EPJ Web of Conferences CONF12

11institutetext: INFN - Istituto Nazionale di Fisica Nucleare - Sezione di
Napoli

# Combination of measurements and the BLUE method

Luca Lista 11<EMAIL_ADDRESS>
###### Abstract

The most accurate method to combine measurements from different experiments is
to build a combined likelihood function and use it to perform the desired
inference. This is not always possible for various reasons, hence approximate
methods are often convenient. Among those, the best linear unbiased estimator
(BLUE) is the most popular, allowing to take into account individual
uncertainties and their correlations. The method is unbiased by construction
if the true uncertainties and their correlations are known, but it may exhibit
a bias if uncertainty estimates are used in place of the true ones, in
particular if those estimated uncertainties depend on measured values. In
those cases, an iterative application of the BLUE method may reduce the bias
of the combined measurement.

##  1 Measurement combination

The most rigorous and accurate method to combine measurements from different
experiments relies on the combination of the individual likelihood functions
that have been used for each measurements. Imagine that an experiment $A$
provides a measured data sample $\vec{x}_{A}$, whose likelihood function,
characterized by a set of parameters $\vec{\theta}_{A}$, is

$L(\vec{x}_{A};\vec{\theta}_{A})\,,$ (1)

and an experiment $B$ provides a sample $\vec{x}_{B}$, whose likelihood
function, characterized by a set of parameters $\vec{\theta}_{B}$, is

$L(\vec{x}_{B};\vec{\theta}_{B})\,.$ (2)

The parameter sets $\vec{\theta}_{A}$ and $\vec{\theta}_{B}$ contain some
experiment-specific parameters and some parameters common to both experiments.
Among the latter, there are physical parameters of interest, nuisance
parameters related to common source of systematic uncertainties, such as
theory uncertainties, accelerator-specific component of luminosity
uncertainty, etc. The global likelihood function, if $A$ and $B$ are
independent experiments, is:

$L(\vec{x}_{A},\vec{x}_{B};\vec{\theta})=L(\vec{x}_{A};\vec{\theta}_{A})\cdot
L(\vec{x}_{B};\vec{\theta}_{B})\,,$ (3)

where $\vec{\theta}$ is the set of all common and non-common parameters. Any
statistical method can be applied at this level to determine a combined
measurement, upper limit and/or significance: a Bayesian or frequentist
inference, profile likelihood, modified frequentist upper limit, etc.

An example of such approach is provided by the combination of measurements of
single top-quark production in the $s$-channel at the Tevatron performed by
the CDF and D0 experiments CDFs1 ; CDFs2 ; D0s .

CDF and D0 measurements of single-top production cross section were combined
using as input the binned distributions of multivariate discriminator outputs
for each individual measurement. Each bin in each data sample was used in a
Bayesian analysis, assuming a Poisson distribution, to extract a central value
of the cross-section estimate. A likelihood-ratio analysis using asymptotic
formulae asymptotic  was instead used to determine the combined significance
of the observed $s$-channel signal.

##  2 Approximate approaches: the BLUE method

In many cases, unless experiments, or even analysis groups within the same
experiment, agree in advance, individual likelihood functions may not be
available, or are available under different software frameworks, etc.
Approximate methods can be used to combine the individual results, which are
usually provided in terms of a central value and an uncertainty:

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\hat{\theta}_{1}\pm\sigma_{1}\,,$ (4) $\displaystyle\theta$
$\displaystyle=$ $\displaystyle\hat{\theta}_{2}\pm\sigma_{2}\,.$ (5)

Correlation between uncertainties, which is related to the non-diagonal
elements of the covariance matrix $\mathbf{V}$

$\rho=\frac{V_{12}}{\sigma_{1}\sigma_{2}}\,,$ (6)

must be properly taken into account.

The most popular method to combine correlated measurements is the best linear
unbiased estimate (BLUE). The method was formulated initially in the ’30s
BLUE0  and proposed in high-energy physics in the ’80s BLUE . By definition,
it is the unbiased linear estimator that provides the smallest possible
variance assuming the true uncertainties and their correlation are known. The
estimator is equivalent to a $\chi^{2}$ minimization which, for Gaussian
distributions, is also equivalent to a maximum-likelihood estimate.

Given two measurements, as in eq. (4) and (5), the BLUE estimate is given by
the following linear combination of the individual measurements:

$\hat{\theta}=\frac{\hat{\theta}_{1}(\sigma_{2}^{2}-\rho\sigma_{1}\sigma_{2})+\hat{\theta}_{2}(\sigma_{1}^{2}-\rho\sigma_{1}\sigma_{2})}{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}+\sigma_{2}^{2}}$
(7)

with variance:

$\sigma_{\hat{\theta}}^{2}=\frac{\sigma_{1}^{2}\sigma_{2}^{2}(1-\rho^{2})}{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}+\sigma_{2}^{2}}\,.$
(8)

Unlike the usual weighted average, for some values of the correlation
coefficient $\rho$, the coefficients of the linear combination that appear in
eq. (7) may be negative.

More in general, for $n$ measurements
$\hat{\theta}_{1}\pm\sigma_{1},\cdots,\hat{\theta}_{n}\pm\sigma_{n}$, with a
covariance matrix $\mathbf{V}$, the BLUE combination in eq. (7) can be
generalized as:

$\hat{\theta}=\sum_{i=1}^{n}w_{i}\hat{\theta}_{i}\,,$ (9)

with variance:

$\sigma_{\hat{\theta}}^{2}=\vec{w}^{\mathrm{T}}\,\mathbf{V}\,\vec{w}\,,$ (10)

The weights in eq. (9) can be computed as:

$\vec{w}=\frac{\mathbf{V}^{-1}\vec{u}}{\vec{u}^{\mathrm{T}}\,\mathbf{V}\,\vec{u}}$
(11)

where $\vec{u}=(1,\cdots,1)$ is the vector with all elements equal to unity.
The normalization condition for weights $\vec{w}$ holds:

$\sum_{i=1}^{n}w_{i}=1\,.$ (12)

In case the weights $w_{i}$ are positive, an interpretation of the BLUE
combination in eq. (7) in term of weighted average is possible by introducing
the “common error” Greenlee , defined as:

$\sigma_{\mathrm{c}}=\rho\sigma_{1}\sigma_{2}\,.$ (13)

The two measurements in eq. (4) and (5) can be rewritten with uncertainties
given by the sum in quadrature of fully uncorrelated contributions
$\sigma_{1}^{\prime}$ and $\sigma_{2}^{\prime}$ and a 100% correlated
contribution $\sigma_{\mathrm{C}}$:

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\hat{\theta}_{1}\pm\sigma_{1}^{\prime}\pm\sigma_{\mathrm{C}}\,,$
(14) $\displaystyle\theta$ $\displaystyle=$
$\displaystyle\hat{\theta}_{2}\pm\sigma_{2}^{\prime}\pm\sigma_{\mathrm{C}}\,,$
(15)

where the uncorrelated uncertainty contributions are defined by:

$\displaystyle\sigma_{1}^{\prime 2}$ $\displaystyle=$
$\displaystyle\sigma_{1}^{2}-\sigma_{\mathrm{C}}^{2}\,,$ (16)
$\displaystyle\sigma_{2}^{\prime 2}$ $\displaystyle=$
$\displaystyle\sigma_{2}^{2}-\sigma_{\mathrm{C}}^{2}\,,$ (17)

The BLUE combination in eq. (7) achieves an expression similar to a regular
weighted average:

$\hat{\theta}=\frac{\displaystyle\frac{\hat{\theta}_{1}}{\sigma_{1}^{\prime
2}}+\frac{\hat{\theta}_{2}}{\sigma_{2}^{\prime
2}}}{\displaystyle\frac{1}{\sigma_{1}^{\prime 2}}+\frac{1}{\sigma_{2}^{\prime
2}}}\,,$ (19)

with weights that only take into account the uncorrelated uncertainty
contributions, but in this case the uncertainty receives an additional
contribution due to the correlated uncertainty, with respect to the
uncertainty of the usual weighted average, and is given by:

$\sigma_{\hat{\theta}}^{2}=\frac{1}{\displaystyle\frac{1}{\sigma_{1}^{\prime
2}}+\frac{1}{\sigma_{2}^{\prime 2}}}+\sigma_{\mathrm{C}}^{2}\,.$ (20)

Cases with negative weights have a less intuitive interpretation than eq.
(19), as will be more evident in section 4.

##  3 Quantifying the importance of individual measurements

In order to quantify the “importance” of each individual measurement used in a
combination, the first approach adopted in literature was to quote the so-
called “relative importance” (RI) of each individual measurement, proportional
to the absolute value of the corresponding weight ‘$w_{i}$, and defined as:

$\mathrm{RI}_{i}=\frac{|w_{i}|}{\displaystyle\sum_{i=1}^{n}|w_{i}|}\,.$ (21)

The definition is chosen in order to have a normalization condition:
$\sum_{i}\mathrm{RI}_{i}=1$. This approach was, for instance, used in
combinations of top-quark mass measurements at Tevatron and at LHC cdfd01 ;
atlascms1 .

This choice is questionable, as observed in ref. Chierici . In fact, imagine
we have three measurement, say $A$, $B_{1}$ and $B_{2}$. The RI of measurement
$A$ changes whether the three measurements are combined all together or if
$B_{1}$ and $B_{2}$ are first combined into $B$, and then $A$ and the partial
combination $B$ are combined together.

