# Probing Top-philic New Physics via Four-Top-Quark Production

Qing-Hong Cao<EMAIL_ADDRESS>Department of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Collaborative Innovation Center of Quantum Matter, Beijing
100871, China  Center for High Energy Physics, Peking University, Beijing
100871, China  Jun-Ning Fu<EMAIL_ADDRESS>Department of Physics and
State Key Laboratory of Nuclear Physics and Technology, Peking University,
Beijing 100871, China  Yandong Liu<EMAIL_ADDRESS>Key Laboratory of Beam
Technology of Ministry of Education, College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  Beijing
Radiation Center, Beijing 100875, China  Xiao-Hu Wang<EMAIL_ADDRESS>Department of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, China  Rui Zhang
<EMAIL_ADDRESS>Theoretical Physics Division, Institute of High Energy
Physics, Beijing 100049, China

###### Abstract

We explore constraints on various new physics resonances from four top-quark
production based on current experimental data. Both light and heavy resonances
are studied in the work. A comparison of full width effect and narrow width
approximation is also made.

1\. Introduction.

Despite its rare rate four top-quark ($t\bar{t}t\bar{t}$) production in hadron
collision was discussed even before the discovery of the top quark
Barger:1991vn . The four-top channel is very special in the Standard Model
(SM) as it involves both the quantum chromodynamics (QCD) and electroweak (EW)
interactions and the strengths of the both interactions are comparable. In
particular, thanks to the heavy top-quark mass, the Yukawa interaction between
the Higgs boson and the top quark is fairly large such that the leading
electroweak contribution from the Higgs-top interaction is as important as the
one from the QCD interaction. These two interactions are interwoven more when
one calculates the higher order quantum corrections; as a result, the
theoretical prediction of the four-top production cross section is sensitive
to the choice of renormalization scale which demands two or more loop
calculation to reduce the theoretical uncertainties Frederix:2017wme . In
addition the four-top channel involves complicated kinematics which enables
the interference between the QCD diagrams and heavy EW resonances to yield a
sizable contribution. Therefore, the four-top production is a good parade
ground to test new physics (NP) beyond the SM.

In addition, the four-top channel serves well for probing both the magnitude
and CP phase of the top-Higgs interaction without assumptions on the decay of
the Higgs boson, e.g. neither the branch ratio of particular decay mode nor
the total decay width of the Higgs boson Cao:2016wib ; Cao:2019ygh . Owing to
the unprecedented colliding energy and the fast accumulation of the integrated
luminosity, the Large Hadron Collider (LHC) is able to measure the
$t\bar{t}t\bar{t}$ production Sirunyan:2017roi ; Sirunyan:2019wxt ;
Aaboud:2018jsj . For example, the $t\bar{t}t\bar{t}$ signal is observed at the
$2.6\sigma$ confidence level at the 13 TeV LHC with an integrated luminosity
of $137~{}{\rm fb}^{-1}$, and the CMS collaboration Sirunyan:2019wxt  has
published new results on the $t\bar{t}t\bar{t}$ cross section,

$\sigma(t\bar{t}t\bar{t})=12.6^{+5.8}_{-5.2}~{}{\rm fb},$ (1)

which is consistent with the tree level prediction in the SM,
$\sigma(t\bar{t}t\bar{t})_{\rm SM}=9.6~{}{\rm fb}$. It yields an upper limit
of $\sigma(t\bar{t}t\bar{t})$ at the $95\%$ confidence level as
$\sigma(t\bar{t}t\bar{t})\leq 22.5~{}{\rm fb}$.

In this work we examine the constraint on various NP resonances from the
$t\bar{t}t\bar{t}$ production. We consider the top-philic NP model in which
the NP resonance ($X$) couples only to the top quark. The NP contribution to
the $t\bar{t}t\bar{t}$ production can be either through the $X\bar{X}$ pair
production or through the $X$ production in association with a top-quark pair
($t\bar{t}X$) with a subsequent decay of $X\to t\bar{t}$ or $X\to tt$; see
Fig. 1. Denote $M$ and $\Gamma$ as the mass and width of the $X$ particle,
respectively, and $\kappa_{X}$ as the coupling strength of $X$ to the top
quark in the top-philic models. The cross section of the $t\bar{t}t\bar{t}$
production can be parametrized as following

$\displaystyle\sigma^{\rm total}=\sigma^{\rm
SM}_{t\bar{t}t\bar{t}}+\kappa_{X}^{2}\sigma^{\rm
Int}_{t\bar{t}t\bar{t}}(M,\Gamma)+\kappa_{X}^{4}\sigma^{\rm
Res}_{t\bar{t}t\bar{t}}(M,\Gamma),$ (2)

where $\sigma_{t\bar{t}t\bar{t}}^{\rm SM}$ ($\sigma_{t\bar{t}t\bar{t}}^{\rm
Int}$, $\sigma_{t\bar{t}t\bar{t}}^{\rm Res}$) denotes the cross section of the
SM contribution, the interference between the SM and NP, and the NP
contribution alone, respectively.

Figure 1: Pictorial diagrams of $t\bar{t}t\bar{t}$ production in NP models:
(a) the associated production of $t\bar{t}X$ with $X\to t\bar{t}$; (b) the
pair production of $X\bar{X}$ with $X\to t\bar{t}$.

The width dependence in Eq. 2 originates from the propagator of $X$ particle
in the intermediate state. One can simplify the cross section when the $X$
particles are on mass shell by using narrow width approximation (NWA)
HPILUHNL:NWA  which enables us to factorize the $t\bar{t}t\bar{t}$ production
induced by NP into the $X$ production process and the $X$ decay process
unambiguously. The interference between the SM and NP contributions is
negligible in the vicinity of $M$, therefore, one can parameterize
$\sigma(t\bar{t}t\bar{t})$ as follows:

$\displaystyle
X\bar{X}~{}:~{}\sigma(t\bar{t}t\bar{t})=\sigma(X\bar{X})\times{\rm
Br}^{2}(X\to t\bar{t}),$ $\displaystyle
t\bar{t}X~{}:~{}\sigma(t\bar{t}t\bar{t})=\sigma(t\bar{t}X)\times{\rm Br}(X\to
t\bar{t}).$ (3)

Note that the resonance $X$ can also decay into a pair of gluons or photons
through a top-quark triangle loop in the top-philic model. The loop-induced
decays are suppressed by a loop factor $\alpha_{s}/4\pi$ or $\alpha_{e}/4\pi$,
resulting in a small branching ratios $\lesssim 1\%$ for a heavy $X$. We
assume ${\rm Br}(X\to t\bar{t})=1$ in the study for simplicity. Therefore,
$\sigma(t\bar{t}t\bar{t})$ of the NP process of $pp\to t\bar{t}X\to
t\bar{t}t\bar{t}$ can be simplified as following:

$\displaystyle\sigma^{\rm NP}_{t\bar{t}X}=\kappa_{X}^{2}\sigma^{\rm
Res}_{t\bar{t}X}(M)\times{\rm Br}(X\to t\bar{t}),$ (4)

while for the process of $pp\to X\bar{X}\to t\bar{t}t\bar{t}$,

$\displaystyle\sigma^{\rm NP}_{X\bar{X}}=\sigma^{\rm
Res}_{X\bar{X}}(M)\times{\rm Br}^{2}(X\to t\bar{t}).$ (5)

The absence of $\kappa_{X}$ dependence in $\sigma^{\rm NP}_{X\bar{X}}$ is due
to the fact that only gauge interaction involves in the $X\bar{X}$ pair
production.

As the width $\Gamma$ increases dramatically with $\kappa_{X}$, the width
effect can be sizable and then the NWA might not be valid. Equations 4 and 5
are meant only to illustrate the dependence of $\sigma(t\bar{t}t\bar{t})$ on
NP parameters, and we keep the width effect of $X$ particle thoroughly in our
calculation. We adopt the CERN LEP line-shape prescription of a resonance
state Cao:2004yy  and write the $X$ propagator as

$\frac{1}{(p^{2}-M^{2})+iM\Gamma\left(p^{2}/{M^{2}}\right)}$ (6)

which makes a distortion on the dispersion relation at $\Gamma^{2}/2M^{2}$
level. A comparison between the full width effect and NWA is present later.

