Angular distribution as an effective probe of new physicsin semi-hadronic three-body meson decays

# Angular distribution as an effective probe of new physics in semi-hadronic three-body meson decays

C. S. Kim Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea    Seong Chan Park Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea    Dibyakrupa Sahoo Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea
July 20, 2019
###### Abstract

We analyze, in a fully model-independent manner, the effects of new physics on a few semi-hadronic three-body meson decays of the type , where are well chosen pseudo-scalar mesons and denote fermions out of which at least one gets detected in experiments. We find that the angular distribution of events of these decays can probe many interesting new physics, such as the nature of the intermediate particle that can cause lepton-flavor violation, or presence of heavy sterile neutrino, or new intermediate particles, or new interactions. We also provide angular asymmetries which can quantify the effects of new physics in these decays. We illustrate the effectiveness of our proposed methodology with a few well chosen decay modes showing effects of certain new physics possibilities without any hadronic uncertainties.

Beyond Standard Model, Heavy Quark Physics, Invisible decays, Rare decays, Lepton flavor violation
###### pacs:
13.20.-v, 14.60.St, 14.80.-j
preprint: LDU-18-004

## I Introduction

New physics (NP), or physics beyond the standard model, involves various models that extend the well verified standard model (SM) of particle physics by introducing a number of new particles with novel properties and interactions. Though various aspects of many of these particles and interactions are constrained by existing experimental data, we are yet to detect any definitive signature of new physics in our experiments. Nevertheless, recent experimental studies in meson decays, such as B2KorKstLL (), Aaij:2015esa (), B2DorDstLN () and Aaij:2017tyk () (where can be or ) have reported anomalous observations raising the expectation of discovery of new physics with more statistical significance. In this context, model-independent studies of such semi-leptonic three-body meson decay processes become important as they can identify generic signatures of new physics which can be probed experimentally. In this paper, we have analyzed the effects of new physics, in a model-independent manner, on the angular distribution of a general semi-hadronic three-body meson decay of the type , where and are the initial and final pseudo-scalar mesons respectively, and denote fermions (which may or may not be leptons but not quarks) out of which at least one gets detected experimentally. Presence of new interactions, or new particles such as fermionic dark matter (DM) particles or heavy sterile neutrinos or long lived particles (LLP) would leave their signature in the angular distribution and we show by example how new physics contribution can be quantified from angular asymmetries. Our methodology can be used for detection of new physics in experimental study of various three-body pseudo-scalar meson decays at various collider experiments such as LHCb and Belle II.

The structure of our paper is as follows. In Sec. II we discuss the most general Lagrangian and amplitude which include all probable NP contributions to our process under consideration. The relevant details of kinematics is then described in Sec. III. This is followed by a discussion on the angular distribution and the various angular asymmetries in Sec. IV. In Sec. V we present a few well chosen examples illustrating the effects of new physics on the angular distribution. In Sec. VI we conclude by summarizing the important aspects of our methodology and its possible experimental realization.

## Ii Most general Lagrangian and Amplitude

Following the model-independent analysis of the decay as given in Ref. Kim:2016zbg () and generalizing it for our process where can be etc. as appropriate and can be , , , , , , , , , , , (with denoting leptons, being sterile neutrino, as fermionic dark matter and as long lived fermions)111It is clear that we can not only analyze processes allowed in the SM but also those NP contributions from fermionic dark matter in the final state as well as including flavor violation. Our analysis as presented in this paper is fully model-independent and general in nature., we can write down the effective Lagrangian facilitating the decay under consideration as follows,

 Leff = JS(¯f1f2)+JP(¯f1 γ5 f2)+(JV)α(¯f1 γα f2) (1) +(JA)α(¯f1 γαγ5 f2)+(JT1)αβ(¯f1 σαβ f2) +(JT2)αβ(¯f1 σαβγ5 f2)+h.c.,

where , , , , , are the different hadronic currents which effectively describe the quark level transitions from to meson. It should be noted that we have kept both and terms. This is because of the fact that the currents and describe two different physics aspects namely the magnetic dipole and electric dipole contributions respectively. In the SM, vector and axial-vector currents (mediated by photon, and bosons) and the scalar current (mediated by Higgs boson) contribute. So every other term in Eq. (1) except the ones with , and 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 NP model, which nevertheless are also useful. It is important to note that , and can also get modified due to NP contributions.

