The Belle Collaboration

# Measurement of the branching fractions for Cabibbo-suppressed decays
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and
${D_{(s)}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ at Belle

L. K. Li  0000-0002-7366-1307 A. J. Schwartz  0000-0002-7310-1983 K. Kinoshita
0000-0001-7175-4182 E. Won  0000-0002-4245-7442 H. Aihara  0000-0002-1907-5964
S. Al Said  0000-0002-4895-3869 D. M. Asner  0000-0002-1586-5790 H. Atmacan
0000-0003-2435-501X V. Aulchenko  0000-0002-5394-4406 T. Aushev
0000-0002-6347-7055 R. Ayad  0000-0003-3466-9290 V. Babu  0000-0003-0419-6912
S. Bahinipati  0000-0002-3744-5332 K. Belous  0000-0003-0014-2589 J. Bennett
0000-0002-5440-2668 M. Bessner  0000-0003-1776-0439 V. Bhardwaj
0000-0001-8857-8621 T. Bilka  0000-0003-1449-6986 A. Bobrov
0000-0001-5735-8386 D. Bodrov  0000-0001-5279-4787 G. Bonvicini
0000-0003-4861-7918 J. Borah  0000-0003-2990-1913 A. Bozek
0000-0002-5915-1319 M. Bračko  0000-0002-2495-0524 P. Branchini
0000-0002-2270-9673 T. E. Browder  0000-0001-7357-9007 A. Budano
0000-0002-0856-1131 M. Campajola  0000-0003-2518-7134 D. Červenkov
0000-0002-1865-741X P. Chang  0000-0003-4064-388X A. Chen  0000-0002-8544-9274
B. G. Cheon  0000-0002-8803-4429 H. E. Cho  0000-0002-7008-3759 K. Cho
0000-0003-1705-7399 S.-J. Cho  0000-0002-1673-5664 S.-K. Choi
0000-0003-2747-8277 Y. Choi  0000-0003-3499-7948 S. Choudhury
0000-0001-9841-0216 D. Cinabro  0000-0001-7347-6585 S. Cunliffe
0000-0003-0167-8641 S. Das  0000-0001-6857-966X N. Dash  0000-0003-2172-3534
G. De Nardo  0000-0002-2047-9675 G. De Pietro  0000-0001-8442-107X R. Dhamija
0000-0001-7052-3163 F. Di Capua  0000-0001-9076-5936 Z. Doležal
0000-0002-5662-3675 T. V. Dong  0000-0003-3043-1939 D. Dossett
0000-0002-5670-5582 D. Epifanov  0000-0001-8656-2693 A. Frey
0000-0001-7470-3874 B. G. Fulsom  0000-0002-5862-9739 R. Garg
0000-0002-7406-4707 V. Gaur  0000-0002-8880-6134 A. Garmash
0000-0003-2599-1405 A. Giri  0000-0002-8895-0128 P. Goldenzweig
0000-0001-8785-847X E. Graziani  0000-0001-8602-5652 D. Greenwald
0000-0001-6964-8399 T. Gu  0000-0002-1470-6536 Y. Guan  0000-0002-5541-2278 K.
Gudkova  0000-0002-5858-3187 C. Hadjivasiliou  0000-0002-2234-0001 K. Hayasaka
0000-0002-6347-433X H. Hayashii  0000-0002-5138-5903 D. Herrmann
0000-0001-9772-9989 W.-S. Hou  0000-0002-4260-5118 C.-L. Hsu
0000-0002-1641-430X K. Inami  0000-0003-2765-7072 A. Ishikawa
0000-0002-3561-5633 R. Itoh  0000-0003-1590-0266 M. Iwasaki
0000-0002-9402-7559 W. W. Jacobs  0000-0002-9996-6336 S. Jia
0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 D. Kalita  0000-0003-3054-1222
A. B. Kaliyar  0000-0002-2211-619X K. H. Kang  0000-0002-6816-0751 T. Kawasaki
0000-0002-4089-5238 C. Kiesling  0000-0002-2209-535X C. H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim
0000-0002-4659-1112 K. T. Kim  0000-0003-2884-6772 Y.-K. Kim
0000-0002-9695-8103 P. Kodyš  0000-0002-8644-2349 T. Konno
0000-0003-2487-8080 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 M. Kumar  0000-0002-6627-9708 R. Kumar
0000-0002-6277-2626 K. Kumara  0000-0003-1572-5365 A. Kuzmin
0000-0002-7011-5044 Y.-J. Kwon  0000-0001-9448-5691 Y.-T. Lai
0000-0001-9553-3421 T. Lam  0000-0001-9128-6806 M. Laurenza
0000-0002-7400-6013 S. C. Lee  0000-0002-9835-1006 J. Li  0000-0001-5520-5394
Y. Li  0000-0002-4413-6247 Y. B. Li  0000-0002-9909-2851 L. Li Gioi
0000-0003-2024-5649 J. Libby  0000-0002-1219-3247 K. Lieret
0000-0003-2792-7511 D. Liventsev  0000-0003-3416-0056 A. Martini
0000-0003-1161-4983 M. Masuda  0000-0002-7109-5583 T. Matsuda
0000-0003-4673-570X D. Matvienko  0000-0002-2698-5448 S. K. Maurya
0000-0002-7764-5777 F. Meier  0000-0002-6088-0412 M. Merola
0000-0002-7082-8108 F. Metzner  0000-0002-0128-264X K. Miyabayashi
0000-0003-4352-734X R. Mizuk  0000-0002-2209-6969 G. B. Mohanty
0000-0001-6850-7666 H. K. Moon  0000-0001-5213-6477 M. Nakao
0000-0001-8424-7075 Z. Natkaniec  0000-0003-0486-9291 A. Natochii
0000-0002-1076-814X L. Nayak  0000-0002-7739-914X M. Nayak
0000-0002-2572-4692 N. K. Nisar  0000-0001-9562-1253 S. Nishida
0000-0001-6373-2346 H. Ono  0000-0003-4486-0064 P. Oskin  0000-0002-7524-0936
P. Pakhlov  0000-0001-7426-4824 G. Pakhlova  0000-0001-7518-3022 S. Pardi
0000-0001-7994-0537 H. Park  0000-0001-6087-2052 S.-H. Park
0000-0001-6019-6218 A. Passeri  0000-0003-4864-3411 S. Patra
0000-0002-4114-1091 S. Paul  0000-0002-8813-0437 T. K. Pedlar
0000-0001-9839-7373 R. Pestotnik  0000-0003-1804-9470 L. E. Piilonen
0000-0001-6836-0748 T. Podobnik  0000-0002-6131-819X E. Prencipe
0000-0002-9465-2493 M. T. Prim  0000-0002-1407-7450 A. Rostomyan
0000-0003-1839-8152 N. Rout  0000-0002-4310-3638 G. Russo  0000-0001-5823-4393
D. Sahoo  0000-0002-5600-9413 S. Sandilya  0000-0002-4199-4369 A. Sangal
0000-0001-5853-349X L. Santelj  0000-0003-3904-2956 V. Savinov
0000-0002-9184-2830 G. Schnell  0000-0002-7336-3246 J. Schueler
0000-0002-2722-6953 C. Schwanda  0000-0003-4844-5028 Y. Seino
0000-0002-8378-4255 K. Senyo  0000-0002-1615-9118 M. E. Sevior
0000-0002-4824-101X M. Shapkin  0000-0002-4098-9592 C. Sharma
0000-0002-1312-0429 J.-G. Shiu  0000-0002-8478-5639 F. Simon
0000-0002-5978-0289 J. B. Singh  0000-0001-9029-2462 A. Sokolov
0000-0002-9420-0091 E. Solovieva  0000-0002-5735-4059 M. Starič
0000-0001-8751-5944 Z. S. Stottler  0000-0002-1898-5333 M. Sumihama
0000-0002-8954-0585 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 M. Uchida  0000-0003-4904-6168 T. Uglov
0000-0002-4944-1830 K. Uno  0000-0002-2209-8198 S. Uno  0000-0002-3401-0480 Y.
Ushiroda  0000-0003-3174-403X Y. Usov  0000-0003-3144-2920 R. van Tonder
0000-0002-7448-4816 G. Varner  0000-0002-0302-8151 K. E. Varvell
0000-0003-1017-1295 A. Vossen  0000-0003-0983-4936 E. Waheed
0000-0001-7774-0363 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 X. L. Wang  0000-0001-5805-1255 S. Watanuki
0000-0002-5241-6628 O. Werbycka  0000-0002-0614-8773 J. Wiechczynski
0000-0002-3151-6072 B. D. Yabsley  0000-0002-2680-0474 W. Yan
0000-0003-0713-0871 S. B. Yang  0000-0002-9543-7971 H. Ye  0000-0003-0552-5490
J. Yelton  0000-0001-8840-3346 J. H. Yin  0000-0002-1479-9349 C. Z. Yuan
0000-0002-1652-6686 Z. P. Zhang  0000-0001-6140-2044 V. Zhilich
0000-0002-0907-5565 V. Zhukova  0000-0002-8253-641X