When the $X$ particle is below the top quark pair mass threshold ($M\leq
340~{}{\rm GeV}$) or too heavy to be produced directly at the LHC, the $X$
particle in the intermediate state can never be on mass shell. Therefore, the
$t\bar{t}t\bar{t}$ production then depends on $\kappa_{X}$ and $M$,

$\displaystyle\sigma^{\rm total}=\sigma^{\rm
SM}_{t\bar{t}t\bar{t}}+\kappa_{X}^{2}\sigma^{\rm
Int}_{t\bar{t}t\bar{t}}(M)+\kappa_{X}^{4}\sigma^{\rm
Res}_{t\bar{t}t\bar{t}}(M).$ (7)

For colored resonances the process in Fig. 1(a) is dominant for a light $X$ as
it consists of more Feynman diagrams than the $X\bar{X}$ pair production (b).
On the other hand, the process in Fig. 1(b) dominates when $m_{X}>2m_{t}$.

2\. New Physics Resonances.

Now consider the LHC search on NP resonances. Rather than focusing on specific
NP models, we consider various simplified NP models which extend the SM with
an additional NP resonance. The effective Lagrangians are listed as follows:

  * •

A color singlet scalar ($\mathcal{S}$), e.g., Dev:2014yca ; Alvarez:2016nrz ;
Alvarez:2019uxp ,

$\displaystyle\mathcal{L}\supset-\bar{t}(a_{t}+i\gamma_{5}b_{t})t\mathcal{S}\,,$
(8)

where $a_{t}=0$ and $b_{t}=0$ corresponds to a CP-odd scalar ($A$) and a CP-
even scalar ($H$), respectively, and the case of $a_{t}\neq 0$ and $b_{t}\neq
0$ represents a CP-mixture scalar;

  * •

a color octet scalar ($\mathcal{S}_{8}$), e.g., sgluon Calvet:2012rk ;
Darme:2018dvz ,

$\displaystyle\mathcal{L}\supset\text{Tr}[(D_{\mu}\mathcal{S}_{8})^{\dagger}(D^{\mu}\mathcal{S}_{8})]-\mathcal{S}_{8}^{A}\overline{t}(g_{8}^{a}+i\gamma_{5}g_{8}^{b})T^{A}t,$
(9)

where $T^{A}$ stands for the $SU(3)_{C}$ generator;

  * •

a color octet vector ($\mathcal{V}_{8\mu}$), e.g., axigluon Antunano:2007da ;
Chivukula:2010fk , color octet boson Ferrario:2008wm ; AguilarSaavedra:2011ck
, KK gluon Guchait:2007jd ,

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\frac{1}{2}D_{\nu}\mathcal{V}_{8\mu}^{A}(D^{\nu}\mathcal{V}_{8}^{\mu
A}-D^{\mu}\mathcal{V}_{8}^{\nu A})$
$\displaystyle-\frac{1}{2}g_{s}f^{ABC}G_{\mu\nu
A}\mathcal{V}_{8\mu}^{B}\mathcal{V}_{8\mu}^{C}$
$\displaystyle+\mathcal{V}_{8\mu}^{A}\overline{t}(g_{8}^{V}\gamma^{\mu}+g_{8}^{A}\gamma^{\mu}\gamma_{5})T^{A}t\,,$
(10)

where $g_{8}^{V}$ and $g_{8}^{A}$ denotes the vector and axial vector
coupling, respectively;

  * •

a color singlet vector ($\mathcal{V}_{\mu}$)Greiner:2014qna ; Kim:2016plm ,

$\displaystyle\mathcal{L}\supset
V_{\mu}\overline{t}(g^{V}\gamma^{\mu}+g^{A}\gamma^{\mu}\gamma^{5})t\,,$ (11)

where $g^{V}$ and $g^{A}$ presents the vector and axial vector coupling,
respectively;

  * •

a color sextet and EW singlet scalar $\mathcal{S}_{6}$ 111The color sextet
scalar could be an EW singlet or triplet under $SU(2)_{L}$, which couples to
right-handed or left-handed fermions in the SM, respectively. As an EW triplet
and color sextet, the scalar could induce rich collider signatures such as
$b\bar{b}\to t\bar{t}$. For simplicity we focus on the case of $S_{6}$ being a
EW singlet here. , e.g., color sextet scalar Chen:2008hh ; Berger:2010fy ;
Zhang:2010kr ,

$\displaystyle\mathcal{L}$
$\displaystyle\supset~{}\text{Tr}[(D_{\mu}\mathcal{S}_{6})^{\dagger}(D^{\mu}\mathcal{S}_{6})]$
$\displaystyle+~{}g_{6}\mathcal{S}_{6}^{A}\overline{t}_{R}\overline{K}^{A}t^{C}_{R}+\mbox{h.c.}\,,$
(12)

where $K^{A}$ stands for the Clebsh-Gordon coefficient.

Equipped with the effective Lagrangians shown above, we are ready to check the
validation of the NWA. The decay width of NP resonances into a pair of top
quarks, $\Gamma(X)\equiv\Gamma(X\to t\bar{t})$ or $\Gamma(X\to tt)$, are

$\displaystyle\Gamma(\mathcal{S})=$
$\displaystyle\frac{3M}{8\pi}\left[(a_{t})^{2}\beta_{t}^{2}+(b_{t})^{2}\right]\beta_{t},$
(13) $\displaystyle\Gamma(\mathcal{S}_{8})=$
$\displaystyle\frac{3M}{48\pi}\left[(g_{8}^{a})^{2}\beta_{t}^{2}+(g_{8}^{b})^{2}\right]\beta_{t},$
(14) $\displaystyle\Gamma(\mathcal{V})=$
$\displaystyle\frac{M}{4\pi}\left[(g^{V})^{2}(1+2\frac{m_{t}^{2}}{M^{2}})+(g^{A})^{2}\beta_{t}^{2}\right]\beta_{t},$
(15) $\displaystyle\Gamma(\mathcal{V}_{8})=$
$\displaystyle\frac{M}{24\pi}\left[(g_{8}^{V})^{2}(1+2\frac{m_{t}^{2}}{M^{2}})+(g_{8}^{A})^{2}\beta_{t}^{2}\right]\beta_{t},$
(16) $\displaystyle\Gamma(\mathcal{S}_{6}^{A})=$
$\displaystyle\frac{g_{6}^{2}M}{8\pi}\left(1-2\frac{m_{t}^{2}}{M^{2}}\right)\beta_{t},$
(17)

where $\beta_{t}=\sqrt{1-4m_{t}^{2}/M^{2}}$ is the velocity of top quark. As
the colored resonance exhibits a narrower width in comparison with the color
neutral objects, e.g. the width of a color-octet (sextet) resonance is $1/6$
($1/3$) of a color neutral resonance, respectively, we examine the width-to-
mass ratio of color-neutral resonances below.

Figure 2 plots the ratio $\Gamma/M$ as a function of $M$ for a color singlet
scalar (red) and color singlet vector (blue). For demonstration we choose two
sets of coupling parameters. When the strength of its coupling to top quark is
large, a heavy color-singlet scalar exhibits a large width, e.g.
$\Gamma/M\simeq 23\%$ in the region of $M\sim 2000~{}{\rm GeV}$; see the red-
solid curve. However, for such a heavy scalar, the $\sigma(t\bar{t}t\bar{t})$
is highly suppressed such that the large width leads to a mild effect in the
constraints on NP resonance. The width effect turns to be sizable in the
$t\bar{t}t\bar{t}$ production in the region of $M\sim 800~{}{\rm GeV}$. On the
other hand, the width-to-mass ratio of the colored scalar and vector do not
exceed $5\%$ such that the NWA works well and the interference effect can be
safely dropped. In this study we include the full width effect in the
calculation to constrain the NP resonance at the LHC and a comparison of the
full width and NWA is made in the end of the section.