In order to get the most general amplitude for our process under consideration, we need to go from the effective quark-level description of Eq. (1) to the meson level description by defining appropriate form factors. It is easy to write down the most general form of the amplitude for the process depicted in Fig. 1 as follows,

 M(Pi→Pff1f2) =FS(¯f1f2)+FP(¯f1 γ5 f2) +(F+Vpα+F−Vqα)(¯f1 γα f2) +(F+Apα+F−Aqα)(¯f1 γα γ5 f2) +FT1 pα qβ(¯f1 σαβ f2) +FT2 pα qβ(¯f1 σαβ γ5 f2), (2)

where , , , , and are the relevant form factors, and are defined as follows,

 ⟨Pf|JS|Pi⟩ =FS, (3a) ⟨Pf|JP|Pi⟩ =FP, (3b) ⟨Pf|(JV)α|Pi⟩ =F+Vpα+F−Vqα, (3c) ⟨Pf|(JA)α|Pi⟩ =F+Apα+F−Aqα, (3d) ⟨Pf|(JT1)αβ|Pi⟩ =FT1 pα qβ, (3e) ⟨Pf|(JT2)αβ|Pi⟩ =FT2 pα qβ, (3f)

with and , in which are the 4-momenta of the and respectively (see Fig. 1). All the form factors appearing in the amplitude in Eq. (2) and as defined in Eq. (3) are, in general, complex and contain all NP information. It should be noted that for simplicity we have implicitly put all the relevant Cabibbo-Kobayashi-Maskawa matrix elements as well as coupling constants and propagators inside the definitions of these form factors. In the SM only and are present. Presence of NP can modify these as well as introduce other form factors222It should be noted that the form factors, especially the ones describing semi-leptonic meson decays, can be obtained by using the heavy quark effective theory HQET (), the lattice QCD Lattice (), QCD light-cone sum rule Light-cone () or the covariant confined quark model CCQM () etc. In this paper we present a very 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 our analysis. Moreover, we shall show, by using certain examples and in a few specific cases, that one can also probe new physics without worrying about the details of the form factors. Nevertheless, when one concentrates on a specific decay mode, considering the form factors in detail is always useful.. These various NP contributions would leave behind their signatures in the angular distribution for which we need to specify the kinematics in a chosen frame of reference.

## Iii Decay Kinematics

We shall consider the decay in the Gottfried-Jackson frame, especially the center-of-momentum frame of the system, which is shown in Fig. 2. In this frame the parent meson flies along the positive -direction with 4-momentum and decays to the daughter meson which also flies along the positive -direction with 4-momentum and to , which fly away back-to-back with 4-momenta and respectively, such that by conservation of 4-momentum we get, , , and . The fermion (which we assume can be observed experimentally) flies out subtending an angle with respect to the direction of flight of the meson, in this Gottfried-Jackson frame. The three invariant mass-squares involved in the decay under consideration are defined as follows,

 s =(k1+k2)2=(k−k3)2, (4a) t =(k1+k3)2=(k−k2)2, (4b) u =(k2+k3)2=(k−k1)2. (4c)

It is easy to show that , where and denote the masses of particles and respectively. In the Gottfried-Jackson frame, the expressions for and are given by

 t =at−bcosθ, (5a) u =au+bcosθ, (5b)

where

 at =m21+m2f+12s(s+m21−m22)(m2i−m2f−s), (6a) au =m22+m2f+12s(s−m21+m22)(m2i−m2f−s), (6b) b =12s√λ(s,m21,m22) λ(s,m2i,m2f), (6c)

with the Källén function defined as,

 λ(x,y,z)=x2+y2+z2−2(xy+yz+zx).

It is clear that , and are functions of only. For the special case of (say) we have and . It is important to note that we shall use the angle in our angular distribution.

## Iv Most general angular distribution and angular asymmetries

Considering the amplitude as given in Eq. (2), the most general angular distribution in the Gottfried-Jackson frame is given by,

 d2Γdsdcosθ=b√s(C0+C1cosθ+C2cos2θ)128π3m2i(m2i−m2f+s), (7)

