# A Possible Optical Quasi-periodic Oscillation of 134 day in the Radio-loud
Narrow-line Seyfert 1 Galaxy TXS 1206+549 at $z=1.34$

Chongyang Ren  Department of Astronomy, School of Physics and Astronomy, Key
Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University,
Kunming 650091, People’s Republic of China<EMAIL_ADDRESS>Sisi Sun
Department of Astronomy, School of Physics and Astronomy, Key Laboratory of
Astroparticle Physics of Yunnan Province, Yunnan University, Kunming 650091,
People’s Republic of China<EMAIL_ADDRESS>Pengfei Zhang  Department
of Astronomy, School of Physics and Astronomy, Key Laboratory of Astroparticle
Physics of Yunnan Province, Yunnan University, Kunming 650091, People’s
Republic of China<EMAIL_ADDRESS>
###### Abstract

Here we report an optical quasi-periodic oscillation (QPO) with a period of
$\sim$134 day detected in g- and r-band light curves of a narrow-line Seyfert
1 galaxy TXS 1206+549 at redshift of 1.34 with the data from the Zwicky
Transient Facility (ZTF) observations. After considering the trial factor, the
significance levels in the two bands are 3.1 $\sigma$ and 2.6 $\sigma$,
respectively. The QPO signal presents about 10 cycles ranging from 2018 March
to 2021 December lasting $\sim$4 year. A near-sinusoidal profile also appears
in the folded light curves by a phase-resolved analysis. Interestingly, in the
simultaneous light curve as the time scale of ZTF observations, a potential
periodic signal with similar period is detected in the o-band light curve from
the Asteroid Terrestrial-Impact Last Alert System data, additionally a weak
peak is also detected at the similar period in the gamma-ray light curve
obtained from the _Fermi_ Gamma-ray Space Telescope data. Some potential
origins of periodicities in active galactic nuclei are discussed for the QPO
reported here.

Active galactic nuclei (16); Time series analysis (1916); Period search (1955)

##  1 Introduction

Active galactic nuclei (AGNs), hosting a supermassive black hole (SMBH) over
$10^{8}$ M⊙, are the most luminous sources of electromagnetic radiation in the
universe, the majority of physical processes are launched from their large-
scale powerful relativistic jets. AGN multi-wavelength electromagnetic
spectral energy distribution (SED) extends from MHz radio frequencies to TeV
gamma-ray energies (Ulrich et al., 1997). And the Narrow-line Seyfert 1
(NLSy1) galaxies are a unique subclass of AGNs (Osterbrock & Pogge, 1985),
they have some properties including of narrow width of the broad Balmer
emission lines and weak forbidden lines (Goodrich, 1989), strong permitted
optical/UV FeII lines (Shuder & Osterbrock, 1981), soft X-ray excess (Ojha et
al., 2020), near-Eddington accretion rates (Mineshige et al., 2000; Rakshit et
al., 2017), and smaller black hole (BH) masses. Their amplitudes of optical
light curves are usually lower comparing with other broad-line AGNs (Rakshit &
Stalin, 2017), and the variabilities may be resulted in the dynamical
processes close to BHs, which play an important role in revealing AGN
radiation mechanisms and structures.

Recently many interesting quasi-periodic oscillation (QPO) cases are detected
in the electromagnetic emissions of AGNs at multiwavelength from radio
frequencies to GeV $\gamma$-rays, the scale of periods of the QPO signals span
the timescale ranges from minutes to years (King et al., 2013; Bhatta, 2017;
Li et al., 2017; Bhatta, 2018; Ren et al., 2021; Zhang & Wang, 2021; Kidger et
al., 1992; Fan & Lin, 2000; Valtonen et al., 2006, 2008; Zhang et al., 2014;
Sandrinelli et al., 2014; Valtonen et al., 2023; Gierliński et al., 2008; Lin
et al., 2013; Pan et al., 2016; Zhang et al., 2017b, 2018; Ackermann et al.,
2015; Sandrinelli et al., 2016; Zhang et al., 2017c, d, a; Yan et al., 2018;
Zhou et al., 2018; Zhang et al., 2020; Yang et al., 2021; Zhang et al., 2021;
Gong et al., 2022; Chen et al., 2022; Gong et al., 2023, and references
therein). Several special cases of them are summarized here, for example,
Zhang & Wang (2021) claimed a discovery of a 176 day QPO of the NLSy1 galaxy
J0849+5108 in the 11 yr radio light curve obtained with the Owens Valley Radio
Observatory 40 m telescope at 15 GHz. Up to now, it is the longest duration
QPO signal found in AGNs, has been presented for 21 cycles. Recently, Zhang et
al. (2022) reported double yearly QPOs of a Seyfert type galaxy NGC 1275 (or
3C 84 in radio) in 1.3 mm wavelength light curve monitored by the
Submillimeter Array. Moreover, even in the optical polarized emission of the
blazar sources, the QPO signals are reported in blazars B2 1633+38 and PKS
1222+216 with periods of years (Otero-Santos et al., 2020; Zhang & Wang,
2022). Due to the current limited understanding of the variability
characteristics of AGNs, there is a considerable debate about the real
statistical significance of these QPO findings. A huge sample of the long-term
light curves for AGNs have been provided by the Zwicky Transient Facility
(ZTF) observations over four years with measurements made every few days
(Graham et al., 2019; Bellm et al., 2019), which provide us a great
convenience to study QPO signals in the long baseline time series. Indeed some
QPO cases have been detected in Zhang (2022); Banerjee et al. (2022) from the
ZTF survey.

TXS 1206+549 is a radio and $\gamma$-ray loud NLSy1 galaxy and has a flat
radio spectrum (Rakshit et al., 2021). As the blazars and other $\gamma$-ray
NLSy1 galaxies, their broad-band SED shows the typical two-bump structure.
They exhibit strong variability in the universal electromagnetic emissions in
the optical, infrared, and $\gamma$-ray bands. Up to now, TXS 1206+549 is the
most distant NLSy1 galaxy at $z=$1.344 having $\gamma$-ray emissions (Rakshit
et al., 2021), while other (only about a dozen) $\gamma$-ray NLSy1 galaxies
were detected at $z<$1\. Recently, we conducted a study on the phenomena of
$\gamma$-ray eruptions exhibiting higher flux and shorter duration in _Fermi_
-LAT observations, then we focused on the target TXS 1206+549. During the
multi-wavelength studies on this source, we accidentally discovered that its
optical light curves may exhibit periodic behavior. Subsequently, we carried
out a timing analysis for the $\sim$4 year optical light curve of TXS 1206+549
with the g- and r-band data from ZTF survey. And a QPO signal was detected at
both bands having a similar period of $\sim$134 day presenting about 10
cycles. After considering the trial factor, the significance levels in the two
bands are 3.1 $\sigma$ and 2.6 $\sigma$, respectively. Interestingly, in the
simultaneous light curve as the time scale of ZTF, a potential QPO signal with
a similar period was also found in the o-band light curve from the Asteroid
Terrestrial-impact Last Alert System (ATLAS; Tonry et al., 2018). TXS
1206+549’s $\gamma$-ray emissions have been detected by the Large Area
Telescope (LAT) onboard the _Fermi_ Gamma-ray Space Telescope (_Fermi_ -LAT;
Atwood et al., 2009) since 2008. And a weak signal was also detected in the
nearly simultaneous light curve as ZTF observations at a similar period in
0.1–500.0 GeV.

We describe the data analysis of the ZTF optical light curve, ATLAS o-band
light curve, _Fermi_ -LAT $\gamma$-ray data, and the main results in Section
2. Results for periodic analysis are shown in Section 3. Summary and
discussion are shown in Section 4. In the following text, numbers in
parentheses represent the corresponding uncertainties on the last digit.

##  2 Data Preparation and Reduction

###  2.1 Light curves from ZTF

Zwicky Transient Facility is a new 48 inch Samuel Oschin Telescope at Palomar
Observatory with a 47 square degree field of view camera in optical wavelength
(Graham et al., 2019; Bellm et al., 2019). It has scanned the entire Northern
sky every two days started in 2018, and released high quality data products
every two months. Such an observation cadence makes ZTF archival data very
suitable for time-domain science, especially for the exploring of periodic
variabilities in the timescales of sub-year-long ones. The ZTF released data
can be obtained from the website of the archive science
data111https://irsa.ipac.caltech.edu/Missions/ztf.html.

The source TXS 1206+549 (at the same coordinate as it in 4FGL) is one of ZTF
targets. It has been monitored more than 1300 day spanning from 2018 March 23
to to 2021 December 2 (MJD 58,200-59,550) in the g-, r-, and i-bands. We show
the optical light curves of g-, r-, and i-bands by ZTF in the panels A, B, and
C of Figure 1, respectively. The median and minimum time intervals of two
adjacent data points of g-band (r-band) are $\sim 0.927$ (0.829) and 0.001
(0.001) day, respectively, the maximum interval is 126.266 (126.262) day at
MJD 59,454 (marked with a pink star in Figure 1). For g-band, the brightest
and faintest epochs of target are at MJD 58,561.3 and 59,265.4 with the
magnitudes of 19.1 (indicated with a red arrow) and 20.8 (a green arrow),
respectively. The weighted magnitude was calculated and weighted by the errors
of data points with a formula of $\overline{M}=\sum{w_{i}*M_{i}}$, where
$w_{i}=\frac{1}{\sigma^{2}_{i}}\frac{1}{W}$,
$W=\sum{\frac{1}{\sigma^{2}_{i}}}$, and $\sigma_{i}$ indicating the errors.
And the weighted standard deviation ($\rm\sigma_{d}$) was calculated by
$\rm\sigma_{d}$=$\sum{w_{i}*(M_{i}-\overline{M})^{2}}$, its value is 34.4%.
For r-band, the corresponding epochs are MJD 58,306.2 and 58,227.3 at the
magnitudes of 18.5 and 20.3 shown with red and green arrows in B panel of
Figure 1. Its values of $\overline{M}$ and $\rm\sigma_{d}$ are 19.53 and
38.0%. We show that weighted results of g- and r-bands as horizontal gray
lines and yellow shaded regions, respectively.