Figure 2: Ratio $\Gamma/M$ as a function of $M$ for a color-neutral scalar
(red) and color-neutral vector (blue).  Figure 3: The $t\bar{t}t\bar{t}$
production cross section as a function of $\kappa_{X}$ in the extended
simplified models for $M=300~{}{\rm GeV}$. The solid lines represent the
scalar or vector couplings while the dashed curves the pseudo-scalar or axial-
vector couplings.

Next we show the comparison of the four top quark production rate in those
simplified models. First we focus on a light resonance, namely its mass is
smaller than $2m_{t}$ and we fix the mass of 300 GeV. For such a light
resonance, it can only contribute to four top quark production via off-shell
effects and its width effect can be neglected in the cross section
calculation. It means we can parametrize the cross section in terms of its
couplings. We assume that both the SM and NP processes exhibit the same
$K$-factor of 1.58 Frederix:2017wme . The cross section of $t\bar{t}t\bar{t}$
production induced by 300 GeV NP resonances at the 13 TeV LHC are given by

$\displaystyle\sigma^{\rm total}_{\mathcal{S}}=$ $\displaystyle
9.608+1.414(a_{t})^{2}+3.999(b_{t})^{2}+$ $\displaystyle
3.724(a_{t})^{4}+10.771(a_{t}b_{t})^{2}+8.750(b_{t})^{4},$
$\displaystyle\sigma^{\rm total}_{\mathcal{V}}=$ $\displaystyle
9.608-7.728(g^{V})^{2}+17.394(g^{A})^{2}+$ $\displaystyle
34.604(g^{V})^{4}+32.648(g^{V}g^{A})^{2}+75.216(g^{A})^{4},$
$\displaystyle\sigma^{\rm total}_{\mathcal{S}_{8}}=$ $\displaystyle
9.608+0.378(g_{8}^{a})^{2}+0.623(g_{8}^{b})^{2}+$ $\displaystyle
1.522(g_{8}^{a})^{4}+6.017(g_{8}^{a}g_{8}^{b})^{2}+3.244(g_{8}^{b})^{4},$
$\displaystyle\sigma^{\rm total}_{\mathcal{V}_{8}}=$ $\displaystyle
9.608+25.895(g_{8}^{V})^{2}+0.412(g_{8}^{A})^{2}+$ $\displaystyle
63.398(g_{8}^{V})^{4}+82.039(g_{8}^{V}g_{8}^{A})^{2}+34.292(g_{8}^{A})^{4},$
$\displaystyle\sigma^{\rm total}_{\mathcal{S}_{6}}=$ $\displaystyle
9.608+0.373(g_{6})^{2}+16.435(g_{6})^{4}.$

Throughout the paper all the cross sections are in the unit of femtobarn (fb).
The magnitude of the coefficients before effective coupling combinations
reveals the relative size of interference and NP contribution in comparison
with the SM prediction (i.e. the constant term in above equation). We also
plot the $\sigma(t\bar{t}t\bar{t})$ as a function of effective couplings
(denoted by $\kappa_{X}$) in Fig. 3, and for simplicity we consider one
coupling at a time. The solid lines represent the scalar or vector couplings
while the dashed curves denote the pseudo-scalar or axial-vector couplings.
Obviously, when the coupling approaches to 1, the NP contribution alone, i.e.
the terms proportional to $\kappa_{X}^{4}$, tends to dominate the production
cross section. For a medium $\kappa_{X}$, the axial-vector and pseduo-scalar
couplings enhance the cross section sizably owing to the interference with the
SM contribution.

Second, we consider the production of an 800 GeV resonance in the
$t\bar{t}t\bar{t}$ production. For a color neutral resonance, the
$t\bar{t}t\bar{t}$ production is dominated by the $t\bar{t}X$ association
production with $X\to t\bar{t}$; while for a colored resonance, the $X\bar{X}$
pair production through pure QCD with $X\to t\bar{t}$ in consequence
overwhelmingly dominates the $t\bar{t}t\bar{t}$ production assuming ${\rm
Br}(X\to t\bar{t})=1$. Under the NWA approximation, the $t\bar{t}t\bar{t}$
production cross sections read as

$\displaystyle\sigma^{\rm total}_{\mathcal{S}}=$ $\displaystyle
9.608+5.926(a_{t})^{2}+7.076(b_{t})^{2},$ (18) $\displaystyle\sigma^{\rm
total}_{\mathcal{V}}=$ $\displaystyle
9.608+35.804(g^{V})^{2}-1.830g^{V}g^{A}+$ $\displaystyle 35.804(g^{A})^{2},$
(19) $\displaystyle\sigma^{\rm total}_{\mathcal{S}_{8}}=$ $\displaystyle
189.477+20.166(g_{8}^{a})^{2}+19.4288(g_{8}^{b})^{2},$ (20)
$\displaystyle\sigma^{\rm total}_{\mathcal{V}_{8}}=$ $\displaystyle
6792.253+196.729(g_{8}^{V})^{2}+189.482(g_{8}^{A})^{2},$ (21)
$\displaystyle\sigma^{\rm total}_{\mathcal{S}_{6}}=$ $\displaystyle
349.305+63.197(g_{6})^{2}.$ (22)

The cross section depends only on the quadratic power of the effective
coupling $\kappa_{X}$ as the decay branching ratio of $X\to t\bar{t}$ is
assumed to be 1 and independent of $\kappa_{X}$. The constant terms represent
the sum of the SM contribution and the $X\bar{X}$ production, if applicable.

Now we are ready to explore constraints on various top-philic resonances from
the recent result of the $t\bar{t}t\bar{t}$ production reported by the CMS
collaboration, $\sigma(t\bar{t}t\bar{t})=12.6^{+5.8}_{-5.2}~{}{\rm fb}$
Sirunyan:2019wxt . As the sensitivity of the $t\bar{t}t\bar{t}$ production to
$\kappa_{X}$ highly depends on the production channel and the mass of the
resonance $M$, we consider both light resonances and heavy resonances in this
study.

Figure 4: Allowed region for a color-neutral vector $\mathcal{V}$ (a, b) and
scalar $\mathcal{S}$ (c, d). The red (blue, yellow) region in (a, c) presents
the resonance with a mass of 100 (200, 300) GeV, respectively, and in (b, d)
with a mass of 350 (800, 1000) GeV, respectively.

We begin with the case of a color singlet vector boson $\mathcal{V}$. Fig.
4(a) displays the allowed parameter region in the plane of ($g^{A}$, $g^{V}$)
for a light color-neutral vector. For illustration we consider three benchmark
masses, 100 GeV (red), 200 GeV (blue) and 300 GeV (yellow). The axial-vector
coupling is constrained more than the vector coupling as the axial-vector
contribution is enhanced by $m_{t}^{2}/m_{\mathcal{V}}^{2}$ for a light
resonance; for example, $|g^{A}|\lesssim 0.2$ and $|g^{V}|\lesssim 1$. Figure
4(b) shows the allowed parameter space of a heavy resonance for three
benchmark masses, 350 GeV (red), 800 GeV (blue) and 1000 GeV (yellow). The
major contribution to the $t\bar{t}t\bar{t}$ production is from the
$t\bar{t}\mathcal{V}$ production whose cross section depends on the quadratic
power of $g^{V/A}$’s; see $\sigma^{\rm total}_{\mathcal{V}}$ for $M=800~{}{\rm
GeV}$ in Eq. 19. It leads to a circle parameter space centering around
$g^{A}=g^{V}=0$. The $\sigma(t\bar{t}t\bar{t})$ decreases dramatically with
$M$ so as to weaken the bound; for example, $|g^{V,A}|\lesssim 0.5$ for
$M=1000~{}{\rm GeV}$ while $|g^{V,A}|\lesssim 0.2$ for $M=350~{}{\rm GeV}$.