where , and are functions of and are given by,

 C0 =2(−∣∣FT1∣∣2(−Σm212s2+2Σm212(Σm2)ifs +(Δm2)212s−Δa2tus−2(Δm2)212(Σm2)if −(Δm2)2ifΣm212+2Δatu(Δm2)12(Δm2)if) −2Im(F+VF∗T1)(−Σm12s2+2Σm12(Σm2)ifs +Δm12(Δm2)12s−2Δm12(Δm2)12(Σm2)if −(Δm2)2ifΣm12+ΔatuΔm12(Δm2)if) +∣∣FT2∣∣2(Δm212s2−2Δm212(Σm2)ifs−(Δm2)212s +Δa2tus+2(Δm2)212(Σm2)if+Δm212(Δm2)2if −2Δatu(Δm2)12(Δm2)if) −2Im(F+AF∗T2)(Δm12s2−2Δm12(Σm2)ifs −Δatu(Δm2)ifΣm12+Δm12(Δm2)2if) +∣∣F+A∣∣2(s2−2(Σm2)ifs−Σm212s +2Σm212(Σm2)if+(Δm2)2if−Δa2tu) +∣∣F+V∣∣2(s2−2(Σm2)ifs−Δm212s +2Δm212(Σm2)if+(Δm2)2if−Δa2tu) +∣∣F−A∣∣2(Σm212s−(Δm2)212) −2Re(FPF−∗A)(Σm12s−Δm12(Δm2)12) −∣∣F−V∣∣2((Δm2)212−Δm212s) −2Re(FSF−∗V)((Δm2)12Σm12−Δm12s) −|FS|2(Σm212−s)−|FP|2(Δm212−s) +2Re(F+AF−∗A)((Δm2)ifΣm212−Δatu(Δm2)12) −2Re(FPF+∗A)((Δm2)ifΣm12−ΔatuΔm12) −2Re(FSF+∗V)(ΔatuΣm12−Δm12(Δm2)if) +2Re(F+VF−∗V)(Δm212(Δm2)if−Δatu(Δm2)12)), (8a) C1 =8b(Δm12(Im(F−VF∗T1)s−Re(FPF+∗A)) +Σm12(−Im(F−AF∗T2)s+Re(FSF+∗V) −(Δm2)ifIm(F+AF∗T2)) +(Im(FSF∗T1)+Im(FPF∗T2))s +(Δm2)12(Re(F+VF−∗V)+Re(F+AF−∗A)) +(Δm2)ifΔm12Im% (F+VF∗T1)), (8b) C2 =8b2((∣∣FT2∣∣2+∣∣FT1∣∣2)s−∣∣F+V∣∣2−∣∣F+A∣∣2), (8c)

with

 Δatu =at−au, (9a) Δm12 =m1−m2, (9b) Δmif =mi−mf, (9c) Σm12 =m1+m2, (9d) Σmif =mi+mf, (9e) (Δm2)12 =Δm12Σm12=m21−m22, (9f) (Δm2)if =ΔmifΣmif=m2i−m2f, (9g) (Σm2)if =m2i+m2f. (9h)

In the limit , which happens when or etc., our expressions for the angular distribution matches with the corresponding expression in Ref. Kim:2016zbg (). It is important to remember that in the SM we come across scalar, vector and axial vector currents only. Therefore, in the SM, , which implies that,

 CSM0= 2(∣∣(F+A)SM∣∣2(s2−2(Σm2)ifs−Σm212s +2Σm212(Σm2)if+(Δm2)2if−Δa2tu) +∣∣(F+V)SM∣∣2(s2−2(Σm2)ifs−Δm212s +2Δm212(Σm2)if+(Δm2)2if−Δa2tu) +∣∣(F−A)SM∣∣2(Σm212s−(Δm2)212) −∣∣(F−V)SM∣∣2((Δm2)212−Δm212s) −∣∣(FS)SM∣∣2(Σm212−s) +2Re((F+A)SM(F−A)∗SM)((Δm2)ifΣm212 −Δatu(Δm2)12) +2Re((F+V)SM(F−V)∗SM)((Δm2)ifΔm212 −Δatu(Δm2)12)), (10a) CSM1= 8b(Δatu(∣∣(F+V)%SM∣∣2+∣∣(F+A)SM∣∣2) +Re((F+A)SM(F−A)∗SM))), (10b) CSM2= −8b2(∣∣(F+V)SM∣∣2+∣∣(F+A)SM∣∣2). (10c)

It is interesting to note that in the special case of , such as in , we always have . For specific meson decays of the form allowed in the SM, one can write down , and , at least in principle. The SM prediction for the angular distribution can thus be compared with corresponding experimental measurement. In order to quantitatively compare the theoretical prediction with experimental measurement, we define the following three angular asymmetries which can precisely probe , and individually,

