting new physics, such as the nature of the intermediate particlethat can cause lepton-flavor violation, or presence of heavy sterile neutrino, or new intermediate particles, ornew interactions. We also provide angular asymmetries which can quantify the e ff ects of new physics in thesedecays. We illustrate the e ff ectiveness of our proposed methodology with a few well chosen decay modesshowing e ff ects of certain new physics possibilities without any hadronic uncertainties. PACS numbers: 13.20.-v, 14.60.St, 14.80.-jKeywords: Beyond Standard Model, Heavy Quark Physics, Invisible decays, Rare decays, Lepton flavor violation
I. INTRODUCTION
New physics (NP), or physics beyond the standard model,involves various models that extend the well verified standardmodel (SM) of particle physics by introducing a number ofnew particles with novel properties and interactions. Thoughvarious aspects of many of these particles and interactions areconstrained by existing experimental data, we are yet to detectany definitive signature of new physics in our experiments.Nevertheless, recent experimental studies in B meson decays,such as B → K ( ∗ ) (cid:96) − (cid:96) + [1], B s → φ(cid:96) − (cid:96) + [2], B → D ( ∗ ) (cid:96)ν [3] and B c → J /ψ(cid:96)ν [4] (where (cid:96) can be e , µ or τ ) have re-ported anomalous observations raising the expectation of dis-covery of new physics with more statistical significance. Inthis context, model-independent studies of such semi-leptonicthree-body meson decay processes become important as theycan identify generic signatures of new physics which can beprobed experimentally. In this paper, we have analyzed thee ff ects of new physics, in a model-independent manner, onthe angular distribution of a general semi-hadronic three-bodymeson decay of the type P i → P f f f , where P i and P f arethe initial and final pseudo-scalar mesons respectively, and f , denote fermions (which may or may not be leptons butnot quarks) out of which at least one gets detected experimen-tally. Presence of new interactions, or new particles such asfermionic dark matter (DM) particles or heavy sterile neutri-nos or long lived particles (LLP) would leave their signaturein the angular distribution and we show by example how newphysics contribution can be quantified from angular asym-metries. Our methodology can be used for detection of newphysics in experimental study of various three-body pseudo-scalar meson decays at various collider experiments such asLHCb and Belle II. ∗ E-mail at: [email protected] † E-mail at: [email protected] ‡ E-mail at: [email protected]. and D.S. are the corresponding authors.
The structure of our paper is as follows. In Sec. II wediscuss the most general Lagrangian and amplitude whichinclude all probable NP contributions to our process underconsideration. The relevant details of kinematics is then de-scribed in Sec. III. This is followed by a discussion on theangular distribution and the various angular asymmetries inSec. IV. In Sec. V we present a few well chosen examplesillustrating the e ff ects of new physics on the angular distribu-tion. In Sec. VI we conclude by summarizing the importantaspects of our methodology and its possible experimental re-alization. II. MOST GENERAL LAGRANGIAN AND AMPLITUDE
Following the model-independent analysis of the decay B → D (cid:96) − (cid:96) + as given in Ref. [5] and generalizing it for ourprocess P i → P f f f where P i , f can be B , B s , B c , D , K , π etc.as appropriate and f f can be (cid:96) − (cid:96) + , (cid:96) ¯ (cid:96) (cid:48) , (cid:96)ν (cid:96) , (cid:96)ν S , (cid:96) f DM , ν (cid:96) ν (cid:96) , ν S ν (cid:96) , ν (cid:96) ν S , ν S ν S , f DM ¯ f DM , f DM f DM , f LLP f LLP (with (cid:96), (cid:96) (cid:48) = e , µ, τ denoting leptons, ν S being sterile neutrino, f DM , as fermionic dark matter and f LLP , as long lived fermions) ,we can write down the e ff ective Lagrangian facilitating thedecay under consideration as follows, L e ff = J S (cid:16) ¯ f f (cid:17) + J P (cid:16) ¯ f γ f (cid:17) + ( J V ) α (cid:16) ¯ f γ α f (cid:17) + ( J A ) α (cid:16) ¯ f γ α γ f (cid:17) + (cid:0) J T (cid:1) αβ (cid:16) ¯ f σ αβ f (cid:17) + (cid:0) J T (cid:1) αβ (cid:16) ¯ f σ αβ γ f (cid:17) + h.c. , (1)where J S , J P , ( J V ) α , ( J A ) α , (cid:0) J T (cid:1) αβ , (cid:0) J T (cid:1) αβ are the di ff erenthadronic currents which e ff ectively describe the quark leveltransitions from P i to P f meson. It should be noted that It is clear that we can not only analyze processes allowed in the SM butalso those NP contributions from fermionic dark matter in the final state aswell as including flavor violation. Our analysis as presented in this paperis fully model-independent and general in nature. a r X i v : . [ h e p - ph ] D ec we have kept both σ αβ and σ αβ γ terms. This is because ofthe fact that the currents ¯ f σ αβ f and ¯ f σ αβ γ f describetwo di ff erent physics aspects namely the magnetic dipole andelectric dipole contributions respectively. In the SM, vec-tor and axial-vector currents (mediated by photon, W ± and Z bosons) and the scalar current (mediated by Higgs boson)contribute. So every other term in Eq. (1) except the ones with J S , ( J V ) α and ( J A ) α can appear in some specific NP model.Since, in this paper, we want to concentrate on a fully model-independent analysis to get generic signatures of new physics,we shall refrain from venturing into details of any specific NPmodel, which nevertheless are also useful. It is important tonote that J S , ( J V ) α and ( J A ) α can also get modified due to NPcontributions. P i ( k ) f ( k ) f ( k ) P f ( k ) FIG. 1. Feynman diagram for P i → P f f f considering f as aparticle and f as an anti-particle. Here the blob denotes the e ff ectivevertex and includes contributions from all the form factors definedin Eq. (3). In order to get the most general amplitude for our processunder consideration, we need to go from the e ff ective quark-level description of Eq. (1) to the meson level description bydefining appropriate form factors. It is easy to write downthe most general form of the amplitude for the process P i → P f f f depicted in Fig. 1 as follows, M (cid:16) P i → P f f f (cid:17) = F S (cid:16) ¯ f f (cid:17) + F P (cid:16) ¯ f γ f (cid:17) + (cid:0) F + V p α + F − V q α (cid:1) (cid:16) ¯ f γ α f (cid:17) + (cid:0) F + A p α + F − A q α (cid:1) (cid:16) ¯ f γ α γ f (cid:17) + F T p α q β (cid:16) ¯ f σ αβ f (cid:17) + F T p α q β (cid:16) ¯ f σ αβ γ f (cid:17) , (2)where F S , F P , F ± V , F ± A , F T and F T are the relevant formfactors, and are defined as follows, (cid:104) P f | J S | P i (cid:105) = F S , (3a) (cid:104) P f | J P | P i (cid:105) = F P , (3b) (cid:104) P f | ( J V ) α | P i (cid:105) = F + V p α + F − V q α , (3c) (cid:104) P