Figure 4(c) and (d) shows the allowed parameter space for a light and heavy
color-neutral scalar $\mathcal{S}$, respectively. First, we observe a similar
pattern as the color-neutral vector but with weaker bounds. Note that the
contribution of the scalar interaction ($a_{t}\neq 0$, $b_{t}=0$) is
suppressed by the top-quark velocity, therefore, the scalar interaction is
less constrained. Second, owing to the small production rate, the TeV scalar
is loosely bounded, e.g. $\sqrt{a^{2}_{t}+b^{2}_{t}}\sim 1.5$; see the yellow
region in Fig. 4(d). The NP scalar exhibits a large width for such a large
coupling, e.g. $\Gamma/M\sim 30\%$ and cannot be treated as a fundamental
particle.

Figure 5: Allowed region for a light (a) and heavy (b) color-octet vector
boson $\mathcal{V}_{8}$. The red (blue, yellow) region in (a) denotes a vector
boson with mass of 100 (200, 300) GeV, respectively. The black curve in (b)
denotes $\sigma(t\bar{t}t\bar{t})$ as a function of $m_{\mathcal{V}_{8}}$
while the red curve represents the exclusion limit on
$\sigma(t\bar{t}t\bar{t})$ at the $2\sigma$ confidence level.

Next, we consider a color octet vector boson $\mathcal{V}_{8}$. Figure 5(a)
plots the allowed parameter space in the plane of ($g_{8}^{V}$, $g_{8}^{A}$)
for three benchmark masses of a light $\mathcal{V}_{8}$. The red (blue,
yellow) region in (a) denotes a vector boson with mass of 100 (200, 300) GeV,
respectively. The contribution of the axial-vector current interaction in the
four-top production is enhanced by $m_{t}^{2}/m_{\mathcal{V}_{8}}^{2}$,
therefore, the lighter the $\mathcal{V}_{8}$ is, the larger the
$\sigma(t\bar{t}t\bar{t})$ can be. For example, the bound of a 100 GeV vector
(red) is much tighter than the bound of a 300 GeV vector (yellow). However,
the enhancement disappears when $m_{\mathcal{V}_{8}}\sim 300~{}{\rm GeV}$ and
the interference with the SM gluon contribution, i.e. the vector coupling,
plays a leading role. As a result, the $g_{8}^{V}$ is constrained more tightly
than the $g_{8}^{A}$ for a 300 GeV resonance. When the vector boson mass
exceeds $2m_{t}$, as the vector boson carries color charge, the four top
production is dominated by the $\mathcal{V}_{8}\mathcal{V}_{8}$ production
with a subsequent decay of $\mathcal{V}_{8}\to t\bar{t}$. The production rate
depends only on the color charge of $\mathcal{V}_{8}$ but not on the
$g_{8}^{V}$ and $g_{8}^{A}$. The black curve in Fig. 5(b) shows
$\sigma(t\bar{t}t\bar{t})$, i.e. $\sigma(\mathcal{V}_{8}\mathcal{V}_{8})$ with
${\rm Br}(\mathcal{V}_{8}\to t\bar{t})=1$, at the LHC. The red line denotes
the exclusion bound at the $2\sigma$ confidence level which shows that the
mass of $\mathcal{V}_{8}$ is larger than 1.82 TeV.

Now we consider a color octet scalar $\mathcal{S}_{8}$. Figure 6(a) plots the
allowed parameter space of $g_{8}^{a}$ and $g_{8}^{b}$ where the red (blue,
yellow) region represents the $\mathcal{S}_{8}$ mass of 100 (200, 300) GeV,
respectively. When $m_{\mathcal{S}_{8}}\sim 300~{}{\rm GeV}$, the QCD pair
production becomes important and enlarges the production cross section,
yielding a smaller region; see the yellow oval. Once going above the top-quark
pair threshold, say $m_{\mathcal{S}_{8}}>2m_{t}$, the current data demands
$m_{\mathcal{S}_{8}}>1.19~{}{\rm TeV}$; see Fig. 6(b).

Figure 6: Allowed region for a light (a) and heavy (b) color octet scalar
$\mathcal{S}_{8}$. The red (blue, yellow) region represents the mass of 100
(200, 300) GeV, respectively. The black curve in (b) denotes
$\sigma(t\bar{t}t\bar{t})$ as a function of $M_{\mathcal{S}_{8}}$ while the
red curve represents the exclusion limit on $\sigma(t\bar{t}t\bar{t})$ at the
$2\sigma$ confidence level.

Last but not least, we discuss the case of a color sextet scalar
$\mathcal{S}_{6}$ Chen:2008hh ; Berger:2010fy ; Zhang:2010kr . The production
depends on $m_{\mathcal{S}_{6}}$ and $g_{6}$. Again, we consider both light
and heavy scalars. Figure 7(a) displays the bound on a light color-sextet
scalar where the region above the black curve is excluded by the current data.
Once $m_{\mathcal{S}_{6}}>2m_{t}$ the
$\mathcal{S}_{6}\mathcal{S}^{\dagger}_{6}$ pair production dominates the
$t\bar{t}t\bar{t}$ production such that the $\sigma(t\bar{t}t\bar{t})$ does
not depend on $g_{6}$ at all; see the black curve in Fig. 7(b). To respect the
current bound on $\sigma(t\bar{t}t\bar{t})$ (red), the mass of
$\mathcal{S}_{6}$ has to be larger than 1.38 TeV.

Figure 7: Limits on the color sextet scalar coupling and mass from current
four top quark production constraints.

Finally, we comment on the difference between the full width effect and NWA.
As shown in Fig. 2, for given mass and couplings, the width of a color neutral
resonance is larger than a colored object, we focus on the color singlet
scalar and vector. To emphasize the width effect, we choose a benchmark mass
for both the scalar and vector as $M=800~{}{\rm GeV}$. Figure 8 displays the
allowed parameter space in the plane of effective couplings for a color
neutral scalar (a) and vector (b), where the red solid curve represents the
bound with the full width effects while the blue dashed curve the bound with
the NWA. When the full width is incorporated into the calculation, the cross
section of the $t\bar{t}t\bar{t}$ production is slightly enlarged to yield
more stringent parameter space.

Figure 8: Comparison between the full width (red) and the NWA (blue dashed):
(a) color singlet scalar, (b) color singlet vector. For simplicity, the mass
of the scalar and the vector are chosen both as 800 GeV.

3\. High Dimensional Operators.

In this section we consider two samples of high dimension operators which
contribute to four top quark production. One is the dimension-6 four top-quark
operator in composite top-quark models Eichten:1983hw ; Lillie:2007hd ;
Pomarol:2008bh ; Kumar:2009vs ; Zhou:2012dz  in which the right-handed top
quark is composite. The four top quark contact operator is Zhang:2017mls

$\displaystyle\mathcal{O}_{t\bar{t}t\bar{t}}=\frac{g_{4}}{\Lambda^{2}}(\overline{t}_{R}\gamma_{\mu}t_{R})(\overline{t}_{R}\gamma^{\mu}t_{R}),$
(23)

where $\Lambda$ denotes the NP scale. It yields the cross section of
$t\bar{t}t\bar{t}$ production as following:

$\displaystyle\sigma^{\rm
total}=9.608-1.637~{}g_{4}\left(\frac{\mathrm{TeV}}{\Lambda}\right)^{2}+4.664~{}g_{4}^{2}\left(\frac{\mathrm{TeV}}{\Lambda}\right)^{4},$
(24)

where the constant term denotes the SM contribution, the linear term of
$g_{4}$ is the interference between the SM and
$\mathcal{O}_{t\bar{t}t\bar{t}}$, and the quadratic term of $g_{4}$ represents
the contribution from $\mathcal{O}_{t\bar{t}t\bar{t}}$ alone. The pure QCD
corrections to the interference and quadratic term are calculated in
Degrande:2020evl , namely 0.57 and 0.93 respectively. We end up with a
constraint on the $g_{4}$ coupling from the current data as