Ref. Chierici  proposes alternatives to RI based on the Fisher information,
which is defined as:

${\cal{J}}_{ij}=\left<\frac{\partial\ln
L}{\partial\theta_{i}}\frac{\partial\ln L}{\partial\theta_{j}}\right>\,,$ (22)

the average being performed over all possible measurements, hence ${\cal J}$
does not depend on a specific measurement, but only on the form of the
likelihood function and on the parameters choice. Fisher information sets a
lower bound to the variance of an unbiased estimator Cramer ; Rao :

$\sigma_{\hat{\theta}_{i}}^{2}\geq{\cal{J}}_{ii}^{-1}\,,$ (23)

and for a single parameter, the Fisher information is given by:

${\cal{J}}={\cal{J}}_{11}={\vec{u}^{\mathrm{T}}\,\mathbf{V}\,\vec{u}}=\frac{1}{\sigma_{\hat{\theta}}^{2}}\,.$
(24)

The alternative quantities to RI proposed in ref. Chierici  are the intrinsic
information weight (IIW), defined as:

$\mathrm{IIW}_{i}=\frac{1/\sigma_{i}^{2}}{1/\sigma_{\hat{\theta}}^{2}}=\frac{1/\sigma_{i}^{2}}{\cal{J}}$
(25)

and the marginal information weight (MIW), defined as follows:

$\mathrm{MIW}_{i}=\frac{\Delta\cal{J}_{i}}{\cal{J}}=\frac{{\cal{J}}-{\cal{J}}_{\left\\{1,\cdots,n\right\\}-\\{i\\}}}{\cal{J}}\,,$
(26)

i.e.: the relative difference of the Fisher information of the combination and
the Fisher information of the combination excluding the $i^{\mathrm{th}}$
measurement. Both IIW and MIW do not obey a normalization condition. For IIW
the quantity $\mathrm{IIW}_{\mathrm{corr}}$ can be defined such that:

$\sum_{i=1}^{n}\mathrm{IIW}_{i}+\mathrm{IIW}_{\mathrm{corr}}=1\,.$ (27)

$\mathrm{IIW}_{\mathrm{corr}}$ represents the weight assigned to the
correlation interplay, not assignable to individual measurements, and is given
by:

$\mathrm{IIW}_{\mathrm{corr}}=\frac{1/\sigma_{\hat{\theta}^{2}}-\sum_{i=1}^{n}1/\sigma_{i}^{2}}{1/\sigma_{\hat{\theta}}^{2}}=\frac{{\cal{J}}-\sum_{i=1}^{n}1/\sigma_{i}^{2}}{\cal{J}}\,.$
(28)

Table 1 summarizes the properties of the different proposed quantities.

Table 1: Properties of the different quantities proposed for the BLUE combination Weight type | $\geq 0$ | $\sum_{i}=1$ |  | Consistent with
---
partial combination
BLUE coefficient | $w_{i}$ | ✗ | ✓ | ✓
Relative importance | $|w_{i}|/\sum_{i=1}^{n}|w_{i}|$ | ✓ | ✓ | ✗
Intrinsic Information Weight | $\mathrm{IIW}_{i}$ | ✓ | ✗ | ✓
Marginal Information Weight | $\mathrm{MIW}_{i}$ | ✓ | ✗ | ✓

IIW and MIW are reported, instead of RI, in recent papers about combination of
LHC and Tevatron measurements related to top-quark physics IIW1 ; IIW2 ; IIW3
.

##  4 Negative weights

Negative weights are always sign of a high-correlations regime. The maximum
value of the ratio $\sigma_{\hat{\theta}^{2}}/\sigma_{1}^{2}$ (eq. (8)) as a
function of $\rho$ is obtained for $\rho=\sigma_{1}/\sigma_{2}$. For
$\rho>\sigma_{1}/\sigma_{2}$, an increase in correlation implies a decrease of
the uncertainty and a negative weight, and uncertainty becomes strongly
dependent on $\rho$. For $\rho=\sigma_{1}/\sigma_{2}$, in particular, the
weight $w_{2}$ becomes equal to zero, as well as $\mathrm{MIW}_{2}=0$. But
this does not imply that the measurement $\theta_{2}$ is not used in the
combination. Figure 1, from ref. Chierici , shows how the BLUE coefficient
$w_{2}$ and the ratio of uncertainties
$\sigma_{\hat{\theta}}^{2}/\sigma_{1}^{2}$ vary as a function of the
correlation $\rho$ for different fixed values of the ratio
$\sigma_{2}/\sigma_{1}$ (note that the figure uses a different notation with
respect to this text, as specified in the caption).

Figure 1:  BLUE coefficient for the second measurement $w_{2}$ (left;
$\lambda_{B}$ in the original figure notation) and combined BLUE variance
$\sigma_{\hat{\theta}}^{2}$ (right; $\sigma_{\hat{Y}}^{2}$ in the original
figure notation) as a function of the correlation $\rho$ between two
measurements $1$ and $2$ for various fixed values of the ratio
$\sigma_{2}/\sigma_{1}$ ($\sigma_{B}/\sigma_{A}$ in the original figure
notation). The figure is from ref. Chierici .

When the correlation coefficient $\rho$ is not well know, assuming $\rho=1$ is
not always a conservative choice. The assumption $\rho=1$ yields the largest
possible uncertainty only if the uncorrelated contributions to the total
uncertainty dominate. $\rho$ should be accurately determined in case of
negative weights in order to avoid the risk of underestimating uncertainties.
Assume that the two measurements $A$ and $B$ have total uncertainties given by
the sum in quadrature of uncorrelated contributions, $\sigma_{A}$(unc) and
$\sigma_{B}$(unc) respectively, and correlated contributions,
$\sigma_{A}$(cor) and $\sigma_{B}$(cor) respectively, whose correlation
coefficient is $\rho$(cor). Figure 2 shows the most “conservative” value of
the correlation coefficient $\rho$(cor), which is equal to 1 only for
$\sigma_{B}$(cor)$/\sigma_{A}$(cor)$<(\sigma_{A}/\sigma_{A}$(cor))2.

Figure 2: The most “conservative” value of an unknown correlation $\rho$(cor)
between $\sigma_{A}$(cor) and $\sigma_{A}$(cor) as a function of
$\sigma_{A}$(cor)$/\sigma_{A}$, for several values of
$\sigma_{B}$(cor)$/\sigma_{A}$(cor)$\geq 1$. The figure is from ref. Chierici
.

##  5 Bias with the BLUE method

The BLUE method provides an unbiased estimate only if uncertainties and their
correlation are known exactly. This is not always a realistic scenario, since
estimates of uncertainties and their correlation, and not their true values,
are available in most of the cases. Moreover, the uncertainty estimates may
depend on the assumed central value. One example of such a case in which the
BLUE method provides a bias is the combination of two Poissonian estimates,
each of which has an uncertainty estimate that depends on the central value
through a square-root relation:

$\displaystyle\hat{n}_{1}$ $\displaystyle=$
$\displaystyle\hat{n}_{1}\pm\sqrt{\hat{n}_{1}}\,,$ (29)
$\displaystyle\hat{n}_{2}$ $\displaystyle=$
$\displaystyle\hat{n}_{2}\pm\sqrt{\hat{n}_{2}}\,.$ (30)

A maximum-likelihood estimate, using Poissonian distributions, would produce
the following unbiased combined estimate:

$\hat{n}=\frac{1}{2}\left(\hat{n}_{1}+\hat{n}_{2}\pm\sqrt{\hat{n}_{1}+\hat{n}_{2}}\right)\,.$
(31)

The BLUE combination, instead, gives weights proportional to $1/\hat{n}_{i}$,
which correspond to a harmonic average:

$\hat{n}=\frac{2\hat{n}_{1}\hat{n}_{2}}{\hat{n}_{1}+\hat{n}_{2}}\pm\sqrt{\frac{\hat{n}_{1}\hat{n}_{2}}{\hat{n}_{1}+\hat{n}_{2}}}\,.$
(32)

Equation (32), compared to eq. (31), exhibits a bias because, due to the
dependence of uncertainties on the measured values, downward measurement
fluctuations achieve larger weights pulling down the combination, while upward
fluctuations produce a smaller opposite effect.

In many cases, relative uncertainty estimates are available, like for
uncertainty contributions due to luminosity, efficiencies, etc. When
performing a combination, the best estimate of the central value improves the
individual measurements, hence one may argue whether the assumed uncertainties
should change accordingly, re-evaluating them using the central-value estimate
from the BLUE combination Lyons ; Lista . The method can be applied
iteratively until the combination converges to a stable value. Namely:

$\displaystyle\
\hat{\sigma}_{i}^{(0)}=\hat{\sigma}_{i}(\hat{\theta}_{i})\,,\,\,\hat{\rho}_{ij}^{(0)}=\hat{\rho}(\hat{\theta}_{i},\hat{\theta}_{j})$
$\displaystyle\xrightarrow{\,\,\,\mathrm{BLUE}\,\,\,}$
$\displaystyle\hat{\theta}^{(1)}$ (33)
$\displaystyle\hat{\sigma}_{i}^{(1)}=\hat{\sigma}_{i}(\hat{\theta}^{(1)})\,,\,\,\hat{\rho}_{ij}^{(1)}=\hat{\rho}(\hat{\theta}^{(1)},\hat{\theta}^{(1)})$
$\displaystyle\xrightarrow{\,\,\,\mathrm{BLUE}\,\,\,}$
$\displaystyle\hat{\theta}^{(2)}$ (34)
$\displaystyle\hat{\sigma}_{i}^{(2)}=\hat{\sigma}_{i}(\hat{\theta}^{(2)})\,,\,\,\hat{\rho}_{ij}^{(2)}=\hat{\rho}(\hat{\theta}^{(2)},\hat{\theta}^{(2)})$
$\displaystyle\xrightarrow{\,\,\,\mathrm{BLUE}\,\,\,}$
$\displaystyle\hat{\theta}^{(3)}$ $\displaystyle\cdots$

In practice, convergence only needs few iterations in most of the cases.