Figure 1: Optical and $\gamma$-ray brightness measurements for TXS 1206+549.
Optical g-, r-, and i-band light curves monitored by the ZTF observations are
shown in panels A, B, and C, respectively. The orange dashed and blue dashed-
dotted curves (in panels A and B) stand for the sinusoidal models in g- and
r-bands, respectively, their parameters were given by the LSP code. O-band
light curves from the ATLAS observations is shown in panel D. The nearly
simultaneous 0.1–500 GeV light curves with time-bin of 30 day with
observations by the _Fermi_ –LAT shown in panel E.

###  2.2 Light curve from ATLAS

Asteroid Terrestrial-impact Last Alert System222https://fallingstar-
data.com/forcedphot/ consists of four 0.5 m telescopes with two of them in
Hawaii, one each in Chile and South Africa (Tonry et al., 2018), which
provides public access to photometric measurements over its full history
survey. TXS 1206+549 is one of the objects monitored by ATLAS. We obtained its
o-band light curve from the website of archive data333https://fallingstar-
data.com/forcedphot/queue/. ATLAS observations for TXS 1206+549 span from 2015
December 9 to 2023 March 28 (MJD 57,365.6-60,031.4), we show the o-band light
curve with a simultaneous observations of ZTF in D panel of Figure 1. For the
simultaneous o-band light curve, the median, minimum and maximum time
intervals of two adjacent data points are 1.6, 0.0005 and 146.3 day. Its
minimum, maximum, and median magnitudes are $\sim$15.3, 25.6 and 19.7
respectively. It should be noted that data quality of ATLAS light curve is
poorer than that obtained by ZTF (see D panel of Figure 1), and the ATLAS
light curve is used only as an independent check for that QPO detected by ZTF
observations.

###  2.3 $\gamma$-ray light curve from Fermi-LAT

The source TXS 1206$+$549 was cataloged in the fourth Fermi Large Area
Telescope catalog (4FGL-DR3, Fermi-LAT collaboration et al., 2022) and named
as $\gamma$-ray point source 4FGL J1208.9$+$5441 with a coordinate (R. A. =
$\rm 12^{h}08^{m}54^{s}.264$, decl. =
$+54^{\circ}41^{{}^{\prime}}58^{{}^{\prime\prime}}.208$). In the $\gamma$-ray
data analysis, the Pass 8 $Front+Back$ SOURCE class events (i.e., evclass=128
and evtype=3) were selected with the energy range of 0.1–500 GeV spanning from
2008 August 4 to 2022 June 22 centered at target’s position within a
$20^{\circ}\times 20^{\circ}$ region of interest (RoI). The events were
reduced by a selection of the zenith angle $\leq 90$ and an expression of
”DATA_QUAL$>$0&&LAT_CONFIG==1” to obtain those high-quality events in the good
time intervals. Then a binned maximum likelihood analysis was performed
between the whole data and a model file. The model contains the parameters of
the flux normalizations and spectral shapes for all known 4FGL sources in the
RoI. In 4FGL-DR3, the $\gamma$-ray emissions from TXS 1206$+$549 was described
by a log-parabolic spectrum model with a form of
$dN/dE=N_{0}(E/E_{b})^{-\alpha-\beta\log(E/E_{b})}$. We obtained an average
photon flux of $3.13(2)\times 10^{-8}$ photons cm-2 s-1 in the 0.1–500 GeV
energy range with a test statistic (TS) value of 2,810.18. Its best-fit
spectral parameters $\alpha$, $\beta$, and $E_{b}$ are 2.53(1), 0.07(1), and
0.42(1) GeV, respectively. Those values are in agreement with those reported
in 4FGL-DR3. All the corresponding best-fit parameters were saved as a new
model file, and the following light curve analysis was carried out based on
this model.

Then a monthly light curve with 0.1–500.0 GeV energy range was constructed by
performed an unbinned likelihood analysis during the corresponding
observations. In this analysis, we only freed the flux normalizations for the
sources within $5^{\circ}$ RoI, others were fixed at their best-fit values. In
order to catch the variations as complete as possible, the data points with TS
$\geqslant$ 4 are included in the following periodic analysis. We show flux
variations and their corresponding TS values for each time-bin as black
crosses and pink shaded regions in Figure 2, respectively. And the nearly
simultaneous light curve with ZTF observations is shown in E panel of Figure 1
to facilitate comparison of multi-wavelength variabilities. The average photon
flux over the whole _Fermi_ -LAT observations in 0.1–500.0 GeV is shown as a
blue dashed-dotted line.

Figure 2: TXS 1206+549 $\gamma$-ray light curve in 0.1–500 GeV with _Fermi_
–LAT during the whole observations, data points with TS $\geqslant$ 4 are
selected. The blue dashed-dotted line represents the average integrated photon
flux. The vertical dashed line indicates the starting epoch of ZTF
observations. The corresponding TS values for data points are shown with pink
bars.

##  3 Periodic analysis and Results

###  3.1 Periodicity Search

Two different mathematical techniques, the generalized Lomb-Scargle
periodogram (LSP; Lomb, 1976; Scargle, 1982; Zechmeister & Kürster, 2009) and
the weighted wavelet Z-transform (WWZ; Foster, 1996), were employed in the
periodic variability analysis for optical and $\gamma$-ray light curves. These
independent techniques were used to check each other results. In the optical
g- and r-bands data from ZTF observations, we found a periodic signal with a
sharp peak at the similar period in their power spectra.

Figure 3: TXS 1206+549 power spectra by constructed with the techniques of LSP
and WWZ for the optical light curves of g- and r-bands covering the time range
of MJD $\sim$58200–59550. The LSP, time-averaged WWZ, and WWZ power spectra
are shown with red histogram, black curve, and color-scaled WWZ power,
respectively. The spectra of g- and r-bands are shown in the left and right
panels respectively. A obvious power peak is shown in the LSP and WWZ power
spectra with a similar frequency corresponding to $\sim$134 day. The green
dashed curves stand for 4$\sigma$ significance level derivd from the
artificial light curves.

For the g-band light curve, we constructed its LSP power spectrum and show it
in left panel of Figure 3 with red histogram in log-log units. From it, a
significant QPO signal presents in the LSP power spectrum. The periodical
variations can be described by a sinusoidal function with a form of
$A\sin([2\pi(t-t_{0})/P])+A_{0}$. Their parameters and uncertainties were
derived by the LSP code, and their values are 0.300$\pm$0.016, 132.8$\pm$0.7
day, 58197.5$\pm$1.2, and 19.99$\pm$0.01 for $A$, $P$, $t_{0}$ and $A_{0}$,
respectively. We show the sine in A panel of Figure 1 with an orange dashed
sine. From it, about ten cycles of periodic modulation appears in the light
curve.

Table 1: PSD best-fit parameters

| $K$ | $\alpha$ | $\beta$ | $f_{\rm bend}$ | $c$
---|---|---|---|---|---
| $\times 10^{-5}$ |  |  |  $\times 10^{-2}$ day-1 | $\times 10^{-6}$
g-band | 1.88(41) | 1.00(17) | 2.47(22) | 4.63(147) | 1.02(40)
r-band | 2.72(93) | 1.00(14) | 2.39(33) | 2.15(107) | 9.45(354)

Notes. Parameters of the underlying best-fit PSD obtained by the latest
implementation (https://github.com/lena-lin/emmanoulopoulos).

A periodic signal or statistical fluctuations in a time-series may lead to a
power peak in a power spectrum. In order to determine the significance level
for periodicity signal, we employed a program for simulating light curve
basing the power spectral density (PSD) and probability density function (PDF)
of variations derived from the original light curve, as that provided by
Emmanoulopoulos et al. (2013). In general, the power spectra of AGN light
curves are typically described using a power-law (PL) or a smoothly bending
power law plus a constant (SPL) models (González-Martín & Vaughan, 2012). In
our analysis, the SPL model yielded the better maximum likelihood fitting
results than the PL one. So, in the subsequent simulated light curve, we
adopted the SPL model as the underlying best-fit PSD. We used the SPL,
$PSD(f)=Kf^{-\alpha}[1+(f/f_{\rm bend})^{\beta-\alpha}]^{-1}+c$, to estimate
the underlying PSD (González-Martín & Vaughan, 2012). Then the maximum
likelihood fitting was employed to obtain their best-fit parameters for the
PSD (Barret & Vaughan, 2012). In the fitting, we make the assumption that the
components of the periodogram asymptotically follow a $\chi^{2}$ distribution
around the underlying PSD model. Employing maximum likelihood estimation and
minimizing the negative logarithmic likelihood function, we obtained the best-
fit parameters. We present the best-fit parameters along with their associated
relative uncertainties in Table 1. Based on them, the program generated the
artificial light curves. The artificial light curves have similar PSD and PDF
comparing with them derived from the original light curve. Using the
simulation code, we generated $10^{6}$ artificial light curves for g-band
data. The 4$\sigma$ significance curve built based on these light curves is
shown in left panel of Figure 3 with a green dashed line. And the significance
level of the QPO signal in the g-band is $\sim 4.4\sigma$ (with $p$-value of
$1\times 10^{-5}$).

The same process was carried out for r-band data, and the results are shown in
right panel of Figure 3. A power peak presents in its power spectra with a
similar period as that detected in g-band. The sine parameters and
uncertainties are 0.334$\pm$0.016, 133.9$\pm$0.6 day, 58196.1$\pm$1.1, and
19.58$\pm$0.01 corresponding to $A$, $P$, $t_{0}$ and $A_{0}$, respectively.
We show it in B panel of Figure 1 with a blue dashed-dotted sine and the
4$\sigma$ significance curve in right panel of Figure 3 with a green dashed
line. Its significance of the periodic signal is $\sim 4.1\sigma$ (with
$p$-value of $4.5\times 10^{-5}$).