f | ( J A ) α | P i (cid:105) = F + A p α + F − A q α , (3d) (cid:104) P f | (cid:0) J T (cid:1) αβ | P i (cid:105) = F T p α q β , (3e) (cid:104) P f | (cid:0) J T (cid:1) αβ | P i (cid:105) = F T p α q β , (3f) with p ≡ k + k and q ≡ k − k = k + k , in which k , k , k , k are the 4-momenta of the P i , f , f and P f respec-tively (see Fig. 1). All the form factors appearing in theamplitude in Eq. (2) and as defined in Eq. (3) are, in gen-eral, complex and contain all NP information. It should benoted that for simplicity we have implicitly put all the rel-evant Cabibbo-Kobayashi-Maskawa matrix elements as wellas coupling constants and propagators inside the definitionsof these form factors. In the SM only F ± V and F ± A are present.Presence of NP can modify these as well as introduce otherform factors . These various NP contributions would leavebehind their signatures in the angular distribution for whichwe need to specify the kinematics in a chosen frame of refer-ence. III. DECAY KINEMATICS zP i ( k ) f ( k ) f ( k ) P f ( k ) θ FIG. 2. Decay of P i → P f f f in the Gottfried-Jackson frame. We shall consider the decay P i → P f f f in the Gottfried-Jackson frame, especially the center-of-momentum frame ofthe f , f system, which is shown in Fig. 2. In this frame theparent meson P i flies along the positive z -direction with 4-momentum k = ( E , k ) = ( E , , , | k | ) and decays to the daugh-ter meson P f which also flies along the positive z -directionwith 4-momentum k = ( E , k ) = ( E , , , | k | ) and to f , f which fly away back-to-back with 4-momenta k = ( E , k )and k = ( E , k ) respectively, such that by conservation of 4-momentum we get, k + k = , k = k , and E = E + E + E .The fermion f (which we assume can be observed experi-mentally) flies out subtending an angle θ with respect to the It should be noted that the form factors, especially the ones describingsemi-leptonic B meson decays, can be obtained by using the heavy quarke ff ective theory [6], the lattice QCD [7], QCD light-cone sum rule [8] orthe covariant confined quark model [9] etc. In this paper we present avery general analysis which is applicable to a diverse set of meson decays.Hence we do not discuss any specifics of the form factors used in ouranalysis. Moreover, we shall show, by using certain examples and in afew specific cases, that one can also probe new physics without worryingabout the details of the form factors. Nevertheless, when one concentrateson a specific decay mode, considering the form factors in detail is alwaysuseful. direction of flight of the P i meson, in this Gottfried-Jacksonframe. The three invariant mass-squares involved in the decayunder consideration are defined as follows, s = ( k + k ) = ( k − k ) , (4a) t = ( k + k ) = ( k − k ) , (4b) u = ( k + k ) = ( k − k ) . (4c)It is easy to show that s + t + u = m i + m f + m + m , where m i , m f , m and m denote the masses of particles P i , P f , f and f respectively. In the Gottfried-Jackson frame, the ex-pressions for t and u are given by t = a t − b cos θ, (5a) u = a u + b cos θ, (5b)where a t = m + m f + s (cid:16) s + m − m (cid:17) (cid:16) m i − m f − s (cid:17) , (6a) a u = m + m f + s (cid:16) s − m + m (cid:17) (cid:16) m i − m f − s (cid:17) , (6b) b = s (cid:113) λ (cid:16) s , m , m (cid:17) λ (cid:16) s , m i , m f (cid:17) , (6c)with the K¨all´en function λ ( x , y , z ) defined as, λ ( x , y , z ) = x + y + z − xy + yz + zx ) . It is clear that a t , a u and b are functions of s only. For thespecial case of m = m = m (say) we have a t = a u = (cid:16) m i + m f + m − s (cid:17) and b = (cid:113)(cid:0) − m / s (cid:1) λ (cid:16) s , m i , m f (cid:17) .It is important to note that we shall use the angle θ in ourangular distribution. IV. MOST GENERAL ANGULAR DISTRIBUTION ANDANGULAR ASYMMETRIES
Considering the amplitude as given in Eq. (2), the mostgeneral angular distribution in the Gottfried-Jackson frame isgiven by, d Γ ds d cos θ = b √ s (cid:16) C + C cos θ + C cos θ (cid:17) π m i (cid:16) m i − m f + s (cid:17) , (7)where C , C and C are functions of s and are given by, C = (cid:32) − (cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) (cid:18) − Σ m s + Σ m (cid:16) Σ m (cid:17) i f s + (cid:16) ∆ m (cid:17) s − ∆ a tu s − (cid:16) ∆ m (cid:17) (cid:16) Σ m (cid:17) i f − (cid:16) ∆ m (cid:17) i f Σ m + ∆ a tu (cid:16) ∆ m (cid:17) (cid:16) ∆ m (cid:17) i f (cid:19) − (cid:16) F + V F ∗ T (cid:17) (cid:18) − Σ m s + Σ m (cid:16) Σ m (cid:17) i f s + ∆ m (cid:16) ∆ m (cid:17) s − ∆ m (cid:16) ∆ m (cid:17) (cid:16) Σ m (cid:17) i f − (cid:16) ∆ m (cid:17) i f Σ m + ∆ a tu ∆ m (cid:16) ∆ m (cid:17) i f (cid:19) + (cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) (cid:18) ∆ m s − ∆ m (cid:16) Σ m (cid:17) i f s − (cid:16) ∆ m (cid:17) s + ∆ a tu s + (cid:16) ∆ m (cid:17) (cid:16) Σ m (cid:17) i f + ∆ m (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:16) ∆ m (cid:17) (cid:16) ∆ m (cid:17) i f (cid:19) − (cid:16) F + A F ∗ T (cid:17) (cid:18) ∆ m s − ∆ m (cid:16) Σ m (cid:17) i f s − (cid:16) ∆ m (cid:17) Σ m s + (cid:16) ∆ m (cid:17) Σ m (cid:16) Σ m (cid:17) i f − ∆ a tu (cid:16) ∆ m (cid:17) i f Σ m + ∆ m (cid:16) ∆ m (cid:17) i f (cid:19) + (cid:12)(cid:12)(cid:12) F + A (cid:12)(cid:12)(cid:12) (cid:18) s − (cid:16) Σ m (cid:17) i f s − Σ m s + Σ m (cid:16) Σ m (cid:17) i f + (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:19) + (cid:12)(cid:12)(cid:12) F + V (cid:12)(cid:12)(cid:12) (cid:18) s − (cid:16) Σ m (cid:17) i f s − ∆ m s + ∆ m (cid:16) Σ m (cid:17) i f + (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:19) + (cid:12)(cid:12)(cid:12) F − A (cid:12)(cid:12)(cid:12) (cid:18) Σ m s − (cid:16) ∆ m (cid:17) (cid:19) − (cid:0) F P F −∗ A (cid:1) (cid:16) Σ m s − ∆ m (cid:16) ∆ m (cid:17) (cid:17) − (cid:12)(cid:12)(cid:12) F − V (cid:12)(cid:12)(cid:12) (cid:18)(cid:16) ∆ m (cid:17) − ∆ m s (cid:19) − (cid:0) F S F −∗ V (cid:1) (cid:16)(cid:16) ∆ m (cid:17) Σ m − ∆ m s (cid:17) − | F S | (cid:16) Σ m − s (cid:17) − | F P | (cid:16) ∆ m − s (cid:17) + (cid:0) F + A F −∗ A (cid:1) (cid:18)(cid:16) ∆ m (cid:17) i f Σ m − ∆ a tu (cid:16) ∆ m (cid:17) (cid:19) − (cid:0) F P F + ∗ A (cid:1) (cid:18)(cid:16) ∆ m (cid:17) i f Σ m − ∆ a tu ∆ m (cid:19) − (cid:0) F S F + ∗ V (cid:1) (cid:18) ∆ a tu Σ m − ∆ m (cid:16) ∆ m (cid:17) i f (cid:19) + (cid:0) F + V F −∗ V (cid:1) (cid:18) ∆ m (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:16) ∆ m (cid:17) (cid:19) (cid:33) , (8a) C = b (cid:32) ∆ m (cid:16) Im (cid:16) F − V F ∗ T (cid:17) s − Re (cid:0) F P F + ∗ A (cid:1)(cid:17) + Σ m (cid:16) − Im (cid:16) F − A F ∗ T (cid:17) s + Re (cid:0) F S F + ∗ V (cid:1) − (cid:16) ∆ m (cid:17) i f Im (cid:16) F + A F ∗ T (cid:17) (cid:17) + ∆ a tu (cid:18)(cid:12)(cid:12)(cid:12) F + V (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F + A (cid:12)(cid:12)(cid:12) (cid:19) + (cid:16) Im (cid:16) F S F ∗ T (cid:17) + Im (cid:16) F P F ∗ T (cid:17)(cid:17) s + (cid:16) ∆ m (cid:17) (cid:0) Re (cid:0) F + V F −∗ V (cid:1) + Re (cid:0) F + A F −∗ A (cid:1)(cid:1) + (cid:16) ∆ m (cid:17) i f ∆ m Im (cid:16) F + V F ∗ T (cid:17) (cid:33) , (8b) C = b (cid:18)(cid:18)(cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) (cid:19) s − (cid:12)(cid:12)(cid:12) F + V (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) F + A (cid:12)(cid:12)(cid:12) (cid:19) , (8c)with ∆ a tu = a t − a u , (9a) ∆ m = m − m , (9b) ∆ m i f = m i − m f , (9c) Σ m = m + m , (9d) Σ m i f = m i + m f , (9e) (cid:16) ∆ m (cid:17) = ∆ m Σ m = m − m , (9f) (cid:16) ∆ m (cid:17) i f = ∆ m i f Σ m i f = m i − m f , (9g) (cid:16) Σ m (cid:17) i f = m i + m f . (9h)In the limit m = m , which happens when f f = (cid:96) − (cid:96) + , νν, or f DM ¯ f DM etc., our expressions for the angular distributionmatches with the corresponding expression in Ref. [5]. It isimportant to remember that in the SM we come across scalar,vector and axial vector currents only. Therefore, in the SM, F SM P = F SM T = F SM T =
0, which implies that, C SM0 = (cid:32) (cid:12)(cid:12)(cid:12)(cid:0) F + A (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:18) s − (cid:16) Σ m (cid:17) i f s − Σ m s + Σ m (cid:16) Σ m (cid:17) i f + (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:19) + (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:18) s − (cid:16) Σ m (cid:17) i f s − ∆ m s + ∆ m (cid:16) Σ m (cid:17) i f + (cid:16) ∆ m (cid:17) i f − ∆ a tu (cid:19) + (cid:12)(cid:12)(cid:12)(cid:0) F − A (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:18) Σ m s − (cid:16) ∆ m (cid:17) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:0) F − V (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:18)(cid:16) ∆ m (cid:17) − ∆ m s (cid:19) − | ( F S ) SM | (cid:16) Σ m − s (cid:17) + (cid:16)(cid:0) F + A (cid:1) SM (cid:0) F − A (cid:1) ∗ SM (cid:17) (cid:18) (cid:16) ∆ m (cid:17) i f Σ m − ∆ a tu (cid:16) ∆ m (cid:17) (cid:19) + (cid:16)(cid:0) F + V (cid:1) SM (cid:0) F − V (cid:1) ∗ SM (cid:17) (cid:18) (cid:16) ∆ m (cid:17) i f ∆ m − ∆ a tu (cid:16) ∆ m (cid:17) (cid:19)(cid:33) , (10a) C SM1 = b (cid:32) ∆ a tu (cid:18)(cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) F + A (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:19) + (cid:16) ∆ m (cid:17) (cid:18) Re (cid:16)(cid:0) F + V (cid:1) SM (cid:0) F − V (cid:1) ∗ SM (cid:17) + Re (cid:16)(cid:0) F + A (cid:1) SM (cid:0) F − A (cid:1) ∗ SM (cid:17) (cid:19)(cid:33) , (10b) C SM2 = − b (cid:18)(cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:0) F + A (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:19) . (10c)It is interesting to note that in the special case of m = m ,such as in P i → P f (cid:96) + (cid:96) − , we always have C SM1 =
0. Forspecific meson decays of the form P i → P f f f allowed in the SM, one can write down ( F S ) SM , (cid:16) F ± V (cid:17) SM and (cid:16) F ± A (cid:17) SM , atleast in principle. The SM prediction for the angular distribu-tion can thus be compared with corresponding experimentalmeasurement. In order to quantitatively compare the theo-retical prediction with experimental measurement, we definethe following three angular asymmetries which can preciselyprobe C , C and C individually, A ≡ A ( s ) = − (cid:18)(cid:82) − / − − (cid:82) + / − / + (cid:82) + + / (cid:19) d Γ ds d cos θ d cos θ d Γ / ds = C / (6 C + C ) , (11a) A ≡ A ( s ) = − (cid:18)(cid:82) − − (cid:82) + (cid:19) d Γ ds d cos θ d cos θ d Γ / ds = C / (6 C + C ) , (11b) A ≡ A ( s ) = (cid:18)(cid:82) − / − − (cid:82) + / − / + (cid:82) + + / (cid:19) d Γ ds d cos θ d cos θ d Γ / ds = C / (6 C + C ) . (11c)The angular asymmetries of Eq. (11) are functions of s and itis easy to show that A = / − A ). We can do the inte-gration over s in Eq. (7) and define the following normalizedangular distribution,1 Γ d Γ d cos θ = T + T cos θ + T cos θ, (12)where T j = c j / (6 c + c ) , (13)for j = , , c j = (cid:90) ( m i − m f ) ( m + m ) b √ s C j π m i (cid:16) m i − m f + s (cid:17) ds . (14)From Eq. (13) it is easy to show that T = / − T ) whichalso ensures that integration over cos θ on Eq. (12) is equalto 1. It is interesting to note that the angular distribution ofEq. (12) can be written in terms of the orthogonal Legendrepolynomials of cos θ as well,1 Γ d Γ d cos θ = (cid:88) i = (cid:104) G ( i ) (cid:105) P i (cos θ ) . (15)Here we have followed the notation of Ref. [10] which alsoanalyzes decays of the type P i → P f f f , with only leptonsfor f , , in a model-independent manner but using a gener-alized helicity amplitude method. The observables (cid:104) G ( i ) (cid:105) ofEq. (15) are related to T , T and T of Eq. (12) as follows, (cid:104) G (0) (cid:105) = T + T / = / , (16a) (cid:104) G (1) (cid:105) = T , (16b) (cid:104) G (2) (cid:105) = T / . (16c)These angular observables (cid:104) G ( i ) (cid:105) ’s can be obtained by usingthe method of moments [10, 11]. Another important way todescribe the normalized angular distribution is by using a flatterm F H / A FB [12] asfollows,1 Γ d Γ d cos θ = F H + A FB cos θ +
34 (1 − F H ) (cid:16) − cos θ (cid:17) . (17)This form of the angular distribution has also been used inthe experimental community [13] in the study of B → K (cid:96) + (cid:96) − .The parameters F H and A FB are related to T , T and T asfollows, F H = T + T ) = − T , (18a) A FB = T . (18b)Thus we have shown that Eqs. (12), (15) and (17) are equiv-alent to one another. In this paper, we choose to work usingthe normalized angular distribution in terms of T , T and T as shown in Eq. (12). This is because the terms T , T and T can be easily determined experimentally by using the t -vs- u Dalitz plot which does not depend on any specific frameof reference. This Dalitz plot can be easily divided into foursegments I , II , III and IV as shown in Fig. 3. The segmentsare decided as follows,Segment I : − (cid:54) cos θ (cid:54) − . II : − . < cos θ (cid:54) II : 0 < cos θ (cid:54) . IV : 0 . < cos θ (cid:54) T , T and T can thus be expressed in terms of thefollowing asymmetries, T = − (cid:32) N I − N II + N III ) + N IV N I + N II + N III + N IV (cid:33) , (19a) T = ( N I + N II ) − ( N III + N IV ) N I + N II + N III + N IV , (19b) T = (cid:32) N I − ( N II + N III ) + N IV N I + N II + N III + N IV (cid:33) , (19c)where N j denotes the number of events contained in the seg-ment j . Since the t -vs- u Dalitz plot does not depend on theframe of reference, we need not constraint ourselves to theGottfried-Jackson frame of Fig. 2 and can work in the labora-tory frame as well. Furthermore, we can use the expressionsin Eq. (19) to search for NP.