Let us assume that a contribution to the total uncertainty is known as
relative uncertainty. In this case, uncertainties can be written as the sum in
quadrature of a contribution that does not depend on the central value and
another contribution that is proportional to the central value111 This was not
the case for the combination of Poissonian measurements from eq. (29), (30),
where the dependence on central value was not linear, but it is a convenient
assumption in many realistic cases. :

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\hat{\theta}_{1}\pm\sigma_{1}\pm r_{1}\hat{\theta}_{1}\,,$ (36)
$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\hat{\theta}_{2}\pm\sigma_{2}\pm r_{2}\hat{\theta}_{2}\,.$ (37)

The covariance matrix estimate can be written as follows:

$\hat{\mathbf{V}}=\left(\begin{array}[]{cc}\sigma_{1}^{2}+(r_{1}\hat{\theta}_{1})^{2}&\rho_{0}\sigma_{1}\sigma_{2}+\rho_{r}r_{1}r_{2}\hat{\theta}_{1}\hat{\theta}_{2}\\\
\rho_{0}\sigma_{1}\sigma_{2}+\rho_{r}r_{1}r_{2}\hat{\theta}_{1}\hat{\theta}_{2}&\sigma_{2}^{2}+(r_{2}\hat{\theta}_{2})^{2}\end{array}\right)$
(38)

where $\rho_{0}$ and $\rho_{r}$ represent the correlation coefficients of the
uncertainty contributions that do not depend on the central value and of the
ones that are proportional to the central value, respectively.

A special case is when uncertainties are fully proportional to the central
values, i.e.: $\sigma_{1}=0$ and $\sigma_{2}=0$. In that case, the iterative
BLUE method converges in two iterations to the following central value:

$\hat{\theta}=\frac{(r_{2}^{2}-\rho_{r}r_{1}r_{2})\hat{\theta}_{1}+(r_{1}^{2}-\rho_{r}r_{1}r_{2})\hat{\theta}_{2}}{q_{1}^{2}-2\rho_{r}r_{1}r_{2}+r_{2}^{2}}\,,$
(39)

which is similar to eq. (7) but relative uncertainties are used in place of
the absolute ones.

A numerical Monte Carlo study Lista  shows that in most of the cases the bias
in the combination can be mitigated by applying the iterative procedure.
Assuming for simplicity, and without loss of generality, a true value
$\theta=1$, uncertainties and their correlations are chosen randomly by
spanning a wide ranges of values in the boundaries
$\sigma_{1},\,\sigma_{2},\,r_{1}\,,r_{2}\,,<1$ and $-1<\rho_{0},\,\rho_{r}<1$.
For each set of randomly-extracted uncertainties and correlation values, 500
000 random values of $\hat{\theta}_{1}$ and $\hat{\theta}_{2}$ are generated
using the proper two-dimensional correlated Gaussian distribution. The BLUE
method is then applied in its standard formulation and iteratively. Pulls
distributions allow to study the bias and the correctness of uncertainty
estimates. Bias is in general mitigated with the iterative BLUE method while
the standard BLUE method tends to underestimate the central value, as shown in
Fig. 3. The iterative BLUE application may provide overestimates in few of the
cases with large uncertainties. More detailed comparisons of the two method
are available in ref. Lista . In particular, the combination uncertainty may
be overestimated (and in fewer cases underestimated) in both standard and
iterative BLUE, but the iterative BLUE estimates tend to be more conservative,
as visible in Fig. 4. A dedicated toy Monte Carlo may be useful in case of
large individual uncertainties in order to achieve a proper combined
uncertainty estimate.

Figure 3: Distribution of the average value of the standard (top and middle,
left) and iterative (top and middle, right) BLUE estimates for different
limits on $r_{1,\,2}$ and for $\rho_{0}\geq 0$ and $\rho_{r}\geq 0$ (top) and
for $\rho_{0}<0$ and $\rho_{r}<0$ (middle). Bottom plots show the difference
of measured absolute value of the bias for the standard and iterative BLUE
estimates for different limits on $r_{1,\,2}$ and for $\rho_{0}\geq 0$ and
$\rho_{r}\geq 0$ (left) and for $\rho_{0}<0$ and $\rho_{r}<0$ (right).
Positive values indicate a smaller bias in the iterative method compared to
the standard method. Underestimate of the uncertainty (negative values) occur
in fewer cases with the iterative BLUE method than with the standard BLUE
method. The figure is from ref. Lista .

Figure 4: Distribution of the average value of the pull standard deviation for
the standard (left) and iterative (right) BLUE estimates for different limits
on $r_{1,\,2}$. The figure is from ref. Lista .

##  6 Conclusions

Combination of measurements is a crucial task to improve the precision on the
knowledge of important parameters. Combining individual likelihood functions
before applying any statistical method is the most rigorous and precise
approach, but often the full likelihood function of individual measurements is
not available. When uncertainties and their correlations are available, the
BLUE method is a simple and powerful tool to combine measurements. But BLUE
may have counterintuitive behaviors such as negative weights or uncertainties
that may decrease for increasing correlation. Correlation needs to be
accurately evaluated in those cases, and an assumption of a 100% correlation
is not necessary a conservative choice. In case relative uncertainty
contributions are known, or more in general when uncertainty estimates depend
on the central values, the BLUE method may exhibit a bias that can be
mitigated, in most of the cases, with an iterative application of the method
where uncertainties are rescaled at each iteration to the combined central
value.

## References

  * (1) CDF Collaboration, Phys. Rev. Lett 112, 231804 (2014)
  * (2) CDF Collaboration, Phys. Rev. Lett 112, 231805 (2014)
  * (3) D0 Collaboration, Phys. Lett B 726, 656 (2013)
  * (4) G. Cowan et al., Eur. Phys. J. C 71, 1554 (2011)
  * (5) J. Aitken, Proc. Roy. Soc. Edinburgh 55, 42 (1935)
  * (6) L. Lyons et al. Nucl. Instr. Meth. A 270, 110 (1988)
  * (7) H. Greenlee, Fermilab Workshop on Confidence Limits (2000)
  * (8) CDF and D0 Collaborations, arXiv:1107.5255 (2011)
  * (9) ATLAS and CMS Collaborations, ATLAS-CONF-2012-095, CMS-PAS-TOP-12-001 (2012)
  * (10) A. Valassi, R. Chierici, Eur. Phys. J. C 74 2717 (2014)
  * (11) H. CramÃr, Mathematical Methods of Statistics (Princeton Univ. Press, Princeton, NJ, 1946)
  * (12) C. R. Rao, Bulletin of the Calcutta Mathematical Society, 37, 81â89 (1945)
  * (13) ATLAS and CMS Collaborations, ATLAS-CONF-2013-102, CMS-PAS-TOP-13-005 (2013)
  * (14) ATLAS and CMS Collaborations, ATLAS-CONF-2014-012, CMS-PAS-TOP-14-006 (2014)
  * (15) ATLAS, CMS, CDF and D0 Collaborations, ATLAS-CONF-2014-008, CDF-NOTE-11071, CMS-PAS-TOP-13-014, D0-NOTE-6416, FERMILAB-TM-2582, arXiv:1403.4427 (2014)
  * (16) L. Lyons, A. J. Martin, D. H. Saxon, Phys. Rev. D 41 982â985 (1990)
  * (17) L. Lista, Nucl. Instr. Meth. A 764 82-93 (2014); corrig. ibid. A 773 87-96 (2015)


# A topological $L_{\omega_{1},\omega}$-invariant.

B.Zilber

###### Abstract

We suggest to look at formal sentences describing complex algebraic varieties
together with their universal covers as topological invariants. We prove that
for abelian varieties and Shimura varieties this is indeed a complete
invariant, i.e. it determines the variety up to complex conjugation.

##  1 Introduction

The universal (and more general) cover of a complex algebraic variety as an
abstract structure has been a subject of study in model theory in recent
years, see e.g. [1], [2], [4], [5]. More specifically, given a smooth complex
variety ${\mathbb{X}}$ and its cover $\pi:{\mathbb{U}}\to{\mathbb{X}}$
presented in a natural universal language, the aim has been to find an
$L_{\omega_{1},\omega}$-sentence $\Sigma_{\mathbb{X}}$ which axiomatises
$\pi:{\mathbb{U}}\to{\mathbb{X}}$ categorically, that is determines the
isomorphism type of an uncountable model of $\Sigma_{\mathbb{X}}$ uniquely in
a given cardinality, up to an abstract isomorphism (ignoring topology). This
proved to be the case in the classes where one has enough knowledge of the
Galois action and arithmetic of ${\mathbb{X}},$ see references above.

