# Uncovering the High Scale Higgs Singlet Model

Sally Dawson  Department of Physics, Brookhaven National Laboratory, Upton,
N.Y., 11973, U.S.A.  Pier Paolo Giardino  Instituto Galego de Física de Altas
Enerxías, Universidade de Santiago de Compostela,
15782 Santiago de Compostela, Galicia, Spain Samuel Homiller  Department of
Physics, Harvard University, Cambridge, MA, 02138, U.S.A.

###### Abstract

The scalar singlet model extends the Standard Model with the addition of a new
gauge singlet scalar. We re-examine the limits on the new scalar from oblique
parameter fits and from a global fit to precision electroweak observables and
present analytic expressions for our results. For the case when the new scalar
is much heavier than the weak scale, we map the model onto the dimension-six
Standard Model effective field theory (SMEFT) and review the allowed parameter
space from unitarity considerations and from the requirement that the
electroweak minimum be stable. A global fit to precision electroweak data,
along with LHC observables, is used to constrain the parameters of the high
scale singlet model and we determine the numerical effects of performing the
matching at both tree level and 1-loop.

##  I Introduction

The Higgs singlet model Bowen:2007ia ; O'Connell:2006wi ; Dawson:2015haa ;
Muhlleitner:2020wwk ; Robens:2015gla ; Costa:2015llh ; Chen:2014ask  has been
extensively studied as a simple extension of the Standard Model (SM)
containing only one new particle. Depending on the potential parameters, the
model can lead to a first order electroweak phase transition Huber:2006wf ;
Profumo:2007wc ; Espinosa:2011ax ; Barger:2011vm ; Profumo:2014opa ;
Curtin:2014jma ; Kotwal:2016tex ; Huang:2017jws ; Chen:2017qcz ; Kurup:2017dzf
; Li:2019tfd , making it highly motivated in addressing the problem of
baryogenesis. It can also arise as the limiting case of many interesting
models addressing the hierarchy problem Craig:2013xia ; Curtin:2015bka  or
even dark matter Silveira:1985rk ; McDonald:1993ex ; Burgess:2000yq ;
Menon:2004wv ; He:2008qm ; Gonderinger:2009jp ; Mambrini:2011ik . When the
mass of the new scalar becomes much larger than the weak scale, the theory can
be mapped onto an effective field theory. The utility and simplicity of the
model thus makes it an ideal candidate for exploring the limits of an
effective field theory framework in reproducing the features of the underlying
UV models Dawson:2020oco ; Brehmer:2015rna ; Henning:2014gca ; Henning:2014wua
; Gorbahn:2015gxa ; Ellis:2018gqa ; Ellis:2020unq ; Anisha:2020ggj .

In the full UV complete singlet model, restrictions on the parameters can be
found from fits to precision electroweak observables as well as LHC data.
These limits can then be compared with limits found in the context of a low
energy effective field theory. We consider an effective field theory in which
the SM Higgs doublet is constrained to be an $SU(2)$ doublet, the Standard
Model effective field theory (SMEFT). At tree level, the singlet model
generates only two SMEFT coefficients when matched at the UV scale
Henning:2014wua ; Egana-Ugrinovic:2015vgy . The aim of this work is to examine
to what extent the extraction of SMEFT coefficients from global fits at the
weak scale gives information on the parameters of the UV complete singlet
model Ellis:2018gqa ; Ellis:2020unq ; Dawson:2020oco ; Bakshi:2020eyg ;
Kribs:2017znd ; Falkowski:2015iwa ; Gorbahn:2015gxa . The focus is on
understanding the numerical importance of various choices made when performing
the low energy fits and to this end, we implement both tree and 1-loop
matching Jiang:2018pbd ; Haisch:2020ahr ; Cohen:2020fcu  at the UV scale. We
find that the effects of the 1-loop matching are typically rather small.
Effects of ${\cal{O}}(10\%)$ can be obtained only for rather large values of
certain dimensionless parameters in the Lagrangian.

Section II contains a recap of the model and restrictions on the model
parameters from unitarity and the minimization of the potential. Analytic
results for electroweak precision observables in the singlet model are found
in Section III along with a comparison between a global fit to electroweak
precision observables (EWPOs) and a fit to the oblique parameters, and
restrictions from unitarity and the minimization of the potential are in
Section IV. The SMEFT matching with the singlet model at both tree and loop
level is studied in Section V and a global fit to electroweak precision
observables, Higgs, and di-boson data is presented. Section VI has some
conclusions.

##  II Basics

The singlet model we consider contains the SM Higgs doublet, $\Phi$, and a
scalar gauge singlet, $S$. The most general scalar potential is,

$\displaystyle V(\Phi,S)=$
$\displaystyle-\mu_{H}^{2}\Phi^{\dagger}\Phi+\lambda_{H}(\Phi^{\dagger}\Phi)^{2}$
$\displaystyle+\frac{m_{\xi}}{2}\Phi^{\dagger}\Phi
S+\frac{\kappa}{2}\Phi^{\dagger}\Phi S^{2}$
$\displaystyle+t_{S}S+\frac{M^{2}}{2}S^{2}+\frac{m_{\zeta}}{3}S^{3}+\frac{\lambda_{S}}{4}S^{4}\,.$
(1)

The parameters can be redefined such that $\langle{S}\rangle\equiv x=0$. After
spontaneous symmetry breaking, the 2 neutral scalars, $\Phi_{0}$ and ${S}$,
mix to form the physical scalars, $h$ and $H$,

$\displaystyle h$ $\displaystyle=$
$\displaystyle\cos\theta\,{\Phi_{0}}+\sin\theta\,{S}$ $\displaystyle H$
$\displaystyle=$ $\displaystyle-\sin\theta\,{\Phi_{0}}+\cos\theta\,{S}\,,$ (2)

with the physical masses, $m_{h}=125.1\,\textrm{GeV}$ and $M_{H}$. The
parameters of the model can be taken as,

$m_{h},\quad M_{H},\quad v=246\,\textrm{GeV},\quad\sin\theta,\quad
x=0,\quad\kappa,\quad m_{\zeta},\quad\lambda_{S}\,.$ (3)

The other parameters of the Lagrangian are determined in the singlet model
by:111We note that for $\kappa v\gg M$, the mass of the new scalar, $M_{H}$,
comes from electroweak symmetry breaking and in this case the theory cannot be
mapped onto the SMEFT Buchalla:2016bse ; Cohen:2020fcu . Additionally, the
kinematic distributions for $hh$ production in this limit are quite different
from those where $M_{H}$ primarily depends on $M$ Dawson:2015oha .

$\displaystyle m_{\xi}$ $\displaystyle=$
$\displaystyle\frac{m_{h}^{2}-M_{H}^{2}}{v}\sin 2\theta\,,$ $\displaystyle
M^{2}$ $\displaystyle=$ $\displaystyle
m_{h}^{2}\sin^{2}\theta+M_{H}^{2}\cos^{2}\theta-\frac{\kappa}{2}v^{2}\,,$
$\displaystyle\lambda_{H}$ $\displaystyle=$
$\displaystyle\frac{m_{h}^{2}\cos^{2}\theta+M_{H}^{2}\sin^{2}\theta}{2v^{2}}\,.$
(4)

The $Z_{2}$ symmetric case has $m_{\zeta}=t_{S}=m_{\xi}=0$ and $x\neq 0$.

The couplings of $h$ to SM fermions and gauge bosons are suppressed relative
to the SM Higgs couplings by a factor of $\cos\theta$, while the $H$ couplings
are suppressed by $\sin\theta$. We can thus immediately find a trivial limit
on $\cos\theta$ from Higgs production to SM particles $X$, (assuming no decays
to invisible particles),222If $2M_{H}<m_{h}$ then the decay $h\rightarrow HH$
is allowed, altering the limit on $\cos~{}\theta$.