$\displaystyle-1.34<g_{4}\left(\frac{\mathrm{TeV}}{\Lambda}\right)^{2}<1.55\,.$
(25)

The other sample is top quark dipole operator given by Malekhosseini:2018fgp

$\displaystyle\mathcal{L}\supset\frac{g_{s}}{m_{t}}\overline{t}T^{A}(d_{V}+id_{A}\gamma_{5})i\sigma_{\mu\nu}tG^{\mu\nu
A}.$ (26)

where $\sigma_{\mu\nu}\equiv[\gamma_{\mu},\gamma_{\nu}]/2\,$. The dipole
operator interferes with the SM diagram in a complicated manner. For example,
its interference between the electric dipole operator and the SM depends
mainly on the even power of the Wilson coefficient as the electric dipole
operator violates the CP parity. It thus yields a symmetric bound on $d_{A}$.
On the other hand, the interference between the magnetic dipole operator and
the SM depends on the odd power of Wilson coefficient and yields an asymmetric
constraint on the $d_{V}$. Figure 9 displays the allowed parameter space in
the plane of $d_{V}$ and $d_{A}$ with respect to the current LHC data.

Figure 9:  Allowed region for the Wilson coefficients of the top color
electric dipole operator and color magnetic dipole operator.

4\. A Broad Resonance.

At last, we investigate a special case of a broad vector resonance
$\rho^{\mu}$ proposed in Ref. Liu:2019bua . The resonance decays mainly into a
pair of top quarks or bottom quarks, and the Lagrangian reads

$\displaystyle\mathcal{L}\supset
g_{t}(\overline{t}_{L}\gamma_{\mu}t_{L}-\overline{b}_{L}\gamma_{\mu}b_{L})\rho^{\mu}\,,$
(27)

where the coupling $g_{t}$ describes strong dynamics and can be fairly large.
Such a large $g_{t}$ inevitably generates a very broad width of $\rho^{\mu}$.
It is necessary to modify the Breit-Wigner distribution of the $\rho^{\mu}$
propagator to take care of the broad width. We modify the broad resonance
propagator utilizing the CERN LEP line-shape scheme Cao:2004yy . To be
distinct from the previous study of the NP resonance with narrow width, we
focus on the large $g_{t}$ and broad width here. Figure 10 displays the
exclusion limit of $g_{t}$ as a function of $M_{\rho}$, which shows that
$g_{t}$ increases with $M_{\rho}$ linearly for a heavy $\rho$ with a mass of
several TeVs. This is because the heavy resonance contribution to the
$t\bar{t}t\bar{t}$ production depends on the quadratic and quartic powers of
coupling-mass-ratio, e.g. $g_{t}/M_{\rho}$, after utilizing the modified
propagator.

Figure 10: The excluded parameter region for the broad vector resonance model.
Only the large coupling region is shown.

5\. Discussion and Conclusion.

Despite the rare rate at the LHC, four-top production is the best channel to
probe the so-called top-philic resonances that couple only to top quark. We
studied various new physics resonances and, based on the current data at the
LHC, explored the constraints on the mass of new resonance and the effective
coupling of new resonance to the top quark. When the effective coupling is
small, one can use narrow width approximation to simplify the study. However,
when the coupling is large, the width effect is no longer negligible. A
comparison of the full width effect and narrow width approximation is made in
the study of the color-neutral scalar and vector. A special case of strong
dynamics model is also addressed.

Note added: While finalizing the manuscript a nice work dealing with the same
topic appears online Darme:2021xxu .

Acknowledgements

The work is supported in part by the National Science Foundation of China
under Grant Nos. 11725520, 11675002, 11635001, 11805013, 12075257, the
Fundamental Research Funds for the Central Universities under Grant No.
2018NTST09, the funding from the Institute of High Energy Physics, Chinese
Academy of Sciences (Y6515580U1) and the funding from Chinese Academy of
Sciences (Y8291120K2).

## References

  * (1) V. D. Barger, A. L. Stange and R. J. N. Phillips, Phys. Rev. D 44, 1987-1996 (1991) doi:10.1103/PhysRevD.44.1987
  * (2) R. Frederix, D. Pagani and M. Zaro, JHEP 02, 031 (2018) doi:10.1007/JHEP02(2018)031 [arXiv:1711.02116 [hep-ph]].
  * (3) Q. H. Cao, S. L. Chen and Y. Liu, Phys. Rev. D 95, no.5, 053004 (2017) doi:10.1103/PhysRevD.95.053004 [arXiv:1602.01934 [hep-ph]].
  * (4) Q. H. Cao, S. L. Chen, Y. Liu, R. Zhang and Y. Zhang, Phys. Rev. D 99, no.11, 113003 (2019) doi:10.1103/PhysRevD.99.113003 [arXiv:1901.04567 [hep-ph]].
  * (5) A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 78, no.2, 140 (2018) doi:10.1140/epjc/s10052-018-5607-5 [arXiv:1710.10614 [hep-ex]].
  * (6) A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 80, no.2, 75 (2020) doi:10.1140/epjc/s10052-019-7593-7 [arXiv:1908.06463 [hep-ex]].
  * (7) M. Aaboud et al. [ATLAS], Phys. Rev. D 99, no.5, 052009 (2019) doi:10.1103/PhysRevD.99.052009 [arXiv:1811.02305 [hep-ex]].
  * (8) H. Pilkuhn, The Interactions of Hadrons, (North-Holland, Amsterdam, 1967).
  * (9) Q. H. Cao and C. P. Yuan, Phys. Rev. Lett. 93, 042001 (2004) doi:10.1103/PhysRevLett.93.042001 [arXiv:hep-ph/0401026 [hep-ph]].
  * (10) P. S. Bhupal Dev and A. Pilaftsis, JHEP 12, 024 (2014) [erratum: JHEP 11, 147 (2015)] doi:10.1007/JHEP12(2014)024 [arXiv:1408.3405 [hep-ph]].
  * (11) E. Alvarez, D. A. Faroughy, J. F. Kamenik, R. Morales and A. Szynkman, Nucl. Phys. B 915, 19-43 (2017) doi:10.1016/j.nuclphysb.2016.11.024 [arXiv:1611.05032 [hep-ph]].
  * (12) E. Alvarez, A. Juste and R. M. S. Seoane, JHEP 12, 080 (2019) doi:10.1007/JHEP12(2019)080 [arXiv:1910.09581 [hep-ph]].
  * (13) S. Calvet, B. Fuks, P. Gris and L. Valery, JHEP 04, 043 (2013) doi:10.1007/JHEP04(2013)043 [arXiv:1212.3360 [hep-ph]].
  * (14) L. Darmé, B. Fuks and M. Goodsell, Phys. Lett. B 784, 223-228 (2018) doi:10.1016/j.physletb.2018.08.001 [arXiv:1805.10835 [hep-ph]].
  * (15) O. Antunano, J. H. Kuhn and G. Rodrigo, Phys. Rev. D 77, 014003 (2008) doi:10.1103/PhysRevD.77.014003 [arXiv:0709.1652 [hep-ph]].
  * (16) R. S. Chivukula, E. H. Simmons and C. P. Yuan, Phys. Rev. D 82, 094009 (2010) doi:10.1103/PhysRevD.82.094009 [arXiv:1007.0260 [hep-ph]].
  * (17) P. Ferrario and G. Rodrigo, Phys. Rev. D 78, 094018 (2008) doi:10.1103/PhysRevD.78.094018 [arXiv:0809.3354 [hep-ph]].
  * (18) J. A. Aguilar-Saavedra and J. Santiago, Phys. Rev. D 85, 034021 (2012) doi:10.1103/PhysRevD.85.034021 [arXiv:1112.3778 [hep-ph]].
  * (19) M. Guchait, F. Mahmoudi and K. Sridhar, Phys. Lett. B 666, 347-351 (2008) doi:10.1016/j.physletb.2008.07.085 [arXiv:0710.2234 [hep-ph]].
  * (20) N. Greiner, K. Kong, J. C. Park, S. C. Park and J. C. Winter, JHEP 04, 029 (2015) doi:10.1007/JHEP04(2015)029 [arXiv:1410.6099 [hep-ph]].
  * (21) J. H. Kim, K. Kong, S. J. Lee and G. Mohlabeng, Phys. Rev. D 94, no.3, 035023 (2016) doi:10.1103/PhysRevD.94.035023 [arXiv:1604.07421 [hep-ph]].
  * (22) C. R. Chen, W. Klemm, V. Rentala and K. Wang, Phys. Rev. D 79, 054002 (2009) doi:10.1103/PhysRevD.79.054002 [arXiv:0811.2105 [hep-ph]].
  * (23) E. L. Berger, Q. H. Cao, C. R. Chen, G. Shaughnessy and H. Zhang, Phys. Rev. Lett. 105, 181802 (2010) doi:10.1103/PhysRevLett.105.181802 [arXiv:1005.2622 [hep-ph]].
  * (24) H. Zhang, E. L. Berger, Q. H. Cao, C. R. Chen and G. Shaughnessy, Phys. Lett. B 696, 68-73 (2011) doi:10.1016/j.physletb.2010.12.005 [arXiv:1009.5379 [hep-ph]].
  * (25) E. Eichten, K. D. Lane and M. E. Peskin, Phys. Rev. Lett. 50, 811-814 (1983) doi:10.1103/PhysRevLett.50.811
  * (26) B. Lillie, J. Shu and T. M. P. Tait, JHEP 04, 087 (2008) doi:10.1088/1126-6708/2008/04/087 [arXiv:0712.3057 [hep-ph]].
  * (27) A. Pomarol and J. Serra, Phys. Rev. D 78, 074026 (2008) doi:10.1103/PhysRevD.78.074026 [arXiv:0806.3247 [hep-ph]].
  * (28) K. Kumar, T. M. P. Tait and R. Vega-Morales, JHEP 05, 022 (2009) doi:10.1088/1126-6708/2009/05/022 [arXiv:0901.3808 [hep-ph]].
  * (29) N. Zhou, D. Whiteson and T. M. P. Tait, Phys. Rev. D 85, 091501 (2012) doi:10.1103/PhysRevD.85.091501 [arXiv:1203.5862 [hep-ph]].
  * (30) C. Zhang, Chin. Phys. C 42, no.2, 023104 (2018) doi:10.1088/1674-1137/42/2/023104 [arXiv:1708.05928 [hep-ph]].
  * (31) C. Degrande, G. Durieux, F. Maltoni, K. Mimasu, E. Vryonidou and C. Zhang, Phys. Rev. D 103, no.9, 096024 (2021) doi:10.1103/PhysRevD.103.096024 [arXiv:2008.11743 [hep-ph]].
  * (32) M. Malekhosseini, M. Ghominejad, H. Khanpour and M. Mohammadi Najafabadi, Phys. Rev. D 98, no.9, 095001 (2018) doi:10.1103/PhysRevD.98.095001 [arXiv:1804.05598 [hep-ph]].
  * (33) D. Liu, L. T. Wang and K. P. Xie, Phys. Rev. D 100, no.7, 075021 (2019) doi:10.1103/PhysRevD.100.075021 [arXiv:1901.01674 [hep-ph]].
  * (34) L. Darmé, B. Fuks and F. Maltoni, [arXiv:2104.09512 [hep-ph]].