###### Abstract

We present measurements of the branching fractions for the singly Cabibbo-
suppressed decays ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and the doubly
Cabibbo-suppressed decay ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$,
based on 980 $\rm fb^{-1}$ of data recorded by the Belle experiment at the
KEKB $e^{+}e^{-}$ collider. We measure these modes relative to the Cabibbo-
favored modes ${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. Our results for the
ratios of branching fractions are
${\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}={(11.32\pm
0.13\pm 0.26)\%}$,
${\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}={(1.68\pm
0.11\pm 0.03)\%}$, and
${\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})}={(17.13\pm
0.62\pm 0.51)\%}$, where the uncertainties are statistical and systematic,
respectively. The second value corresponds to ${(5.83\pm
0.42)\times\tan^{4}\theta_{C}}$, where $\theta_{C}$ is the Cabibbo angle; this
value is larger than other measured ratios of branching fractions for a doubly
Cabibbo-suppressed charm decay to a Cabibbo-favored decay. Multiplying these
results by world average values for
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})$ and
$\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})$ yields
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})={(7.08\pm
0.08\pm 0.16\pm 0.20)\times 10^{-3}}$,
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})={(1.05\pm
0.07\pm 0.02\pm 0.03)\times 10^{-3}}$, and
$\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})={(9.44\pm
0.34\pm 0.28\pm 0.32)\times 10^{-3}}$, where the third uncertainty is due to
the branching fraction of the normalization mode. The first two results are
consistent with, but more precise than, the current world averages. The last
result is the first measurement of this branching fraction.

##  I Introduction

Cabibbo-suppressed (CS) hadronic decays of charm mesons provide a powerful
means to search for new physics [1]. Because such decays are suppressed in the
Standard Model, their decay rates are especially sensitive to small new-
physics contributions to the amplitudes. Thus, it is important to measure such
decays with high precision. It is notable that the only observation of $C\\!P$
violation in charm decays, possibly arising from new physics [1], was made
with the singly Cabibbo-suppressed (SCS) decays
${D^{0}\\!\rightarrow\\!K^{+}K^{-}}$ and
${D^{0}\\!\rightarrow\\!\pi^{+}\pi^{-}}$ [2]. Experimentally, CS decays can be
challenging to measure, as they typically have higher background levels than
those for Cabibbo-favored (CF) decays.

In this paper, we present measurements of the branching fractions for the SCS
decays ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and the doubly
Cabibbo-suppressed (DCS) decay
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$. Throughout this paper,
charge-conjugate modes are implicitly included. The branching fractions are
measured relative to those for the well-measured CF modes
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. The branching fraction
of a DCS decay relative to its CF counterpart is expected to be approximately
$\tan^{4}\theta_{C}\\!=\\!0.29\%$ [3], where $\theta_{C}$ is the Cabibbo
angle. The SCS decay ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, and
the DCS decay ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, were
recently observed by the BESIII experiment [4, 5, 6]. The decay
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ has not yet been
observed. The average of the absolute branching fractions measured at BESIII
[5, 6] is
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})\\!=\\!{(1.18\pm
0.07)\times 10^{-3}}$; this gives a ratio of branching fractions
${\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}\\!=\\!{(1.89\pm
0.12)\%}$, which corresponds to $(6.56\pm 0.42)\times\tan^{4}\theta_{C}$. This
value is larger than other measured ratios of DCS to CF branching fractions,
which are in the range ${(0.7\text{--}1.8)\times\tan^{4}\theta_{C}}$ [7]. To
investigate this further, we use the full Belle data set to measure these
decay modes with high precision.