V. ILLUSTRATING THE EFFECTS OF NEW PHYSICS ONTHE ANGULAR DISTRIBUTIONA. Classification of the P i → P f f f decays It should be emphasized that for our methodology to work,we need to know the angle θ in the Gottfried-Jackson frame, B → D µ − µ + B → K ν ¯ ν u (cid:0) G e V (cid:1) − − . . c o s θ IIIIIIIVIIIIIIIV u (cid:0) G e V (cid:1) t (cid:0) GeV (cid:1) IIIIIIIV
FIG. 3. Two examples depicting the variation of cos θ in the interiorregion of the t -vs- u Dalitz plot. The interior of the Dalitz plot can bedivided into four segments, I , II , III and IV , as shown here. or equivalently the t -vs- u Dalitz plot, which demand that 4-momenta of the final particles be fully known. Usually, the4-momenta of the initial and final pseudo-scalar mesons aredirectly measured experimentally. However, depending onthe detection possibilities of f and f we can identify threedistinct scenarios for our process P i → P f f f . We introducethe notations f (cid:51) i and f (cid:55) i to denote whether the fermion f i getsdetected ( (cid:51) ) or not ( (cid:55) ) by the detector. Using this notationthe three scenarios are described as follows.(S1) P i → P f + f (cid:51) + f (cid:51) ≡ P f + ‘visible’. Here both f and f are detected, e.g. when f f = (cid:96) − (cid:96) + or (cid:96) ¯ (cid:96) (cid:48) .(S2) P i → P f + f (cid:51) + f (cid:55) P f + f (cid:55) + f (cid:51) ≡ P f + ‘visible’ + ‘invisible’.Here either f or f gets detected, e.g. when f f = (cid:96)ν (cid:96) , (cid:96)ν S , (cid:96) f DM , (cid:96) f LLP .(S3) P i → P f + f (cid:55) + f (cid:55) ≡ P f + ‘invisible’. Here neither f nor f gets detected, e.g. when f f = ν (cid:96) ν (cid:96) , ν (cid:96) ν S , ν S ν (cid:96) , ν S ν S , f DM ¯ f DM , f DM f DM , f LLP f LLP etc.It should be noted that the above classification is based on ourexisting experimental explorations. What is undetected todaymight get detected in future with advanced detectors. In sucha case we can imagine that, in future, the modes grouped inS2 might migrate to S1 and those in S3 might be groupedunder S2. Below we explore each of the above scenarios inmore details. B. Exploration of new physics e ff ects in each scenario The first scenario (S1) is an experimenter’s delight as inthis case all final 4-momenta can be easily measured and the t -vs- u Dalitz plot can be obtained. Here, our methodology canbe used to look for the possible signature of new physics inrare decays such as B → D (cid:96) − (cid:96) + (which can be found in [5]) orstudy the nature of new physics contributing to lepton-flavorviolating processes such as B → P (cid:96) ± (cid:96) (cid:48)∓ where P = π, K , D , (cid:96) (cid:44) (cid:96) (cid:48) and (cid:96), (cid:96) (cid:48) = e , µ, τ . Let us consider a few NP possi-bilities mediating this lepton-flavor violating decay. There isno contribution within the SM to such decays. Therefore, allcontribution to these decays comes from NP alone. It is veryeasy to note that for the decay B → P (cid:96) − (cid:96) (cid:48) + , from Eqs. (8) and(12) we get,1 Γ d Γ d cos θ = , only scalar orpseudo-scalarinteraction T + T cos θ, (cid:18) only tensorialinteraction (cid:19) T + T cos θ + T cos θ, only vector oraxial-vectorinteraction (20)where T = / − T ) with the quantities T , T and T be-ing easily obtainable from the Dalitz plot distribution by usingEq. (19). It is clear from Eq. (20) that scalar or pseudo-scalarinteraction would give rise to a uniform (or constant) angulardistribution, while tensorial interaction gives a non-uniformdistribution which is symmetric under cos θ ↔ − cos θ andfor this T (cid:54) /
2. On the other hand vector or axial-vectorinteraction can only be described by the most general formof the angular distribution, with its signature being T (cid:44) B → P (cid:96) − (cid:96) (cid:48) + , it is importantto note that T ∝ (cid:16) m (cid:96) − m (cid:96) (cid:48) (cid:17) , where m (cid:96) , m (cid:96) (cid:48) denote the massesof the charged leptons (cid:96) − and (cid:96) (cid:48) + respectively. Therefore, weshould observe an increase in the value of T when going from B → P µ − e + to B → P τ − µ + to B → P τ − e + . This would naildown the vector or axial vector nature of the NP, if it is theonly NP contributing to these decays. Thus far we have ana-lyzed the first scenario (S1) in which the relevant decays canbe easily probed with existing detectors.The second scenario (S2) can also be studied experimen-tally with existing detectors. In this case, the missing 4-momentum can be fully deduced using conservation of 4-momentum. Thus the t -vs- u Dalitz plot can readily be ob-tained. Using our methodology the signatures of NP can thenbe extracted. One promising candidate for search for NP inthis kind of scenario is in the decay B → P (cid:96) N where P = π , K or D and N can be an active neutrino ( ν (cid:96) ) or sterile neutrino( ν S ) or a neutral dark fermion ( f DM ) or a long lived neutralfermion ( f LLP ) which decays outside the detector. These S2decay modes o ff er an exciting opportunity for study of NPe ff ects.The third scenario (S3), which has the maximum number ofNP possibilities, is also the most challenging one for the cur- rent generation of experimental facilities, due to lack of infor-mation about the individual 4-momentum of f and f . Thisimplies that we can not do any angular analysis for these kindof decays unless by some technological advancement such asby using displaced vertex detectors we can manage to makemeasurement of the 4-momentum or the angular informationof at least one of the final fermions. Getting 4-momenta ofboth the fermions would be ideal, but knowing 4-momentumof either one of them would su ffi ce for our purpose. We areoptimistic that the advancement in detector technology wouldpush the current S3 decay modes to get labelled as S2 modesin the foreseeable future. It is important to note that oncethe current S3 modes enter the S2 category, we can cover thewhole spectrum of NP possibilities in the P i → P f f f de-cays. Below we make a comprehensive exploration of NPpossibilities in the generalized S2 decay modes, which in-cludes the current S2 and S3 modes together. C. Probing e ff ects of new physics in the (S2)and generalized (S2) scenarios In the generalized S2 (GS2) scenario we have decays ofthe type P i → P f + f (cid:51) + f (cid:55) P f + f (cid:55) + f (cid:51) ≡ P f + ‘visible’ + ‘invisible’,where the detected ( (cid:51) ) or undetected ( (cid:55) ) nature is not con-strained by our existing detector technology. In some cases,even with advanced detectors, either of the fermions f , f might not get detected simply because its direction of flightlies outside the finite detector coverage, especially when thedetector is located farther from the place of origin of the par-ticle. Such possibilities are also included here. As notedbefore, measuring the 4-momentum of either of the finalfermions would su ffi ce to carry out the angular analysis fol-lowing our approach.In this context let us analyze the following decays.