Considering the observation cadence performed by ZTF for the target, in our
analysis, we searched the QPO signal in a scale of frequencies between $f_{\rm
max}$ = 1/7 and $f_{\rm min}$ = 1/1350 day-1 by employed the LSP method, where
1350 day is approximately the total length of the data. In order to evaluate
the _precise_ significance for the QPO signal, we take a trial factor into our
significance level calculation. The false-alarm probability (FAP) is derived
by a form of FAP=1-(1-$p$)N, where $N=192$ is the number of independent
frequencies in the frequency range. The trial number is calculated by a form
of $N=\frac{f_{\rm max}-f_{\rm min}}{\delta f}$, where $\delta f$ is
determined by the mean total length of the g- and r-band data (i.e. the
frequency resolution). After considering that the trial factor in the periodic
signal searching, the significance of the QPO in the g-band is 3.1$\sigma$,
while in the r-band, it is 2.6$\sigma$.

Then a independent method of WWZ was used to check the results obtained by the
LSP. And a time–frequency WWZ power spectra were obtained for g- and r-band
light curves, the suitable WWZ power values were calculated by changing the
parameters of wavelets to match the LSP results. In g- and r-band data, a QPO
signal presents in time–frequency power spectra over the whole-data time
range, we show them in Figure 3 by scaled the power values with a colorbar. A
time-averaged power spectra along the time-dimensionality were also created
for WWZ power spectra, we show them in Figure 3 with a black curves, and a QPO
signal presents in them too, which has the same shape and period as the LSP
power spectra.

For the i-band light curve, a weak peak was also seen in its power spectra
with a frequency corresponding to the periods determined from the g- and
r-band light curves. It is worth mentioning that the i-band light curve
comprises only 133 data points, which are concentrated in three time intervals
with large gaps between them, namely 2018 June, 2019 May, and 2020 May (see C
panel in Figure 1). It is possible that the properties of QPO variabilities
may not be fully captured by i-band light curve because of these intervals.
These causes may result in the QPO signal not being statistically significant
in i-band.

Figure 4: Power spectra of TXS1206+549 constructed with the o-band and
$\gamma$-ray light curves with the simultaneous time range of ZTF
observations. A strong and weak QPO signals at $\sim$134 day are shown in
ATLAS and _Fermi_ -LAT observations. We use a dashed-dotted line to stand for
the QPO signal that detected in ZTF light curves.

By the same procedure, we searched the periodicity in the o-band light curve
from the whole ATLAS observations, and no significant power peak in the power
spectra was found. When we focused on the simultaneous light curve as the time
scale of ZTF observations, a potential QPO signal with a similar period as
that found in ZTF reveals in power spectra. We show them in left panel of
Figure 4. The same case was also shown in the gamma-ray light curve, when we
focuses on the simultaneous time scale of ZTF, a weak periodic signal was
detected at similar period. We show the $\gamma$-ray results in right panel of
Figure 4. For comparison, we show the QPO found in ZTF data with a blue
dashed-dotted line in Figure 4.

###  3.2 Phase-resolved Results

Figure 5: Pulse profiles of the optical light curves in ZTF g- and r-bands. We
folded the data points with the period cycles of 134.82 and 133.94 day with
phase zero at MJD 58200 for g- and r-bands, respectively. Two cycles are
plotted for clarity. The red dashed lines represent the weighted magnitudes by
the data point errors. Lower panel: Data points were folded with the
corresponding periods of g- and r-bands. Upper panel: The folded data points
were averaged into 30 phase bins in one cycle, their error-bars were
calculated by the weighted standard deviations of the magnitudes and errors.

Based on the results of the periodic analysis, we folded the light curves with
the respective periods that detected in g- and r-bands. The folded light
curves have a same phase zero at MJD 58200.0, which corresponding to the start
time of ZTF observations. We show the folded data points with the raw data in
lower panel of Figure 5. Based on the lower panel, we defined 30 phase bins to
average the weighted magnitudes for each data point which falls into the
corresponding phase bin. The weighted magnitudes and their errors were
calculated with the same procedure presented in Section 2.1. In lower panel,
for g-band (r-band), the minimum, maximum, and mean values are
$\sim$19.55(19.10), 20.31(19.98), and 20.01(19.61) Mag, respectively. From the
folded light curves, we can see that a near-sinusoidal profile show in it.

##  4 Summary and Discussion

TXS 1206+549 is a radio and $\gamma$-ray loud AGN classed as NLSy1 galaxy at
redshift of $z$=1.344 (Rakshit et al., 2021). Here, we report an optical QPO
signal with a period of $\sim$134 day detected in the g- and r-band light
curves from the ZTF observations spanning approximately 4 yr, the signal
presents about 10 cycles. The QPO behavior is depicted in Figure 1 using a
sinusoidal function. We observe that some data points do not align with the
periodic model. This leads us to suspect that the target’s variability is not
solely characterized by the QPO behavior, it also encompasses variations
originating from other electromagnetic radiation mechanisms across the entire
frequency spectrum, while the periodic model only represents the QPO signal at
one certain single frequency. This may be the reason of the deviations between
data points and the periodic model. After considering that the trial factor in
the timing analysis, the significance of the QPO in the g-band is 3.1$\sigma$,
while in the r-band, it is 2.6$\sigma$. Interestingly, by the same procedure,
a potential QPO signal with a similar period as that detected in ZTF was found
in the simultaneous light curve of o-band from the ATLAS observations, and a
weak signal also presented in the nearly simultaneous light curve in _Fermi_
-LAT observations. Based on the period detected by ZTF observations, we folded
the light curves of g- and r-bands, and a near-sinusoidal pulse profile show
in them (see Figure 5).

Recently many periodicity phenomena have been reported in the electromagnetic
emissions from AGNs. And many mechanisms causing these periodicities are also
proposed (Ackermann et al., 2015, and references therein), while they are
still poorly understood. The main kinds of models for explaining these QPOs
generally including (i) jet precession/rotation or helical structure, (ii)
pulsational accretion flow instabilities, and (iii) binary SMBH system.

TXS 1206+549 is a $\gamma$-ray emitting NLSy1 galaxy (Rakshit et al., 2021)
and displays a large amplitude variabilities in $\gamma$-rays (as shown in
Figure 2), which make it to be one of $\gamma$-ray blazar-like sources, they
generally host powerful relativistic jets. And indeed a probable kpc-scale jet
reveals in the Very Large Array Sky Survey (VLASS) data (Pajdosz-Śmierciak et
al., 2022). If TXS 1206+549’s jet is precessing, or an internal helical
configuration presents in its jet, the QPO variations could arise from a
varying Doppler boosting for a periodically changing viewing angle (Rieger,
2004; Komossa & Zensus, 2016). In the optical light curves, TXS 1206+549 has a
observed period $T_{\rm obs}\sim$ 4 month, and the intrinsic periods $T_{\rm
int}$ is calculated by $T_{\rm int}=T_{\rm obs}/(1+z)\sim$2 month, this period
is in agreement with that a monthly QPO found in PKS 2247–131 by Zhou et al.
(2018), they also claimed that the QPO can be explained by the model of a
helical structure in the jet.

For the scenario of binary SMBHs, we can estimate the parameters of the binary
system, e.g. the separation ($D$) between the two SMBHs and the merging
timescale ($t_{\rm m}$) of the two SMBH. As the equation (26) shown in
Sobacchi et al. (2017), the separation between the two SMBH can be estimated
by

$D=1.3\times 10^{16}(\frac{1+q}{q})\frac{T_{\rm int}}{2~{}\rm yr}~{}\rm cm,$
(1)

where $q$ stands for the mass ratio of the secondary SMBH to the primary one.
Using $T_{\rm int}\sim 0.16$ (134 day) and $q\sim 1$, we obtain $D\sim 0.0006$
pc. The separation of binary SMBH system might be too small. Owing to emission
of gravitational waves, the two SMBHs will merge at the timescale of

$t_{\rm m}=3.9\times 10^{4}q(\frac{q}{1+q})^{3}\frac{T_{\rm int}}{2~{}\rm
yr}~{}\rm yr.$ (2)

We obtain $t_{\rm m}\sim 380$ yr. This value is slightly early than a normal
timescale for the binary SMBHs merging (Peters, 1964; Yu, 2002). In addition,
only one radio core in TXS 1206+549 was detected in the radio VLASS
observations (Pajdosz-Śmierciak et al., 2022). In consideration of above
factors, we suspect that the scenario of binary SMBHs may not be suitable for
this QPO case reported here.

We know that the NLSy1 galaxies generally have high accretion rates. The model
of pulsational accretion flow instabilities possibly induces quasi-periodic
variabilities in jet emissions. While they are generally believed to result in
the short periods (Honma et al., 1992) at a scale of

$T_{\rm int}\sim 1\times\frac{M_{\rm SMBH}}{10^{8}~{}M_{\odot}}~{}\rm day.$
(3)

Using the mass of SMBH of a value of $1.2\times 10^{8}~{}M_{\odot}$ for TXS
1206+549 reported in Rakshit et al. (2021), we obtained $T_{\rm int}\sim$1.2
day, this value is significantly lower than the period of QPO signal found
here. Therefore, this scenario may be not suitable for the QPO reported here.

It should be kept in mind that many other scenarios also have been proposed to
explain the QPOs detected in AGNs (Komossa & Zensus, 2016), more researches of
the searchings for the multifrequency QPOs should be encouraged to probe the
underlying physical mechanism of periodicity phenomena in AGNs.