##  II Detector and data set

Our analysis uses the full data set of the Belle experiment, which corresponds
to an integrated luminosity of 980 $\mbox{\,fb}^{-1}$ collected at or near the
$\Upsilon(nS)$ ($n=1,\,2,\,3,\,4,\,5$) resonances. The Belle experiment ran at
the KEKB asymmetric-energy $e^{+}e^{-}$ collider [8, 9]. The Belle detector is
a large-solid-angle magnetic spectrometer consisting of a silicon vertex
detector (SVD), a $50$-layer central drift chamber (CDC), an array of aerogel
threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-
flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL)
comprising CsI(Tl) crystals located inside a superconducting solenoid coil
providing a $1.5$ T magnetic field. An iron flux-return located outside the
coil is instrumented to detect $K_{L}^{0}$ mesons and to identify muons (KLM).
A detailed description of the detector is given in Ref. [9, 10].

We use Monte Carlo (MC) simulated events to optimize selection criteria, study
sources of background, and calculate selection efficiencies. Signal MC events
are generated using EvtGen [11] and propagated through a detector simulation
based on Geant3 [12]. Final-state radiation from charged particles is
simulated using Photos [13]. Four-body decays are generated to decay uniformly
in phase space without intermediate resonances. An MC sample of generic
$e^{+}e^{-}$ collisions corresponding to the same integrated luminosity as the
data sample is used to develop selection criteria.

##  III Event selection

To ensure that tracks are well reconstructed, each final-state charged
particle is required to have at least two SVD hits in each of the longitudinal
and azimuthal measuring coordinates. Charged particles are identified by
calculating likelihoods $\mathcal{L}_{i}$ for specific particle hypotheses,
where $i=\pi,\,K,\,p,\,\mu,\,e$. These likelihoods are based on information
from various detectors: photon yield in the ACC, $dE/dx$ information from the
CDC, time-of-flight information from the TOF, energy in the ECL, and hits in
the KLM [14, 15, 16]. Tracks with
$\mathcal{L}_{K}/(\mathcal{L}_{K}+\mathcal{L}_{\pi})>0.6$ are identified as
kaon candidates; otherwise, tracks are considered pion candidates. Kaon
candidates must also satisfy
$\mathcal{L}_{p}/(\mathcal{L}_{p}+\mathcal{L}_{K})<0.95$. Tracks that satisfy
$\mathcal{L}_{e}/(\mathcal{L}_{e}+\mathcal{L}_{\rm hadron})>0.95$ or
$\mathcal{L}_{\mu}/(\mathcal{L}_{\mu}+\mathcal{L}_{\pi}+\mathcal{L}_{K})>0.95$
are rejected, where $\mathcal{L}_{e}$, $\mathcal{L}_{\rm hadron}$, and
$\mathcal{L}_{\mu}$ are determined mainly using information from the ECL and
KLM detectors [15, 16]. These requirements have an efficiency of about $90\%$
for kaons and $95\%$ for pions.

Photon candidates are identified from energy clusters in the ECL that are not
associated with any charged track. The photon energy is required to be greater
than 50 MeV in the barrel region (covering the polar angle
$32^{\circ}<\theta<129^{\circ}$), and greater than 100 MeV in the endcap
region (covering $12^{\circ}<\theta<31^{\circ}$ or
$132^{\circ}<\theta<157^{\circ}$). The ratio of the energy deposited in the
3$\times$3 array of crystals centered on the crystal with the highest energy,
to the energy deposited in the corresponding 5$\times$5 array of crystals, is
required to be greater than 0.80. Candidate
${\pi^{0}\\!\rightarrow\\!\gamma\gamma}$ decays are reconstructed from photon
pairs having an invariant mass satisfying $115~{}{\rm
MeV}/c^{2}<M(\gamma\gamma)<150~{}{\rm MeV}/c^{2}$; this region corresponds to
about $3\sigma$ in $M(\gamma\gamma)$ resolution.

A $D^{+}_{(s)}$ candidate is reconstructed by combining $K^{-}K^{+}\pi^{+}$ or
$K^{\pm}\pi^{\mp}\pi^{+}$ track combinations with a $\pi^{0}$ candidate. A
vertex fit is performed for the three charged tracks and its fit quality is
defined as $\chi^{2}_{\rm vtx}$. The coordinates of the fitted vertex are
assigned as the $D^{+}_{(s)}$ decay vertex position.

Final selection criteria are determined by maximizing a figure-of-merit (${\rm
FOM}$), which is defined as either $S/\sqrt{S+B}$ for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ or $S/\sqrt{B}$ for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, where $S$ and $B$
are the numbers of signal and background events, respectively, expected in a
region $-30~{}{\rm MeV}/c^{2}<M(D)-m_{D}<20~{}{\rm MeV}/c^{2}$. In this
expression, $M(D)$ is the invariant mass of a reconstructed $D^{+}$ or
$D^{+}_{s}$ candidate, and $m_{D}$ is the known $D^{+}$ or $D^{+}_{s}$ mass
[7]. This region corresponds to about $2.5\sigma$ in the $M(D)$ resolution.
The FOM for ${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ is
different because the branching fraction for this mode has not yet been
measured.

Pairs of $\gamma$ candidates are subjected to a fit in which the $\gamma$’s
are constrained to originate from the $D^{+}_{(s)}$ decay vertex, and their
invariant mass is constrained to the nominal $\pi^{0}$ mass [7]. The resulting
fit quality ($\chi^{2}_{\pi^{0}}$) is required to satisfy
$\chi^{2}_{\pi^{0}}<8$. To improve the momentum resolution of the $\pi^{0}$,
the $\gamma$ energies are updated from this fit; the resulting $\pi^{0}$
momentum is required to be greater than 0.40 GeV/$c$. We veto
$D^{+}_{(s)}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}$ candidates satisfying
$|M(\pi^{+}\pi^{-})-m_{K^{0}_{S}}|<10~{}{\rm MeV}/c^{2}$, where
$m_{K^{0}_{S}}$ is the nominal $K^{0}_{S}$ mass [7], to suppress peaking
backgrounds such as $D^{+}_{(s)}\\!\rightarrow\\!K^{+}K^{0}_{S}\pi^{0}$. This
region corresponds to about $3\sigma$ in mass resolution.

The $D^{+}_{(s)}$ production vertex is determined by fitting the $D^{+}_{(s)}$
trajectory to the $e^{+}e^{-}$ interaction point (IP), which is determined
from the beam profiles. This vertex fit quality is defined as $\chi^{2}_{\rm
IP}$. The sum of vertex fit qualities $\chi^{2}_{\rm vtx}+\chi^{2}_{\rm IP}$
is required to be less than 14 for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ decays, and less than 10
for the other signal modes. This requirement has a signal efficiency of
80%–82% while rejecting 60%–80% of background.