(i) S2 decay: B → P (cid:96) − f (cid:55) where P can be π or D and f (cid:55) is a neutral fermion. In the SM this process is mediatedby W − boson and we have f (cid:55) = ν (cid:96) . Presence of NP canimply f (cid:55) being a sterile neutrino ν S or a fermionic darkmatter particle f DM or a long lived fermion f LLP , withadditional non-SM interactions.(ii) GS2 decay: B → K f (cid:51) f (cid:55) where f (cid:51) and f (cid:55) are bothneutral fermions. In the SM this process is mediated by Z boson and we have f f = ν (cid:96) ν (cid:96) . However, in case ofNP contribution we can get pairs of sterile neutrinos orfermionic dark matter or fermionic long lived particlesetc. along with nonstandard interactions as well. Herewe are assuming that either of the final neutral fermions There are many existing proposals for such displaced vertex studies fromother theoretical and experimental considerations (see Refs. [14, 15] andreferences contained therein for further information). leaves a displaced vertex signature in an advanced de-tector so that its 4-momentum or angular informationcould be obtained.
1. New physics e ff ects in the S2 decay B → P (cid:96) − f (cid:55) Analyzing the B → P (cid:96) − f (cid:55) decay in the SM we find thatonly vector and axial vector currents contribute and F ± A = − F ± V while other form factors are zero. Also considering theanti-neutrino to be massless, i.e. m = a t = m (cid:96) + m P + (cid:16) s + m (cid:96) (cid:17) (cid:16) m B − m P − s (cid:17) / (2 s ) , a u = m P + (cid:16) s − m (cid:96) (cid:17) (cid:16) m B − m P − s (cid:17) / (2 s ) , b = (cid:16) s − m (cid:96) (cid:17) (cid:113) λ (cid:16) s , m B , m P (cid:17) / (2 s ) , where m (cid:96) , m P and m B denote the masses of the charged lepton (cid:96) − , mesons P and B respectively. Substituting these informa-tion in Eqs. (10) and in Eq. (7) we get, d Γ SM ds d cos θ = b √ s (cid:16) C SM0 + C SM1 cos θ + C SM2 cos θ (cid:17) π m B (cid:16) m B − m P + s (cid:17) , (21)where C SM0 = (cid:32) (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) (cid:18) λ (cid:16) s , m B , m P (cid:17) − m (cid:96) (cid:16) s − (cid:16) m B − m P (cid:17)(cid:17) − m (cid:96) (cid:16) m B − m P (cid:17) / s (cid:19) + (cid:12)(cid:12)(cid:12)(cid:0) F − V (cid:1) SM (cid:12)(cid:12)(cid:12) m (cid:96) (cid:16) s − m (cid:96) (cid:17) + (cid:16)(cid:0) F + V (cid:1) SM (cid:0) F − V (cid:1) ∗ SM (cid:17) m (cid:96) (cid:16) m B − m P (cid:17) − m (cid:96) s (cid:33) , (22a) C SM1 = m (cid:96) b (cid:32) m B − m P s (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) + Re (cid:16)(cid:0) F + V (cid:1) SM (cid:0) F − V (cid:1) ∗ SM (cid:17) (cid:33) , (22b) C SM2 = − b (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) . (22c)It is important to notice that in Eq. (22) we have many termsin the expression for C SM0 that are proportional to some powerof the lepton mass, while the entire C SM1 is directly propor-tional to m (cid:96) . If we compare the m (cid:96) dependent and m (cid:96) indepen-dent contributions in C SM0 we find that the dependent terms aresuppressed by about a factor of O (cid:16) m (cid:96) / m B (cid:17) which is roughly8 × − for muon and 2 × − for electron. Thus we can ne-glect these m (cid:96) dependent terms in comparison with mass inde-pendent terms. Equivalently, we can consider the charged lep-tons such as electron and muon as massless fermions, whencompared with the B meson mass scale. In the limit m (cid:96) → d Γ SM ds d cos θ = b √ s π m B (cid:16) m B − m P + s (cid:17) (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) sin θ. (23) Independent of the expression for (cid:16) F + V (cid:17) SM , it is easy to showthat the normalized angular distribution is given by,1 Γ SM d Γ SM d cos θ =
34 sin θ, (24)which implies that T = / = − T , T =
0. Since thedistribution of events in the Dalitz plot is symmetric undercos θ ↔ − cos θ , we have N I = N IV and N II = N III whichautomatically satisfies the condition T =
0. If we solve T = / = − T , we find that the number of events in the di ff erentsegments of the Dalitz plot (equivalently the number of eventsin the four distinct bins of cos θ ) are related to one another by N I N II = = N IV N III . (25)Any significant deviation from this would imply presence ofNP e ff ects. To illustrate the e ff ects of NP on the angular dis-tribution in these types of decays, we consider two simple andspecific NP possibilities. Here we assume the charged leptonto be massless ( m (cid:96) =
0) and the undetected fermion ( f (cid:55) ) tohave mass m (cid:44) a. Scalar type new physics:
Considering the simplestscalar type NP scenario, with F S (cid:44) F P = F ± V = F ± A = F T = F T =
0, we get C NP0 = (cid:16) s − m (cid:17) | F S | , C NP1 = = C NP2 . In other words, there is no angular dependence at all here, i.e. d Γ NP ds d cos θ = b √ s π m B (cid:16) m B − m P + s (cid:17) (cid:16) s − m (cid:17) | F S | , where b = (cid:16) s − m (cid:17) (cid:113) λ (cid:16) s , m B , m P (cid:17) / (2 s ) and m (cid:54) s (cid:54) ( m B − m P ) . If we do the integration over s , then the nor-malized angular distribution is given by,1 Γ NP d Γ NP d cos θ = . In fact, if such a new physics were present, our observation of B → P + (cid:96) − + f (cid:55) would have the following angular distribution, d Γ d cos θ = Γ SM (cid:32)
34 sin θ + (cid:15) (cid:33) , where we have parametrized the new physics contribution interms of (cid:15) , (cid:15) = Γ NP / Γ SM . Doing integration over cos θ we get, Γ = Γ SM (1 + (cid:15) ) = Γ SM + Γ NP . This implies 1 Γ d Γ d cos θ = θ + (cid:15) + (cid:15) ) . (26)This angular distribution is shown in Fig. 4 where we havevaried (cid:15) in the range [0 , θ there is no di ff erence between the standard modelprediction alone and the combination of standard model andnew physics contributions. These two points can be easilyobtained by equating Eqs. (24) and (26), and then solving forcos θ gives us cos θ = ± / √ ≈ ± . . (27)This corresponds to the angle θ ≈ . ◦ . At these two pointsin cos θ , the normalized uni-angular distribution always hasthe value 0 .