$\displaystyle\cos^{2}\theta$ $\displaystyle=$
$\displaystyle\mu\equiv\frac{\sigma\cdot\textrm{BR}}{(\sigma\cdot\textrm{BR})_{\textrm{SM}}}$
(5)

Naively combining the combined ATLAS results with $80\,\textrm{fb}^{-1}$
Aad:2019mbh  and the CMS combined limits with $139\,\textrm{fb}^{-1}$
CMS:2020gsy ,

$\mu[\textrm{ATLAS}]=1.11^{+.09}_{-.08},\qquad\mu[\textrm{CMS}]=1.02^{+.07}_{-.06}$
(6)

we find at $95\%$ C.L.,

$|\sin\theta\,|<0.2\quad\text{for}\quad m_{h}<2M_{H}\,.$ (7)

For $m_{h}>2M_{H}$, the naive limit of Eq. (7) does not apply because the $h$
decays to $HH$ must be included and this branching ratio is sensitive to the
other parameters of the scalar potential. Limits on the singlet model from
resonant double Higgs production are beginning to be competitive with those
from single Higgs production for $M_{H}\lesssim 700\,\textrm{GeV}$ Aad:2019uzh
, although our primary focus here will be on $M_{H}\sim(1-2)\,\textrm{TeV}$.

##  III Restrictions on Model Parameters

The parameters of the singlet model can be limited by a fit to the $Z$\- and
$W$-pole observables (we term this the EWPO fit):

$\displaystyle
M_{W},~{}~{}\Gamma_{W},~{}~{}\Gamma_{Z},~{}~{}\sigma_{h},~{}~{}R_{b},~{}~{}R_{c},~{}~{}R_{l},~{}~{}A_{FB,b},~{}~{}A_{FB,c},~{}~{}A_{FB,l},~{}~{}A_{b},~{}~{}A_{c},~{}~{}A_{l}\,.$
(8)

The SM results for these observables are well known Hollik:1988ii ;
Freitas:2014hra . In a previous study, Ref. Dawson:2019clf , we computed the
limits on the coefficients of an effective field theory that result from a fit
to the observables of Eq. (8) computed to NLO in both QCD and electroweak
interactions, and we apply an identical calculational framework here. The
observables and SM theory numbers used in the current study can be found in in
Table III of Ref. Dawson:2019clf . We take as our input parameters:
$G_{\mu}=1.1663787(6)\times 10^{-5}\,\textrm{GeV}^{-2}$,
$M_{Z}=91.1876\pm.0021\,\textrm{GeV}$, $1/\alpha={137.035999139(31)}$,
$\Delta\alpha_{\rm had}^{(5)}=0.02764\pm 0.00009$,
$\alpha_{s}(M_{Z})=0.1181\pm 0.0011$, $m_{h}=125.10\pm 0.14\,\textrm{GeV}$,
$m_{b}=4.18\,\textrm{GeV}$ and $M_{t}=172.9\pm 0.5\,\textrm{GeV}$ .

The one loop relation between the Fermi constant $G_{\mu}$ and the vacuum
expectation value $v$ is, as usual,

$G_{\mu}=\frac{1}{\sqrt{2}v^{2}}(1+\Delta r)$ (9)

where,

$\Delta r=\Delta r_{\textrm{SM}}+\Delta r_{\textrm{singlet}}.$

In computing $\Delta r$, we use
${\hat{M}}_{W}^{2}\equiv(M_{Z}^{2}/2)\bigg{(}1+\sqrt{1-\frac{\sqrt{8}\pi\alpha}{G_{\mu}M_{Z}^{2}}}\bigg{)}$
calculated from our inputs. For simplicity, we define $h\equiv m_{h}^{2}$,
$H\equiv M_{H}^{2}$, $z\equiv M_{Z}^{2}$ and $w\equiv{\hat{M}}_{W}^{2}$ and
obtain the simple form,333The function $A_{0}$ is defined as
$A_{0}(m^{2})=\int\frac{d^{d}k}{(2\pi)^{2}}\frac{1}{k^{2}-m^{2}},$ where we
calculate in $d=4-2\epsilon$ dimensions. $B_{0}$ is the Passarino- Veltman
$2$-point function,
$B_{0}(m_{1}^{2},m_{2}^{2},p^{2})=\int\frac{d^{d}k}{(2\pi)^{n}}\frac{1}{[k^{2}-m_{1}^{2}][(k+p)^{2}-m_{2}^{2}]}\,.$
The Passarino-Veltman functions are evaluated using QCDLOOPS Carrazza:2016gav
.

$\displaystyle\Delta r_{\textrm{singlet}}$ $\displaystyle=\Delta
r_{\textrm{singlet}}(h,H)-\Delta r_{\textrm{singlet}}(H,h)$
$\displaystyle\Delta r_{\textrm{singlet}}(h,H)$
$\displaystyle={\sqrt{2}\sin^{2}\theta G_{\mu}\over
16\pi^{2}}\biggl{\\{}-{h\over
2}+{3wA_{0}(h)\over(h-w)}+{3whA_{0}(w)\over(H-w)(h-w)}\biggr{\\}}\,.$ (10)

We find the one-loop prediction for $M_{W}$ in the singlet model,

$\displaystyle M_{W}$ $\displaystyle=$ $\displaystyle
M_{W}^{\textrm{SM}}+F_{W}(h,H)-F_{W}(H,h)$ $\displaystyle F_{W}(h,H)$
$\displaystyle=$ $\displaystyle{\alpha\sin^{2}\theta\over 8\pi
M_{W}}\biggl{\\{}{hz\over 24(2w-z)}+{A_{0}(h)\over
12(h-w)w(2w-z)}\biggl{(}hw(w-4z)+12w^{2}z+h^{2}(z-w)\biggr{)}$ (11)
$\displaystyle+{hzA_{0}(w)\over
12(h-w)w(H-w)(z-w)(2w-z)}\biggl{(}(2w-z)\biggl{[}hH-w(h+H)\biggr{]}+w^{2}(8z-7w)\biggr{)}$
$\displaystyle+{hwA_{0}(z)\over 12(z-w)(z-2w)}-{zB_{0}(w,h,w)\over
12w(w-z)}\biggl{(}h^{2}-4hw+12w^{2}\biggr{)}$
$\displaystyle+{wB_{0}(z,h,z)\over
12(z-2w)(z-w)}\biggl{(}h^{2}-4hz+12z^{2}\biggr{)}\biggr{\\}}\,.$

This is in agreement with Ref. Lopez-Val:2014jva . For a massless $b$ quark,
the total $W$ decay width is,

$\displaystyle\Gamma_{W}=$
$\displaystyle~{}~{}\Gamma_{W}^{\textrm{SM}}+G_{W}(h,H)-G_{W}(H,h)$
$\displaystyle G_{W}(h,H)=$
$\displaystyle-{3\sqrt{w}G_{F}^{2}\sin^{2}\theta\over
32\pi^{3}}\biggl{\\{}{wh(-hH-8w^{2}+4wh+4wH)\over 2(h-4w)(H-4w)}$
$\displaystyle-{A_{0}(h)\over
h(h-4w)(h-w)}\biggl{(}h^{4}-7h^{3}w+21h^{2}w^{2}-32hw^{3}+8w^{4}\biggr{)}$
$\displaystyle+{hA_{0}(w)\over(h-4w)(h-w)(H-w)(H-4w)}\biggl{(}4w^{2}(H^{2}+h^{2})-5whH(H+h)$
$\displaystyle+h^{2}H^{2}-4w^{3}(H+h)+18w^{2}hH-36w^{4}\biggr{)}$
$\displaystyle+{B_{0}(w,h,w)\over(h-4w)}\biggl{(}h^{3}-7h^{2}w+20hw^{2}-28w^{3}\biggr{)}\biggr{\\}}\,.$
(12)

Analytic expressions for the remaining observables of Eq. (8) are given in the
supplemental material attached to this note.