The dominant source of background is random combinations of particles produced
in $e^{+}e^{-}\\!\rightarrow\\!c\bar{c}$ events or in $B$ decays. To suppress
this background, the momentum of the $D^{+}$ or $D^{+}_{s}$ candidate in the
$e^{+}e^{-}$ center-of-mass frame is required to be greater than 2.5 GeV/$c$
or 2.9 GeV/$c$, respectively. To further suppress backgrounds, we calculate
the significance of the $D^{+}_{(s)}$ decay length $L/\sigma_{L}$, where $L$
is the projection of the vector running from the production vertex to the
$D^{+}_{(s)}$ decay vertex onto the momentum direction. The corresponding
uncertainty $\sigma_{L}$ is calculated by propagating uncertainties in the
vertices and the $D^{+}_{(s)}$ momentum, including their correlations. We
subsequently require $L/\sigma_{L}\\!>\\!4.0$ for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, $L/\sigma_{L}\\!>\\!9.0$
for ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and
$L/\sigma_{L}\\!>\\!2.5$ for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$. The resulting signal
efficiencies are 58%–77%, while more than 93%–99.8% of background is rejected.

The CF normalization modes
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ are selected with the
same criteria as those used to select the signal modes, to minimize systematic
uncertainties. For both signal and normalization modes, we retain events that
satisfy $-70~{}{\rm MeV}/c^{2}<M(D)-m_{D}<60~{}{\rm MeV}/c^{2}$.

After applying all selection criteria, about 10% of events for $D^{+}$ decay
modes and 15% of events for $D_{s}^{+}$ decay modes have multiple signal
candidates. For these events, the average multiplicity is about 2.2 candidates
for each channel. We select a single candidate by choosing the one with the
smallest value of the sum $\chi^{2}_{\pi^{0}}+\chi_{\rm vtx}^{2}+\chi_{\rm
IP}^{2}$. Based on MC simulation, this criterion selects the correct signal
candidate 68% of the time.

There are backgrounds from $D^{*+}$ decays in which the final state particles
are the same as those for the signal or normalization modes. These are as
follows:

  * •

for ${D^{+}\\!\rightarrow\\!K^{-}h^{+}\pi^{+}\pi^{0}}$ decays, where
$h^{+}\\!=\\!K^{+}$ or $\pi^{+}$, there is background from
${D^{*+}\\!\rightarrow\\!D^{0}\pi^{+}}$,
${D^{0}\\!\rightarrow\\!K^{-}h^{+}\pi^{0}(\pi^{0})}$.To reject this
background, we require
$M({K^{-}h^{+}\pi^{+}\pi^{0}})-M({K^{-}h^{+}\pi^{0}})-m_{\pi^{+}}\\!>\\!20~{}{\rm
MeV}/c^{2}$.

  * •

for ${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, there is
background from ${D^{*+}\\!\rightarrow\\!D^{0}\pi^{+}}$,
${D^{0}\\!\rightarrow\\!K^{-}K^{+}\pi^{0}}$, and from
${D^{*+}\\!\rightarrow\\!D^{+}\pi^{0}}$,
${D^{+}\\!\rightarrow\\!K^{-}K^{+}\pi^{+}}$. To reject these, we require
$M({K^{-}K^{+}\pi^{+}\pi^{0}})-M({K^{-}K^{+}\pi^{+}})-m_{\pi^{0}}\\!>\\!10~{}{\rm
MeV}/c^{2}$, and also
$M(K^{-}K^{+}\pi^{+}\pi^{0})-M(K^{-}K^{+}\pi^{0})-m_{\pi^{+}}\\!>\\!10~{}{\rm
MeV}/c^{2}$.

  * •

for ${D^{+}_{(s)}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, there is
background from ${D^{*-}\\!\rightarrow\\!\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}}$, ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{0}}$,
with the $\pi^{-}$ replaced by a random $\pi^{+}$. To suppress this
background, we require
$M(K^{+}\pi^{-}\pi^{+}\pi^{0})-M(K^{+}\pi^{+}\pi^{0})-m_{\pi^{-}}\\!>\\!40~{}{\rm
MeV}/c^{2}$.

These requirements reject only 1%–3% of signal decays but reduce $D^{*+}$
backgrounds to a negligible level.

##  IV Yield extraction

We determine signal yields by performing an extended unbinned maximum-
likelihood fit to the $M(D)$ distributions. The probability density function
(PDF) describing signal decays is taken to be the sum of a Crystal Ball
function [17] and three asymmetric Gaussians (AG), which are Gaussian
functions with different widths on the left- and right-hand sides of the peak
position. This position is denoted by the parameter $\mu$, and all four
functions are required to have a common value of $\mu$. An additional term
($\mathcal{P}_{\rm FSR}$) is included to describe signal decays with final-
state radiation (FSR). For this term, the sum of a CB function and a Gaussian
is used; the parameters of $\mathcal{P}_{\rm FSR}$ and its ratio to the total
signal yield ($f_{\rm FSR}$) are fixed to MC values. The overall PDF is

$\displaystyle\mathcal{P}_{\rm sig}$ $\displaystyle=$ $\displaystyle(1-f_{\rm
FSR})\bigg{[}f_{3}\Big{[}f_{2}\big{[}f_{1}\cdot{\rm
AG}(\mu,\sigma_{1},\delta_{1})$ (1) $\displaystyle\hskip 79.49744pt+\
(1-f_{1})\cdot{\rm AG}(\mu,\sigma_{2},\delta_{2})\big{]}$ $\displaystyle\hskip
65.04256pt+\ (1-f_{2})\cdot{\rm AG}(\mu,\sigma_{3},\delta_{3})\Big{]}$
$\displaystyle\hskip 65.04256pt+\ (1-f_{3})\cdot{\rm
CB}(\mu,\sigma_{4},\alpha_{\rm cb},n_{\rm cb})\bigg{]}$ $\displaystyle\hskip
65.04256pt+\ f_{\rm FSR}\cdot\mathcal{P}_{\rm FSR}\,,$

where $\sigma_{i+1}\\!=\\!r_{i}\sigma_{i}$ ($i\\!=\\!1,\,2,\,3$) with a
scaling factor $r_{i}$, and the left-side ($L$) and right-side ($R$) widths of
the asymmetric Gaussians are specified by the parameter $\delta_{i}$:
$\sigma_{i}^{L,R}=\sigma_{i}(1\pm\delta_{i})$. The parameters $\mu$ and
$\sigma_{1}$ are free to vary, which allows for a difference in resolution
between data and MC simulation; all other parameters are fixed to MC values.
The mode ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ is fitted
simultaneously with the normalization mode
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$, as both modes share
signal shape parameters.