#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
New results of $\gamma$ measurements in ADS and GLW-like decays at LHCb

Seophine Stanislaus on behalf of the LHCb collaboration
<EMAIL_ADDRESS>Department of Physics, University of Oxford,
Oxford, United Kingdom

(March 28, 2024)

###### Abstract

Recent LHCb measurements of the CKM angle $\gamma$ in ADS and GLW-like decays
are presented. One measurement considers $B^{0}\rightarrow DK^{*0}(892)$
decays with two- and four-body $D$-decay final states and another considers
$B^{\pm}\rightarrow Dh^{\pm}$ decays with $D\rightarrow K\pi\pi\pi$ final
states. The former supersedes a previous LHCb measurement and was presented
for the first time at the 12th Workshop on the CKM Unitarity Triangle. The
latter analysis measures the largest $C\\!P$ asymmetry seen at LHCb by
splitting the $D$-decay phase space into bins.

## Introduction

The Standard Model describes $C\\!P$-violation in the quark sector using the
Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2] which can be represented using
the Unitarity Triangle [3]. In this triangle, measurements of the CKM angle
$\gamma$ are particularly important as it is the only angle that can be
measured using only tree-level processes which are assumed not to include
physics beyond the Standard Model [4, 5]. Therefore, direct measurements of
$\gamma$ can be compared to indirect determinations which are sensitive to
loop contributions from new physics to test the Standard Model. Measurements
of CKM angle $\gamma$ are also useful as they have negligible theoretical
uncertainty since all hadronic parameters are determined from data [6].

The LHCb collaboration has performed numerous measurements of $\gamma$ where
direct $C\\!P$-violation is measured using the interference of $b\rightarrow
u\bar{c}s$ and $b\rightarrow c\bar{u}s$ quark transitions in, for example,
${B^{\pm}\rightarrow DK^{\pm}}$ decays. The weak interaction vertices involved
allow access to $\gamma$ which can be approximated to be
$\arg\left[V_{cb}V_{us}^{*}/V_{ub}V_{cs}^{*}\right]$ [7]. The LHCb detector
[8, 9] is well-suited for this topology of decay. The Vertex Locator (VELO)
provides excellent vertex resolution to identify displaced $B$ and $D$
vertices. Particle identification is also a vital component and is achieved
using the VELO, tracking system and, most importantly, the RICH detectors.

Having studied numerous charged and neutral $B$ decays, the latest combination
of $\gamma$ measurements from LHCb produced a value of
$\gamma=(63.8^{+3.5}_{-3.7})^{\circ}$ [10]. To fully exploit a $B$ decay, it
is important to consider different $D$ final states since each gives different
sensitivity to different hadronic parameters. This work presents two recent
results from LHCb [11, 12] which studied two- and four-body $D$ final states
and $C\\!P$-violation is measured using the ADS [13, 14] and GLW [15, 16]
methods.

##  $B^{0}\rightarrow DK^{*0}(892)$ Decays

A measurement of $C\\!P$-violation in $B^{0}\rightarrow DK^{*0}(892)$ decays
with two- and four-body $D$ final states was recently completed by LHCb [11].
The measurement supersedes results from a previous LHCb Run 1 and partial Run
2 analysis [17], which corresponded to a total integrated luminosity of 4.8
fb-1. The measurement presented here uses the full 9 fb-1 Run 1 and Run 2
dataset.

When considering $B^{0}\rightarrow DK^{*0}(892)$ decays, the charge on the
kaon from the $K^{*0}$ meson is used to tag the flavour of the $B$ meson at
decay. This results in a time-integrated measurement with no effects from
$B^{0}$–$\overline{B^{0}}$ mixing. In the case of $B^{0}\rightarrow
DK^{*0}(892)$ decays, although the $K^{*0}$ resonance is selected, there are
also non-resonant contributions. This means that a coherence factor, $\kappa$,
is necessary to account for the small contribution from other intermediate
states. The value for this is taken from Ref. [18], and therefore the same
selection for the $K^{*0}$ meson is used. Details on the other selections can
be found in Ref. [11].