The finite $b$ mass contribution to $Z$ decays to bottom pairs is sensitive to
the Higgs-$b$ Yukawa coupling and generates non-oblique contributions. We
compute $R_{l},R_{b},R_{c}$ and $\Gamma_{Z}$ for $m_{b}\neq 0$ and find that
the numerical effect is less than $\sim 2\%$ for $M_{H}>20\,\textrm{GeV}$,
rising to $\sim 5\%$ for $M_{H}\sim 10\,\textrm{GeV}$, justifying the neglect
of $b$ mass effects in our fits.

We perform a fit, including correlations, to the observables of Eq. (8) to
determine the maximum allowed value of $\sin\theta$ for a given value of
$M_{H}$ including all one-loop contributions. It is of interest to compare the
complete EWPO fit with the results using the oblique parameters only. Using
the results of Dawson:2009yx ; Englert:2020gcp , we find that the differences
between the Peskin-Takeuchi Peskin:1991sw  variables in the Higgs singlet
model and the SM take the form,

$\displaystyle\Delta S$
$\displaystyle=\frac{\sin^{2}\theta}{12\pi}(\mathcal{G}(H,z)-\mathcal{G}(h,z))$
(13) $\displaystyle\Delta T$ $\displaystyle=\frac{3\sin^{2}\theta}{16\pi
s_{W}^{2}c_{W}^{2}}(\mathcal{K}(H)-\mathcal{K}(h))$ (14) $\displaystyle\Delta
S+\Delta U$ $\displaystyle=\frac{\sin^{2}\theta}{12\pi
c_{W}^{2}}(\mathcal{G}(H,w)-\mathcal{G}(h,w)),$ (15)

where $s_{W}^{2}=1-c_{W}^{2}=1-w/z$ is $\sin^{2}\theta_{W}$ of the electroweak
mixing angle and we define,

$\displaystyle\mathcal{K}(h)$
$\displaystyle=h(\frac{(z-w)}{(h-w)(h-z)}A_{0}(h)+\frac{A_{0}(w)}{(h-w)}-\frac{A_{0}(z)}{(h-z)})$
(16) $\displaystyle\mathcal{G}(h,z)$
$\displaystyle=\frac{h}{2}+\mathcal{F}(h,z)(A_{0}(h)-A_{0}(z)-(h-z)B_{0}(z,h,z))$
(17) $\displaystyle\mathcal{F}(h,z)$
$\displaystyle=\frac{h^{2}-4hz+12z^{2}}{z(h-z)}\,.$ (18)

We fit to the values in Zyla:2020zbs ,

$\displaystyle\Delta S$ $\displaystyle=-0.01\pm 0.10$ $\displaystyle\Delta T$
$\displaystyle=0.02\pm 0.12$ $\displaystyle\Delta U$ $\displaystyle=0.02\pm
0.11\,$ (19)

with the correlation matrix,

$\rho=\left(\begin{array}[]{c c c}1.&0.92&-0.80\\\ 0.92&1.&-0.93\\\
-0.80&-0.93&1.\end{array}\right)\,.$ (20)

In Fig. 1 we report the results corresponding to different sets of
observables:

  * •

Only $M_{W}$

  * •

The $Z$ pole observables alone

  * •

Oblique parameters only

  * •

EWPOs given in Eq. (8).

Figure 1:  Maximum allowed $\sin\theta$ in the Higgs singlet model at $95\%$
confidence level based on fits to electroweak precision observables (EWPO),
the $W$ mass measurement only, the $Z$ pole observables only, and to the $STU$
parameters as described in the text. A naive limit from the Higgs coupling
measurements is shown on the RHS for comparison.

The results for the fit to $M_{W}$ alone are in agreement with those of Ref.
Ilnicka:2018def  and are a good approximation to the complete EWPO fit. The
EWPO fit limits are in agreement with Ref. Falkowski:2015iwa  after adjusting
for the different input parameters. It is interesting that the current limits
from Higgs couplings give better bounds for all $M_{H}\lesssim
1\,\textrm{TeV}$ as given in Eq. (7). The limits obtained from oblique
parameters are in approximate agreement with those from the EWPO fit for a
heavier second Higgs boson, although slightly different data sets and
approximations were used in the numbers we fit to.444 We find rough agreement
with Refs. Profumo:2014opa ; Falkowski:2015iwa  (the differences can be
explained by the different numerical values of the input parameters) and
disagree with the oblique parameter limits of Fig. 1 of Chalons:2016lyk . We
note that the curve labelled “Exact Singlet” on the RHS of Fig. 1 of Ref.
Dawson:2020oco  is the $STU$ result and has used a slightly different fit to
the oblique parameters deBlas:2017wmn  from the PDG Zyla:2020zbs  results used
here. The curve labelled Higgs in that plot is the prediction from fitting
Higgs data within the context a SMEFT fit and thus differs from the SM Higgs
coupling fit shown in Fig. 1. For the case where the second Higgs is light,
$M_{H}<m_{h}$, the limits obtained from the oblique parameters are not a good
approximation of the complete EWPO fit.

##  IV Theoretical Constraints

In Section V, we will match the singlet model with a very heavy $H$ to the
SMEFT. Before we do so, we consider the theoretical restrictions on the
singlet model parameters that are relevant for the matching.

###  IV.1 Vacuum Structure of the Potential

The first set of theoretical constraints on the singlet model come from
requiring a suitable vacuum structure of the potential Espinosa:2011ax ;
Chen:2014ask ; Robens:2015gla ; Robens:2016xkb . Demanding that the potential
is stable at large field values leads to the requirement
$\lambda_{H},\lambda_{S}>0$, and $\kappa\geq-2\sqrt{\lambda_{H}\lambda_{S}}$
Chen:2014ask , where $\lambda_{H}$ is determined by Eq. (4). Additional bounds
result from requiring that the electroweak minimum be the global minimum of
the potential. Following Chen:2014ask , we compute these bounds by finding all
the extrema of the potential expanded around the electroweak vev as a function
of $(v,x)$, and then checking whether or not the value of the potential at
$(v=246\,\textrm{GeV},x=0)$ is the global minimum.

The extrema of the potential can be divided into two classes: extrema where
$v\neq 0$, and those where $v=0$. In the former case, the new extrema are
denoted $(v_{\pm},x_{\pm})$ as in ref. Chen:2014ask , and tend to bound lower
values of $\kappa$. In the latter case, the extrema are denoted by
$(0,x^{0}_{\pm})$, and these tend to limit both large values of $\kappa$ as
well as large values of $m_{\zeta}$. An example of the vacuum structure is
shown in Fig. 2, where we illustrate the regions excluded by the emergence of
different global minima as well as the condition from vacuum stability. Fig. 3
illustrates how these bounds change as a function of the physical parameters.
In particular, we see that for larger masses, $M$, the bounds on $m_{\zeta}/M$
from the $v=0$ minima become constant as a function of $\kappa$, depending
only on $\lambda_{S}$. It is interesting that quite large values of $\kappa$
are allowed in all scenarios. The upper bound on $m_{\zeta}/M$ never exceeds
${\cal{O}}(2-3)$.

Figure 2: Demonstration of the bounds from the appearance of other global
minima in the $\kappa$ vs. $m_{\zeta}/M$ plane for $M=2\,\textrm{TeV}$,
$\sin\theta=0.25$, and $\lambda_{S}=1.0$ in the singlet model.

Figure 3: Regions in the singlet model where the electroweak minimum is the
global minimum of the potential as a function of $\kappa$ and $m_{\zeta}/M$,
varying the other physical parameters.