We suggest here that such a $\Sigma_{\mathbb{X}}$ can serve as a complete
topological invariant of respective ${\mathbb{X}}.$ More precisely, assuming
that in a given class of varieties ${\mathbb{X}}$ such a categorical
$L_{\omega_{1},\omega}$-sentence $\Sigma_{\mathbb{X}}$ exists, we expect:

If, for a complex variety ${\mathbb{X}}^{\prime},$ its cover structure
$\pi^{\prime}:{\mathbb{U}}^{\prime}\to{\mathbb{X}}^{\prime}$ is a model of the
same sentence $\Sigma_{\mathbb{X}}$ then there is a biholomorphic isomorphism
either between ${\mathbb{X}}$ and ${\mathbb{X}}^{\prime}$ or between
${\mathbb{X}}^{c},$ the complex conjugate of ${\mathbb{X}}$, and
${\mathbb{X}}^{\prime}.$

(Compare this with speculations in a similar vein presented by M.Gavrilovich
in [7].)

In this note the cases of abelian varieties and Shimura varieties are
elaborated.

I am grateful to Chris Daw and Martin Bays for valuable comments and
corrections.

##  2 The language of covers

###### 2.1

Let $\mathrm{k}\subseteq{\mathbb{C}},$ a countable subfield,
$\\{{\mathbb{X}}_{i}:i\in I\\}$ a collection of non-singular irreducible
complex algebraic varieties (of $\dim>0$) defined over $\mathrm{k}$ and
$I:=(I,\geq)$ a lattice with the minimal element $\emptyset$ determined by
unramified $\mathrm{k}$-rational epimorphisms ${\rm
pr}_{i^{\prime},i}:{\mathbb{X}}_{i^{\prime}}\to{\mathbb{X}}_{i},$ for
$i^{\prime}\geq i.$ Let ${\mathbb{U}}({\mathbb{C}})$ be a connected complex
manifold and $\\{\pi_{i}:i\in I\\}$ a collection of holomorphic covering maps
(local bi-holomorphismss)

$\pi_{i}:{\mathbb{U}}({\mathbb{C}})\twoheadrightarrow{\mathbb{X}}_{i}({\mathbb{C}}),\
\ {\rm pr}_{i^{\prime},i}\circ\pi_{i^{\prime}}=\pi_{i}.$

as illustrated by the picture:

${\mathbb{X}}_{n}({\mathbb{C}})$${\rm
pr}_{m,n}$${\mathbb{X}}_{m}({\mathbb{C}})$${\mathbb{X}}({\mathbb{C}})$${\rm
pr}_{n,0}$${\rm pr}_{m,0}$$\mathbb{U}({\mathbb{C}})$$\pi_{m}$$\pi_{n}$

###### 2.2

The langauge of covers which we use is introduced in full generality in [12]
based on the simple idea first formulated by M.Gavrilovich in [7].

Definition. Given $S\subset{\mathbb{U}}^{m}$ we say that $S$ is
$\mathcal{L}$-primitive if there are $I_{S}\subseteq I$ and Zariski closed
$Z_{i}\subseteq{\mathbb{X}}_{i}^{m},$ $i\in I_{S},$ defined over $\mathrm{k},$
such that

$S=\bigcap_{i\in I_{S}}\pi_{i}^{-1}(Z_{i}).$

It is easy to see that the one-sorted structure $({\mathbb{U}};\mathcal{L}),$
interprets any finite cover ${\mathbb{X}}_{i}$ of ${\mathbb{X}}$ as the
structure with $m$-ary relations given by Zariski closed subsets of
${\mathbb{X}}_{i}^{m}$ over $\mathrm{k}.$

###### 2.3

Examples.

1\. Let $x_{0}\in{\mathbb{X}}_{i}(\mathrm{k})$ be a rational point. Then
$\pi_{i}^{-1}(x_{0})$ is definable over $\mathrm{k}.$ Each point
$u\in\pi_{i}^{-1}(x_{0})$ is type-definable over $\bar{\mathrm{k}},$ the
algebraic closure of $\mathrm{k}.$

2\. Let $\Gamma$ be the fundamental group of ${\mathbb{X}}({\mathbb{C}})$
acting on ${\mathbb{U}},$ $\gamma\in\Gamma.$ Then

$\mathrm{graph}(\gamma)=\\{\langle u,\gamma\cdot u\rangle:u\in{\mathbb{U}}\\}$

is an analytic component of
$\pi^{-1}(\mathrm{Diag}({\mathbb{X}}\times{\mathbb{X}})),$ the pre-image of
the diagonal. It is type-definable in the language $\mathcal{L}$ using
parameters from algebraic closure of $\mathrm{k}.$ In fact, it is
$\L_{\omega_{1},\omega}$-definable using parameters from ${\mathbb{U}}.$

3\. Let ${\mathbb{U}}({\mathbb{C}})={\mathbb{C}}$ and ${\mathbb{X}}$ an
elliptic curve over $\mathrm{k}$ or $\mathbb{G}_{m},$ $\pi=\exp,$ the
respective exponentiation. By definition $\exp(0)$ is a $\mathrm{k}$-rational
point. It follows (using the fact that the $\pi(u)$ can be taken just to be
$\exp(\frac{u}{n})$) that $\\{0\\}$ is definable. And so is the the ternary
relation $u_{1}+u_{2}-u_{3}=0$ on ${\mathbb{C}}.$

4\. For the case of elliptic curves, the group $\Lambda\otimes{\mathbb{Q}}$ is
endowed with an alternating bilinear form corresponding to the Weil pairing on
the torsion of the elliptic curve. It induces a definable relation on
$\Lambda$ which is described in [1], 4.4.

###### 2.4

Structures in this language were studied in [12] under the assumptions of
o-minimal representation of the cover (assumptions 2.2 in [12]). These are
obviously satisfied when ${\mathbb{X}}({\mathbb{C}})$ is compact.

##  3 Abelian varities

Recall that by [1] for each elliptic non-CM curve $\mathrm{E}$ over a field
$\mathrm{k}\subset{\mathbb{C}}$ there is a categorical
$L_{\omega_{1},\omega}$-sentence $\Sigma_{\mathrm{E}}$ in the language with
parameters $C$ interpreted as elements of $\mathrm{k},$ which models the cover
structure $(\pi:{\mathbb{C}}\to\mathrm{E}({\mathbb{C}})):$

$(\pi:{\mathbb{C}}\to\mathrm{E}({\mathbb{C}}))\vDash\Sigma_{\mathrm{E}}.$

The same is true for the general case of abelian varieties $\mathbb{A}$ by
[2]. The statement $\Sigma_{\mathbb{A}}(\bar{\lambda})$ there is proved over
the language naming generators $\bar{\lambda}$ of $\Lambda$ but this can be
eliminated by taking instead

$\Sigma^{\sharp}_{\mathbb{A}}:=\exists\bar{\lambda}\Sigma_{\mathbb{A}}(\bar{\lambda})\
\&\ \mathrm{qftp}(\pi({\mathbb{Q}}\otimes\Lambda)).$

(The $\Sigma_{\mathrm{E}}$ in [1] is more precisely determined which is
important in the context of this paper.)

We may assume that $\Sigma_{\mathbb{A}}$ is stronger than the first order
theory $\mathrm{Th}_{\mathbb{A}}$ in the same language with parameters
$C=\\{c_{a}:a\in\mathrm{k}\\},$ and so any Zariski closed subset definable
over $\mathrm{k}$ is first-order definable in this theory. Recall the
following folklore fact mentioned e.g. in [13] and proved in detail in [1],
Fact A.21.

Fact. Let $\mathbb{F}$ be an algebraically closed field of characteristic 0
and ${\mathbb{X}}(\mathbb{F})$ be $\mathbb{F}$-points an algebraic variety
over $\mathrm{k}\subset\mathbb{F}$ considered as the structure with relations
corresponding to $\mathrm{k}$-definable Zariski closed subsets of
${\mathbb{X}}^{m}(\mathbb{F}),$ $m\in{\mathbb{N}}.$ Then
$(\mathbb{F};+,\cdot,c_{a})_{a\in\mathrm{k}}$ is definable in the structure.

Consequently, if ${\mathbb{X}}^{\prime}$ is a variety over
$\mathrm{k}^{\prime}\subset\mathbb{F}$ and ${\mathbb{X}}^{\prime}(\mathbb{F})$
is a model of the theory $\mathrm{Th}_{\mathbb{X}}$ then there is an
automorphism $\sigma$ of field $\mathbb{F}$ such that
$\mathrm{k}^{\sigma}=\mathrm{k}$ and
${\mathbb{X}}^{\sigma}(\mathbb{F})={\mathbb{X}}^{\prime}(\mathbb{F}).$

Let ${\mathbb{X}}_{1}$ and ${\mathbb{X}}_{2}$ be two complex algebraic
varieties over $\mathrm{k}_{1}$ and $\mathrm{k}_{2}$ respectively. We will say
that ${\mathbb{X}}_{1}$ and ${\mathbb{X}}_{2}$ have the same
$L_{\omega_{1},\omega}$-type if there is an uncountably categorical
$L_{\omega_{1},\omega}$-sentence $\Sigma$ over parameters $C$ interpreted as
elements of $\mathrm{k}_{1}$ in ${\mathbb{X}}_{1}$ and of $\mathrm{k}_{2}$ in
${\mathbb{X}}_{2}$ such that the cover structures of both
${\mathbb{X}}_{1}({\mathbb{C}})$ and ${\mathbb{X}}_{2}({\mathbb{C}})$ are
models of $\Sigma.$

Theorem. Let $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ be two complex abelian
varieties.