The background shapes are described by second-order Chebyshev polynomials for
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and a third-order
Chebyshev polynomial for ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$.
All parameters of these shapes are free to vary.

Projections of the $M(D)$ fits are shown in Fig. 1 for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and its normalization mode
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$; in Fig. 2 for
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ and its normalization
mode ${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$; and in Fig. 3 for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ and its normalization
mode ${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. Also plotted are
the pulls, defined as $(N_{\rm data}-N_{\rm fit})/\sigma$, where $\sigma$ is
the uncertainty on $N_{\rm data}$. The pull distributions show that the fits
describe the data satisfactorily. The signal and background yields ($N_{\rm
sig}$ and $N_{\rm bkg}$) obtained from the fits for the signal region, $\pm
20$ MeV/$c^{2}$ around the nominal $D^{+}_{(s)}$ mass, are listed in Table 1.

The statistical significance of a signal yield is evaluated as the difference
in the log likelihoods obtained from fits performed with and without a signal
PDF. For ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, we obtain
$\Delta\ln\mathcal{L}=177$; as the number of degrees of freedom for the fit
without a signal component is one less than that with a signal component, this
value corresponds to a statistical significance of greater than $10\sigma$.
For ${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, we obtain
$\Delta\ln\mathcal{L}=594$. In this case, the number of degrees of freedom
without a signal component is three less than that with a signal component
(parameters $N_{\rm sig}$, $\mu$, and $\sigma_{1}$ are dropped), and this
value of $\Delta\ln\mathcal{L}$ corresponds to a statistical significance of
greater than $10\sigma$. This measurement constitutes the first observation of
this $D^{+}_{s}$ decay.

\begin{overpic}[width=195.12767pt]{MassDp_DpCSall_unblind_floatBkg_2MeV.pdf}\put(20.0,70.0){${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$}
\end{overpic}

\begin{overpic}[width=195.12767pt]{MassDp_DpCFall_normSCS_unblind_floatBkg.pdf}\put(20.0,70.0){${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$}
\end{overpic}

Figure 1:  Fit projections for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, and its normalization mode
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$. Data are plotted as
filled circles with error bars. The red solid, red dashed, and blue dashed
curves denote the overall fit result, the signal component, and the background
component, respectively.

\begin{overpic}[width=195.12767pt]{MassDp_simultaneous_DpDCS_unblind.pdf}\put(20.0,70.0){${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$}
\end{overpic}

\begin{overpic}[width=195.12767pt]{MassDp_simultaneous_DpCF_unblind.pdf}\put(20.0,70.0){${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$}
\end{overpic}

Figure 2:  Fit projections for
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and its normalization
mode ${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$. The signal PDFs are
the same for the two modes (see text). Data are plotted as filled circles with
error bars. The red solid, red dashed, and blue dashed curves denote the
overall fit result, the signal component, and the background component,
respectively.

\begin{overpic}[width=195.12767pt]{MassDs_DsCSall_unblind_floatBkg.pdf}
\put(20.0,70.0){${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$}
\end{overpic}

\begin{overpic}[width=195.12767pt]{MassDs_DsCFall_normSCS_unblind_floatBkg.pdf}
\put(20.0,70.0){${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$}
\end{overpic}

Figure 3:  Fit projections for ${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and its normalization mode ${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. Data are plotted as filled circles with error bars. The red solid, red dashed, and blue dashed curves denote the overall fit result, the signal component, and the background component, respectively. Table 1:  Fitted signal yields ($N_{\rm sig}$) and background yields ($N_{\rm bkg}$) in the region $\pm 20~{}{\rm MeV}/c^{2}$ around the nominal $D^{+}_{(s)}$ mass, and efficiency-corrected signal yields ($N_{\rm sig}^{\rm corr}$) for (1) ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$; (2) ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$; and (3) ${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$. The normalization modes are ${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ for (1) and (2), and ${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ for (3). The $N_{\rm sig}$ values are not directly used to calculate $N_{\rm sig}^{\rm corr}$, which is calculated using Eq. (2). The normalization mode ${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ is used for both (1) and (2) but selected with different criteria in the two cases. Thus, the yields of this mode for (1) and (2) differ. Also listed are ratios of branching fractions (see text). The event yields are listed with their statistical uncertainty; the branching fraction ratios are listed with both statistical and systematic uncertainties. The right-most column lists the current world average (WA) [4, 7, 5, 6]. Decay mode | $N_{\rm sig}$ | $N_{\rm bkg}$ | $N_{\rm sig}^{\rm corr}(\times 10^{6})$ | Branching fraction ratio | Current WA
---|---|---|---|---|---
(1) | ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ | $49\,798\pm 564$ | $50\,463\pm 214$ | $2.352\pm 0.025$ | $(11.32\pm 0.13\pm 0.26)\%$ | $(10.6\pm 0.6)\%$
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ | $602\,463\pm 1\,614$ | $197\,151\pm 570$ | $20.77\pm 0.07$
(2) | ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ | $3\,631\pm 198$ | $18\,879\pm 101$ | $0.303\pm 0.020$ | $(1.68\pm 0.11\pm 0.03)\%$ | $(1.89\pm 0.12)\%$
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$ | $208\,118\pm 707$ | $22\,327\pm 212$ | $18.06\pm 0.10$
(3) | ${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ | $26\,150\pm 1\,442$ | $277\,160\pm 582$ | $1.464\pm 0.052$ | $(17.13\pm 0.62\pm 0.51)\%$ | …
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ | $110\,261\pm 735$ | $71\,425\pm 263$ | $8.547\pm 0.059$

##  V Branching fractions

To determine the branching fractions, we divide the signal yields by their
respective reconstruction efficiencies. However, the reconstruction efficiency
for a decay can vary across the four-body phase space, and the distribution in
phase space of these decays is unknown. To reduce systematic uncertainty
arising from the unknown decay distribution (which often contains intermediate
resonances), we correct the signal yields for reconstruction efficiencies in
bins of phase space as follows.