###  IV.2 Unitarity

The next set of theoretical constraints come from the requirements of tree-
level perturbative unitarity Lee:1977yc ; Lee:1977eg ; Dawson:2017vgm ;
Robens:2015gla . The simplest constraints come from $hh\to hh$ and $HH\to HH$
scattering, where the spin-$0$ partial waves in the high energy limit are

$\displaystyle a_{0}(hh\to hh)\Big{|}_{s\gg m_{h}^{2}}$
$\displaystyle=-\frac{3}{16\pi}\left(\lambda_{H}\cos^{4}{\theta}+\kappa\sin^{2}{\theta}\cos^{2}{\theta}+\lambda_{S}\sin^{4}{\theta}\right)$
(21) $\displaystyle a_{0}(HH\to HH)\Big{|}_{s\gg m_{H}^{2}}$
$\displaystyle=-\frac{3}{16\pi}\left(\lambda_{S}\cos^{4}{\theta}+\kappa\sin^{2}{\theta}\cos^{2}{\theta}+\lambda_{H}\sin^{4}{\theta}\right)$
(22)

For $\sin\theta\ll 1$, requiring $|a_{0}|<1/2$ sets the bounds
$\lambda_{S},\lambda_{H}\lesssim 8\pi/3$. This bound on $\lambda_{H}$
indirectly bounds $M_{H}$ as a function of $\sin\theta$:

$\displaystyle M_{H}^{2}\,\sin^{2}\theta$ $\displaystyle\lesssim$
$\displaystyle\frac{16\pi}{3}v^{2}-m_{h}^{2}\,\cos^{2}\theta$ $\displaystyle
M_{H}$ $\displaystyle\lesssim$ $\displaystyle
7\,\textrm{TeV}\quad{\rm{for}}\cos\theta=0.99\,.$ (23)

The similar bound from $hH\to hH$ scattering only restricts $|\kappa|\lesssim
8\pi$.

##  V One-loop Matching of the Singlet Model to SMEFT

###  V.1 One-loop Matching

When the mass of the heavy scalar is much larger than the weak scale and any
relevant energy scales, the singlet model can be modeled by an effective field
theory,

$\mathcal{L}_{\textrm{singlet}}~{}\xrightarrow[M_{H}\to\infty]{}~{}\mathcal{L}_{\textrm{SM}}+\sum_{i}\frac{C_{i}(M)}{M^{2}}\mathcal{O}_{i}^{(6)}+\dots$
(24)

with coefficients matched to the singlet model at the high scale, $M$. We
retain only the dimension-6 operators, $\mathcal{O}_{i}^{(6)}$ ,and use the
Warsaw basis Buchmuller:1985jz  with the notation of Ref. Dedes:2017zog .

The global fits of Ref. Dawson:2020oco  were performed using tree level
matching at the scale $M$.555Ref. Brehmer:2015rna  noted that better agreement
between the SMEFT and singlet model predictions for $hh$ production are
obtained when the matching is performed at the physical mass, $M_{H}$. The
one-loop matching would then contain terms proportional to $\log(M_{H}/M)$
that we have omitted. It is of interest to implement the one-loop matching for
the case of the singlet model and examine the numerical impacts. The
coefficients at the matching scale, $M$, generically take the form,

$C_{i}(M)=c_{i}(M)+\frac{d_{i}(M)}{16\pi^{2}}\,,$ (25)

where $c_{i}(M)$ is the tree level result and $d_{i}(M)/(16\pi^{2})$ is the
one-loop contribution at the matching scale. When the renormalization group
evolution to the low scale $\mu_{R}$ is included,

$C_{i}(\mu_{R})=c_{i}(M)+{d_{i}(M)\over 16\pi^{2}}+{\gamma_{ij}\over
32\pi^{2}}c_{ij}(M)\log\biggl{(}{\mu_{R}^{2}\over M^{2}}\biggr{)}\,.$ (26)

In the case of the singlet model only two coefficients are generated at tree
level Egana-Ugrinovic:2015vgy ; Henning:2014wua ; deBlas:2017xtg ;
Dawson:2017vgm ,

$\displaystyle c_{H\square}$ $\displaystyle=$ $\displaystyle-{m_{\xi}^{2}\over
8M^{2}}$ (27) $\displaystyle c_{H}$ $\displaystyle=$
$\displaystyle{m_{\xi}^{2}\over 8M^{2}}\biggl{(}{m_{\xi}m_{\zeta}\over
3M^{2}}-\kappa\biggr{)}\,,$ (28)

with all other $c_{i}(M)=0$. However, there are many coefficients generated at
one-loop at the matching scale, $M$ Jiang:2018pbd ; Cohen:2020fcu ;
Haisch:2020ahr . The majority of these coefficients are proportional to the
tree level coefficient, $c_{H\square}$. We use the shorthand
$C_{Hu}\rightarrow C_{Hu},C_{Hc},C_{Ht}$, etc., and take
$y_{u}=y_{c}=0,~{}y_{t}=M_{t}\sqrt{2}/v$ (similarly we set all other
$y_{i}=0$) and we further assume that
$C_{Hq}^{(1)},~{}C_{Hq}^{(3)},~{}C_{Hl}^{(1)}$, and $C_{Hl}^{(3)}$ are flavor
diagonal and use an analogous shorthand. For convenience, we list the results
of Ref. Jiang:2018pbd  in our notation:666Since $y_{t}$ is the only non-zero
Yukawa that we include, $O_{2y}=y_{t}^{2}{\overline{t}}t{\overline{t}}t$.

$\displaystyle d_{HD}$ $\displaystyle=$ $\displaystyle{31g^{\prime~{}2}\over
9}c_{H\square}$ $\displaystyle d_{HW}$ $\displaystyle=$
$\displaystyle-{g^{2}\over 6}c_{H\square}$ $\displaystyle d_{HB}$
$\displaystyle=$ $\displaystyle-{g^{\prime~{}2}\over 6}c_{H\square}$
$\displaystyle d_{HWB}$ $\displaystyle=$ $\displaystyle-{gg^{\prime}\over
3}c_{H\square}$ $\displaystyle d_{Hu}$ $\displaystyle=$ $\displaystyle{1\over
108}(34g^{\prime~{}2}-135y_{u}^{2})c_{H\square}$ $\displaystyle d_{Hd}$
$\displaystyle=$ $\displaystyle{1\over 3}d_{He}={2\over
3}d_{Hl}^{(1)}=-{17g^{\prime~{}2}\over 108}c_{H\square}$ $\displaystyle
d_{Hq}^{(1)}$ $\displaystyle=$ $\displaystyle{1\over
216}(17g^{\prime~{}2}+135y_{u}^{2})c_{H\square}$ $\displaystyle d_{Hq}^{(3)}$
$\displaystyle=$ $\displaystyle{1\over
72}\biggl{[}17g^{~{}2}-45y_{u}^{2}\biggr{]}c_{H\square}$ $\displaystyle
d_{Hl}^{(3)}$ $\displaystyle=$ $\displaystyle{17g^{2}\over 72}c_{H\square}$
$\displaystyle d_{2y}$ $\displaystyle=$ $\displaystyle-{1\over
3}c_{H\square}\,.$ (29)

The one-loop contribution $d_{tH}$ can be written in terms of $c_{H\square}$
and $C_{H}$ and is,

$d_{tH}=y_{t}\bigg{[}-{1\over 18}(45y_{t}^{2}-31g^{2})c_{H\square}+{3\over
2}c_{H}-{29\over 3}\lambda\,c_{H\square}\bigg{]}\,,$ (30)

where in the SMEFT the physical Higgs mass is determined in terms of the
potential parameters to ${\cal{O}}(v^{2}/M^{2})$ by Dedes:2017zog ,

$\displaystyle{m_{h}^{2}\over 2v^{2}}$ $\displaystyle=$
$\displaystyle\lambda_{H}\biggl{(}1+{2v^{2}\over
M^{2}}c_{H\square}\biggr{)}-{3\over 2}{v^{2}\over M^{2}}c_{H}$ (31)

and we define,

$\displaystyle\lambda$ $\displaystyle\equiv$
$\displaystyle{2\lambda_{H}}\biggl{(}1+{2v^{2}\over
M^{2}}c_{H\square}\biggr{)}$ $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{m_{h}^{2}}{v^{2}}+3\frac{v^{2}}{M^{2}}c_{H}+\mathcal{O}\bigg{(}\frac{v^{4}}{M^{4}}\bigg{)}\,,$
(32)

where we note that Ref. Jiang:2018pbd  absorbs the factor of $c_{H\square}$
into the definition of $\lambda$ used in the matching conditions, along with a
relative factor of $2$ in the definition of the quartic terms in the
potential.777 We drop the $c_{HD}$ term in Eq. (31) since it doesn’t occur in
the singlet model. Eq. (31) represents the dimension-6 SMEFT limit of Eq. (4)
for the relationship between the parameters of the potential and $m_{h}$.

