Rare radiative charm decays within the standard model and beyond
Abstract
We present standard model (SM) estimates for exclusive processes in heavy quark and hybrid frameworks. Measured branching ratios are at or somewhat exceeding the upper range of the SM and suggest slow convergence of the expansion. Modelindependent constraints on dipole operators from data are obtained. Predictions and implications for leptoquark models are worked out. While branching ratios are SMlike CP asymmetries can be induced. In SUSY deviations from the SM can be even larger with CP asymmetries of . If baryons are produced polarized, such as at the pole, an angular asymmetry in decays can be studied that is sensitive to chiralityflipped contributions.
I Introduction
A multitude of radiative charm decays is accessible at current and future high luminosity flavor facilities Hewett:2004tv (); Aushev:2010bq (). In anticipation of the new data we revisit opportunities to test the standard model (SM) with transitions Burdman:1995te (); Khodjamirian:1995uc (); Greub:1996wn (); Fajfer:1997bh (); Fajfer:1998dv (); Fajfer:1999dq (); Fajfer:2000zx (); Isidori:2012yx (), complementing studies with dileptons, e.g., Paul:2011ar (); Fajfer:2001sa (); deBoer:2015boa (). To estimate the beyond the standard model (BSM) reach we detail and evaluate exclusive decay amplitudes, where is a light vector meson. We employ two frameworks, one based on the heavy quark expansion and QCD, adopting expressions from physics Bosch:2001gv (), and a hybrid phenomenological one, updating Fajfer:1997bh (); Fajfer:1998dv (). The latter combines chiral perturbation and heavy quark effective theory and vector meson dominance (VMD). Both frameworks have considerable systematic uncertainties, leaving individual charm branching ratios without clearcut interpretation unless the deviation from the SM becomes somewhat obvious. On the other hand, considering several observables, correlations can shed light on hadronic parameters or on the electroweak model Fajfer:2000zx (). The interpretation of asymmetries is much easier, as (approximate) symmetries of the SM make them negligible compared to the experimental precision for a while. In particular, we discuss implications of the recent measurements by Belle Abdesselam:2016yvr ()
(1) 
where the CP asymmetry is defined as^{1}^{1}1The CP asymmetry of is mostly direct, analogous to the timeintegrated CP asymmetry in Aaij:2016cfh (). We thank Alan Schwartz for providing us with this information. In this work, we refer to as the direct CP asymmetry, neglecting the small indirect contribution.
(2) 
We compare data (1) to the SM predictions and derive modelindependent constraints on BSM couplings. We further discuss two specific BSM scenarios, leptoquark models and the minimal supersymmetric standard model with flavor mixing (SUSY). For the former we point out that large logarithms from the leading 1loop diagrams with leptons and leptoquarks require resummation. The outcome is numerically of relevance for the interpretation of radiative charm decays.
We further obtain analytical expressions for the contributions from the QCDpenguin operators to the effective dipole coefficient at 2loop QCD. This extends the description of radiative and semileptonic processes at this order Greub:1996wn (); deBoer:2015boa (); deBoer:2016dcg ().
While one expects the heavy quark and expansion to perform worse than in physics an actual quantitative evaluation of the individual contributions in radiative charm decays has not been done to date. Our motivation is to fill this gap and detail the expansion’s performance when compared to the hybrid model, and to data. In view of the importance of charm for probing flavor in and beyond the SM seeking after opportunities for any, possibly datadriven improvement of the theorydescription is worthwhile.
The organization of this paper is as follows: In section II we calculate weak annihilation and hard scattering contributions to decay amplitudes. In section III we present SM predictions for branching ratios and CP asymmetries in this approach and in the hybrid model. We present modelindependent constraints on BSM physics and look into leptoquark models and SUSY within the mass insertion approximation in section IV. Section V is on decays and the testability of a polarized induced angular asymmetry at future colliders Zhao:2002zk (); dEnterria:2016fpc (). In section VI we summarize. In appendix A and B we give the numerical input and form factors used in our analysis. Amplitudes in the hybrid model are provided in appendix C. Details on the 2loop contribution from QCDpenguin operators are given in appendix D.
Ii in effective Theory framework
The effective weak Lagrangian and SM Wilson coefficients are discussed in section II.1. We work out and provide a detailed breakdown of the individual contributions to amplitudes in the heavyquark approach. We work out weak annihilation and hard gluon exchange corrections in section II.2, with contributions from the gluon dipole operator given in section II.3. In section II.4 we consider weak annihilation induced modes.
ii.1 Generalities
The effective weak Lagrangian can be written as deBoer:2015boa ()
(3) 
where is the Fermi constant, are CKM matrix elements and the operators read
(4) 
where denote the electromagnetic, gluonic field strength tensor, respectively, and are the generators of QCD. In the following all Wilson coefficients are understood as evaluated at the charm scale of the order of the charm mass , and unless otherwise explicitly stated.