### GLW-like decays

One way to measure $C\\!P$-violation is using $D$-decay final states which are
$C\\!P$-eigenstates. These are studied using the GLW [15, 16] method. In
addition, quasi-GLW decays refer to final states which have a significant
$C\\!P$ content, meaning the amplitude for the $D^{0}$ and $\overline{D^{0}}$
decays are roughly, but not exactly, equal. This is quantified by the hadronic
parameter, $r_{D}$, the ratio of the two amplitudes, and the strong-phase
difference between them, $\delta_{D}$. The amplitude of the ${B^{0}\rightarrow
D^{0}K^{*0}}$ decay is smaller than that of the
${B^{0}\rightarrow\overline{D^{0}}K^{*0}}$ decay, and this is quantified by
the hadronic parameter, $r_{B}$. The strong and weak phase differences between
the amplitudes are given by $\delta_{B}$ and $\gamma$, respectively. This
dependence on $\gamma$ indicates that interference is required to measure it.
The sensitivity to $\gamma$ is proportional to the size of $r_{B}$, which for
${B^{0}\rightarrow DK^{*0}(892)}$ decays is around 0.25 [10]. To measure
$\gamma$, the ratio of the rate compared to a control mode, which are Cabibbo-
favoured modes, and the asymmetry between the $B^{0}$ and ${\overline{B^{0}}}$
decays are measured. These are defined as follows and related to the physical
parameters ($r_{B}$, $\delta_{B}$, and $\gamma$) as

$R_{CP}\equiv\frac{\Gamma(B^{0}\rightarrow[h^{+}h^{-}]_{D}K^{*0})+\Gamma(\overline{B^{0}}\rightarrow[h^{+}h^{-}]_{D}\overline{K^{*0}})}{\Gamma(B^{0}\rightarrow[K^{-}\pi^{+}]_{D}K^{*0})+\Gamma(\overline{B^{0}}\rightarrow[K^{+}\pi^{-}]_{D}\overline{K^{*0}})}\frac{\mathcal{B}(D^{0}\rightarrow
K^{-}\pi^{+})}{\mathcal{B}(D^{0}\rightarrow h^{+}h^{-})}$
$=\frac{1+r_{B}^{2}+2r_{B}\kappa\cos(\delta_{B})\cos(\gamma)}{1+r_{B}^{2}r_{D}^{2}+2r_{B}r_{D}\kappa\cos(\delta_{B}-\delta_{D})\cos(\gamma)},$
(1)
$A_{CP}\equiv\frac{\Gamma(B^{0}\rightarrow[h^{+}h^{-}]_{D}K^{*0})-\Gamma(\overline{B^{0}}\rightarrow[h^{+}h^{-}]_{D}\overline{K^{*0}})}{\Gamma(B^{0}\rightarrow[h^{+}h^{-}]_{D}K^{*0})+\Gamma(\overline{B^{0}}\rightarrow[h^{+}h^{-}]_{D}\overline{K^{*0}})}=\frac{2r_{B}\kappa\sin(\delta_{B})\sin(\gamma)}{1+r_{B}^{2}+2\kappa\cos(\delta_{B})\cos(\gamma)}.$
(2)

The ratios and asymmetries are determined by fitting the mass distribution of
the $DK\pi$ system from either $B^{0}$ or $\overline{B^{0}}$ decays. Results
of these fits for the control modes are shown in Fig. 1. The asymmetry,
$A_{K\pi}=0.033\pm 0.017\pm 0.015$, is consistent with no asymmetry as
expected. This is since the interference in this control mode is smaller than
the experimental sensitivity.

For these asymmetries, there are contributions from not only interference but
also production and detection asymmetries, for which corrections must be
applied. The value for the production asymmetry is taken from Ref. [19],
$A_{\text{prod}}=(-8\pm 5)\times 10^{-3}$. The detection asymmetry is
determined using the difference in detection asymmetries for kaons and pions.
This is estimated using calibration data for $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{+}$ and $D^{+}\rightarrow K_{S}^{0}\pi^{+}$ decays. The
asymmetries are then weighted to match the signal kinematic distributions
resulting in $A_{\text{det}}=(-9.8\pm 5.5)\times 10^{-3}$. The detection
asymmetry correction is then applied depending on the difference between the
number of kaons and pions of the same sign in the final state [11].

There are also efficiency corrections to consider. Since the $C\\!P$
observables are calculated from ratios, most efficiencies cancel but two
corrections must be applied. Firstly, due to the selection. This is determined
by applying the selection to simulation. Secondly, charge-dependent PID
efficiencies. These are estimated using calibration data for kaons and pions
from $D^{*}$ decays. These vary with track momentum, and therefore
efficiencies determined using calibration data are then weighted to match the
kinematics of the signal decays using simulation.

An example of the results for GLW-like decays is shown in Fig. 1 for
$D\rightarrow K^{+}K^{-}$ decays. Here, the asymmetry is small,
$A_{{C\\!P}}^{KK}=-0.047\pm 0.063\pm 0.015$, but it is the rate measurement
which is important, $R_{{C\\!P}}^{KK}=0.817\pm 0.057\pm 0.017$. This would be
close to 1 if the interference effect were small. But the rate deviates from 1
and therefore contributes significant information towards $\gamma$. Alongside
the $D\rightarrow K^{+}K^{-}$ mode, the $D\rightarrow\pi^{+}\pi^{-}$ mode is
also measured and results are in Ref. [11].

Figure 1: Results (from top to bottom) from the $D\rightarrow
K^{\pm}\pi^{\mp}$ control mode, $D\rightarrow K^{\pm}K^{\mp}$ GLW mode,
$D\rightarrow\pi^{\pm}K^{\mp}$ ADS mode, $D\rightarrow\pi\pi\pi\pi$ GLW mode,
and $D\rightarrow\pi K\pi\pi$ ADS mode for (left) $\overline{B^{0}}$ decays
and (right) $B^{0}$ decay.

### ADS decays

The measurement in Ref. [11] also studies ADS-like decays. These are final
states which, depending on the $D$ flavour, occur via favoured or doubly-
Cabibbo suppressed decays. This means that the two interfering amplitudes (the
total amplitudes including the $B$ and $D$ decays) are of similar magnitudes,
resulting in maximal interference and greater sensitivity to $\gamma$. These
decays are measured using the ADS [13, 14] method where the $C\\!P$
observables are ratios of the rate of the suppressed to the favoured modes.
These are defined as,

$R^{+}_{\pi
K}\equiv\frac{\Gamma(\overline{B^{0}}\rightarrow[\pi^{+}K^{-}]_{D}\overline{K^{*0}})}{\Gamma(\overline{B^{0}}\rightarrow[K^{+}\pi^{-}]_{D}\overline{K^{*0}})}=\frac{r_{B}^{2}+r_{D}^{2}+2r_{B}r_{D}\kappa\cos(\delta_{B}+\delta_{D}+\gamma)}{1+r_{B}^{2}r_{D}^{2}+2r_{B}r_{D}\kappa\cos(\delta_{B}-\delta_{D}+\gamma)},$
(3) $R^{-}_{\pi
K}\equiv\frac{\Gamma(B^{0}\rightarrow[\pi^{-}K^{+}]_{D}K^{*0})}{\Gamma(B^{0}\rightarrow[K^{-}\pi^{+}]_{D}K^{*0})}=\frac{r_{B}^{2}+r_{D}^{2}+2r_{B}r_{D}\kappa\cos(\delta_{B}+\delta_{D}-\gamma)}{1+r_{B}^{2}r_{D}^{2}+2r_{B}r_{D}\kappa\cos(\delta_{B}-\delta_{D}-\gamma)}.$
(4)

The results from this mode with $D\rightarrow\pi^{\pm}K^{\mp}$ decays are
shown in Fig. 1 where $R^{+}_{\pi K}=0.069\pm 0.013\pm 0.005$ and $R^{-}_{\pi
K}=0.093\pm 0.013\pm 0.005$. To leading order, $R^{\pm}\approx r_{B}^{2}$.
Therefore, larger rates imply a larger $r_{B}$.