Finally, the coefficients generated at tree level also receive one-loop
corrections,

$\displaystyle d_{H\square}$ $\displaystyle=-\frac{9}{2}\lambda
c_{H\square}+{31\over 36}(3g^{2}+g^{\prime~{}2})c_{H\square}+{3\over
2}c_{H}+\delta d_{H\square}+\delta d_{H\square}^{shift}$ $\displaystyle d_{H}$
$\displaystyle=\lambda\biggl{[}\biggl{(}{62g^{2}-336\lambda\over
9}\biggr{)}c_{H\square}+6c_{H}\biggr{]}+\delta d_{H}+\delta d_{H}^{shift}\,.$
(33)

where,

$\displaystyle\delta d_{H}=$
$\displaystyle-\frac{\kappa^{3}}{12}+\frac{m_{\xi}}{4M^{2}}\bigg{(}m_{\xi}\big{(}9\lambda^{2}-12\kappa\lambda+\frac{11}{2}\kappa^{2}-3\kappa\lambda_{S}\big{)}-\kappa^{2}m_{\zeta}\bigg{)}$
$\displaystyle+\frac{m_{\xi}^{2}}{6M^{4}}\bigg{(}\frac{m_{\xi}^{2}}{16}\big{(}39\kappa-18\lambda_{S}-36\lambda\big{)}+m_{\xi}m_{\zeta}\big{(}9\lambda-\frac{15}{2}\kappa+3\lambda_{S}\big{)}+3\kappa
m_{\zeta}^{2}\bigg{)}$
$\displaystyle+\frac{m_{\xi}^{3}}{12M^{6}}\bigg{(}-\frac{1}{8}m_{\xi}^{3}-\frac{9}{8}m_{\xi}^{2}m_{\zeta}+3m_{\xi}m_{\zeta}^{2}-2m_{\zeta}^{3}\bigg{)}\,,$
(34) $\displaystyle\delta d_{H\square}=$
$\displaystyle-\frac{\kappa^{2}}{24}+\frac{m_{\xi}}{12M^{2}}\bigg{(}\frac{m_{\xi}}{2}\big{(}17\kappa-\frac{27}{2}\lambda-18\lambda_{S}\big{)}-5\kappa
m_{\zeta}\bigg{)}$
$\displaystyle+\frac{m_{\xi}^{2}}{24M^{4}}\bigg{(}\frac{13}{8}m_{\xi}^{2}-8m_{\xi}m_{\zeta}+11m_{\zeta}^{2}\bigg{)}\,.$
(35)

The terms of Eqs. (34) and (35) can be written in terms of
$c_{H},c_{H\square}$ along with $m_{\zeta},\lambda_{S}$ and $\kappa$,
($m_{\xi}$ can be written in terms of these parameters). The one-loop shift
terms from canonically normalizing the Higgs kinetic energy are,

$\displaystyle\delta d_{H}^{shift}$ $\displaystyle=$ $\displaystyle
3c_{H\square}c_{H}\,,$ $\displaystyle\delta d_{H\square}^{shift}$
$\displaystyle=$ $\displaystyle 2(c_{H\square})^{2}\,.$ (36)

The one-loop shift terms are ${\cal{O}}({v^{4}/M^{4}})$ and can be neglected,
since we consistently work to linear order in the coefficient functions.

After performing the one-loop matching at $M$, the renormalization group is
used to evolve the coefficients to $M_{Z}$, where the resulting coefficients
can be compared with data.888 A more consistent approach would employ the
2-loop anomalous dimensions, however, these are not available for the SMEFT.
The complete set of one-loop anomalous dimension matrices can be found in
Refs. Jenkins:2013zja ; Jenkins:2013wua ; Alonso:2013hga . The inclusion of
the one-loop matching makes a relatively minor difference in the evolution of
$C_{H}$ and $C_{H\square}$, as seen in Fig. 4 where we evolve from
$2\,\textrm{TeV}$ (note that $M_{H}$ is related to $M$ by Eq. (4)). In Fig. 5,
we show the effect of the one-loop matching on the evolution of $C_{HD}$. In
this case, since $C_{HD}$ is zero at tree level, the contributions from the
$1-$loop matching and the renormalization group running are of the same order
of magnitude and the effects are more significant. In Fig. 6 we show the
relative size of the $1-$loop matching compared to the tree level matching as
the matching scale $M$ is increased and the overall effects are between
$10-30\%$. The size of the effects for $C_{H\square}$ and $C_{H}$ increase
dramatically as the matching scale rises over a few TeV. This is due to the
logarithmic running becoming large and in the case of $C_{H}$, the 1-loop
matching terms become of the same order as the tree level terms, implying that
the perturbative expansion is no longer valid.

Figure 4: Renormalization group evolution of coefficient functions from the
matching scale, $M$, to $\mu_{R}$ when the matching is done at tree level and
at one-loop for coefficients that are generated at tree level. The
coefficients are evaluated as a function of the running scale, $\mu_{R}$.
Figure 5:  Renormalization group evolution of the coefficient function from
the matching scale, $M$, to $\mu_{R}$ for $C_{HD}$, which is generated only by
the renormalization group running in the singlet model. $C_{HD}$ is evaluated
as a function of the running scale, $\mu_{R}$. Figure 6:  Shift in the
coefficient functions at $M_{Z}$ as a function of the matching scale, $M$,
when the matching is done at tree level and at one-loop.

###  V.2 Global Fit

Following Ref. Dawson:2020oco , we perform a global fit to the parameters of
the non-$Z_{2}$ symmetric singlet model. At the matching scale, $M$, only the
tree level coefficients $c_{H}$ and $c_{H\square}$ are non-zero and other
coefficients are generated at $M_{Z}$ from the renormalization group running.
With tree level matching, the results can be expressed in terms of $c_{H}(M)$
and $c_{H\square}(M)$. Using the 1-loop matching at $M$ described in the
previous section, additional coefficients are generated with a distinctive
pattern. The 1-loop matching introduces a dependence on three additional
parameter combinations beyond those at generated by the tree level matching
and we take as our 5 unknown input parameters, $M_{H},\sin\theta,m_{\zeta}$,
$\lambda_{S}$, and $\kappa$.999 The results used to include the effects of
$C_{H}$ require $|C_{H}|\lesssim(5-6)\big{(}M/\textrm{TeV}\big{)}^{2}$
Degrassi:2017ucl ; Degrassi:2016wml . The matching scale, $M$, is then
calculated using Eq. (4). We match the SMEFT coefficients at $M$ and use the
1-loop renormalization group equations to evolve the SMEFT coefficients to
$M_{Z}$ where we fit to data.

The included data are identical to that of Ref. Dawson:2020oco  and include
Higgs coupling strengths from ATLAS Aad:2019mbh , CMS Higgs coupling strengths
CMS:2020gsy , $W^{+}W^{-}$, $W^{\pm}Z$, $Wh$ and $Zh$ differential
measurements including QCD effects as in Baglio:2019uty ; Baglio:2020oqu , and
precision electroweak measurements including QCD and electroweak NLO effects
from Table III of Ref. Dawson:2019clf . We determine the $95\%$ confidence
level limits using a $\chi^{2}$ fit, including the new physics effects at
linear order in the SMEFT coefficients.