5, even if there is some scalar new physics con-tributing to our process under consideration. . . . . − − . . Γ d Γ d cos θ cos θ SM 00 . . . . (cid:15) FIG. 4. Normalized uni-angular distribution showing the e ff ect ofa scalar new physics contribution to B → P (cid:96) − f (cid:55) where we haveneglected the mass of the charged lepton (cid:96) = e , µ . This also showsthe normalized uni-angular distribution showing the e ff ect of a scalarnew physics contribution to B → K f (cid:51) f (cid:55) considering the m = m case only. From Eq. (26) it is clear that despite the scalar NP e ff ect,the distribution is still symmetric under cos θ ↔ − cos θ , andsolving for the number of events in the four segments of theDalitz plot (equivalently the four cos θ bins) we get, N I N II = + (cid:15) + (cid:15) = N IV N III . (28)It is easy to see that when (cid:15) = b. Tensor type new physics:
Let us consider a tensortype of new physics possibility in which F T (cid:44) C NP0 = m (cid:16) s − m (cid:17) λ (cid:16) s , m B , m P (cid:17) s (cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) , C NP1 = , C NP2 = (cid:16) s − m (cid:17) λ (cid:16) s , m B , m P (cid:17) s (cid:12)(cid:12)(cid:12) F T (cid:12)(cid:12)(cid:12) . It is easy to notice that in the limit m → C → C (cid:54)→
0. If we do the integration over s , then the normalizedangular distribution is given by,1 Γ NP d Γ NP d cos θ = T NP0 + T NP2 cos θ, (29)where T NP2 = (cid:16) / − T NP0 (cid:17) and T NP0 = c / (6 c + c ) with c j = (cid:90) ( m B − m P ) m b √ s C NP j π m B (cid:16) m B − m P + s (cid:17) ds . Thus in the limit m → T =
0. If such a newphysics were present, our observation of B → P (cid:96) − f (cid:55) wouldhave the following angular distribution, d Γ d cos θ = Γ SM (cid:32)
34 sin θ + (cid:32) T NP0 + (cid:32) − T NP0 (cid:33) cos θ (cid:33) (cid:15) (cid:33) , (30)where (cid:15) = Γ NP / Γ SM is the NP parameter which can vary inthe range [0 ,
1] denoting the possibility that the NP contribu-tion can be as large as that of the SM, and T NP0 acts as a freeparameter here which can vary in the range [0 , /
4] in which d Γ NP / d cos θ (cid:62) θ . Doing integrationover cos θ we get Γ = Γ SM (1 + (cid:15) ) = Γ SM + Γ NP . This implies1 Γ d Γ d cos θ = + T NP0 (cid:15) − (cid:16) T NP0 (cid:15) − (cid:15) + (cid:17) cos θ + (cid:15) ) . (31)This angular distribution is shown in Fig. 5 in which wehave considered nine values of T NP0 and varied (cid:15) in the range[0 , T NP0 = / θ at which there is no di ff erencebetween the SM prediction alone and the combination of SMand NP contributions. These two points can be easily com-puted by equating Eqs. (24) and (31), and then solving forcos θ we once again find that,cos θ = ± / √ ≈ ± . , (32)which corresponds to the angle θ ≈ . ◦ . At these twopoints in cos θ , the normalized uni-angular distribution al-ways has the value 0 .
5, even if there is some tensor newphysics contributing to our process under consideration. Itshould be noted that these are also the same points where thescalar new physics contribution shows similar e ff ect.It is also easy to notice that the angular distribution as givenin Eq. (31) is symmetric under cos θ ↔ − cos θ , and solvingfor the number of events in the four segments of the Dalitzplot (equivalently the four cos θ bins) we get, N I N II = + (cid:15) (cid:16) − T NP0 (cid:17) + (cid:15) (cid:16) + T NP0 (cid:17) = N IV N III . (33)It is easy to see that when (cid:15) = T NP0 = / . . . . . . . . . . . . − − . . − − . . − − . . Γ d Γ d cos θ SM 00 . . . . (cid:15) T NP0 = T NP0 = . T NP0 = . Γ d Γ d cos θ SM T NP0 = . T NP0 = . T NP0 = . Γ d Γ d cos θ cos θ SM T NP0 = . θ SM T = . θ SM T NP0 = . FIG. 5. Normalized uni-angular distribution showing the e ff ect of a tensor new physics contribution to B → P (cid:96) − f (cid:55) where we have neglectedthe mass of the charged lepton (cid:96) = e , µ . These set of plots can also describe the e ff ect of a vector new physics contribution to B → K f (cid:51) f (cid:55) when the final fermions are equally massive. Finally we analyze new physics possibilities in the decaysbelonging to the GS2 category. Due to the very nature of theGS2 decay modes, the following discussion of NP e ff ects pre-sumes usage of advanced detector technology to get angularinformation.