## acknowledgments

We thank anonymous referees for their comments that helped improving this
work. This work is supported in part by the National Natural Science
Foundation of China No. 12163006 and 12233006, the Basic Research Program of
Yunnan Province No. 202201AT070137, and the joint foundation of Department of
Science and Technology of Yunnan Province and Yunnan University No.
202201BF070001-020.

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CALT 68-2902

# Global CKM Fits with the Scan Method

Gerald Eigen  University of Bergen, 5007 Bergen, Norway  Gregory Dubois-
Felsmann  SLAC National Accelerator Laboratory, Stanford, California 94309,
USA  David G. Hitlin  Frank C. Porter  California Institute of Technology,
Pasadena, California 91125, USA

###### Abstract

We present results of unitary triangle fits based on the scan method. This
frequentist approach employs Gaussian uncertainties for experimental
quantities, but avoids assumptions about the distribution of theoretical
errors. Instead, we perform a large number of fits, scanning over regions of
plausible theory errors for each quantity. We retain those fits meeting a
specific confidence level criterion, thereby constructing a region in the
$\bar{\rho}-\bar{\eta}$ plane using the “standard” measurements (CKM matrix
elements, $\sin 2\beta,B^{0}_{d,s}$ mixing, $\epsilon_{K}$). In addition we
use branching fraction and $C\\!P$ asymmetry measurements of $B$ decays to
pseudoscalar pseudoscalar, pseudoscalar vector, vector vector and $a_{1}$
pseudoscalar final states to determine $\alpha$, $D^{(*)}K^{(*)}$ modes to
determine $\gamma$, and $D^{(*)}\pi$ and $D\rho$ modes to determine
$2\beta+\gamma$. We parameterize individual decay amplitudes in terms of
color-allowed tree, color-suppressed tree, penguin, singlet penguin,
electroweak penguin, as well as $W$-exchange and $W$-annihilation amplitudes.
With this parameterization, we obtain a good fit to the measured branching
fractions and $C\\!P$ asymmetries within the Standard Model ansatz, with no
new physics contributions. This simultaneous fit allows us to determine, for
the first time in a global fit, the correlation between $\alpha$ and $\beta$,
as well as between $\gamma$ and $\beta$.

###### pacs:

12.15.Hh, 13.25.Hw, 14.65.Fy

##  I Introduction

The phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix ckm  is responsible
for $C\\!P$ violation in the Standard Model (SM). The unitarity relations
within the CKM matrix provide an excellent laboratory to test this prediction,
the relation ${V_{ub}}^{*}V_{ud}+{V_{cb}}^{*}V_{cd}+{V_{tb}}^{*}V_{td}=0$
being particularly useful, since many measurements and theory inputs in the
$B$ and $K$ systems can be combined for this test. Tests (including
simultaneous estimation of the fundamental CKM parameters) such as those by
the CKMfitter ckmfitter  and UT${}_{\it fit}\ $utfit  groups are in common
circulation. The former is a frequentist technique, the latter Bayesian. The
scan method scan  presented herein is a frequentist-based fitting technique to
determine the parameters of the CKM matrix and test consistency with the
standard model. The scan method takes a different approach in its treatment of
theoretical uncertainties, as well as in the construction of confidence sets.

We first describe the scan method in section II. The results for the global
fits are presented in section III. We begin with a comparison with CKMfitter
and UTfit, employing the scan method to extract CKM parameters using inputs
standardized for the book Physics of the $B$ Factories pbf . Then we look at
the current situation using inputs from PDG12 pdg12  and HFAG HFAG ,
investigating the question of consistency of the results with SM expectations.
We call such fits “baseline” fits. Their characteristic is that the $\chi^{2}$
function has of the order of ${\cal O}(20)$ terms and ${\cal O}(10)$ fit
parameters including explicit inputs for $\alpha$ and $\gamma$. To take into
account the correlations between $\alpha$ and $\beta$ or $\gamma$ and $\beta$
in the extraction of the parameters of the unitarity triangle, we perform a
fit in which we replace the inputs for $\alpha$ and $\gamma$ by branching
fractions and $C\\!P$ asymmetries of $B$ decays to pseudoscalar pseudoscalar
($PP$), pseudoscalar vector ($PV$), vector vector ($VV$) and $a_{1}$
pseudoscalar ($a_{1}P$) final states and $B$ decays to $D^{(*)}K^{(*)}$,
$D^{(*)}\pi$, and $D\rho$ final states. We refer to the latter as “full” fits.
Thus, we are able for the first time to include the correlations between
$\alpha$ and $\beta$ and $\gamma$ and $\beta$ in the extraction of the
parameters of the unitarity triangle.

In section IV, we perform stand-alone determinations of the angles $\alpha$
and $\gamma$, which are then used in the baseline fits. First, we determine
the correlations between $\alpha$ and $\beta$ by performing fits to
measurements of $B$ decays to $PP,PV,VV$, and $a_{1}P$ final states. Next, we
determine the correlations between $\gamma$ and $\beta$ by performing fits to
measurements of $B$ decays to $D^{(*)}K^{(*)}$, $D^{(*)}\pi$ and $D\rho$ final
states. Finally, the results are summarized in section V.

##  II Fit Methodology

The scan method accounts for the theoretical uncertainties in the QCD
parameters $f_{B_{s}}$, $\xi_{f}=f_{B_{s}}/f_{B_{d}}$, $B_{B_{s}}$,
$\xi_{b}=B_{B_{s}}/B_{B_{d}}$, and $B_{K}$ laiho10 ; laiho2  and the CKM
parameters $|V_{ub}|$ and $|V_{cb}|$ by scanning over the range allowed by
theory uncertainties using fixed grid or Monte Carlo (MC) methods. In the
baseline fit, we combine measurements of $\Delta m_{B_{d}},\Delta
m_{B_{s}},\epsilon_{K},|V_{cb}|,|V_{ub}|,|V_{ud}|$, $|V_{us}|$, $|V_{cd}|$,
$|V_{cs}|$, $|V_{tb}|$, $\sin 2\beta,\alpha$ and $\gamma$ in the $\chi^{2}$
function, Eq. II. The angle brackets ($<>$) here indicate the experimental
averages.

$\displaystyle\chi^{2}(\bar{\rho},\bar{\eta},p_{i},t_{j})=$
$\displaystyle\Bigl{(}\frac{\langle\Delta m_{B}{{}_{d,s}}\rangle-\Delta
m_{B_{d,s}}(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{\Delta
m_{B_{d,s}}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle|V_{cb,ub,ud,us}|\rangle-|V_{cb,ub,ud,us}|(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{|V_{cb,ub,ud,us}|}}\Bigr{)}^{2}$
$\displaystyle+$
$\displaystyle\Bigl{(}\frac{\langle|\epsilon_{K}|\rangle-\epsilon_{K}(\bar{\rho},\bar{\eta},p_{i},t_{j})}{\sigma_{\epsilon_{K}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle
S_{\psi K^{0}}\rangle-\sin
2\beta(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{S_{\psi
K^{0}}}}\Bigr{)}^{2}+\Bigl{(}\frac{\langle\alpha\rangle-\alpha(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{\alpha}}\Bigr{)}^{2}$
$\displaystyle+$
$\displaystyle\Bigl{(}\frac{\langle\gamma\rangle-\gamma(\bar{\rho},\bar{\eta},p_{i})}{\sigma_{\gamma}}\Bigr{)}^{2}+\sum_{k}\Bigl{(}\frac{\langle{\cal
M}_{k}\rangle-{\cal M}_{k}(p_{i})}{\sigma_{{\cal
M}_{k}}}\Bigr{)}^{2}+\sum_{n}\Bigl{(}\frac{\langle{\cal T}_{n}\rangle-{\cal
T}_{n}(p_{i},t_{j})}{\sigma_{{\cal T}_{n}}}\Bigr{)}^{2}.$ (1)

Table 1: Observables used in the baseline fits and full fits. Fit type I are baseline fits with inputs specified by CKMfitter and UTfit (July 2012) to compare the fit results among the different fit methodologies. The values of $|V_{cb}|$ and $|V_{ub}|$ have only a total uncertainty. We list a second set in which experimental and theory uncertainties are separated allowing scans over the theoretical component of the uncertainties in $|V_{cb}|$ and $|V_{ub}|$. Fit type II are the baseline fits using the most recent input values to test the SM with and without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$. The first set of $\alpha$ and $\gamma$ values has been fixed by CKMfitter and UTfit, while the second set results from our global fits to branching fractions and $C\\!P$ asymmetries of $B$ decay modes. Fit type III represents the full fits in which $\alpha$ and $\gamma$ are replaced by branching fraction and $C\\!P$ asymmetry measurements of the $B$ decay modes. fit type | $m_{t}^{\rm pole}~{}\rm[GeV/c^{2}]$ | $\overline{m}_{c}(m_{c})~{}\rm[GeV/c^{2}]$ | $\Delta m_{B_{d}}~{}\rm[ps^{-1}]$ |  $\Delta m_{B_{s}}~{}\rm[ps^{-1}]$
---|---|---|---|---
I |  $173.2\pm 0.9$ pbf  |  $1.275\pm 0.025$ pdg12  |  $0.508\pm 0.004$ pdg12  |  $17.719\pm 0.042$ pdg12
II, III |  $173.5\pm 0.9$ pdg12  | same | same | same
fit type | $|V_{cb}|$ | $|V_{ub}|$ | $|V_{ud}|$ |  $|V_{us}|$
I |  $(4.16\pm 0.038\pm 0.05)\times 10^{-2}$ pbf  |  $(3.95\pm 0.38\pm 0.39)\times 10^{-3}$ pbf  |  $\ 0.97425\pm 0.0002\ $ colangelo  |  $\ 0.2208\pm 0.0039$ colangelo
II, III |  $(4.09\pm 0.07\pm 0.09)\times 10^{-2}$ pdg12  |  $(4.15\pm 0.31\pm 0.39)\times 10^{-3}$ pdg12  | same | same
fit type | $\epsilon_{K}$ | $\sin 2\beta$ | $\alpha$ |  $\gamma$
I |  $(2.228\pm 0.0011)\times 10^{-3}$ pdg12  |  $0.0677\pm 0.020$ HFAG  |  $(88.0\pm 5.0)^{\circ}$ pbf  |  $(67.0\pm 11)^{\circ}$ pbf
II, III | same | same |  $(84.6\pm 2.1)^{\circ}$ 111Input from global fit to Unitarity Triangle angles (see below); not used in the full fit (III). |  $(79.7\pm 4.2)^{\circ}$${\rm\ {}^{a}}$
fit type | $|V_{cd}|$ | $|V_{cs}|$ | $|V_{tb}|$ |  ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$
I | not used | not used | not used |  $(1.15\pm 0.23)\times 10^{-4}$
II, III |  $0.23\pm 0.011$ pdg12  |  $1.023\pm 0.036$ pdg12  |  $0.97\pm 0.08$ pdg12  | same
Table 2: QCD parameters used in the baseline fits. The first row shows the inputs for fit type I that were the averages listed by the Lattice group in July 2012. The second row shows the values with separate “statistical” and theory uncertainties used in the baseline fit II and the full fit (III).  fit type |  $f_{B_{s}}$ [MeV] | $\xi_{f}$ | $B_{B_{s}}$ | $\xi_{b}$ | $B_{K}$ | Ref.
---|---|---|---|---|---|---
I | $\ 227.6\pm 2.2\pm 4.5$ | $\ 1.201\pm 0.012\pm 0.012$ | $\ 1.33\pm 0.06$ | $1.05\pm 0.07$ | $\ 0.7643\pm 0.0034\pm 0.0091$ |  laiho2 ; laiho10
II, III | same | same | $\ 1.33\pm 0.018\pm 0.06$ | $1.05\pm 0.025\pm 0.07$ | same |  gamiz
fit type | $\eta_{cc}$ | $\eta_{ct}$ | $\eta_{tt}$ | $\eta_{b}$ |  |
I, II, III | $\ 1.39\pm 0.35$ | $\ 0.47\pm 0.04$ | $\ 0.5765\pm 0.0065$ | $\ 0.551\pm 0.007$ |  |  nierste