Suppose $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$ have the same
$L_{\omega_{1},\omega}$-type. Then there is a biholomorphic isomorphism

$\varphi:\mathbb{A}_{1}({\mathbb{C}})\to\mathbb{A}_{2}({\mathbb{C}}),\mbox{ or
}\varphi:\mathbb{A}_{1}({\mathbb{C}})\to\mathbb{A}_{2}^{c}({\mathbb{C}})$

where $\mathbb{A}_{2}^{c}$ is the elliptic curve obtained by complex
conjugation from $\mathbb{A}_{2}.$

Proof. We assume that $\mathbb{A}_{1}$ is over $\mathrm{k}_{1}$ and
$\mathbb{A}_{2}$ over $\mathrm{k}_{2},$ subfields of ${\mathbb{C}}.$ We will
write ${\mathbb{U}}({\mathbb{C}})$ for the cover sort for both abelian
varieties, and $\pi_{1}:{\mathbb{U}}\to\mathbb{A}_{1},$
$\pi_{2}:{\mathbb{U}}\to\mathbb{A}_{2}$ the respective covers.

By categoricity and the Fact above there is an abstract isomorphism $\phi$
between the cover structures, which have the restrictions to subsets,

$\phi_{\mathrm{k}}:\mathrm{k}_{1}\to\mathrm{k}_{2};$

$\phi_{\Lambda}:\langle\Lambda_{1}\rangle\to\langle\Lambda_{2}\rangle$

where $\langle\Lambda_{1}\rangle=\Lambda_{1}\otimes{\mathbb{Q}},$
$\langle\Lambda_{2}\rangle=\Lambda_{2}\otimes{\mathbb{Q}}$ and

$\phi_{\pi\Lambda}:\pi_{1}\langle\Lambda_{1}\rangle\to\pi_{2}\langle\Lambda_{2}\rangle.$

For $\mathrm{k}_{1}$-interpretable varieties $W$ and $V$ in $\mathbb{A}_{1}$
and a $\mathrm{k}_{1}$-rational map $f:W\to V$ we define $f^{\phi}:W^{\phi}\to
V^{\phi}$ the $\mathrm{k}_{2}$-rational map between the varieties $W^{\phi},$
$V^{\phi}$ $\mathrm{k}_{2}$-interpretable in $\mathbb{A}_{2},$ given by
application of $\phi.$

Note that $\langle\Lambda_{1}\rangle$ is a topological metric space in the
metric of ${\mathbb{Q}}$-vector spaces. It is metrically dense in
${\mathbb{U}}({\mathbb{C}})$ and so is $\pi_{1}\langle\Lambda_{1}\rangle$ in
$\mathbb{A}_{1}({\mathbb{C}}),$ and the same holds for $\mathbb{A}_{2}$ and
its cover structure.

Note also that $\phi_{\Lambda}$ is continuous and so is $\phi_{p\Lambda},$
since $\pi_{1},$ $\pi_{2}$ are continuous.

Now we define

$\bar{\phi}:\left\\{\begin{array}[]{ll}\
{\mathbb{U}}({\mathbb{C}})\to{\mathbb{U}}({\mathbb{C}})\\\
\mathbb{A}_{1}({\mathbb{C}})\to\mathbb{A}_{2}({\mathbb{C}})\end{array},\right.$

the maps which are defined by continuous extension of $\phi_{\Lambda}$ and of
$\phi_{p\Lambda},$ that is, for $z=\lim z_{i},$
$z_{i}\in\langle\Lambda_{1}\rangle,$

$\bar{\phi}(z)=\lim\phi_{\Lambda}(z_{i})$

and for $z=\lim z_{i},$ $z_{i}\in\pi_{1}\langle\Lambda_{1}\rangle,$

$\bar{\phi}(z)=\lim\phi_{p\Lambda}(z_{i}).$

Thus $\bar{\phi}$ is a continuous map between the two complex structures the
restrictions $\bar{\phi}_{\Lambda}$ and $\bar{\phi}_{\pi\Lambda}$ of which
coinside with the restrictions of the isomorphism $\phi.$ We are going to
prove that it is an isomorphism of the cover structures.

Let $q:\mathbb{A}_{1}({\mathbb{C}})\to{\mathbb{C}}$ be a regular non-constant
map over $\mathrm{k}_{1},$ defined on a Zariski open subset
$D_{1,q}\subset\mathbb{A}_{1}({\mathbb{C}}).$ Let
$q^{\phi}:\mathbb{A}_{2}({\mathbb{C}})\to{\mathbb{C}}$ be the respective map
over $\mathrm{k}_{2}.$

Note that $\pi_{1}\langle\Lambda_{1}\rangle\cap D_{1,q}$ is dense in $D_{1,q}$
and so $q(\pi_{1}\langle\Lambda_{1}\rangle)$ is dense in ${\mathbb{C}}$ since
$q$ is continuous and its image contains a Zariski open subset of
${\mathbb{C}}.$ Similarly, $q^{\phi}(\pi_{2}\langle\Lambda_{2}\rangle)$ is
dense in ${\mathbb{C}}.$ Let

$\phi_{q}:q(\pi_{1}\langle\Lambda_{1}\rangle)\to
q^{\phi}(\pi_{2}\langle\Lambda_{2}\rangle)$

be the restriction of the isomorphism $\phi$ on the subsets.

The map $\phi_{q}$ is a homeomorphism since $\phi_{\pi\Lambda}$ is.

Thus the restriction of the isomorphism $\phi$ on the subsets of the form
$q(\pi_{1}\langle\Lambda_{1}\rangle)$ is conituous.

Define $\bar{\phi}$ on ${\mathbb{C}}$ by extending $\phi_{q}$ continuously.
This also defines $\bar{\phi}$ on ${\mathbb{C}}.$

If $q^{\prime}:\mathbb{A}_{1}\to{\mathbb{C}}$ is another such function then
$\phi$ is continuous on $q(\pi_{1}\langle\Lambda_{1}\rangle)\cup
q^{\prime}(\pi_{1}\langle\Lambda_{1}\rangle).$ Define $\bar{\phi}^{\prime}$ by
extending $\phi$ from the union continuously. We will have
$\bar{\phi}^{\prime}=\bar{\phi},$ since the two coinside on the dense subset
$q(\pi_{1}\langle\Lambda_{1}\rangle).$ Thus, $\bar{\phi}$ does not depend on
the choice of $q.$

More generally, using the fact that
$q(\pi_{1}\langle\Lambda_{1}\rangle)\subseteq\mathrm{dcl}(\Lambda_{1})$ we
get:

$\bar{\phi}:{\mathbb{C}}\to{\mathbb{C}}$

is the unique continuous map such that for any
$\omega\in\mathrm{dcl}(\Lambda_{1})\cap{\mathbb{C}}$

$\bar{\phi}(\omega)=\phi(\omega).$

Claim. Let $g:{\mathbb{C}}^{m}\to{\mathbb{C}}$ be a $\mathrm{k}_{1}$-rational
map. Then, for all $z\in{\mathbb{C}}^{m},$

$\bar{\phi}(g(z))=g^{\phi}(\bar{\phi}(z))).$

Proof. Let $z=\lim z_{i}$ for some sequence
$z_{i}\in\mathrm{dcl}(\Lambda_{1}).$ Since $g$ is continuous, $g(z)=\lim
g(z_{i})$ and since $\bar{\phi}$ is continuous on ${\mathbb{C}}$

$\bar{\phi}(g(z))=\lim\bar{\phi}(g(z_{i}))=\lim\phi(g(z_{i}))=\lim
g^{\phi}(\phi(z_{i}))=g^{\phi}(\lim\phi(z_{i}))=g^{\phi}(\bar{\phi}(z)))$

Corollary. $\bar{\phi}$ is a continuous isomorphism of ${\mathbb{C}},$ that is
an identity or complex conjugation $z\mapsto z^{c}.$ In particular
$\phi_{\mathrm{k}}:\mathrm{k}_{1}\to\mathrm{k}_{2}$ is an identity or complex
conjugation.

Indeed, take $g(z_{1},z_{2}):=z_{1}*z_{2}$ where $*$ is + or $\cdot$. Then

$\bar{\phi}(z_{1}*z_{2})=\bar{\phi}(z_{1})*\bar{\phi}(z_{2}).$

$\Box$

##  4 Shimura varieties

Let ${\mathbb{X}}_{1},$ ${\mathbb{X}}_{2}$ be connected Shimura varieties (see
[9], Def.4.4) over their reflex fields $\mathrm{k}_{1},\mathrm{k}_{2}$ given
by the Shimura datum $(\mathrm{G}_{1},\mathcal{H}_{1}),$
$(\mathrm{G}_{2},\mathcal{H}_{2})$ and congruence subgroups
$\Gamma_{1}\subset\mathrm{G}_{1}({\mathbb{Q}}),$
$\Gamma_{2}\subset\mathrm{G}_{2}({\mathbb{Q}}),$ respectively.