For a $D^{+}_{(s)}$ decay to four pseudoscalar particles in the final state,
the phase space is five-dimensional (5D). We thus correct the data for
acceptance and reconstruction efficiency in bins of 5D phase space, where the
bins are taken to be the invariant masses squared of five pairs of final-state
particles 111This parameterization assumes that the detector acceptance and
reconstruction efficiency are independent of the signs of $K$ and $\pi$
momenta in the $D$ rest frame (i.e., symmetric under a parity transformation).
We have checked that any such dependence on the signs of momenta is
negligible.. These are calculated from fits subject to the mass constraint
$M(D)\\!=\\!m_{D^{+}}$ or $m_{D^{+}_{s}}$. The reconstruction efficiency is
determined. The efficiency-corrected signal yield is calculated as

$\displaystyle N_{\rm sig}^{\rm corr}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{N_{i}^{\rm data}-N_{\rm bkg}\cdot f_{i}^{\rm
bkg}}{\varepsilon_{i}}\,,$ (2)

where $N_{i}^{\rm data}$, $f_{i}^{\rm bkg}$, and $\varepsilon_{i}$ are the
number of data events, the fraction of background events, and the
reconstruction efficiency for bin $i$. The summation runs over all bins. The
uncertainties on each term in Eq. (2), for each bin $i$, are propagated to
obtain the overall uncertainty on $N^{\rm corr}_{\rm sig}$. The bin sizes are
chosen to minimize efficiency variations within the bins. There are 576 bins
for ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ (i.e.,
4$\times$4$\times$3$\times$4$\times$3); 243 bins for
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$; 768 bins for
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$; 432 bins for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$; and 576 bins for
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. Invariant mass squared
distributions for different combinations of final-state particles (i.e.,
projections of the five-dimensional distribution) are shown in Fig. 4 for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, Fig. 5 for
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and Fig. 6 for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$.

The reconstruction efficiencies $\varepsilon_{i}$ are determined from a large
sample of MC events. These efficiencies include a correction for particle
identification, to account for small differences observed between data and MC
simulation. The correction, typically 0.93–1.03, is determined from a sample
of $D^{*+}\\!\rightarrow\\!D^{0}\pi^{+},\,D^{0}\\!\rightarrow\\!K^{-}\pi^{+}$
decays. The fraction of background events in the $i$th bin ($f_{i}^{\rm bkg}$)
is obtained from the 5D distribution of events in the $M(D)$ sidebands
$-70~{}{\rm MeV}/c^{2}<M(D)-m_{D}<-50~{}{\rm MeV}/c^{2}$ and $40~{}{\rm
MeV}/c^{2}<M(D)-m_{D}<60~{}{\rm MeV}/c^{2}$. An MC study shows that background
in the signal region is well-described by background in the sidebands. The
background fractions must satisfy the constraint $\sum_{i}f^{\rm bkg}_{i}=1$.
The efficiency-corrected signal yields obtained using Eq. (2) are listed in
Table 1.

\begin{overpic}[width=429.28616pt]{Eff_Dalitz_DpCSaddDlzMC_bkgFromSB}\end{overpic}
Figure 4: Invariant-mass-squared distributions for different combinations of
final-state particles (i.e., projections of the five-dimensional distribution)
for ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ decays. Events in the
$M(D)$ signal region are plotted as filled black circles; events in the $M(D)$
sideband are plotted as blue dashed histograms; and signal efficiencies are
plotted as red hollow circles. Black vertical lines denote the boundaries of
the bins used for the efficiency correction.
\begin{overpic}[width=429.28616pt]{Eff_Dalitz_DpDCS_fulldata_bkgFromSB}\end{overpic}
Figure 5:  Invariant-mass-squared distributions for different combinations of
final-state particles for ${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$
decays. Events in the $M(D)$ signal region are plotted as filled black
circles; events in the $M(D)$ sideband are plotted as blue dashed histograms;
and signal efficiencies are plotted as red hollow circles. Black vertical
lines denote the boundaries of the bins used for the efficiency correction.
\begin{overpic}[width=429.28616pt]{Eff_Dalitz_DsCS_fulldata_bkgFromSB}\end{overpic}
Figure 6:  Invariant-mass-squared distributions for different combinations of
final-state particles for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ decays. Events in the
$M(D)$ signal region are plotted as filled black circles; events in the $M(D)$
sideband are plotted as blue dashed histograms; and signal efficiencies are
plotted as red hollow circles. Black vertical lines denote the boundaries of
the bins used for the efficiency correction.

The ratio of the efficiency-corrected yield for a CS mode to that for a CF
mode is equal to the ratio of branching fractions: $N_{\rm sig}^{\rm corr}(\rm
CS)/N_{\rm sig}^{\rm corr}(\rm CF)=\mathcal{B}(\rm CS)/\mathcal{B}(\rm CF)$.
Inserting values from Table 1, we obtain

$\displaystyle\frac{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(11.32\pm 0.13)\%$ (3)
$\displaystyle\frac{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(1.68\pm 0.11)\%$ (4)
$\displaystyle\frac{\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(17.13\pm 0.62)\%\,,$ (5)

where the uncertainties listed are statistical.

##  VI Systematic uncertainties

Because we measure the ratio of branching fractions for decays with similar
final states, most systematic uncertainties cancel. The remaining
uncertainties are listed in Table 2 and evaluated as follows.

Table 2:  Fractional systematic uncertainties (in %) for the following ratios
of branching fractions:
(a)
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})$;
(b)
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})$;
and
(c) $\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})$. Sources |  (a) |  (b) |  (c)
---|---|---|---
PID efficiency correction |  $1.7$ |  … |  $1.7$
Multiple-candidate selection |  $1.1$ |  $1.3$ |  $1.2$
Signal parameterization |  $0.5$ |  $0.5$ |  $1.1$
$M(D)$ resolution |  $0.5$ |  … |  $1.4$
Binning |  $0.6$ |  $0.5$ |  $0.7$
Background $M^{2}(p_{i}p_{j})$ distribution |  $0.1$ |  $0.1$ |  $0.3$
Efficiency correction bias |  $0.6$ |  $0.8$ |  $1.1$
Total uncertainty |  $2.3$ |  $1.9$ |  $3.0$

As a correction accounting for the difference in particle identification (PID)
efficiencies between data and MC is included in Eq. (2), we evaluate the
uncertainty on this correction. Signal and normalization modes differ by the
flavor of at most one track in the final state. We evaluate the uncertainty
introduced by this difference using a sample of
${D^{*+}\\!\rightarrow\\!D^{0}\pi^{+}}$,
${D^{0}\\!\rightarrow\\!K^{-}\pi^{+}}$ decays. The resulting uncertainties are
0.9% for ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and 0.8% for
${D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}}$; 0.8% for
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ and 0.9% for
${D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$. We thus assign 1.7% as
the systematic uncertainty for the ratios of their branching fractions.