The dependence of the predicted values on the quantities
$\bar{\rho},\bar{\eta},p_{i}$, and $t_{j}$ is described in detail in the
Appendix. The $p_{i}$ are measured inputs to these predicted values, including
the Wolfenstein parameters wolf  $A$ and $\lambda$ and the quark masses and
$B$ meson masses. We add terms in the $\chi^{2}$, denoted generically by
${\cal M}_{k}$, accounting for the contributions from the uncertainties in the
$p_{i}$. Note that the dependence on these terms introduces correlations in
the $\chi^{2}$ expression. The $t_{j}$ represent parameters having a
theoretical uncertainty, e.g. the QCD parameters as well as $|V_{ub}|$ and
$|V_{cb}|$.

We are careful to distinguish among different kinds of uncertainties.
Observables with experimental errors only (statistical and systematic) are
assumed to be Gaussian-distributed. Theoretical quantities such as lattice-
derived QCD parameters, and inputs to $|V_{ub}|$ and $|V_{cb}|$, typically
have two types of uncertainties. The first type of error is of a “statistical”
nature, resulting from an input with statistical uncertainties or from Monte
Carlo statistics in lattice calculations. We assume this error to be Gaussian-
distributed and add corresponding terms to the $\chi^{2}$, denoted by ${\cal
T}_{n}$. The second type of uncertainty is a theory error with no known
underlying statistical distribution. We therefore make no assumption as to the
distribution of these errors, and instead perform a scan over a large range of
plausible values, doing a $\chi^{2}$ minimization at each point. Our use of
such fit results is described in the next section.

The scan includes the QCD parameters ($f_{B_{s}}$, $\xi_{f}$, $B_{B_{s}}$,
$\xi_{b}$ and $B_{K}$) and the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$.
The QCD corrections $\eta_{cc},\ \eta_{ct}$ and $\eta_{tt}$ used in the
determination of $\epsilon_{K}$ and $\eta_{b}$ that appear in the prediction
of $\Delta m_{B_{d}}$ also have theory errors. Although the fit methodology is
able to scan over them, we do not do so, since this is unnecessary at the
current level of precision. We parametrize $\eta_{cc}$ in terms of
$\overline{m}_{c}(m_{c})$ and $\alpha_{s}$ nierste . Tables 1 and 2 summarize
the input parameters for our baseline fits. We are preparing a more detailed
article that includes a fuller discussion of all input values, provides a
study of the correlations among the theory uncertainties and shows further
results unfit .

##  III Results of the Global Fits

We present in this section the results for the global fits according to the
scan method, beginning with a comparison with the results for CKMfitter and
UT${}_{\it fit}\ $ for a common set of inputs. Then we look at the scan method
results for a more current set of inputs, including both a stand-alone
determination of the angles $\alpha$ and $\gamma$ and a fit explicitly
incorporating the measurements that enter into the $\alpha$ and $\gamma$
determination.

###  III.1 Comparison to CKMfitter and UT${}_{\it fit}\ $

Figure 1: Overlay of $68\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the inputs of Physics of the $B$ Factories with scanning over
$f_{B_{s}}$, $\xi_{f}$ and $B_{K}$. The black points show the central values
of each accepted fit.

Figure 2: Overlay of $68\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the inputs of Physics of $B$ the Factories with scanning over
$f_{B_{s}},\xi_{f},B_{K},|V_{ub}|$ and $|V_{cb}|$. The green ellipses show
selected contours of accepted fits.

Table 3: Comparison of unitarity triangle parameters for different fitting techniques using inputs for the book Physics of the $B$ Factories. The second and third columns show the fit results from CKM fitter and UTfit. The fourth column shows our fit result if no scanning over theory parameters is performed and experimental uncertainties and theory uncertainties are added in quadrature. The fifth column shows the fit results if we scan over the QCD parameters $f_{B_{s}},\xi_{f}{\rm and\ }B_{K}$. The sixth column shows the fit results if we scan, in addition, over the theory errors associated with $|V_{ub}|$ and $|V_{cb}|$. |  |  | Our fit | Scan method
---|---|---|---|---
Parameter | CKMfitter | UT${}_{\it fit}\ $ | no scan | scan over | scan over
| pbf  | pbf  |  | $\ f_{B_{s}},\xi_{f},B_{K}\ $ |  $\ f_{B_{s}},\xi_{f},B_{K}$, $|V_{cb}|$, $|V_{ub}|\ $
$\ \bar{\rho}$ | $\ 0.129^{+0.027}_{-0.022}$ | $\ 0.132\pm 0.020\ $ | $\ 0.134\pm 0.041\ $ | $0.132^{+0.048}_{-0.042}$ |  $0.139^{+0.048}_{-0.052}$
$\ \bar{\eta}$ | $\ 0.345\pm 0.014\ $ | $\ 0.348\pm 0.013$ | $\ 0.348^{+0.024}_{-0.023}$ | $0.348^{+0.026}_{-0.025}$ |  $0.341^{+0.034}_{-0.025}$
$\ \beta~{}[^{\circ}]$ | $21.6^{+0.8}_{-0.7}$ | $21.8^{+0.8}_{-0.7}$ | $\ 21.9^{+2.4}_{-2.2}$ | $21.8^{+2.5}_{-2.1}$ |  $21.6^{+2.6}_{-2.2}$
$\ \alpha~{}[^{\circ}]$ | $88.8^{+4.2}_{-3.6}$ | $88.6\pm 2.9$ | $\ 89.2^{+1.7}_{-2.9}$ | $89.0^{+2.9}_{-3.6}$ |  $90.6^{+2.8}_{-5.9}$
$\ \gamma~{}[^{\circ}]$ | $68.96^{+3.5}_{-4.2}$ | $69.4\pm 3.1$ |  $68.9^{+5.1}_{-4.1}$ | $69.2^{+5.5}_{-5.2}$ |  $67.8^{+7.4}_{-5.1}$

To begin, we compare the performance of the scan method with that of CKMfitter
and UT${}_{\it fit}\ $using 19 input measurements ($|V_{ud}|$, $|V_{us}|$,
$|V_{cb}|$, $|V_{ub}|$, $\epsilon_{K}$, $\Delta m_{B_{d}}$, $\Delta
m_{B_{s}}$, $\sin 2\beta$, $\alpha$, $\gamma$, $f_{B_{s}}$, $B_{B_{s}}$,
$\xi_{f}$, $\xi_{b}$, $B_{K}$, $m_{t}^{\rm pole}$, $\overline{m}_{c}(m_{c})$,
$m_{B_{d}}$, $m_{B_{s}}$), choosing values specified for the book Physics of
the $B$ Factories pbf  to fit 13 parameters ($\bar{\rho}$, $\bar{\eta}$, $A$,
$\lambda$, $f_{B_{s}}$, $B_{B_{s}}$, $\xi_{f}$, $\xi_{b}$, $B_{K}$,
$m_{t}^{\rm pole}$, $\overline{m}_{c}(m_{c})$, $m_{B_{d}}$, $m_{B_{s}}$). In
these fits, which we call fit type I in Tables 1 and 2, we use the central
values and measurement errors for $\alpha$ and $\gamma$ in the $\chi^{2}$
function. We compute $\overline{m}_{t}(m_{t})$, which enters into the Inami-
Lim functions inami  for $\Delta m_{B_{d}}$, $\Delta m_{B_{s}}$ and
$\epsilon_{K}$, in the $\overline{MS}$ scheme from the pole mass $m_{t}^{\rm
pole}$ at three loop level for six quarks melnikov00 ; gray90 ; broadhurst91 .
We plot $1\sigma$ contours in the $\bar{\rho}-\bar{\eta}$ plane according to
the prescription described below (see Eq. 2). For the central value, we select
the fit with the highest $P(\chi^{2})$. We take the $\pm 1\sigma$
uncertainties from the maximum and minimum values of the envelope of all
contours. We perform three different fits. In the first, we combine theory and
experimental uncertainties, treating these as Gaussian. In the second, we scan
over the theory uncertainties in $f_{B_{s}},B_{B_{s}},B_{K}$. In the third, we
separate theory uncertainties from experimental uncertainties in $|V_{ub}|$
and $|V_{cb}|$ (see Table 1) thus scanning over theory uncertainties in
$|V_{ub}|,|V_{cb}|$, as well as those for $f_{B_{s}},\xi_{f}$ and $B_{K}$. We
performed the separation according to the ratio of experimental to theory
uncertainties listed in PDG10 pdg10  keeping the total uncertainty unchanged.
Figure 2 shows the overlay of $68\%$ confidence level contours in the
$\bar{\rho}-\bar{\eta}$ plane for the second fit. Since values of $|V_{ub}|$
and $|V_{cb}|$, which have substantial theoretical uncertainties, are not
scanned the accepted contours are similar. Figure 2 shows the comparable
results for the third fit, i.e, the most complete scan. Table 3 lists our
results in comparison to those from CKMfitter and UTfit. The three methods
yield broadly similar results. Scanning over theoretical errors increases the
uncertainties, often substantially. With this methodology the resulting
uncertainties are typically around a factor of two larger than those from
CKMfitter and UTfit. Large differences may be expected with UTfit, as UTfit is
a Bayesian approach and includes prior distributions for the theoretically
uncertain quantities.