### Multi-body $D$ decays

In this measurement [11], multi-body $D$-decay final states are also studied.
The GLW-like decay is $D\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$, where the
$C\\!P$-even fraction is $F_{+}=0.735\pm 0.015\pm 0.005$ [20]. This alters the
$C\\!P$ observables by a factor of $(2F_{+}-1)$, for example,

$A_{CP}=\frac{2r_{B}\kappa{\color[rgb]{1,0,0}(2F_{+}-1)}\sin(\delta_{B})\sin(\gamma)}{1+r_{B}^{2}+2r_{B}\kappa\cos(\delta_{B})\cos(\gamma)}.$
(5)

For the ADS mode, $D\rightarrow\pi^{-}K^{+}\pi^{-}\pi^{+}$ decays are used and
this requires an additional coherence factor, $R_{K3\pi}$, due to intermediate
resonances. The value for this is taken from Ref. [21], and is
$R_{K3\pi}=0.44^{+0.10}_{-0.09}$. This alters the decay rates, for example,

$\Gamma(B^{0}\rightarrow[K^{+}\pi^{-}\pi^{+}\pi^{-}]_{D}K^{*0})\propto
r_{B}^{2}+r_{D}^{2}+2r_{B}r_{D}\kappa{\color[rgb]{1,0,0}R_{K3\pi}}\cos(\delta_{B}+\delta_{D}-\gamma),$
(6)

and therefore reduces the interference by a factor of 2.

The results from the GLW mode are shown in Fig. 1, where again a small
asymmetry is seen, $A_{{C\\!P}}^{4\pi}=0.014\pm 0.087\pm 0.016$, and the rate
is $R_{{C\\!P}}^{4\pi}=0.882\pm 0.086\pm 0.033$. The results from the ADS mode
are shown in Fig. 1, where $R^{+}_{\pi K\pi\pi}=0.060\pm 0.014\pm 0.005$ and
$R^{-}_{\pi K\pi\pi}=0.038\pm 0.014\pm 0.005$.

### Results

In general the statistical uncertainties on the $C\\!P$ observables from Ref.
[11] have improved by 60% compared to the previous analysis [17] due to the
increased signal yield. For both the two- and four-body $D$-decays, the
dominant source of systematic uncertainty in the GLW asymmetries is the
asymmetry correction, and in the GLW ratios it is due to external measurements
of the $D^{0}$ branching fractions. In the ADS mode there is no dominant
source of systematic uncertainty. Overall, the analysis is statistically
limited.

The $C\\!P$ observables can be interpreted in terms of the physical
observables using the procedure detailed in Ref. [22]. The trigonometric
functions in the relations of the $C\\!P$ observables and the low sensitivity
to asymmetries mean that there is a four-fold degeneracy, as shown in Fig. 2.
The preferred solution of $\gamma$, as shown on the right of Fig. 2, is at
$172\pm 6^{\circ}$ which includes both statistical and systematic
uncertainties. However, an alternative solution, which is consistent with the
world average of direct measurements, is at $62\pm 8^{\circ}$. Results from
the $B^{0}\rightarrow DK^{*0}$ analysis with $D\rightarrow K_{s}^{0}\pi\pi$
and $D\rightarrow K_{s}^{0}KK$ final states [23] are needed to break this
degeneracy.

Figure 2: The 68.3% and 95.5% confidence levels from Ref. [11] in (left)
$r_{B^{0}}^{DK^{*}}$ and $\gamma$, and (right) $\gamma$.

##  $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ decays
with $D\rightarrow K\pi\pi\pi$

A measurement of $C\\!P$-violation in $B^{\pm}\rightarrow DK^{\pm}$ and
$B^{\pm}\rightarrow D\pi^{\pm}$ decays with $D\rightarrow K\pi\pi\pi$ final
states was performed by LHCb [12]. The larger yields in $B^{\pm}\rightarrow
DK^{\pm}$ compared to $B^{0}\rightarrow DK^{*0}$ mean that the sensitivity to
$\gamma$ can be improved by splitting the $D$-decay phase space into bins
where the coherence factor is larger. This was proposed in Ref. [24] and the
binning determined in Ref. [21] is used.

### Results

The asymmetries are determined using fits to mass spectra of the $DK^{+}$ or
$DK^{-}$ system. Results for this in each bin are shown in Fig. 3, where bin 2
shows the largest $C\\!P$ asymmetry seen at LHCb. The bins labelled in these
plots are ordered by the average value of $\delta_{D}$. This means that from
bin 1 to 4, the asymmetry can be seen to rise and fall.

Using these measurements, $\gamma$ is found to be,
$\gamma=(54.8^{+6.0}_{-5.8}$${}^{+0.6}_{-0.6}$${}^{+6.7}_{-4.3})^{\circ}$,
where the first uncertainty is statistical, the second is systematic and the
third is due to external inputs. The last uncertainty is slightly larger than
the statistical uncertainty and updated external inputs are needed to improve
this.

Figure 3: Binned mass distributions in $m(DK^{-})$ and $m(DK^{+})$ from Ref.
[12].

## Conclusion

Two recent LHCb measurements of the CKM angle $\gamma$ which study ADS and
GLW-like decays have been presented. These measurements along with other
recent analyses from LHCb show that the target precision on $\gamma$ of
$4^{\circ}$ has been surpassed. This will continue to improve as on-going
analyses are completed and Run 3 data is analysed.

## References

  * [1] N. Cabibbo Phys. Rev. Lett., vol. 10, pp. 531–533, 1963.
  * [2] M. Kobayashi and T. Maskawa Prog. Theor. Phys., vol. 49, pp. 652–657, 1973.
  * [3] L. Wolfenstein Phys. Rev. Lett., vol. 51, no. 21, p. 1945, 1983.
  * [4] J. Brod, A. Lenz, G. Tetlalmatzi-Xolocotzi, and M. Wiebusch Phys. Rev., vol. D92, no. 3, p. 033002, 2015.
  * [5] A. Lenz and G. Tetlalmatzi-Xolocotzi JHEP, vol. 07, p. 177, 2020.
  * [6] J. Brod and J. Zupan JHEP, vol. 01, p. 51, 2014.
  * [7] Y. Grossman and M. Savastio JHEP, vol. 2014, no. 3, 2014.
  * [8] A. A. Alves Jr. et al. JINST, vol. 3, no. LHCb-DP-2008-001, p. S08005, 2008.
  * [9] R. Aaij et al. Int. J. Mod. Phys., vol. A30, p. 1530022, 2015.
  * [10] LHCb collaboration Tech. Rep. LHCb-CONF-2022-003, 2022.
  * [11] R. Aaij et al. no. LHCb-PAPER-2023-040, 2023.  submitted to JHEP.
  * [12] R. Aaij et al. JHEP, vol. 2023, no. 7, 2023.
  * [13] D. Atwood, I. Dunietz, and A. Soni Phys. Rev. Lett., vol. 78, pp. 3257–3260, 1997.
  * [14] D. Atwood, I. Dunietz, and A. Soni Phys. Rev. D, vol. 63, p. 036005, 2001.
  * [15] M. Gronau and D. Wyler Phys. Lett. B, vol. 265, no. 1, pp. 172–176, 1991.
  * [16] M. Gronau and D. London Phys. Lett. B, vol. 253, no. 3, pp. 483–488, 1991.
  * [17] R. Aaij et al. JHEP, vol. 08, p. 041, 2019.
  * [18] R. Aaij et al. Phys. Rev., vol. D93, p. 112018, 2016.
  * [19] R. Aaij et al. Phys. Lett. B, vol. 774, p. 139–158, 2017.
  * [20] M. Ablikim et al. Phys. Rev. D, vol. 106, p. 092004, 2022.
  * [21] M. Ablikim et al. JHEP, vol. 2021, no. 5, 2021.
  * [22] R. Aaij et al. JHEP, vol. 12, p. 087, 2016.
  * [23] R. Aaij et al. Eur. Phys. J. C, vol. 84, no. 206, 2023.
  * [24] T. Evans, J. Libby, S. Malde, and G. Wilkinson Phys. Lett. B, vol. 802, p. 135188, 2020.