${\mathbb{X}}_{1}=\Gamma_{1}\backslash\mathcal{H}_{1}\mbox{ and
}{\mathbb{X}}_{2}=\Gamma_{2}\backslash\mathcal{H}_{2}$

with respective covering maps

$j_{1}:\mathcal{H}_{1}\to{\mathbb{X}}_{1}\mbox{ and
}j_{2}:\mathcal{H}_{2}\to{\mathbb{X}}_{2}.$

A Shimura datum $(\mathrm{G},\mathcal{H}),$ also associates with each point
$h\in\mathcal{H}_{1}$ a homomorphisms
$h:U(1)\to\mathrm{G}^{\mathrm{ad}}_{\mathbb{R}}$ from the $1$-dimensional real
torus $U(1),$ and all the homomorphisms are linked by conjugation by elements
of $\mathrm{G}^{\mathrm{ad}}({\mathbb{R}})^{+}.$ The image $h(U(1))=T_{h}$ is
a torus subgroup of $\mathrm{G}^{\mathrm{ad}}_{\mathbb{R}}$ which can be
identified as a subgroup fixing the point $h.$ Since there is only one non-
identity continuous automorphism of $U(1),$ $u\mapsto u^{-1},$ there are
exactly two $\mathrm{G}^{\mathrm{ad}}_{\mathbb{R}}$-conjugacy classes of
homomorphisms $h:U(1)\to\mathrm{G}^{\mathrm{ad}}_{\mathbb{R}}$ which can be
called $\mathcal{H}^{+}$ and $\mathcal{H}^{-}.$

It follows from Shimura theory, see [3], that ${\mathbb{X}},$ a complex
Shimura variety and ${\mathbb{X}}^{c}$ its complex conjugate,

${\mathbb{X}}=\Gamma\backslash\mathcal{H}^{+}\Leftrightarrow{\mathbb{X}}^{c}=\Gamma\backslash\mathcal{H}^{-}$
(1)

and the action of complex conjugation corresponds to the automorphism $u\to
u^{-1}$ on $U(1).$

Below we use the notation $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ for sets and
$\mathcal{H}_{1}^{\pm}$ and $\mathcal{H}_{2}^{\pm}$ for their presentations as
classes of homomomorphisms from $U(1).$

Fact (Th 5.4,[9]). Let $\mathrm{G}$ be a connected algebraic group over
${\mathbb{Q}}.$ Then $\mathrm{G}({\mathbb{Q}})$ is dense in
$\mathrm{G}({\mathbb{R}}).$

Theorem. Suppose there is a categorical $L_{\omega_{1},\omega}$-sentence
$\Sigma$ over parameters $C$ such that both structures
$(\mathrm{G}_{1},\mathcal{H}_{1},\Gamma_{1},{\mathbb{X}}_{1},j_{1})$ and
$(\mathrm{G}_{2},\mathcal{H}_{2},\Gamma_{2},{\mathbb{X}}_{2},j_{2})$ are
models of $\Sigma$ when one interprets $C$ as $\mathrm{k}_{1}$ and
$\mathrm{k}_{2}$ respectively. Then there is a biholomorphic isomorphism

$\varphi:{\mathbb{X}}_{1}({\mathbb{C}})\to{\mathbb{X}}_{2}({\mathbb{C}}),\mbox{
or
}\varphi:{\mathbb{X}}_{1}({\mathbb{C}})\to{\mathbb{X}}_{2}^{c}({\mathbb{C}}).$
(2)

Proof. First we note that it follows from the categoricity assumption and the
Fact of section 3 that there exists an abstract isomorphism $\phi$ between the
two structures which is realised as ismorphisms

$\phi:\mathrm{k}_{1}\to\mathrm{k}_{2}$ (3)
$\phi:{\mathbb{X}}_{1}\to{\mathbb{X}}_{2}$ (4)
$\phi:\mathcal{H}_{1}\to\mathcal{H}_{2}$ (5)
$\phi:\mathrm{G}_{1}({\mathbb{Q}})\to\mathrm{G}_{2}({\mathbb{Q}});$ (6)
$\phi:\Gamma_{1}\to\Gamma_{2}$ (7) $\forall
g\in\mathrm{G}_{1}({\mathbb{Q}}),\forall x\in\mathcal{H}_{1}\ \ \phi(g\cdot
x)=\phi(g)\cdot\phi(x)$ (8)

and

$j_{2}\circ\phi=\phi\circ j_{1}.$ (9)

Since $\mathrm{G}_{1}({\mathbb{Q}})$ is dense in
$\mathrm{G}_{1}({\mathbb{R}})$ and $\mathrm{G}_{2}({\mathbb{Q}})$ in
$\mathrm{G}_{2}({\mathbb{R}})$ there is a unique extension of $\phi$ to the
continuous isomorphism

$\phi_{\mathbb{R}}:\mathrm{G}_{1}({\mathbb{R}})\to\mathrm{G}_{2}({\mathbb{R}}),\
\mathrm{G}^{\mathrm{ad}}1({\mathbb{R}})\to\mathrm{G}^{\mathrm{ad}}2({\mathbb{R}})$

together with the isomorphism of the action (8) since it is continuous both in
$g$ and $x:$

$\forall g\in\mathrm{G}_{1}({\mathbb{R}}),\forall x\in\mathcal{H}_{1}\ \
\phi_{\mathbb{R}}(g\cdot x)=\phi_{\mathbb{R}}(g)\cdot\phi_{\mathbb{R}}(x)$

(where $\mathrm{G}^{\mathrm{ad}}({\mathbb{R}})=\mathrm{G}({\mathbb{R}})/Z,$
$Z$ subgroup acting trivially).

Thus, $\phi$ induces

$\phi_{\pm}:\\{h:U(1)\to\mathrm{G}_{1}^{\mathrm{ad}}({\mathbb{R}})\\}\to\\{\phi_{\pm}(h):U(1)\to\mathrm{G}_{2}^{\mathrm{ad}}({\mathbb{R}})\\}$

Since $h^{-1}\circ\phi_{\mathbb{R}}^{-1}\circ\phi_{\pm}(h):U(1)\to U(1)$ is
either identity or $u\to u^{-1},$ there are two possibilities:

$\phi_{\pm}:\mathcal{H}_{1}^{+}\to\mathcal{H}_{2}^{+}$

and

$\phi_{\pm}:\mathcal{H}_{1}^{-}\to\mathcal{H}_{2}^{+}$

Respectively, by fact (1), we have two biholomorphic isomorphisms (2). $\Box$

## References

  * [1] M.Bays, Categoricity results for exponential maps of 1-dimensional algebraic groups and Schanuel Conjectures for Powers and the CIT, DPhil Thesis, Oxford, 2009
  * [2] M.Bays, B.Hart and A.Pillay, Universal covers of commutative finite Morley rank groups, Journal of the Institute of Mathematics of Jussieu, April 2018
  * [3] D. Blasius and L. Guerberoff, Complex conjugation and Shimura varieties, Algebra and Number Theory,v.11, (2017), no 10
  * [4] C.Daw and A.Harris, Categoricity of modular and Shimura curves, Journal of the Institute of Mathematics of Jussieu, v. 16, 5, 2017, pp.1075 –1101
  * [5] S.Eterovich, Categoricity of Shimura Varieties, arXiv:1803.06700v2
  * [6] M.Gavrilovich, Model theory of the universal covering spaces of complex algebraic varieties PhD Thesis, Oxford, 2006
  * [7] M.Gavrilovich, Standard conjectures in model theory, and categoricity of comparison isomorphisms, arXiv:1808.09332
  * [8] A.Harris, Categoricity and covering spaces, arXiv:1412.3484v1
  * [9] J.Milne, Introduction to Shimura varieties, In Harmonic Analysis, the trace formula and Shimura varieties, Clay Math. Proc., 2005, pp.265–378
  * [10] B.Zilber, Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety. In Model Theory and Applications, pp.427-458, Quaderni di matematica, v.11, Napoli, 2005
  * [11] B.Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero, J. London Math. Soc. (2), 74(1):41–58, 2006
  * [12] B.Zilber, Non-elementary categoricity and projective locally o-minimal classes, arxiv 2023
  * [13] B.Zilber, Algebraic Geometry via Model Theory In: Contemp.Math. 131(1992) (part 3), 523 - 537


# Higgs exotic decays in general NMSSM with self-interacting dark matter

Wenyu Wang1 Mengchao Zhang2 Jun Zhao1,2 1 Institute of Theoretical Physics,
College of Applied Science, Beijing University of Technology, Beijing 100124,
China
2 Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

Under current LHC and dark matter constraints, the general NMSSM can have
self-interacting dark matter to explain the cosmological small structure. In
this scenario, the dark matter is the light singlino-like neutralino ($\chi$)
which self-interacts through exchanging the light singlet-like scalars
($h_{1},a_{1}$). These light scalars and neutralinos inevitably interact with
the 125 GeV SM-like Higgs boson ($h_{SM}$), which cause the Higgs exotic
decays $h_{SM}\to h_{1}h_{1},a_{1}a_{1},\chi\chi$. We first demonstrate the
parameter space required by the explanation of the cosmological small
structure and then display the Higgs exotic decays. We find that in such a
parameter space the Higgs exotic decays can have branching ratios of a few
percent, which should be accessible in the future $e^{+}e^{-}$ colliders.