The differences with CKMfitter are more subtle, as both are frequentist
approaches. The essential difference between the two interval estimation
methods is the following: CKMfitter uses the conventional change in $\chi^{2}$
from its minimum value to determine confidence regions. The motivation for the
scan method is to test the hypothesis that the standard model is correct,
against the alternative that it is incorrect. The statistical test adopted is
the $\chi^{2}$ test. Hence, this motivation is reflected in our confidence
region determination, in which we use the method of inverting a test
acceptance region (as described in standard statistics texts, for example
Shao2003 ). That is, the confidence regions are determined by comparing
$\chi^{2}$ values with a critical value instead of looking for a change in
$\chi^{2}$.

Specifically, the algorithm for a $1-\alpha$ confidence region in $d$
dimensions of a $p$ dimensional parameter space with $n$ measurements is as
follows. First, determine the acceptance region, at the $\alpha$ significance
level, by determining the critical value $\chi_{c}^{2}$ such that

$P(\chi^{2}\geq\chi_{c}^{2};n-p+d|H_{0})\geq\alpha,$ (2)

where $H_{0}$ is the hypothesis of the standard model, and $n-p+d$ is the
number of degrees of freedom. The $d$ is added back here because the critical
region for the test is constructed for each point in the $d$-dimensional
subspace of the full parameter space. The confidence region is then given by
all those points in the $d$ dimensional parameter subspace for which
$\chi^{2}\leq\chi^{2}_{c}$, under $H_{0}$.

For the theoretical uncertainties, CKMfitter makes an implicit scan by
incorporating a term in the likelihood function for each theoretically
uncertain quantity. The term equals 1 or 0 depending on whether the theory
parameter is in the theoretically “allowed” region or not. A modified
algortihm is also available, which makes a smooth transition between 1 and 0.
In the scan method, the scan over theoretical parameter space is explicit. No
theoretical term is included in the likelihood function. Instead, each point
in the theoretical parameter space of interest is treated as a possible value,
and the fit procedure is performed at each such point.

Depending on the value of the minimum $\chi^{2}$, the regions determined by
either method may be larger or smaller. If the best fit gives a high
$p$-value, then the scan method regions will be larger. On the other hand, if
the best fit gives a low $p$-value, the CKMfitter regions will be larger. In
the present instance, the best fit $p$-value is 0.76, hence the scan method
confidence intervals will tend to be larger than those from CKMfitter.
Reflecting its origins in goodness-of-fit testing, the scan method region goes
to null in the limit where the fit fails at the specified confidence level. In
the limit of no theoretical uncertainties, both methods give valid frequentist
confidence regions, with different properties. However, this comparison points
out the importance of methodology in forming conclusions, and hence of
examining the problem with multiple approaches.

Table 4: The $95\%~{}{\rm CL}$ ranges for unitarity triangle parameters from our baseline fit scans without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$, from our baseline fit scans with the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ and our full fit scans without the inclusion of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$. The values of $\alpha$ and $\gamma$ are, in these cases, computed from the values of $\bar{\rho}$ and $\bar{\eta}$. Parameter | $\bar{\rho}$ | $\bar{\eta}$ | $\beta~{}[^{\circ}]$ | $\alpha~{}[^{\circ}]$ |  $\gamma~{}[^{\circ}]$
---|---|---|---|---|---
baseline fit: scan without $B^{+}\rightarrow\tau^{+}\nu\ $ | $\ 0.069-0.147\ $ | $\ 0.319-0.395\ $ | $\ 19.0-24.7\ $ | $\ 82.7-88.5\ $ |  $\ 68.8-77.9\ $
baseline fit: scan with $B^{+}\rightarrow\tau^{+}\nu$ | $0.073-0.147$ | $0.324-0.396$ | $19.3-24.8$ | $82.8-88.4$ |  $68.8-77.4$
full fit: scan without $B^{+}\rightarrow\tau^{+}\nu$ | $0.070-0.151$ | $0.318-0.395$ | $18.8-24.8$ | $82.4-89.0$ |  $67.9-77.9$

Figure 3: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the baseline fit scan with 22 measurements without including ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.

Figure 4: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for the baseline fit scan with 23 measurements with inclusion of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.

###  III.2 Fit results with direct inputs of $\alpha$ and $\gamma$

To test the SM with the scan method, we perform the same baseline fits with
updated input parameters also listed in Tables 1 and 2. We refer to these fits
as type II fits. We use PDG12 $|V_{ub}|$ and $|V_{cb}|$ averages pdg12 . Since
the $|V_{ub}|$ and $|V_{cb}|$ results from exclusive modes are significantly
lower than those from inclusive modes, the PDG uses scaling factors on the
total errors of 2.6 and 2.0, respectively. If we have any fit satisfying
$P(\chi^{2})>5\%$, the SM is deemed to be compatible with the data at the
present level of theoretical uncertainties. We compute 95% CL contours
according to the inversion of a test acceptance region prescription described
above. That is, we compute contours based on Eq. 2, with d=1, such that taking
the extrema of a contour along a parameter axis provides a 95% CL interval for
the parameter. In the limit of no theoretical uncertainties, the procedure has
the stated coverage. Figure 4 shows the overlay of $95\%~{}{\rm CL}$ contours
in the $\bar{\rho}-\bar{\eta}$ plane for all accepted baseline fits using 22
measurements to fit 13 parameters, as in Section III. To obtain a range of
values for a given parameter, we take the extrema of the union of the contours
attached to accepted fits. Table 4 shows the range of the unitarity triangle
parameters thus obtained (hereinafter referred to as the $95\%~{}{\rm CL}$
range).

The $B^{+}\rightarrow\tau^{+}\nu$ branching fraction conjugate , measured by B
AB AR babar11  and Belle belle11 , is very sensitive to contributions from a
charged Higgs boson. The PDG average ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.65\pm 0.34)\times 10^{-4}$ pdg12 ; HFAG  is
larger than the SM prediction of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.2\pm 0.25)\times 10^{-4}$ silverman . Even
with these high values of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$, we obtain a
sizeable allowed $\bar{\rho}-\bar{\eta}$ region; there is no conflict with the
SM. Belle has now presented a new measurement of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(0.72^{+0.27}_{-0.25}({\rm stat})\pm 0.11\
({\rm sys}))\times 10^{-4}$ belle12  that reduces the world average to ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=(1.14\pm 0.22)\times 10^{-4}$. Figure 4 shows
our results in the $\bar{\rho}-\bar{\eta}$ plane for this world average and
Table 4 summarizes the $95\%~{}{\rm CL}$ ranges of unitarity parameters. The
inclusion of the present ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ world average
has hardly any impact on the $\bar{\rho}-\bar{\eta}$ plane.

Figure 5: Overlay of $95\%~{}{\rm CL}$ contours in the $\bar{\rho}-\bar{\eta}$
plane for fits with 257 measurements without ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$. The green ellipses show selected contours of
accepted fits.
Figure 6: The $95\%~{}{\rm CL}$ contour in the $\alpha-\beta$ plane from a fit
including the $B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ and
$B\rightarrow Pa_{1}$ modes. The black dot shows the central value from the
fit. The vertical band shows the $68\%~{}{\rm CL}$ range for $\beta$ obtained
from $\sin 2\beta$ measurements HFAG .

###  III.3 Fit Results using Individual Measurements of Branching Fractions
and $C\\!P$ Asymmetries of Various Decays

We also perform fits, called type III fits, in which, instead of treating the
values of $\alpha$ and $\gamma$ as inputs, we directly include the
measurements that determine them. This allows us to determine the correlations
in the extraction of the various unitarity triangle angles from the fit. We
omit ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ in these fits; this has little
effect on the results.

We replace the direct $\alpha$ measurement term in the $\chi^{2}$ function by
all measured branching fractions and $C\\!P$ asymmetries in $B\rightarrow PP$,
$B\rightarrow PV$, $B\rightarrow VV$ and $B\rightarrow a_{1}P$ modes in the
fit. Following the Gronau-Rosner approach gronau , we parametrize amplitudes
in terms of tree, color-suppressed tree, penguin, singlet penguin,
$W$-annihilation/$W$-exchange, electroweak and color-suppressed electroweak
diagrams (up to $\lambda^{2}$ beyond leading order). Thus for order
$\lambda^{2}$, we consider SU(3) corrections for the tree, color-suppressed
tree and penguin diagrams. Since calculations of branching fractions use the
$B^{+}$ and $B^{0}_{d}$ lifetimes, we add $\chi^{2}$ terms for these lifetimes
in the fit.