For the SM Wilson coefficients of and the effective coefficient of the chromomagnetic dipole operator at leading order in one obtains deBoer:2015boa (); deBoer:2016dcg (), respectively,
(5) 
where is varied within . is strongly GIM suppressed in the SM and negligible therein. and the color suppressed coefficient of the weak annihilation contribution introduced in section II.2 ,
(6) 
are subject to a large scaleuncertainty. Note, at nextto leading order, . In this work first (second) entries in intervals correspond to the lower (upper) value of within .
The effective coefficient of including the matrix elements of at twoloop QCD, see Greub:1996wn (); deBoer:2015boa (); deBoer:2016dcg () and appendix D for details, is in the range
(7) 
and . Here, we give contributions to the imaginary parts separately: The ones with subscript "" correspond to strong phases, whereas the ones with label "CKM" stem from the weak phases in the CKM matrix. As a new ingredient we provide in this work the 2loop QCD matrix element of , see appendix D for details. Numerically, , that is, negligible due to small SM Wilson coefficients and the GIM suppression.
The decay rate can be written as Isidori:2012yx ()
(8) 
where the parity conserving (PC) and parity violating (PV) amplitudes read
(9) 
times for . Here, and denote the mass of the and the vector meson, respectively, and is a tensor form factor, see appendix B for details. We stress that the dominant SM contribution to branching ratios is independent of . Furthermore,
(10) 
where the ellipses indicate additional contributions from within and outside the SM. Corrections from within the SM are obtained in sections II.2, II.3 and II.4. BSM coefficients and amplitudes are denoted by and , respectively. Note, and do not mix in eq. (8).
ii.2 Corrections
In this section we calculate the hard spectator interaction (HSI) and weak annihilation (WA) contributions shown in figure 1 as corrections to the leftmost diagram in the figure.
The leading () hard spectator interaction within QCD factorization adopted from physics Bosch:2001gv () (also Ali:2001ez (); Beneke:2001at (); Beneke:2004dp ()) can be written as
(11) 
where we consistently use at leading order in due to additional nonfactorizable diagrams at higher order and . Furthermore,
(12) 
and and the decay constants are given in appendix A. We use . As induced HSI contributions are negligible in the SM it follows that is driven by . The transverse distribution at leading twist is to first order in Gegenbauer polynomials
(13) 
Numerical input on the Gegenbauer moments is given in appendix A.
The parameter is defined as
(14) 
that is the first negative moment of the leading twist distribution amplitude of the lightcone momentum fraction of the spectator quark within the meson. In physics, the first negative moment of the meson lightcone distribution amplitude, at 90% C.L. Heller:2015vvm (), a positive lightcone wave function yields Korchemsky:1999qb () and by means of lightcone sum rules (LCSR) Ball:2003fq (); Khodjamirian:2005ea (), where and at oneloop QCD Pilipp:2007sb (). We use .
Taking , varying the Gegenbauer moments and decay constants (but not the form factor as it cancels in the amplitude) we find
(15) 
We neglect isospin breaking in the Gegenbauer moments of the . Contributions induced by are discussed in section II.3.
The leading () weak annihilation contribution to , and can be inferred from physics Bosch:2001gv (); Bosch:2004nd (). We obtain
(16) 
where , and we consistently use at leading order in . We neglect weak annihilation contributions from as the corresponding Wilson coefficients in the SM are strongly GIM suppressed. The minus sign for is due to isospin.
Varying the decay constants and within we find
(17) 
Note that nonfactorizable power corrections (inducing ) could in principle be calculated with LCSR, see, e.g., Ball:2006eu () and that nonlocal corrections to weak annihilation by means of QCD sum rules are additionally power suppressed Kagan:2001zk ().
To summarize, we observe the following hierarchies among the SM contributions to
(18) 
The leading SM uncertainties are therefore those stemming from the WAamplitudes, that is, the scale and uncertainties, followed by the parameters entering HSIamplitudes, i.e., Gegenbauer moments, decay constants and the scale. The latter we fixed for simplicity.
Contributions to arise in the SM from a induced quark loop with a soft gluon as a power correction Grinstein:2004uu ()
(19) 
which is if the expansion coefficients of in are order one. Note that the process induces as well a contribution to , and that the induced quark loop is additionally color suppressed. Note also that could in principle be calculated with LCSR, see e.g., Ball:2006eu (), yet corrections vanish at leading twist in the limit of massless quarks in the meson Grinstein:2000pc (). To be specific, and in absence of further calculations, we limit the size of the chiralityflipped SM amplitudes in our numerical analysis as
(20) 
and take the structure of the weak phases as in (19) into account.
ii.3 Contributions from
We detail here the contributions from to modes. While in the SM they are negligibly small they can be relevant in BSM scenarios.