###### pacs:

14.80.Cp, 12.60.Fr, 11.30.Qc, 98.80.-k

##  I Introduction

The $\Lambda$CDM cosmological model ($\Lambda$ Cold Dark Matter) can
successfully describe the large-scale structure of the Universe ($>1$ Mpc) and
the Cosmic Microwave Background (CMB). Despite the success in describing the
large-scale structure, several recent observations at galactic or smaller
scales fail to be explained Bringmann:2013vra , such as the problems of
missing satellites Klypin:1999uc , cusp vs core deNaray:2011hy  and too big to
fail BoylanKolchin:2011de . These issues strongly indicate that dark matter
(DM) should not be composed of cold collisionless particles, and instead may
have a richer structure involving nontrivial self-interactions. In particular,
the self-interacting dark matter with a light force carrier ($<$$\sim$ 100
MeV) may have a non-trivial velocity-dependent scattering cross section, which
gives rise to the appropriate cross sections for all of the small scales (the
dwarf size, the Milky Way size as well as the galaxy cluster size)
Tulin:2012wi ; Ko:2014nha ; Laha:2013gva . Note that there could be small-
scale power suppression due to self-interacting dark radiation (which could
also lead to dark acoustic oscillations) and/or late-kinetic decoupling.
darkradiation

Such a self-interacting dark matter can be naturally achieved in the general
singlet extension of the MSSM (GNMSSM) fayet ; NMSSM ; Kaminska:2014wia . In
the GNMSSM, the SUSY preserving $\mu$-term is dynamically generated by the
vacuum expectation values (VEV) of a singlet superfield $S$, which can solve
the notorious $\mu$-problem in the MSSM. muew  In the small $\lambda$ limit,
the singlet sector can be almost decoupled from the eletroweak symmetry
breaking sector and becomes a dark sector in the theory. Due to its very weak
interactions with the SM particles, the $O(1)$ GeV singlino-like dark matter
dominantly annihilates into the light singlet-like scalars, which can
correctly produce the DM relic density and a proper Sommerfeld enhancement
factor NMSSM2 ; Hooper:2009gm ; Wang:2009rj . Such a feature can be used to
solve the small cosmological scale anomaly without conflicting with dark
matter direct detection Wang:2014kja . Besides, the light singlet-like scalars
are essential in such a GNMSSM explanation of small structure problem. This
will inevitably lead to various exotic decays of the 125 GeV SM-like Higgs
boson, such as the invisible decay to dark matter and the decays to a pair of
light scalars. Nhung:2013lpa  Therefore, search for these exotic decays at
colliders can allow for a test of the self-interacting dark matter in the
GNMSSM.

In this work, we first revisit the GNMSSM explanation of the small structure
problem under current LHC data and then investigate the 125 GeV Higgs exotic
decays in the allowed parameter space. We organize the content as follows. In
Sec. II, we describe the general self-interacting dark matter interactions for
explaining the small cosmological scale anomaly. In Sec. III we briefly review
the GNMSSM and in Sec. IV we demonstrate the parameter space of the GNMSSM
required by the explanation of the small cosmological scale anomaly. In Sec.
V, we show the Higgs exotic decays in such a parameter space. Sec. VI contains
our conclusions.

##  II Self-interacting dark matter for small scale structure

To explain both large scale and small scale structures of the Universe, one
can introduce the self-interacting DM scenario, in which the interactions
between DM and SM particles can be summarized as (shown in Fig. 1):

  1. 1.

The cross section of the DM annihilation to the SM particles (the left diagram
of Fig. 1), which at high energy determines the relic density of dark matter.
At low energy this cross section can be probed by the indirect detection
experiments like PAMELA pamela  and AMS02 AMS-2013 . When the early universe
was cooling down, the equilibrium between DM and SM particles in the thermal
bath can no longer be maintained. The DM will annihilate to SM particles until
the annihilation rate falls below the expansion rate of the Universe. In our
following calculation, we use the standard method susy-dm-review  to calculate
the relativistic annihilation cross section and the degrees of freedom at the
freeze-out temperature.

  2. 2.

The direct DM elastic scattering off the SM particles (the middle diagram of
Fig. 1), which can be probed by various underground detection experiments,
such as CDMS cdms2 , XENON XENON100  and LUX lux . Due to the high sensitivity
of the spin-independent (SI) measurement, we only consider the SI elastic
interactions of DM (denoted by $\chi$) and nucleon (proton and neutron), which
are dominantly induced by the scalar mediator at tree level, as shown in the
middle diagram of Fig.(1).

  3. 3.

The non-relativistic self-scattering (the right diagram of Fig. 1), where
$\ell=0$ in the partial wave expansion gives the Sommerfeld enhancement
relative to the relativistic annihilation. Such an enhancement was first
proposed to explain some DM indirect detection results, like the positron
excess observed by PAMELA or AMS02.sommerfeld2  On the other hand, when
$\ell\raise 1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}25$,
the anomalies in the small cosmological scales can be successfully accounted
for.

To explain the DM indirect detection results (such as the PAMELA result) and
the small cosmological scale anomalies, the DM scattering cross section should
be calculated at low energies, where the above interactions entangles with
each other. In this case, the complete investigation of the DM properties
needs to consider all the above interactions under various dark matter
detection experiments. Details of this study can be found in Wang:2014kja .
Here we only briefly discuss the calculation of the non-relativistic self-
scattering cross section.

Figure 1: Dark matter interactions: annihilation to SM particles (left panel),
scattering off quarks (middle panel), and self-scattering process (right
panel).

The numerical input for the simulation of small scales is the differential
cross section $d\sigma/d\Omega$, which is a function of the DM relative
velocity $v$. Then the viscosity (or conductivity) cross section $\sigma_{V}$
krstic:1999  can be defined as

$\sigma_{V}=\int d\Omega\,\sin^{2}\theta\,\frac{d\sigma}{d\Omega}\,.$ (1)

This formula is valid for the Majorana-fermion DM candidate. The weight of
$\sin^{2}\theta$ in the integral is needed since both forward and backward
scatterings amplitudes will diverge. Such singularities correspond to the
poles in the $t$\- and $u$-channel diagrams for the identical DM candidate.
Within the resonance region, $\sigma_{V}$ must be computed by solving the
Schrödinger equation with the partial wave expansion method:

$\frac{1}{r^{2}}\frac{d}{dr}\Big{(}r^{2}\frac{dR_{\ell}}{dr}\Big{)}+\Big{(}k^{2}-\frac{\ell(\ell+1)}{r^{2}}-2m_{r}V(r)\Big{)}R_{\ell}=0.$
(2)

The total wave function of the spin-1/2 fermionic DM must be antisymmetric
with respect to the interchange of two identical particles. Then the spatial
wave function should be symmetric (antisymmetric) when the total spin is 0
(1). The viscosity cross section can be reduced to two variables:

$\displaystyle\frac{d\sigma_{VS}}{d\Omega}$ $\displaystyle=$
$\displaystyle\left|f(\theta)+f(\pi-\theta)\right|^{2}=\frac{1}{k^{2}}\left|\sum_{\ell(\mbox{\tiny\rm
EVEN\
number})}^{\infty}(2\ell+1)(\exp(2i\delta_{l})-1)P_{\ell}(\cos\theta)\right|^{2}$
(3) $\displaystyle\frac{d\sigma_{VA}}{d\Omega}$ $\displaystyle=$
$\displaystyle\left|f(\theta)-f(\pi-\theta)\right|^{2}=\frac{1}{k^{2}}\left|\sum_{\ell(\mbox{\tiny\rm
ODD\
number})}^{\infty}(2\ell+1)(\exp(2i\delta_{l})-1)P_{\ell}(\cos\theta)\right|^{2}$
(4)

Using the orthogonality relation of the Legendre polynomials, we can have

$\displaystyle\frac{\sigma_{VS}k^{2}}{4\pi}$ $\displaystyle=$
$\displaystyle\sum_{\ell(\mbox{\tiny EVEN\
number})}^{\infty}4\sin^{2}(\delta_{\ell+2}-\delta_{\ell})(\ell+1)(\ell+2)/(2\ell+3),$
(5) $\displaystyle\frac{\sigma_{VA}k^{2}}{4\pi}$ $\displaystyle=$
$\displaystyle\sum_{\ell(\mbox{\tiny ODD\
number})}^{\infty}4\sin^{2}(\delta_{\ell+2}-\delta_{\ell})(\ell+1)(\ell+2)/(2\ell+3).$
(6)

When the partial wave $\ell$ grows up, it can be seen that $\sigma_{V}$ will
converge to a static value as the phase shift $\delta_{\ell}$ approaches to a
same value. We adopt the numerical method in Tulin:2013teo  to calculate all
these cross sections. In our following analysis, we assume that the DM
scatters randomly. Thus the average cross section will be

$\displaystyle\sigma_{V}=\frac{1}{4}\sigma_{VS}+\frac{3}{4}\sigma_{VA}\;.$ (7)

##  III The general singlet extension of MSSM (GNMSSM)

The superpotential of the general NMSSM is given by Ellwanger:2009dp

$W=W_{\mathrm{Yukawa}}+(\mu+\lambda\widehat{S})\,\widehat{H}_{u}\cdot\widehat{H}_{d}+\xi_{F}\widehat{S}+\frac{1}{2}\mu^{\prime}\widehat{S}^{2}+\frac{\kappa}{3}\widehat{S}^{3}$
(8)

where $\widehat{H}_{u}$ and $\widehat{H}_{d}$ are the two Higgs doublets,
$\widehat{S}$ is the gauge singlet. The terms $\sim\mu$, $\mu^{\prime}$ are
supersymmetric mass terms, and $\xi_{F}$ of dimension $mass^{2}$ parameterizes
the (supersymmetric) tadpole term. The Yukawa couplings of the quark and
lepton superfields are given as