Figure 7:  95% C.L. limits on $\sin\theta$ as a function of $\kappa$ (LHS) and
$\lambda_{S}$ (RHS) for fixed $M_{H}=2\,\textrm{TeV}$ and
$m_{\zeta}=500\,\textrm{GeV}$. The fits with tree-level matching are shown as
dashed curves, with solid curves showing the 1-loop result. The black curves
show the result of a global fit to Higgs, diboson, and electroweak precision
data, while the pink curves only the electroweak precision observables. The
regions to the right of the curves are excluded by the fits. The grey and blue
shaded regions are forbidden by unitarity and electroweak vacuum stability
requirements, respectively.

Figure 8:  As in Fig. 8, now showing limits on $\sin\theta$ as a function of
the heavy Higgs mass, $M_{H}$, with fixed values of $\kappa$, $\lambda_{S}$
and $m_{\zeta}/M_{H}$. Regions above the curves are excluded.

Figs. 8 and 8 contain our major results. In terms of the parameters of the
singlet model given in Eq. (3), we fix $M_{H}=2\,\textrm{TeV}$ and determine
the maximum allowed value of $\sin\theta$ in terms of the other unknown
parameters of the model, $\lambda_{S},~{}\kappa,$ and $m_{\zeta}$. The curves
are relatively insensitive to $m_{\zeta}$ and $\lambda_{S}$ (RHS), and the
major sensitivity is to $\kappa$ (LHS) of Fig. 8. We show the regions excluded
by unitarity bounds and by vacuum stability bounds. The black curves include
the Higgs, diboson, and EWPO data. For $\kappa\lesssim 8$, the inclusion of
the 1-loop matching makes very little difference, but as $\kappa$ becomes
large and approaches the unitarity bound, the difference between tree level
and 1-loop matching can be of ${\cal{O}}(10\%)$. We separately show the limits
from only EWPO limits in magenta and note that the 1-loop matching slightly
improves the bound on $\sin\theta$.

Another interesting way to look at the results is to look at the maximum
allowed value of $\sin\theta$ as a function of the heavy Higgs mass, $M_{H}$,
for fixed values of $\kappa,m_{\zeta}$ and $\lambda_{S}$ as shown in Fig. 8.
We see that including the 1-loop matching changes the bound on $\sin\theta$
only marginally. The effect is larger as $\kappa$ is increased.

Single parameter fits to models with an additional scalar have been presented
in Ref. Ellis:2018gqa  and updated in Ref. Ellis:2020unq  using the dictionary
of Ref. deBlas:2017xtg . Assuming $C_{H\square}=-1/M^{2}$ and $C_{H}=0$, they
find a limit $M_{H}>900\,\textrm{GeV}$ at $2\sigma$. Our tree level matching
result of Ref. Dawson:2020oco  is roughly compatible with this bound, although
we find that the inclusion of the renormalization group running of the
coefficients (in particular $C_{HD}$ which is generated by renormalization
group running) is numerically significant, so the bounds cannot be directly
compared.101010Fits to the singlet model with 1-loop matching, but no
renormalization group running, are given in Ref. Anisha:2020ggj , but the
results are not in a form that we can compare with.

##  VI Conclusions

We have re-examined the sensitivity of a global fit to electroweak precision
observables and to Higgs and diboson data on the parameters of a scalar
singlet model in both the full UV complete model and in the low energy
approximation where the heavy scalar is integrated out and the parameters are
matched to the dimension-6 SMEFT. In the full singlet model, we find
equivalent limits on the allowed mixing angle from the complete EWPO fit and
from the fit to the oblique parameters when $M_{H}$ is heavy. For the case
with the second Higgs boson much lighter than $M_{Z}$, the oblique parameter
limits are not a good approximation to the full fit. When the new scalar is
very heavy, we integrate it out and match to the dimension-6 SMEFT and then
perform the global fit both using tree level and 1-loop matching at the high
scale and derive limits on the parameters of the singlet theory from the SMEFT
fit. We find that the effect on the fit of including the 1-loop matching is
never larger than ${\cal{O}}(10\%)$ and that the results are quite insensitive
to variations in the singlet Lagrangian parameters other than the portal term,
$\kappa$.

Digital data can be found at
https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/singlet_21.

## Acknowledgements

SD is supported by the United States Department of Energy under Grant Contract
DE-SC0012704. The work of PPG has received financial support from Xunta de
Galicia (Centro singular de investigación de Galicia accreditation 2019-2022),
by European Union ERDF, and by “María de Maeztu” Units of Excellence program
MDM-2016-0692 and the Spanish Research State Agency. The work of SH was
supported by DOE Grant DE-SC0013607 and by the Alfred P. Sloan Foundation
Grant No. G-2019-12504.

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# Topological defects in two-dimensional crystals

Yong Chen  Email<EMAIL_ADDRESS>Institute of Theoretical Physics, Lanzhou
University, Lanzhou $730000$, China  Wei-Kai Qi  Email<EMAIL_ADDRESS>Institute of Theoretical Physics, Lanzhou University, Lanzhou $730000$, China

###### Abstract

By using topological current theory, we study the inner topological structure
of the topological defects in two-dimensional (2D) crystal. We find that there
are two elementary point defects topological current in two-dimensional
crystal, one for dislocations and the other for disclinations. The topological
quantization and evolution of topological defects in two-dimensional crystals
are discussed. Finally, We compare our theory with Brownian-dynamics
simulations in 2D Yukawa systems.

###### pacs:

61.72.Cc, 61.72.Lk, 64.70.D-

In 1970’s, Kosterlitz and Thouless construct a detailed and complete theory of
superfluidity on two-dimensions (2D) kt . They indicate vortices pair
unbinding will lead to a second-order transition in superfluid films. Later, a
microscopic scenario of 2D melting has been posited in the form of the
Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory hny . The KTHNY
theory predicts the unbinding of topological defects to break the symmetry in
the two-stage transitions. Topological defects, which are a necessary
consequence of broken continuous symmetry, play a important role in two-
dimensional phase transition Dn ; Cl .

In 2D crystal, the evolution of topological defects have been studied in
experiments and computer simulations. A serial experiments were performed to
calculate dislocations and disclinations dynamic of two-dimensional colloidal
systems, and dissociation of dislocations and disclinations were observed
td001 . During the years, a large number of computer simulations indicated
that exist a two-stage melting as prescribed by KTHNY theory, however, results
are still controversial KnNs . Our previous work found that exist a hexatic-
isotropic liquid phase coexistence during the melting of soft Yukawa systems
Qi . By Voronoi polygons analysis, the behavior of piont defects in the
coexistence is very complicated. The evolution of topological defects during
the melting of two-dimensional system still a open question.

It is interesting to consider the appropriate form for the point defect
densities when expressed in terms of the vector order parameter field
$\phi(\vec{r},t)$. This has been carried out by Halperin Ha , and exploited by
Liu and Mazenko LM . However, their analysis is incomplete Duan00 . In two-
dimensional system, a gauge field-theoretic formalism developed by Kleinert HK
. The gauge theory of topological quantum melting in 2+1 dimensions Bose
system was developed by Nussinov et al, and the Superfluidity and
superconductivity can arise in a strict quantum field-theoretic setting Z1 .

Recently, a topological field theory for topological defect developed by Duan
et al Duan . By using $\phi$-mapping method and topological current theory,
the evolution of topological defect which relate to sigularities of the order-
paramter field, such as vortex in BEC Duan04  and superconductivity Duan05 ,
was studied. In this Letter, we will develop a topological current theory of
discloations and disclination in 2D crystals. By using the topological field
approach, the topological quantization and evolution of topological defects in
2D crystals was discussed. We compared our theory with Brownian-dynamics
simulations in Yukawa systems.