Numerically, we find for the induced hard spectator interaction of eq. (11)
(21) 
Note that QCD factorization breaks down at subleading power for hard spectator interaction due to a logarithmic singularity for a soft spectator quark Kagan:2001zk ().
Alternatively, LCSR yield the gluon spectator interaction (GSI) Dimou:2012un ()
(22) 
The contributions in eqs. (22) and (21) are similar in size for . One may compare these to the induced contribution to eq. (7)
(23) 
BSM values can therefore lift the HSI/GSI contributions and the one to such that
(24) 
ii.4 Weak annihilation induced modes
The contributions to of the weak annihilation induced decays , and are obtained as follows
(25) 
Here we made the flavor structure of the Wilson coefficients explicit, however, since QCD is flavor symmetric, use . While the form factor is processdependent, it cancels together with in the decay amplitude. Numerically, . Varying the decay constants and we find
(26) 
For additional contributions to the decay amplitude can arise, induced by admixture in the or rescattering Isidori:2012yx (). Such effects can be parametrized by as follows
(27) 
To estimate we made CKM factors explicit except for the induced term which can receive large BSM CP violating phases. The amplitudes correspond, in order of appearance, to , and the three contributions eqs. (16), (II.2) and (7) to . Note the minus signs due do the composition. One obtains, modelindependently,
(28) 
Iii SM Phenomenology
We provide SM predictions for various modes and compare to existing data. In addition to the QCDbased approach of the previous section we present branching ratios in the phenomenological approach of Fajfer:1997bh (); Fajfer:1998dv (). This model is a hybrid of factorization, heavy quark effective and chiral perturbation theory, where the flavor symmetry is broken by measured parameters. Compared to Fajfer:1997bh (); Fajfer:1998dv () we rewrite the amplitudes in terms of newly measured parameters and vary (updated) parameters within uncertainties. Analytical expressions for the amplitudes are provided in appendix C. The hierarchies of the various amplitudes are predominantly set by CKM factors and large counting, taken care of in both the heavy quark and the hybrid frameworks.
The SM branching ratios and presently available data are given in table 1.
branching ratio  



twoloop QCD  
HSI+WA  
hybrid  
Fajfer:1997bh (); Fajfer:1998dv ()  
Burdman:1995te ()  
Khodjamirian:1995uc ()^{2}^{2}2Uncertainties not available. We take and Bauer:1986bm ().  –  –  
data  –  – 
We learn the following: The branching ratios induced by hard spectator interaction plus weak annihilation are typically smaller than (similar to) the ones obtained in the hybrid approach for neutral (charged) modes. The branching ratio from twoloop QCD eq. (7) is subleading in each case. The branching ratios in the hybrid approach cover the ranges previously obtained in Burdman:1995te (); Khodjamirian:1995uc (); Fajfer:1997bh (); Fajfer:1998dv (). The measured branching ratio is somewhat above the SM prediction in the hybrid model.
The branching ratios of as a function of are shown in figure 2.
The SM branching ratio is , a measurement would constrain efficiently. Specifically, we find by means of hard spectator interaction plus weak annihilation and in the hybrid model . The branching ratio can be subject to stronger cancellations between the contributions in eq. (18) than in the hybrid model. Assuming that the phase of each amplitude is equal for and reduces the possible isospin breaking to . Note, isospin is already significantly broken by the lifetimes Olive:2016xmw ().
The uncertainties in the hybrid model are dominated by the relative strong phases, followed by the phenomenological fit coefficients , Bauer:1986bm () (also Buras:1994ij (); Cheng:2002wu ()).
The branching ratios of , and are given in table 2.
branching ratio  


WA  
hybrid  
Fajfer:1997bh (); Fajfer:1998dv ()  
Burdman:1995te ()  
Khodjamirian:1995uc ()^{3}^{3}3Uncertainties not available. We use and Bauer:1986bm ().  –  –  –  
Belle Abdesselam:2016yvr ()  –  –  –  
BaBar Aubert:2008ai ()^{4}^{4}4We update the normalization Olive:2016xmw ().  –  –  – 
The measurements by Belle Abdesselam:2016yvr () and BaBar Aubert:2008ai () of differ by , yet both are in the range of the hybrid model predictions. Interpreted in the QCD framework to the order we are working,