We consider uncertainty arising from the multiple-candidate selection
procedure by keeping all candidates without best candidate selection. We refit
the $M(D)$ distributions, redetermine the signal efficiency curves, and obtain
the corrected yields with Eq. (2). The resulting changes in the branching
fractions are assigned as systematic uncertainties.

The uncertainty due to PDF parameters that are fixed in the fit for a signal
yield is evaluated by sampling these parameters from a multivariate Gaussian
distribution that accounts for their uncertainties and correlations, and re-
fitting for the signal yield. The procedure is repeated 1000 times, and the
root-mean-square of the distribution of fitted yields is taken as the
uncertainty due to the fixed parameters.

There are several sources of uncertainty in the efficiency correction
procedure. We first consider effects due to the $M(D)$ resolution. In Eq. (2),
events satisfying $|M(D)-m_{D}|<20~{}{\rm MeV}/c^{2}$ were used; however, data
and MC could have different mass resolutions, and this would bias the
efficiency-corrected signal yield. We evaluate the efficiency of the signal
region requirement ($\varepsilon_{\rm SR}$) by integrating the signal PDFs
over this region. The ratio of the efficiency for a signal mode to that of a
normalization mode is determined, both for data and MC. The ratio of these
ratios $[\varepsilon_{\rm SR}^{\rm norm}/\varepsilon_{\rm SR}^{\rm sig}\,({\rm
data})]/[\varepsilon_{\rm SR}^{\rm norm}/\varepsilon_{\rm SR}^{\rm sig}({\rm
MC})]$ is calculated, and the difference from unity is taken as the systematic
uncertainty due to the $M(D)$ resolution. These uncertainties are $0.5\%$ for
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})$
and $1.4\%$ for
$\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})$.
The uncertainty for
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})/\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})$
is negligible, as the final states have the same particles.

We also consider uncertainty due to binning. We repeat the efficiency
correction with a different number of bins, e.g.,
$4\\!\times\\!4\\!\times\\!4\\!\times\\!4\\!\times\\!4\\!=\\!1024$ bins for
${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$, and take the fractional
change in the ratio of branching fractions as a systematic uncertainty.

We evaluate uncertainties arising from the 5D distribution of background
events by applying a correction to the background distribution. This
correction, obtained from MC, is the ratio of the background distribution for
events having $M(D)$ in the signal region to that for events having $M(D)$ in
the sideband. After applying this correction, the signal yield in each
efficiency bin is recalculated. The fractional change in the overall
efficiency-corrected signal yield is assigned as a systematic uncertainty.

We check for bias in the efficiency correction due to possible intermediate
resonances in the $D^{+}$ or $D^{+}_{s}$ decay, also using MC simulation. The
results for the efficiency-corrected signal yields are all consistent with
input values; the small differences observed are conservatively assigned as
systematic uncertainties.

The total systematic uncertainty is obtained by summing all individual
contributions in quadrature. The results are listed in Table 2.

##  VII Conclusion

In summary, using 980 fb-1 of data collected with the Belle detector, we
observe the SCS decays ${D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}}$ and
${D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$, and the DCS decay
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$. The statistical
significance of each mode is greater than $10\sigma$. The branching fractions
for these decays relative to the branching fractions for topologically similar
CF decays are measured to be

$\displaystyle\frac{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(11.32\pm 0.13\pm 0.26)\%$
$\displaystyle\frac{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(1.68\pm 0.11\pm 0.03)\%$
$\displaystyle\frac{\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})}{\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})}$
$\displaystyle=$ $\displaystyle(17.13\pm 0.62\pm 0.51)\%\,,$

where the uncertainties are statistical and systematic, respectively. Taking
$\sin\theta_{C}=0.2257$ [7], the second result above corresponds to $(5.83\pm
0.42)\times\tan^{4}\theta_{C}$. This value is significantly larger than other
measured ratios of DCS to CF branching fractions, but it is consistent within
$1.2\sigma$ with the large rate of
${D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}}$ measured by BESIII [5,
6].

Inserting world average values for the branching fractions of the
normalization modes
$\mathcal{B}({D^{+}\\!\rightarrow\\!K^{-}\pi^{+}\pi^{+}\pi^{0}})\\!=\\!(6.25\pm
0.18)\%$ [7] and
$\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})\\!=\\!(5.51\pm
0.19)\%$ [7, 18], we obtain

$\displaystyle\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}K^{-}\pi^{+}\pi^{0}})$
$\displaystyle=$ $\displaystyle\hskip-50.58878pt(7.08\pm 0.08\pm 0.16\pm
0.20)\times 10^{-3}$
$\displaystyle\mathcal{B}({D^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})$
$\displaystyle=$ $\displaystyle\hskip-50.58878pt(1.05\pm 0.07\pm 0.02\pm
0.03)\times 10^{-3}$
$\displaystyle\mathcal{B}({D_{s}^{+}\\!\rightarrow\\!K^{+}\pi^{-}\pi^{+}\pi^{0}})$
$\displaystyle=$ $\displaystyle\hskip-50.58878pt(9.44\pm 0.34\pm 0.28\pm
0.32)\times 10^{-3}\,,$

where the uncertainties are statistical, systematic, and from uncertainty in
the branching fractions of the normalization modes, respectively. The first
two results are consistent with recent BESIII results [4, 5, 6] but have
greater precision. The last result is the first measurement of this Cabibbo-
suppressed decay.