In continuum elasticity theory, the homogeneous equation in 2D triangular
solid is given by Landau ; Nelson

$\nabla^{4}\chi/Y=\epsilon_{kl}\partial_{k}\partial_{l}\theta+\epsilon_{ik}\partial_{k}(\epsilon_{jl}\partial_{l}\partial_{j}u_{i})$
(1)

where $Y=4\mu(\mu+\lambda)/(2\mu+\lambda)$ is the two-dimensional Young’s
modulus and $\sigma_{0}=\lambda/(2\mu+\lambda)$ the two-dimensional Poisson
ratio. The defects associated with the continuum elastic theory of a solid are
dislocations and disclinations. In the following we consider only the
triangular lattice since it is the most densely packed one in two-dimensional
and favored by Nature.

Figure 1: Voronoi polygons analysis in 2D Yukawa systems.

Topological defects associated with the continuum elastic theory of a two-
dimensional solid are dislocations and disclinations (see Fig. 1).
Disclinations can be introduced into the theory in a way similar to the
discussion of vortices in two-dimensional XY model QY . Since the bond angle
is defined only up to $2\pi/6$, it implies

$\oint d\theta=\oint\nabla\theta d\vec{s}=\frac{2\pi}{6}W^{6},\quad
W^{6}=0,\pm 1,\ldots,n.$ (2)

Where $W^{6}$ is the winding number of disclinations. Disclinations, which can
influence on the decay of orientational order, have a much higher energy than
dislocations. They are defined in terms of the bond angle field
$\theta(r)\equiv 1/2(\partial u_{y}/\partial x-\partial u_{x}/\partial y),$
which measures the bonds orientational order. It is convenient to define an
order parameter for bond orientations, which for the triangular lattices is
$\psi(r)=\psi_{0}e^{i6\theta(r)}$. However, the bond angle field is undefined
at the disclination cores, i.e., the zero points of the order parameter. We
rewrite the orientation order parameter $\psi(r)=\phi_{6}^{1}+i\phi_{6}^{2}$
instead of $\psi(r)=\psi_{0}e^{i6\theta(r)}$. We can define the unit vector
field $\vec{n}_{6}(\textbf{x})$ as $n_{6}^{a}=\phi_{6}^{a}/||\phi_{6}||$ and
$||\phi_{6}||=\sqrt{\phi_{6}^{a}\phi_{6}^{a}}$, $a=1,2$. We can construct a
topological current of the orientation order parameter field in two-
dimensional crystal, which carries the topological information of
$\vec{\phi}(x)$,

$j_{disc}^{k}=\frac{1}{6}\epsilon^{ijk}\epsilon_{ab}\partial_{i}n_{6}^{a}\partial_{j}n_{6}^{b}.$
(3)

It is the $\phi$-mapping topological current for disclinations. We will see
that this current associate with the singularities of the unit field
$\vec{n}_{6}(x)$. It is easy to see that this topological current is
identically conserved, i.e., $\partial_{k}j_{disc}^{k}=0$.

We consider only the triangular lattice since it is the most densely packed
one in two-dimensional and favored by Nature. The dislocation is characterized
by a Burgers vector $\vec{b}$, which can be determined by considering a loop
enclosing the dislocation. In mathematical language,

$\oint du_{l}=\oint\partial_{i}u_{l}dx_{i}=b_{l}.$ (4)

By introduce two-dimensional local transition order parameters as
$\rho_{x}=\rho_{0x}e^{iG_{x}u_{x}}=\phi_{x}^{1}+i\phi_{x}^{2}$ and
$\rho_{y}=\rho_{0y}e^{iG_{y}u_{y}}=\phi_{y}^{1}+i\phi_{y}^{2}$, where G is any
reciprocal lattice vector and $\rho_{0}=e^{iG_{0}r_{0}}$. We can define a unit
vector field $\vec{n}_{l}(x)$, where
$\vec{n}_{l}(\textbf{x})=e^{iG_{l}u_{l}}=n_{l}^{1}+in_{l}^{2}$ ($l=x,y$). It
can be further expressed as

$n_{l}^{a}(\textbf{x})=\frac{\phi_{l}^{a}(\textbf{x})}{||\phi_{l}(\textbf{x})||},\quad||\phi_{l}(\textbf{x})||=\sqrt{\phi_{l}^{a}(\textbf{x})\phi_{l}^{a}(\textbf{x})},$

obviously, $n_{l}^{a}n_{l}^{a}=1$. We can construct topological currents of
the order parameter field in two-dimensional crystal, which carries the
topological information of $\vec{\phi}_{l}(x)$,

$j_{diso}^{kl}=\frac{1}{G_{l}}\epsilon^{ijk}\epsilon_{ab}\partial_{i}n_{l}^{a}\partial_{j}n_{l}^{b}.$
(5)

where the Burgers vectors satisfy $b_{l}G_{l}=2\pi$. It is the $\phi$-mapping
current for dislocations. Dislocations can be quite effective at breaking up
translational order. However, they are less disruptive of orientational
correlations. The disclination can be seen as a bound of dislocations with
parallel Burger vectors. On the hand, disclinations, which play a important
role in the disruptive of the orientational order in two-dimensional solid,
can be seen as bound disclination and anti-disclination pairs.

From the relationship between the bond angle field and displacement vector
field that $\partial_{i}\theta=-\epsilon_{ab}\partial_{a}\partial_{i}(u^{b})$.
Using the topological current theory, we can obtain

$\partial_{k}j_{diso}^{kl}=-\epsilon_{li}j_{disc}^{i},$ (6)

it indicate that disclinations can be seen as sources for dislocations. Since
there aren’t free disclinations in two-dimensional crystal at the low
temperature phase, the dislocation currents are conserved as
$\partial_{k}J_{k}^{l}=0$.

By using Green function relation and substitute $\phi_{6}^{a}$ to the
homogeneous equation Duan , we can rewrite the homogeneous equation into a
$\delta$-function form as

$\frac{1}{Y}\nabla^{4}\chi=\frac{2\pi}{6}\delta^{2}(\vec{\phi_{6}})J\big{(}\frac{\phi_{6}}{x}\big{)}+\epsilon_{kl}\partial_{k}\frac{2\pi}{G_{l}}\delta^{2}(\vec{\phi_{l}})J\big{(}\frac{\phi_{l}}{x}\big{)}.$
(7)

With the $\delta$-function theory, the homogeneous equation can deduce into

$\frac{\nabla^{4}\chi}{Y}=\sum_{i}\beta_{i}\eta_{i}\delta^{2}(\vec{r}-\vec{z_{i}}(t))+\sum_{j}\beta_{j}^{l}\eta_{j}^{l}\epsilon_{kl}\partial_{k}\delta^{2}(\vec{r}-\vec{z_{j}}(t)),$
(8)

where $\eta_{i}=\pm 1$ is Brouwer degree, the positive integer $\beta_{i}$ is
called the Hopf index of map $x\rightarrow\vec{\phi}$. The topological charge
of disclination and dislication is $\beta_{i}\eta_{i}=2\pi/6$ and
$\beta_{j}^{l}\eta_{j}^{l}=b_{j}^{l}$. The first term is related to
dislocation, and the second term is related to disclination. Also, it can
simply seen as disclination current, if dislocations are thought of as
disclination dipole pairs.