We replace the direct $\gamma$ measurement term in the $\chi^{2}$ function, by
all measured branching fractions and $C\\!P$ asymmetries for $B^{+}\rightarrow
D^{(*)}K^{+}$ and $B^{+}\rightarrow DK^{*+}$ decays analyzed in the Giri-
Grossman-Soffer-Zupan (GGSZ) ggsz , Gronau-London-Wyler (GLW) glw  and Atwood-
Dunietz-Soni (ADS) methods ads . We also include branching fractions and
$C\\!P$ asymmetries for $B^{+}\rightarrow D^{(*)}\pi^{+}$ decays analyzed in
the ADS method. Here, the observables are calculated in terms of $b\rightarrow
c\bar{u}s(d)$ and $b\rightarrow u\bar{c}s(d)$ amplitudes. We also include
time-dependent $C\\!P$ asymmetries in $B^{0}\rightarrow D^{(*)+}\pi^{-}$ and
$B^{0}\rightarrow D^{(*)+}\rho^{-}$ decays that determine
$\sin(2\beta+\gamma)$. Thus, the total number of measurements in the global
fits increases to 257; the fit has 120 parameters. Figure 5 shows the
$95\%~{}{\rm CL}$ contours of all accepted fits in the $\bar{\rho}-\bar{\eta}$
plane. Table 4 summarizes the $95\%~{}{\rm CL}$ ranges for unitarity triangle
parameters. This result shows that the 257 measurements are in good agreement
with the SM. This procedure accounts for possible correlations between
$\alpha,\gamma$ and the other Wolfenstein parameters (e.g., $\beta$).

##  IV Determination of $\alpha$ and $\gamma$

In addition to the global fits of the CKM matrix, we perform separate fits
that determine the unitarity triangle angles $\alpha$ and $\gamma$. These
results are then used as inputs to the baseline fits. They are also used to
investigate the correlation with the angle $\beta$.

###  IV.1 Determination of $\alpha$

For the $\alpha$ determination we combine all measured branching fractions and
$C\\!P$ asymmetries in $B\rightarrow PP$, $B\rightarrow PV$ and $B\rightarrow
VV$ modes. We first perform separate fits for each class of decays. As in
Section IV we parameterize the observables in terms of amplitudes, following
the Gronau-Rosner method. We include $f_{B_{s}}$ and $\xi_{f}$ in the fit, but
we do not scan over them, since these parameters appear only in the
$W$-exchange and $W$-annihilation diagrams that are at order $\lambda^{2}$
with respect to the tree diagram; any variation in these parameters is
absorbed by adjusting the magnitude of the $W$-annihilation/$W$-exchange
diagrams, leaving $\alpha$ and the remaining parameters unchanged. Thus, we
perform a single fit for each class of decays and plot $95\%$ CL
$\alpha-\beta$ contours. Table 5 shows the central values for $\alpha$ and
$\beta$ with uncertainties obtained by changing $\Delta\chi^{2}$ by one for
all fits. In addition, we list the correlation between $\alpha$ and $\beta$
and the fit probability. The correlation coefficients vary between -18% and
5%.

We next perform a combined fit of all measurements in $B\rightarrow
PP,B\rightarrow PV$ and $B\rightarrow VV$ and $B\rightarrow Pa_{1}$ modes to
extract $\alpha$. We use 185 measurements HFAG  to determine 96 parameters.
Figure 6 shows the $95\%~{}{\rm CL}$ contour in the $\alpha-\beta$ plane,
which encompasses the world average $\beta=(21.4\pm 0.8)^{o}$ measured using
$b\rightarrow c\bar{c}s$ modes. The fit probability is $P(\chi^{2})=38.6\%$.
The correlation coefficient is about -4%. These results show that all
measurements in $B\rightarrow PP,B\rightarrow PV$ and $B\rightarrow VV$ and
$B\rightarrow Pa_{1}$ modes are consistent with the SM description and no new
physics amplitudes are required.

###  IV.2 Determination of $\gamma$

Table 5: Measurements of $\alpha$, $\beta$ and $\gamma$ from fits of branching fractions and $C\\!P$ asymmetries in $B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ decays and in a combination of all modes combined plus $B\rightarrow Pa_{1}$ decays ($\alpha$) and $B\rightarrow D^{(*)}K(\pi)+B\rightarrow DK^{*}(\rho)$ decays ($\gamma$). | $B\rightarrow PP$ | $B\rightarrow PV$ | $B\rightarrow VV$ |  $B$ modes combined |  $B\rightarrow D^{(*)}K(\pi)+B\rightarrow DK^{*}(\rho)$
---|---|---|---|---|---
$\ \alpha~{}[^{\circ}]$ | $85.9^{+3.0}_{-2.7}$ | $82.4^{+4.1}_{-4.3}$ | $83.8^{+5.5}_{-5.6}$ | $84.7^{+2.1}_{-2.1}$ | \-
$\ \gamma~{}[^{\circ}]$ | - | - | - | - |  $79.6^{+4.1}_{-4.2}$
$\ \beta~{}[^{\circ}]$ | $20.8^{+2.1}_{-1.9}$ | $20.5^{+3.6}_{-3.4}$ | $24.3^{+6.4}_{-4.9}$ | $21.1^{+1.6}_{-1.6}$ |  $22.8^{+7.7}_{-2.1}$
$\alpha-\beta$ correlation | 0.052 | -0.182 | -0.151 | -0.035 | \-
$\gamma-\beta$ correlation | - | - | - | - | -0.194
p-value | 0.50 | 0.369 | 0.248 | 0.386 | 0.051-0.071

For the $\gamma$ determination, we use branching fraction and $C\\!P$
asymmetries of $B^{+}\rightarrow D^{(*)}K^{+}$ and $B^{+}\rightarrow DK^{*+}$
decays analyzed in the GGSZ ggsz , GLW glw  and ADS ads  methods. We also
include branching fractions and $C\\!P$ asymmetries of $B^{+}\rightarrow
D^{(*)}\pi^{+}$ decays analyzed in the ADS method and time-dependent $C\\!P$
asymmetries in $B^{0}\rightarrow D^{(*)+}\pi^{-}$ and $B^{0}\rightarrow
D^{(*)+}\rho^{-}$ decays that determine $\sin(2\beta+\gamma)$. We separate the
CKM factors, $|V_{us}V_{ub}^{*}|/|V_{cs}V_{cb}^{*}|$ and
$|V_{ud}V_{ub}^{*}|/|V_{cd}V_{cb}^{*}|$ from the ratio of $b\rightarrow u$ to
$b\rightarrow c$ amplitudes. We include the ratios $|V_{us}/V_{ud}|$ and
$|V_{ub}/V_{cb}|$ in the fit, scanning over $|V_{ub}|$ and $|V_{cb}|$. Since
the predictions contain a product of CKM factors and ratios of amplitudes, it
is necessary to constrain the CKM factors in the fit to obtain sensible values
for the amplitude ratios and CKM factors.

We use 56 measurements HFAG  to extract 19 fit parameters, scanning over the
constraint $|V_{ub}|/|V_{cb}|$. The probabilities for these fits range between
$P(\chi^{2})=5.1\%$ and $P(\chi^{2})=7.1\%$. Figure 7 shows the resulting
contours at $95\%~{}{\rm CL}$ in the $\gamma-\beta$ plane. Table 5 lists the
fit results. Again, these 56 modes are well-described within the SM and no new
physics amplitudes are required. The $\gamma$–$\beta$ correlation coefficient
is $-19$%.

Figure 7: Overlay of $95\%~{}{\rm CL}$ contours in the $\gamma-\beta$ plane
for fits including the $B\rightarrow DK^{(*)},DK^{*},D^{(*)}\pi,D\rho$ modes.
The vertical band shows the $68\%~{}{\rm CL}~{}\beta$ region obtained from
$\sin 2\beta$ measurements HFAG .

##  V Conclusion

The three fitting approaches: CKMfitter, UT${}_{\it fit}\ $and the scan method
yield similar central values for $\bar{\rho}$ and $\bar{\eta}$ when presented
with identical inputs. However, the allowed region in the
$\bar{\rho}-\bar{\eta}$ plane is substantially larger in the scan method. This
may be expected from the difference with the Bayesian methodology of
UT${}_{\it fit}\ $, in which prior distributions are assigned for the
theoretical uncertainties. The differences with CKMfitter are more subtle, as
both are frequentist approaches. In fact, the larger scan method intervals is
not a given; with different measurement values the situation could reverse in
the comparison with CKMfitter. This comparison points out the importance of
methodology in forming conclusions, and hence of examining the problem with
multiple approaches.

Using the scan method, we find no tension with the SM even when we include the
current PDG value of ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)=(1.65\pm
0.34)\times 10^{-4}$; the scan yields global fits consistent with the SM at
$95\%~{}{\rm CL}$. When we include the recent Belle result, the ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$ branching fraction has hardly any impact on
the $\bar{\rho}-\bar{\eta}$ plane.