###### Acknowledgements.

This work, based on data collected using the Belle detector, which was
operated until June 2010, was supported by the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the
Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of
Nagoya University; the Australian Research Council including grants
DP180102629, DP170102389, DP170102204, DE220100462, DP150103061, FT130100303;
Austrian Federal Ministry of Education, Science and Research (FWF) and FWF
Austrian Science Fund No. P 31361-N36; the National Natural Science Foundation
of China under Contracts No. 11675166, No. 11705209; No. 11975076; No.
12135005; No. 12175041; No. 12161141008; Key Research Program of Frontier
Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the
Ministry of Education, Youth and Sports of the Czech Republic under Contract
No. LTT17020; the Czech Science Foundation Grant No. 22-18469S; Horizon 2020
ERC Advanced Grant No. 884719 and ERC Starting Grant No. 947006 “InterLeptons”
(European Union); the Carl Zeiss Foundation, the Deutsche
Forschungsgemeinschaft, the Excellence Cluster Universe, and the
VolkswagenStiftung; the Department of Atomic Energy (Project Identification
No. RTI 4002) and the Department of Science and Technology of India; the
Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation
(NRF) of Korea Grant Nos. 2016R1D1A1B02012900, 2018R1A2B3003643,
2018R1A6A1A06024970, RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957,
2021R1F1A1060423, 2021R1F1A1064008, 2022R1A2C1003993; Radiation Science
Research Institute, Foreign Large-size Research Facility Application
Supporting project, the Global Science Experimental Data Hub Center of the
Korea Institute of Science and Technology Information and KREONET/GLORIAD; the
Polish Ministry of Science and Higher Education and the National Science
Center; the Ministry of Science and Higher Education of the Russian
Federation, Agreement 14.W03.31.0026, and the HSE University Basic Research
Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438,
and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos.
J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the
Swiss National Science Foundation; the Ministry of Education and the Ministry
of Science and Technology of Taiwan; and the United States Department of
Energy and the National Science Foundation. These acknowledgements are not to
be interpreted as an endorsement of any statement made by any of our
institutes, funding agencies, governments, or their representatives. We thank
the KEKB group for the excellent operation of the accelerator; the KEK
cryogenics group for the efficient operation of the solenoid; and the KEK
computer group and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 6 (SINET6) for valuable network support.

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$r_{\hbox{\tiny ISCO}}=6m\left\\{1+\frac{na^{2}}{8m^{2}}-{n(49n-22)a^{4}\over
3456m^{4}}+\mathcal{O}\left({a^{6}\over m^{6}}\right)\right\\},$ (9.12)

for the photon sphere and ISCO, respectively.

The surface gravity was calculated to be approximately given by

$\kappa_{H}={1\over 4m}\left\\{1-{n(n-1)a^{4}\over
32m^{4}}+\mathcal{O}\left(\frac{a^{6}}{m^{6}}\right)\right\\},$ (9.13)

and the Hawking temperature is simply $k_{B}T_{H}={1\over
2\pi}\,\hbar\,\kappa_{H}$.

The classical energy conditions were analysed, and it was found that the
regions in which these conditions are satisfied is dependent on the value of
the parameter $n$. It was should that for no value of $n$ (barring the trivial
$n=0$ Schwarzschild spacetime) was there satisfaction of any of the energy
conditions globally in the whole of the spacetime.

In addition to these results, the Regge–Wheeler potential was calculated for
spin 0 and spin 1 excitations.

Somewhat importantly, the spacetimes in this general class could be made
arbitrarily close to the Swarzschild spacetime, and so are indeed able to be
classified as “black hole mimickers”.

###  9.5 Overall summary and outlook

It was the aim of this thesis to show that the issue of black hole
singularities may be resolved in a tractable, classical manner by the concept
of black hole mimickers. Indeed, the results presented in this thesis are far
cleaner and simpler than proposed resolutions of singularities via modified
theories of gravity or quantum gravity candidate theories.

There is still a significant amount of open problems on the subject of black
hole mimickers. A small list of problems are listed below as possible avenues
of future research.

  * •

Construct a regular black hole spacetime model which violates the classical
energy conditions in a ‘minimal’ way.

  * •

Construct a regular black hole spacetime model which is charged and rotating,
but still allows for tractable analysis of key astrophysical observables
(photon spheres, ISCOs, etc.).

  * •

Analyse the possibility of regular black holes as explanation for some of the
currently unexplainable phenomena in the universe (mass-gap black holes,
etc.).

  * •

Analyse specific black hole mimicker spacetime models through the lens of
quantum field theory in curved spacetimes. Specifically, analyse the effect
that regularising black hole singularities has on quantum field theoretic
phenomena, such as Hawking radiation.

In addition to these problems, there are major un-resolved questions in
physics which relate directly to black hole mimickers and the study of black
hole singularities in general:

  * •

Do traversable wormholes really exist, or are they just mathematical
curiosities?

  * •

Will a consistent theory of quantum gravity regularise the singularities at
the core of black holes?

  * •

Is there any black hole remnant left over when a black hole Hawking evaporates
that could be detected astronomically?

Of course, there are many more open problems in this field of research, and
this discussion is in no way exhaustive. However, it is hoped that this thesis
has shown the power in the simplicity of black hole mimickers, and that future
research will continue to be directed towards their analysis. I would once
again like to thank everyone who has helped me in the completion of this
thesis.

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##  Appendix
List of publications

Publications in journals:

  * •

“Thin-shell traversable wormhole crafted from a regular black hole with
asymptotically Minkowski core,” T. Berry, A. Simpson, and M. Visser.
Phys. Rev. D 102, #6, 064054 (2020).
DOI:10.1103/PhysRevD.102.064054, [arXiv:2008.07046 [gr-qc]].

  * •

“Relativistic combination of non-collinear 3-velocities using quaternions,” T.
Berry and M. Visser. Universe 6, #12, 237 (2020).
DOI:10.3390/universe6120237, [arXiv:2002.10659 [gr-qc]].

  * •

“Photon spheres, ISCOs, and OSCOs: Astrophysical observables forregular black
holes with asymptotically Minkowski cores,”
T. Berry, A. Simpson, and M. Visser. Universe 7, #1, 2 (2020).
DOI:10.3390/universe7010002, [arXiv:2008.13308 [gr-qc]].

  * •

“Darboux diagonalization of the spatial 3-metric in Kerr spacetime,”
J. Baines, T. Berry, A. Simpson, and M. Visser.
Gen. Rel. Grav. 53, #1, 3 (2021). DOI:10.1007/s10714-020-02765-0,
[arXiv:2009.01397 [gr-qc]].

  * •

“Unit-lapse versions of the Kerr spacetime,”
J. Baines, T. Berry, A. Simpson, and M. Visser.
Class. Quant. Grav. 38, #5, 055001 (2021).
DOI:10.1088/1361-6382/abd071, [arXiv:2008.03817 [gr-qc]].

Articles under review:

  * •

“Painlevé–Gullstrand form of the Lense–Thirring spacetime,”
J. Baines, T. Berry, A. Simpson, and M. Visser.
[arXiv:2006.14258 [gr-qc]] (2020).

  * •

“Lorentz boosts and Wigner rotations: self-adjoint complexified
quaternions,” T. Berry and M. Visser.
[arXiv:2101.05971 [gr-qc]] (2021).

  * •

“General class of ‘quantum deformed’ regular black holes,”
T. Berry, A. Simpson, and M. Visser.
[arXiv:2102.02471 [gr-qc]] (2021).