2. New physics e ff ects in the GS2 decay B → K f (cid:51) f (cid:55) As mentioned before, the GS2 decay modes are originallypart of S3, i.e. it is extremely di ffi cult to get angular distri-bution for these cases unless we innovate on detector tech-nology. Here we consider such a decay mode B → K f (cid:51) f (cid:55) in which both f , f are neutral fermions who have evaded,till now, all our attempts to detect them near their place oforigin. But probably with displaced vertex detectors or someother advanced detector we could bring at least one of thesefermions (say f ) under the purview of experimental studyand measure its 4-momentum or angular information. Themissing fermion (which is f in our example here) might haveflied in a direction along which there is no detector coverage.To increase the sample size we should include B → K f (cid:55) f (cid:51) events also, provided we know how to ascertain the particleor anti-particle nature of f and f . To illustrate this point,let us consider the possibility f f = ν S ν S . In a displacedvertex detector if we see π + µ − events, they can be attributedto the decay of ν S and similarly π − µ + events would appearfrom the decay of ν S . In this case, we can infer the angle θ by knowing the 4-momentum of either f = ν S or f = ν S (seeFig. 2). If we find that both f and f leave behind their sig-nature tracks in the detector (i.e. f (cid:51) f (cid:51) ) it would be the mostideal situation. But as we have already stressed before, mea-suring 4-momenta of either of the fermions would su ffi ce forour angular studies.In the SM the only contribution to B → K f (cid:51) f (cid:55) and B → K f (cid:55) f (cid:51) would come from B → K ν (cid:96) ν (cid:96) where as inthe case of NP we have a number of possibilities that in-cludes sterile neutrinos, dark matter particles, or some longlived particles in the final state, f f = ν (cid:96) ν S , ν S ν (cid:96) , ν S ν S , f DM ¯ f DM , f DM1 f DM2 , f LLP ¯ f LLP , f LLP1 f LLP2 etc. One can alsoconsider non-standard neutrino interactions also contributingin these cases. To demonstrate our methodology, we shall an-alyze only a subset of these various NP possibilities in which f and f have the same mass, i.e. m = m = m (say), asthis greatly simplifies the calculation. As we shall illustratebelow we can not only detect the presence of NP but ascer-tain whether it is of scalar type or vector type, for example,by analyzing the angular distribution.Before, we go for new physics contributions, let us analyze In addition to the new physics possibilities considered here, there can beadditional contributions to the B → K + ‘invisible’ decay, e.g. from SMsinglet scalars contributing to the ‘invisible’ part as discussed in Ref. [16].As is evident, our analysis is instead focused on a pair of fermions con-tributing to the ‘invisible’ part. B → K ν (cid:96) ν (cid:96) . Here only vector and axial-vector currents contributions, and F ± A = − F ± V . Also the neu-trino and anti-neutrino are massless, i.e. m = = m , whichimplies a t = a u = (cid:16) m B + m K − s (cid:17) and b = (cid:113) λ (cid:16) s , m B , m K (cid:17) ,where m B and m K denote the masses of B and K mesons re-spectively. Substituting these information in Eqs. (10) and inEq. (7) we get, d Γ SM ds d cos θ = b √ s π m B (cid:16) m B − m K + s (cid:17) (cid:12)(cid:12)(cid:12)(cid:0) F + V (cid:1) SM (cid:12)(cid:12)(cid:12) sin θ. (34)Irrespective of the expression for (cid:16) F + V (cid:17) SM , it can be easilyshown that the normalized angular distribution is given by,1 Γ SM d Γ SM d cos θ =
34 sin θ, (35)which implies that T = / = − T , T =
0. Following thesame logic as the one given after Eq. (24), we find that thenumber of events in the di ff erent segments of the Dalitz plot(equivalently the number of events in the four distinct bins ofcos θ ) are related to one another by, N I N II = = N IV N III . (36)This sets the stage for us to explore (i) a scalar type and (ii)a vector type of NP possibility, with final fermions for which m = m = m (cid:44) a. Scalar type new physics:
Once again we considerthe simplest scalar type NP scenario, with F S (cid:44)
0, and otherform factors being zero. This leads us to, C NP0 = (cid:16) s − m (cid:17) | F S | , C NP1 = = C NP2 . In other words, there is no angular dependence at all here, i.e. d Γ NP ds d cos θ = b √ s π m B (cid:16) m B − m K + s (cid:17) (cid:16) s − m (cid:17) | F S | , (37)where b = (cid:18) (cid:112)(cid:0) s − m (cid:1) (cid:113) λ (cid:16) s , m B , m K (cid:17)(cid:19) / (2 √ s ) and 4 m (cid:54) s (cid:54) ( m B − m K ) . If we do the integration over s , then for NPonly the normalized angular distribution is given by,1 Γ NP d Γ NP d cos θ = . Assuming such a NP contributing in addition to the SM, theexperimentally observed angular distribution can be writtenas, d Γ d cos θ = Γ SM (cid:32)
34 sin θ + (cid:15) (cid:33) , where (cid:15) = Γ NP / Γ SM is the new physics parameter which canvary in the range [0 ,
1] if we assume the NP contribution to be as large as that from the SM. Doing integration over cos θ we get, Γ = Γ SM (1 + (cid:15) ) = Γ SM + Γ NP . This implies1 Γ d Γ d cos θ = θ + (cid:15) + (cid:15) ) . (38)Since Eq. (38) is identical to Eq. (26), the angular distributionfor this case is also as shown in Fig. 4 where we have var-ied (cid:15) in the range [0 , θ , namely cos θ = ± / √ ≈ ± . θ ≈ . ◦ , there is no di ff erence betweenthe standard model prediction alone and the combination ofstandard model and scalar new physics contribution. At thesetwo points in cos θ , the normalized uni-angular distributionalways has the value 0 .
5, even if there is some scalar newphysics contributing to our process under consideration.Since the angular distribution as shown in Eq. (38) is fullysymmetric under cos θ ↔ − cos θ , the number of events in thefour segments of the Dalitz plot (equivalently in the four cos θ bins) satisfy the following relationship, N I N II = + (cid:15) + (cid:15) = N IV N III . (39)It is easy to see that (cid:15) = b. Vector type new physics:
Let us now discuss anothernew physics scenario, such as the case of a flavor-changing Z (cid:48) or a dark photon γ D giving rise to the final pair of fermions f f . We assume that for this kind of new physics scenario, F + V = F NP V (cid:44) C NP0 = (cid:12)(cid:12)(cid:12) F NP V (cid:12)(cid:12)(cid:12) λ (cid:16) s , m B , m K (cid:17) , C NP1 = , C NP2 = − b (cid:12)(cid:12)(cid:12) F NP V (cid:12)(cid:12)(cid:12) , where b = (cid:18) (cid:112)(cid:0) s − m (cid:1) (cid:113) λ (cid:16) s , m B , m K (cid:17)(cid:19) / (cid:16) √ s (cid:17) and 4 m (cid:54) s (cid:54) ( m B − m K ) . The angular distribution for the NP alonecontribution can, therefore, be written in terms of T NP0 and T NP2 which are directly proportional to C NP0 and C NP2 respec-tively. It would lead us to describe the complete angular dis-tribution in terms of T NP0 and (cid:15) = Γ NP / Γ S M using Eq. (31)and the angular distribution would look like the one shown inFig. 5. However, it is possible to describe the e ff ects of NPon the angular distribution using a di ff erent set of parametersas well. For this we start a fresh with the angular distributionfor the NP contribution alone, which in our case is given by d Γ NP ds d cos θ = b (cid:12)(cid:12)(cid:12) F NP V (cid:12)(cid:12)(cid:12) λ (cid:16) s , m B , m K (cid:17) (cid:16) s sin θ + m cos θ (cid:17) π m B (cid:16) m B − m K + s (cid:17) √ s . Doing integration over cos θ we obtain, d Γ NP ds = b (cid:12)(cid:12)(cid:12) F NP V (cid:12)(cid:12)(cid:12) λ (cid:16) s , m B , m K (cid:17) π m B (cid:16) m B − m K + s (cid:17) √ s (cid:32) s + m (cid:33) . d Γ NP / ds d Γ NP ds d cos θ = (cid:32) s sin θ + m cos θ s + m (cid:33) . (40)It is interesting to compare this with the standard model ex-pression, 1 d Γ SM / ds d Γ SM ds d cos θ =