Our global fit allows us to determine and incorporate the correlation between
$\alpha$ and $\beta$, as well as between $\gamma$ and $\beta$. Using all
measured branching fractions and $C\\!P$ asymmetries of $B\rightarrow PP$,
$B\rightarrow PV$, $B\rightarrow VV$, $B\rightarrow a_{1}P$ modes and
$B^{+}\rightarrow D^{(*)}K^{+},B\rightarrow DK^{*+}$ modes that are sensitive
to $\alpha$ and $\gamma$, respectively, we observe small changes in the
allowed region in the $\bar{\rho}-\bar{\eta}$ plane. From separate fits of
branching fractions and $C\\!P$ asymmetries in these modes, we determine
$\alpha-\beta$ and $\gamma-\beta$ contours. Though $\alpha$ measurements agree
with each other and $\beta$ results are consistent with $\sin 2\beta$ from
$b\rightarrow c\bar{c}s$ modes, some correlation among Wolfenstein parameters
in the different measurements is observed. The values of $\alpha$, determined
from a fit to all measured branching fractions and $C\\!P$ asymmetries of
$B\rightarrow PP$, $B\rightarrow PV$, $B\rightarrow VV$ and $B\rightarrow
Pa_{1}$ modes, and $\gamma$, extracted from a fit to $B^{+}\rightarrow
D^{(*)}K^{+},B\rightarrow DK^{*+}$, $B\rightarrow D^{(*)}\pi$ and
$B^{0}\rightarrow D^{+}\rho^{-}$ modes, agree with the SM expectations.

## Acknowledgments

This work is supported in part by the U. S. Department of Energy under Grant
DE-FG02-92-ER40701 and by the NFR (Norway).

## Appendix

In the baseline fits, the $\chi^{2}$ function (Eq. II) includes 23 or 22
terms, depending on whether ${\cal B}(B\to\tau\nu)$ is included or not. In
this Appendix, we describe the dependence of the predicted values used in the
$\chi^{2}$ expression on the quantities $\bar{\rho},\bar{\eta},p_{i}$, and
$t_{j}$.

The CKM matrix elements are parameterized in terms of Wolfenstein parameters
$\bar{\rho}$, $\bar{\eta}$, $A$ and $\lambda$ up to order ${\cal
O}(\lambda^{9})$ wolf ; buras94 ; ahn :

$\displaystyle V_{ud}=$ $\displaystyle
1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}-\frac{1}{16}\lambda^{6}\bigl{(}1+8A^{2}(\rho^{2}+\eta^{2})\bigr{)}-\frac{1}{128}\lambda^{8}\bigl{(}5-32A^{2}(\rho^{2}+\eta^{2})\bigr{)},$
$\displaystyle V_{us}=$
$\displaystyle\lambda\bigl{(}1-\frac{1}{2}A^{2}\lambda^{6}(\rho^{2}+\eta^{2})\bigr{)},$
$\displaystyle V_{ub}=$ $\displaystyle A\lambda^{3}(\rho-i\eta),$
$\displaystyle V_{cd}=$
$\displaystyle-\lambda\Bigl{(}1-\frac{1}{2}A^{2}\lambda^{4}\bigl{(}1-2(\rho+i\eta)\bigr{)}-\frac{1}{2}A^{2}\lambda^{6}(\rho+i\eta)\Bigr{)},$
$\displaystyle V_{cs}=$ $\displaystyle
1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}\bigl{(}1+4A^{2}\bigr{)}-\frac{1}{16}\lambda^{6}\bigl{(}1-4A^{2}+16A^{2}(\rho+i\eta)\bigr{)}-\frac{1}{128}\lambda^{8}\bigl{(}5-8A^{2}+16A^{4}\bigr{)},$
$\displaystyle V_{cb}=$ $\displaystyle
A\lambda^{2}(1-\frac{1}{2}A^{2}\lambda^{6}\bigl{(}\rho^{2}+\eta^{2}\bigr{)}),$
$\displaystyle V_{td}=$ $\displaystyle
A\lambda^{3}\bigl{(}1-\rho-i\eta\bigr{)}+\frac{1}{2}A\lambda^{5}\bigl{(}\rho+i\eta\bigr{)}+\frac{1}{8}A\lambda^{7}\bigl{(}1+4A^{2}\bigr{)}\bigl{(}\rho+i\eta\bigr{)},$
$\displaystyle V_{ts}=$
$\displaystyle-A\lambda^{2}\Bigl{(}1-\frac{1}{2}\lambda^{2}\bigl{(}1-2(\rho+i\eta)\bigr{)}-\frac{1}{8}\lambda^{4}-\frac{1}{16}\lambda^{6}\bigl{(}1+8A^{2}(\rho+i\eta)\bigr{)}\Bigr{)},$
$\displaystyle V_{tb}=$ $\displaystyle
1-\frac{1}{2}A^{2}\lambda^{4}-\frac{1}{2}A^{2}\lambda^{6}(\rho^{2}+\eta^{2})-\frac{1}{8}A^{4}\lambda^{8}.$
(3)

The Unitarity Triangle angles $\beta$, $\alpha$ and $\gamma$ are implemented
by

$\displaystyle\sin 2\beta$ $\displaystyle=$
$\displaystyle\frac{2\bar{\eta}(1-\bar{\rho})}{(1-\bar{\rho})^{2}+\bar{\eta}^{2}},$
$\displaystyle\tan\alpha$ $\displaystyle=$
$\displaystyle\frac{\bar{\eta}}{\bar{\eta}^{2}+\bar{\rho}(\bar{\rho}-1)},$
$\displaystyle\tan\gamma$ $\displaystyle=$
$\displaystyle\frac{\bar{\eta}}{\bar{\rho}}.$ (4)

The oscillation frequencies for $B^{0}_{d}\bar{B}^{0}_{d}$ and
$B^{0}_{s}\bar{B}^{0}_{s}$ mixing are computed according to:

$\displaystyle\Delta m_{B_{d}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}}{6\pi^{2}}\eta_{B}m_{B_{d}}\frac{f^{2}_{B_{s}}}{\xi_{f}^{2}}\frac{B_{B_{s}}}{\xi_{b}}m_{W}^{2}S(x_{t})|V_{td}V^{*}_{tb}|^{2},$
$\displaystyle\Delta m_{B_{s}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}_{F}}{6\pi^{2}}\eta_{B}m_{B_{s}}f^{2}_{B_{s}}B_{B_{s}}m_{W}^{2}S(x_{t})|V_{ts}V^{*}_{tb}|^{2},$
(5)

where $G_{F}$ is the Fermi constant, $\eta_{B}$ is a QCD correction, $m_{W}$
is the $W$ mass, $S(x_{t})$ is the Inami-Lim function inami  and
$x_{t}=\overline{m}_{t}^{2}/m^{2}_{W}$ where the top quark mass is calculated
in the $\rm\overline{MS}$ scheme. We have expressed the $B^{0}_{d}$ decay
constant and bag parameters in terms of the $B^{0}_{s}$ decay constant and bag
parameters and their ratios $\xi_{f}$ and $\xi_{b}$ since the latter have
smaller uncertainties. The explicit relation between $\overline{m}_{t}(m_{t})$
and $m_{t}^{\rm pole}$ is given by

$\overline{m}_{t}(m_{t})=m_{t}^{\rm
pole}\bigl{(}1-\frac{4}{3}\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}-9.1253\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}^{2}-80.4045\bigl{(}\frac{\alpha_{s}}{\pi}\bigr{)}^{3}\bigr{)},$
(6)

where $\alpha_{s}(\overline{m}_{t})=0.1068\pm 0.0018$ is calculated in the
$\overline{MS}$ scheme for six quark flavors at the scale of the pole mass
buras97 ; ckmfitter .

$C\\!P$ violation in the $K^{0}\bar{K}^{0}$ system is represented by the
parameter $\epsilon_{K}$. In the SM, this is proportional to the off-diagonal
matrix element of the mixing matrix divided by the $K^{0}_{L}-K^{0}_{S}$ mass
difference $\Delta m_{K}$ yielding buchalla95

$\epsilon_{K}=C_{\epsilon}\kappa_{\epsilon}\exp{i\phi_{\epsilon}}B_{K}\Bigl{(}{\rm
Im}[(V_{cs}V^{*}_{cd})^{2}]\eta_{cc}S(x_{c})+{\rm
Im}[(V_{ts}V^{*}_{td})^{2}]\eta_{tt}S(x_{t})+2{\rm
Im}[V_{cs}V^{*}_{cd}V_{ts}V^{*}_{td}]\eta_{ct}S(x_{c},x_{t})\Bigr{)},$ (7)

where $\kappa_{\epsilon}=0.94\pm 0.02$ buras10 , $\phi_{\epsilon}=(43.5\pm
0.7)^{\circ}$, $x_{c}=\overline{m}_{c}^{2}/m^{2}_{W}$, $\overline{m}_{c}$ is
the charm quark mass in the $\rm\overline{MS}$ scheme. The constant is given
by

$C_{\epsilon}=\frac{G^{2}_{F}f^{2}_{K}m_{K}m^{2}_{W}}{12\pi^{2}\sqrt{2}\Delta
m_{K}},$ (8)

where $m_{K}$ and $f_{K}$ are the kaon mass and kaon decay constant,
respectively. The contribution from the decay rate difference has been
neglected. We use NLO calculations of the QCD parameters; $\eta_{cc}$ and
$\eta_{ct}$ calculations at NNLO brod10 ; brod12  have now been done.

The $B^{+}\rightarrow\tau^{+}\nu$ branching fraction is given by

${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)=\frac{G^{2}_{F}}{8\pi}m_{B^{+}}m^{2}_{\tau}\Bigl{(}1-\frac{m^{2}_{\tau}}{m^{2}_{B^{+}}}\Bigr{)}\frac{f^{2}_{B_{s}}}{\xi_{f}^{2}}|V_{ub}|^{2}\tau_{B^{+}},$
(9)

where $\tau_{B^{+}}$ is the $B^{+}$ lifetime, $m_{B^{+}}$ is the $B^{+}$ mass
and $m_{\tau}$ is the $\tau^{+}$ mass,

We also add Gaussian terms in the $\chi^{2}$ function for the quark masses
$m_{t}^{\rm pole}$ and $\overline{m}_{c}(m_{c})$, meson messes $m_{B^{0}_{d}}$
and $m_{B^{0}_{s}}$, and the Gaussian parts of the QCD parameters $B_{K}$,
$f_{B_{s}}$, $\xi_{f}$, $B_{B_{s}}$, and $\xi_{b}$.

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