 A0≡A0(s) =−16(∫−1/2−1−7∫+1/2−1/2+∫+1+1/2)d2ΓdsdcosθdcosθdΓ/ds =3C0/(6C0+2C2), (11a) A1≡A1(s) =−(∫0−1−∫+10)d2ΓdsdcosθdcosθdΓ/ds =3C1/(6C0+2C2), (11b) A2≡A2(s) =2(∫−1/2−1−∫+1/2−1/2+∫+1+1/2)d2ΓdsdcosθdcosθdΓ/ds =3C2/(6C0+2C2). (11c)

The angular asymmetries of Eq. (11) are functions of and it is easy to show that . We can do the integration over in Eq. (7) and define the following normalized angular distribution,

 1ΓdΓdcosθ=T0+T1cosθ+T2cos2θ, (12)

where

 Tj=3cj/(6c0+2c2), (13)

for and with

 cj=∫(mi−mf)2(m1+m2)2b√sCj128π3m2i(m2i−m2f+s)ds. (14)

From Eq. (13) it is easy to show that which also ensures that integration over on Eq. (12) is equal to . It is interesting to note that the angular distribution of Eq. (12) can be written in terms of the orthogonal Legendre polynomials of as well,

 1ΓdΓdcosθ=2∑i=0⟨G(i)⟩Pi(cosθ). (15)

Here we have followed the notation of Ref. Gratrex:2015hna () which also analyzes decays of the type , with only leptons for , in a model-independent manner but using a generalized helicity amplitude method. The observables of Eq. (15) are related to , and of Eq. (12) as follows,

 ⟨G(0)⟩ =T0+T2/3=1/2, (16a) ⟨G(1)⟩ =T1, (16b) ⟨G(2)⟩ =2T2/3. (16c)

These angular observables ’s can be obtained by using the method of moments Gratrex:2015hna (); Beaujean:2015xea (). Another important way to describe the normalized angular distribution is by using a flat term and the forward-backward asymmetry AngDist:Hiller () as follows,

 1ΓdΓdcosθ=12FH+AFBcosθ+34(1−FH)(1−cos2θ). (17)

This form of the angular distribution has also been used in the experimental community AngDist:Expt () in the study of . The parameters and are related to , and as follows,

 FH =2(T0+T2)=3−4T0, (18a) AFB =T1. (18b)

Thus we have shown that Eqs. (12), (15) and (17) are equivalent to one another. In this paper, we choose to work using the normalized angular distribution in terms of , and as shown in Eq. (12). This is because the terms , and can be easily determined experimentally by using the -vs- Dalitz plot which does not depend on any specific frame of reference. This Dalitz plot can be easily divided into four segments , , and as shown in Fig. 3. The segments are decided as follows,

 Segment I : −1⩽cosθ⩽−0.5, Segment II : −0.5

The terms , and can thus be expressed in terms of the following asymmetries,

 T0 =−16(NI−7(NII+NIII)+NIVNI+NII+NIII+NIV), (19a) T1 =(NI+NII)−(NIII+NIV)NI+NII+NIII+NIV, (19b) T2 =2(NI−(NII+NIII)+NIVNI+NII+NIII+NIV), (19c)

where denotes the number of events contained in the segment . Since the -vs- Dalitz plot does not depend on the frame of reference, we need not constraint ourselves to the Gottfried-Jackson frame of Fig. 2 and can work in the laboratory frame as well. Furthermore, we can use the expressions in Eq. (19) to search for NP.

## V Illustrating the effects of new physics on the angular distribution

### v.1 Classification of the Pi→Pff1f2 decays

It should be emphasized that for our methodology to work, we need to know the angle in the Gottfried-Jackson frame, or equivalently the -vs- Dalitz plot, which demand that 4-momenta of the final particles be fully known. Usually, the 4-momenta of the initial and final pseudo-scalar mesons are directly measured experimentally. However, depending on the detection possibilities of and we can identify three distinct scenarios for our process . We introduce the notations and to denote whether the fermion gets detected (✓) or not (✗) by the detector. Using this notation the three scenarios are described as follows.

1. . Here both and are detected, e.g. when or .

2. . Here either or gets detected, e.g. when , , , .

3. . Here neither nor gets detected, e.g. when , , , , , , etc.

It should be noted that the above classification is based on our existing experimental explorations. What is undetected today might get detected in future with advanced detectors. In such a case we can imagine that, in future, the modes grouped in S2 might migrate to S1 and those in S3 might be grouped under S2. Below we explore each of the above scenarios in more details.

### v.2 Exploration of new physics effects in each scenario

The first scenario (S1) is an experimenter’s delight as in this case all final 4-momenta can be easily measured and the -vs- Dalitz plot can be obtained. Here, our methodology can be used to look for the possible signature of new physics in rare decays such as