We explore what will happen to the disclination at the zero point
$(t^{*},\vec{z_{l}})$. According to the values of the vector Jacobian at zero
points of the order parameter, there are limit points ($J^{0}(\phi_{6}/x)\neq
0$) and bifurcation points ($J^{0}(\phi_{6}/x)=0$). Each kind corresponds to
different cases of branch processes. The limit points are determined by

$\left.J^{0}\left(\frac{\phi_{6}}{x}\right)\right|_{t^{*},\vec{z_{l}}}=0,\quad\left.J^{1}\left(\frac{\phi_{6}}{x}\right)\right|_{t^{*},\vec{z_{l}}}\neq
0.$ (9)

Considering the condition (9) and making use of the implicit function theorem,
the solution in the neighborhood of the point ($t^{*},\vec{z_{l}}$) is
$t=t(x),~{}y=y(x)$, where $t^{*}=t(z_{l}^{1})$. In this case, one can see that
$\left.dx/dt\right|_{(t^{*},\vec{z_{l}})}=\infty$, the Taylor expansion of
$t=t(x)$ at the limit points $(t^{*},\vec{z_{l}})$ is

$t-t^{*}=\left.\frac{1}{2}\frac{d^{2}t}{dx^{2}}\right|_{t^{*},\vec{z_{l}}}(x-z_{l}^{1})^{2},$
(10)

which is a parabola in the x-t plane. From this equation, we can obtain two
solutions $x_{1}(t)$ and $x_{2}(t)$, which give two branch solutions (World
lines of disclinations). If $d^{2}t/dx^{2}|_{(t^{*},\vec{z_{l}})}>0,$ we have
the branch solutions for $t>t^{*}$. It is related to the origin of a
disclination pair. Otherwise, we have the branch solutions for $t<t^{*}$,
which related to the annihilation of a disclination pair. Since the
topological current is identically conserved, the topological charges of these
two generated or annihilated disclinations must be opposite at the limit
points, say $\beta_{1}\eta_{1}+\beta_{2}\eta_{2}=0$. For a limit point it is
required that $J^{1}(\phi/x)|_{t^{*},\vec{z_{l}}}\neq 0$.

At the neighborhood of the limit point,one can obtain the scaling law as Xu
$v\varpropto(t-t^{*})^{-1/2}$. The growth or annihilation is parameterized in
terms of a relevant characteristic length $\xi(t)$, which is also
characteristic the mean separation distance of disclinations pair. The
relevant length $\xi(t)$ obeys that $\xi(t)\sim(t-t^{*})^{1/2}$. It is the
dynamic scaling law of the topological defects pairs. In the low temperature
equilibrium phase has essentially no disclination pairs, and $\xi(t)$ is
infinite. The relationship between the relevant length and the metastable
disclination density which below the critical temperature
is$\xi(t)=\sqrt{1/\rho_{v}}$. then the number of disclinations satisfy the
power law $N\propto t^{-1}$.

Consider in which the restrictions on zero point $(t^{*},\vec{z_{l}})$ are
$J^{k}(\phi_{6}/x)\big{|}_{(t^{*},\vec{z_{l}})}=0,(k=0,1,2)$, which imply an
important fact that the function relationship between t and x or y is not
unique in the neighborhood of the bifurcation point $(t^{*},\vec{z_{l}})$.
With the aim of finding the different directions of all branch curves at the
bifurcation point, we suppose $\partial\phi_{6}^{1}/\partial
y\big{|}_{t^{*},\vec{z_{l}}}\neq 0$. According to the $\phi$-mapping theory,
the Taylor expansion of the solution of the zeros of the order parameter field
in the neighborhood of $(t^{*},\vec{z_{l}})$ can be expressed as

$A(x-z_{l}^{1})^{2}+2B(x-z_{l}^{1})(t-t^{*})+C(t-t^{*})^{2}+\cdots=0,$ (11)

where A, B, and C are constants determined by the order parameter. The
solutions of Eq. (11) give different directions of the branch curves (world
line of disclinations) at the bifurcation point. There are four possible
cases, which will show the physical meanings of the bifurcation points.

First we consider the case that $A\neq 0$. If $\Delta=4(B^{2}-AC)>0$, from
Eq.(11) we get two different motion directions of the core of disclination
$dx/dt\big{|}_{1,2}=(-B\pm\sqrt{B^{2}-AC})/A$, where two topological defects
encounter and the depart at the bifurcation point. However, when
$\Delta=4(B^{2}-AC)=0$, form Eq. (11), we obtain only one motion direction of
the core of disclination$dx/dt\big{|}_{1,2}=-B/A$, which includes three
important cases. (i) Two disclinations tangentially encounter at the
bifurcation point. (ii) Two disclinations merge into one disclination at the
bifurcation point. (iii) One disclinations splits into two disclinations at
the bifurcation point.

Then we consider the case that $A=0$, If $C\neq 0$ and $\Delta=4(B^{2}-AC)=0$,
from Eq. (11) we have $dt/dx\big{|}_{1,2}=(-B\pm\sqrt{B^{2}-AC})/C=0,-(2B)/C$.
There are two important cases. (i) One disclination split into three
disclinations at the bifurcation point. (ii)Three disclinations merge into one
disclination at the bifurcation point. From the above discuss, we can then
obtain the growth or annihilation velocity of the vortices as $v\propto
const$. The approximation asymptotic relation of is $\xi(t)\propto(t-t^{*})$.
If $C=0$, Equations (11) give $dx/dt=0$ or $dt/dx=0$. This case shows that two
worldlines intersect normally at the bifurcation point, which is similar to
the case that $A=0$ and $C\neq 0$. We can obtain $\xi(t)=const,~{}v=0$.

It is obvious that vortices are relatively at rest when $\xi(t)=const$.
Through our topological current theory, we can get the dynamical scaling law.
Moreover, it worth to emphasizing that the dynamical scaling law of vortices,
only depend on topological properties of the order parameter field.

Figure 2: (a)The fractions of $6$-coordinated, $5$-coordinated, and
$7$-coordinated particles. At low temperature the number of defects is very
small. When the temperature above $0.500$, $N_{6}/N$ decreases rapidly.

We performed Brownian dynamics simulations on melting of two-dimensional
colloidal crystal in which particles interact with Yukawa potential Qi . We
have observed the mechanisms of defects in the two-dimensional Yukawa system
mv . Fig. 2 plots the fractions of $6$-coordinated, $5$-coordinated, and
$7$-coordinated particles. At low temperature, all particles are nearly six-
coordinated, and the number of defects is very small. When the temperature
reaches $0.500$, $N_{6}/N$ behaves in a rapidly decreasing. At $T^{*}=6.05$,
almost $20\%$ of particles are attached to the defects, which consisted with
the results in Ref. Ta. As the defects fraction rises above $30\%$, the system
melt into a liquid phase. At the low temperature, the paired dislocation is
formed or annihilated. With the temperature rises to the hexatic phase, the
dislocation pair comes into dissociated. In the solid phase, the unbinding of
dislocation is unstable, and these unbind dislocation will bind soon, whereas
the stable free dislocation is found in the hexatic phase. The above solutions
form topological current theory reveal the evolution of disclinations.

Besides the encountering of disclinations, splitting and merging of
disclinations are also included. When a multicharged disclinations, such as
$8$-coordinated particles, pass the bifurcation point, it may split into
several disclinations with lower value of Burgers vector along different
branch curves. On the other hand, several disclinations can merge into one
disclination at the bifurcation point. As before, since the topological
current of disclinations is identically conserved, the sum of the topological
charge of final disclinations must be equal to that the initial vortices at
the bifurcation point, i.e.
$\sum_{f}\beta_{lf}\eta_{lf}=\sum_{i}\beta_{li}\eta_{li}$ for fixed l. It
indicates that, vortices with a higher value of Burgers vector can evolve to
the lower value of Burgers vector, or vortices with a lower value of Burgers
vector can evolve to the higher value of Burgers vector through the
bifurcation process. But due to Such case was observed in our BD simulation in
Yukawa systems. That is why at the liquid phase the number of $5$-coordinated
particles is much more than $7$-coordinated particles due to the emergence of
the $8$-coordinated particles. The similar experimental result was observed in
melting of two-dimensional tunable-diameter colloidal crystals Han .

###### Acknowledgements.

We would like to thank Y. X. Liu for his helpful discussions. This work was
supported by the National Natural Science Foundation of China and by the SRF
for ROCS, SEM.

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