34 sin θ. (41)Since the range for s is di ff erent in the SM and the NP sce-narios, we can not add Eqs. (40) and (41) directly. Carryingout the integration over s we get, d Γ NP d cos θ = (cid:16) S sin θ + C cos θ (cid:17) , where S = (cid:90) ( m B − m K ) m d Γ NP ds (cid:18) ss + m (cid:19) ds , C = (cid:90) ( m B − m K ) m d Γ NP ds (cid:32) m s + m (cid:33) ds . Doing integration over cos θ we get, Γ NP = S + C / , and hence 1 Γ NP d Γ NP d cos θ = (cid:16) S sin θ + C cos θ (cid:17) S + C ) . For the SM contribution we know that1 Γ SM d Γ SM d cos θ =
34 sin θ. Now the uni-angular distribution for the process B → K f (cid:51) f (cid:55) is given by, d Γ d cos θ = Γ SM (cid:16) (1 + (cid:15) s ) sin θ + (cid:15) c cos θ (cid:17) , where (cid:15) s = S / Γ SM and (cid:15) c = C / Γ SM , are the two parameterswhich describe the e ff ect of vector type NP. It is easy to checkthat, Γ = Γ SM (cid:32)
43 (1 + (cid:15) s ) + (cid:15) c (cid:33) = Γ SM + Γ NP . Therefore, the normalized angular distribution is given by,1 Γ d Γ d cos θ = + (cid:15) s ) sin θ + (cid:15) c cos θ + (cid:15) s ) + (cid:15) c . (42)It is important to note that, if we consider the mass of thefermion f to be zero, i.e. m =
0, then (cid:15) c =
0, since C =
0. Insuch a case the uni-angular distribution is given by,1 Γ d Γ d cos θ =
34 sin θ, (here (cid:15) c =
0) which is same as that of the SM case. This is plausible, asin the SM case also one has m = (cid:54) Γ NP (cid:54) Γ SM , we get0 (cid:54) (cid:15) s + (cid:15) c / (cid:54) . Thus 0 (cid:54) (cid:15) s (cid:54) (cid:54) (cid:15) c (cid:54) − (cid:15) s ).In Fig. 6 we have considered nine values of (cid:15) s and varied (cid:15) c in the range [0 , − (cid:15) s )], to obtain the uni-angular dis-tribution. It is clearly evident in Fig. 6 that (cid:15) c = θ = ± / √ ≈ ± . ff erence betweenthe SM prediction alone and the combination of SM and NPcontributions.It is also easy to notice that the angular distribution as givenin Eq. (42) is symmetric under cos θ ↔ − cos θ , and solvingfor the number of events in the four segments of the Dalitzplot (equivalently the four cos θ bins) we get, N I N II = + (cid:15) s ) + (cid:15) c
11 (1 + (cid:15) s ) + (cid:15) c = N IV N III . (43)It is easy to see that when (cid:15) c = = (cid:15) s we get back the SMprediction of Eq. (36) as expected. D. Discussion
It should be noted that our discussions on the types of NPcontributions to the S2 and GS2 modes, specifically B → P (cid:96) − f (cid:55) and B → K f (cid:51) f (cid:55) respectively, has been fully general.There is no complications arising out of hadronic form factorssince we have considered normalized angular distribution. Itshould be noted that our analysis also does not depend on howlarge or small the masses of the fermions f , f , are, as longas they are non-zero.It is also very interesting to note that both the scalar andtensor type of NP for the B → P (cid:96) − f (cid:55) decays and both thescalar and vector types of NP for the B → K f (cid:51) f (cid:55) decays,exhibit similar behaviour at cos θ = ± / √
3. In order to knowthe real reason behind this we must do a very general analy-sis. Let us assume that the most general angular distributionfor the processes B → P (cid:96) − f (cid:55) and B → K f (cid:51) f (cid:55) is given byEq. (12). If we now equate this distribution to the SM predic-tion of Eq. (24) or Eq. (35), and solve for cos θ after substi-tuting Eq. (13) we find that,cos θ = − c ± (cid:113) c + c + c ) c + c ) , (44)where the c j ’s (for j = , ,
2) are obtained from Eq. (14) withappropriate substitutions of masses and form factors. ThusEq. (44) is the most general solution that we can get for thetwo specific values of cos θ . However, let us look at the spe-cific case when c =
0. Only in this situation do we getcos θ = ± / √ . (45)2 . . . . . . . . . . . . − − . . − − . . − − . . Γ d Γ d cos θ SM 00 . . (cid:15) c (cid:15) s = (cid:15) s = .
125 SM (cid:15) s = . Γ d Γ d cos θ SM (cid:15) s = .
375 SM (cid:15) s = . (cid:15) s = . Γ d Γ d cos θ cos θ SM (cid:15) s = .
75 cos θ SM (cid:15) s = .
875 cos θ SM (cid:15) s = FIG. 6. Normalized uni-angular distribution showing the e ff ect of a vector new physics contribution to B → K f (cid:51) f (cid:55) . Now it is clear that since, in both the scalar and tensor typeof NP considerations for the B → P (cid:96) − f (cid:55) decays and in boththe scalar and vector types of NP considerations for the B → K f (cid:51) f (cid:55) decays, the angular distribution did not have any termdirectly proportional to cos θ (i.e. c = θ = ± / √ . θ = ± / √
3, it implies that c (cid:44) (cid:15) = (cid:15) c + (cid:15) s − (cid:15) c ) = (cid:15) (cid:16) − T NP0 (cid:17) − (cid:15) (cid:16) − T NP0 (cid:17) . (46)In order for (cid:15) to vary in the range [0 ,
1] we find that (i) for0 (cid:54) (cid:15) s (cid:54) (cid:54) (cid:15) c (cid:54) + (cid:15) s ) / (cid:54) (cid:15) (cid:54) (cid:54) T NP0 (cid:54) . In these specific re-gions, therefore, it would not be possible to clearly distin-guish whether scalar or vector or tensor type NP is contribut-ing to our process under consideration. Nevertheless, our ap-proach can be used to constraint these NP hypothesis withoutany hadronic uncertainties. VI. CONCLUSION
We have shown that all NP contributions to three-bodysemi-hadronic decays of the type P i → P f f f , where P i ( f ) denotes appropriate initial (final) pseudo-scalar meson and f , are a pair of fermions, can be codified into the most gen-eral Lagrangian which gives rise to a very general angulardistribution. The relevant NP information can be obtainedby using various angular asymmetries, provided at least oneof the final pair of fermions has some detectable signature,such as a displaced vertex, in the detector. Depending on thedetection feasibility of the final fermions we have groupedthe P i → P f f f decays into three distinct categories: (i) S1where both f and f are detected, (ii) S2 where either f or f gets detected, and (ii) S3 where neither f nor f gets de-tected. We consider the possibility that with advancement indetector technology S3 decays could, in future, be groupedunder S2 category. We analyze some specific NP scenarios ineach of these categories to illustrate how NP a ff ects the an-gular distribution. Specifically we have analyzed (a) lepton-flavor violating S1 decay B → P (cid:96) − (cid:96) (cid:48) + (with P = π, K , D and (cid:96), (cid:96) (cid:48) = e , µ, τ ) showing angular signatures of all generic NPpossibilities, (b) S2 decays of the type B → P (cid:96) − f (where f isnot detected in the laboratory) showing the e ff ect of a scalartype and a tensor type NP on the angular distribution, and fi-nally (c) S3 decays (more correctly generalized S2 decays) ofthe type B → K f ¯ f (where either f or ¯ f gets detected in anadvanced detector) showing the e ff ects of a scalar type anda vector type NP on the angular distribution. The e ff ects onthe angular distribution can be easily estimated from Dalitz3plot asymmetries. The signatures of NP in angular distribu-tion are distinct once the process is chosen carefully. More-over, as shown in our examples it can be possible to do theidentification and quantification of NP e ff ects without worry-ing about hadronic uncertainties. We are optimistic that ourmethodology can be put to use in LHCb, Belle II in the studyof appropriate B meson decays furthering our search for NP. ACKNOWLEDGMENTS
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