$W_{\mathrm{Yukawa}}=h_{u}\,\widehat{Q}\cdot\widehat{H}_{u}\;\widehat{U}^{c}_{R}+h_{d}\,\widehat{H}_{d}\cdot\widehat{Q}\;\widehat{D}^{c}_{R}+h_{e}\,\widehat{H}_{d}\cdot\widehat{L}\;\widehat{E}_{R}^{c}$
(9)

where the Yukawa couplings $h_{u}$, $h_{d}$, $h_{e}$ and the superfields
$\widehat{Q}$, $\widehat{U}^{c}_{R}$, $\widehat{D}^{c}_{R}$, $\widehat{L}$ and
$\widehat{E}_{R}^{c}$ should be understood as matrices and vectors in family
space, respectively. The corresponding soft SUSY breaking masses and couplings
are

$\displaystyle-{\cal L}_{\mathrm{soft}}$ $\displaystyle=$ $\displaystyle
m_{H_{u}}^{2}|H_{u}|^{2}+m_{H_{d}}^{2}|H_{d}|^{2}+m_{S}^{2}|S|^{2}+m_{Q}^{2}|Q^{2}|+m_{U}^{2}|U_{R}^{2}|$
(10)
$\displaystyle+m_{D}^{2}|D_{R}^{2}|+m_{L}^{2}|L^{2}|+m_{E}^{2}|E_{R}^{2}|$
$\displaystyle+(h_{u}A_{u}\;Q\cdot H_{u}\;U_{R}^{c}-h_{d}A_{d}\;Q\cdot
H_{d}\;D_{R}^{c}-h_{e}A_{e}\;L\cdot H_{d}\;E_{R}^{c}$ $\displaystyle+\lambda
A_{\lambda}\,H_{u}\cdot H_{d}\;S+\frac{1}{3}\kappa
A_{\kappa}\,S^{3}+m_{3}^{2}\,H_{u}\cdot H_{d}+\frac{1}{2}m_{S}^{\prime
2}\,S^{2}+\xi_{S}\,S+\mathrm{h.c.})\;.$

Clearly, if $\mu=\mu^{\prime}=\xi_{F}=0$, the $\mathcal{Z}_{3}$ NMSSM
superpotential is obtained as

$W_{\mathrm{NMSSM}}=\lambda\widehat{S}\,\widehat{H}_{u}\cdot\widehat{H}_{d}+\frac{\kappa}{3}\widehat{S}^{3}$
(11)

and the parameters $m_{3}^{2}$, $m_{S}^{\prime 2}$ and $\xi_{S}$ in (10) are
vanishing. Then, a VEV $s$ of $\widehat{S}$ of the order of the weak or SUSY
breaking scale generates an effective $\mu$-term with

$\mu_{\mathrm{eff}}=\lambda s\;$ (12)

The neutral physical Higgs fields (with index $R$ for the CP-even and index
$I$ for the CP-odd states) can be obtained by expanding the full scalar
potential around the real neutral vevs $v_{u}$, $v_{d}$ and $s$

$H_{u}^{0}=v_{u}+\frac{H_{uR}+iH_{uI}}{\sqrt{2}},\quad
H_{d}^{0}=v_{d}+\frac{H_{dR}+iH_{dI}}{\sqrt{2}},\quad
S=s+\frac{S_{R}+iS_{I}}{\sqrt{2}}\;.$ (13)

It is convenient to define, together with $\mu_{\mathrm{eff}}$ as in (12),

$B_{\mathrm{eff}}=A_{\lambda}+\kappa
s,\quad\widehat{m}_{3}^{2}=m_{3}^{2}+\lambda(\mu^{\prime}s+\xi_{F})$ (14)

where we have used the convention $\mu=0$. $B_{\mathrm{eff}}$ plays the role
of the MSSM-like $B$-parameter, and $\widehat{m}_{3}^{2}$ vanishes in the
NMSSM. After the spontaneously symmetry breaking, we can have the mass
spectrum of all the particles. The tree level Higgs mass matrices can be
obtained by expanding the full Higgs potential around the real neutral vevs
$v_{u}$, $v_{d}$ and $s$ as in (13). Then, the elements of the $3\times 3$ CP-
even mass matrix ${\cal M}_{S}^{2}$ in the basis $(H_{dR},H_{uR},S_{R})$ after
the elimination of $m_{H_{d}}^{2}$, $m_{H_{u}}^{2}$ and $m_{S}^{2}$ can be
read as

$\displaystyle{\cal M}_{S,11}^{2}$ $\displaystyle=$ $\displaystyle
g^{2}v_{d}^{2}+(\mu_{\mathrm{eff}}\,B_{\mathrm{eff}}+\widehat{m}_{3}^{2})\,\tan\beta\;,$
$\displaystyle{\cal M}_{S,22}^{2}$ $\displaystyle=$ $\displaystyle
g^{2}v_{u}^{2}+(\mu_{\mathrm{eff}}\,B_{\mathrm{eff}}+\widehat{m}_{3}^{2})/\tan\beta\;,$
$\displaystyle{\cal M}_{S,33}^{2}$ $\displaystyle=$
$\displaystyle\lambda(A_{\lambda}+\mu^{\prime})\frac{v_{u}v_{d}}{s}+\kappa
s(A_{\kappa}+4\kappa s+3\mu^{\prime})-(\xi_{S}+\xi_{F}\mu^{\prime})/s\;,$
$\displaystyle{\cal M}_{S,12}^{2}$ $\displaystyle=$
$\displaystyle(2\lambda^{2}-g^{2})v_{u}v_{d}-\mu_{\mathrm{eff}}\,B_{\mathrm{eff}}-\widehat{m}_{3}^{2}\;,$
$\displaystyle{\cal M}_{S,13}^{2}$ $\displaystyle=$
$\displaystyle\lambda(2\mu_{\mathrm{eff}}\,v_{d}-(B_{\mathrm{eff}}+\kappa
s+\mu^{\prime})v_{u})\;,$ $\displaystyle{\cal M}_{S,23}^{2}$ $\displaystyle=$
$\displaystyle\lambda(2\mu_{\mathrm{eff}}\,v_{u}-(B_{\mathrm{eff}}+\kappa
s+\mu^{\prime})v_{d})\;.$ (15)

After diagonalization, we can have three CP-even mass eigenstates, which are
denoted as $h_{1}$, $h_{2}$, $h_{3}$, respectively. The element of the CP-odd
mass matrix ${\cal M^{\prime}}_{P}^{2}$ is $3\times 3$ matrix in the basis
$(H_{dI},H_{uI},S_{I})$. ${\cal M^{\prime}}_{P}^{2}$ usually contains a
massless Goldstone mode ${G}$. We rotate this mass matrix into the basis
(${A},{G},S_{I}$), where ${A}=\cos\beta\,H_{uI}+\sin\beta\,H_{dI}$. After
absorbing the Goldstone mode, the remaining $2\times 2$ mass matrix ${\cal
M}_{P}^{2}$ in the basis ($A,S_{I}$) has the following elements

$\displaystyle{\cal M}_{P,11}^{2}$ $\displaystyle=$
$\displaystyle\frac{2(\mu_{\mathrm{eff}}\,B_{\mathrm{eff}}+\widehat{m}_{3}^{2})}{\sin
2\beta}\;,$ $\displaystyle{\cal M}_{P,22}^{2}$ $\displaystyle=$
$\displaystyle\lambda(B_{\mathrm{eff}}+3\kappa
s+\mu^{\prime})\frac{v_{u}v_{d}}{s}-3\kappa A_{\kappa}s-2m_{S}^{\prime
2}-\kappa\mu^{\prime}s-\xi_{F}\left(4\kappa+\frac{\mu^{\prime}}{s}\right)-\frac{\xi_{S}}{s}\;,$
$\displaystyle{\cal M}_{P,12}^{2}$ $\displaystyle=$
$\displaystyle\lambda(A_{\lambda}-2\kappa s-\mu^{\prime})\,v\;.$ (16)

Similarly, we can obtain two mass eigenstates denoted as ($a_{1}$, $a_{2}$).
Note that we can set $A_{\lambda}-2\kappa s-\mu^{\prime}=0$ to forbid the
mixing between the doublet and the singlet. In the neutralino sector, the
neutral gaugino $\lambda_{1}$ and $\lambda_{2}^{3}$ mix with the neutral
higgsinos $\psi_{d}^{0},\psi_{u}^{0},\psi_{S}$, which generates a symmetric
$5\times 5$ mass matrix ${\cal M}_{0}$. In the basis
$\psi^{0}=(-i\lambda_{1},-i\lambda_{2}^{3},\psi_{d}^{0},\psi_{u}^{0},\psi_{S})$,
the resulting mass terms in the Lagrangian can be read as

${\cal L}=-\frac{1}{2}(\psi^{0})^{T}{\cal M}_{0}(\psi^{0})+\mathrm{h.c.}$ (17)

where

${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&-\frac{g_{1}v_{d}}{\sqrt{2}}&\frac{g_{1}v_{u}}{\sqrt{2}}&0\\\
&M_{2}&\frac{g_{2}v_{d}}{\sqrt{2}}&-\frac{g_{2}v_{u}}{\sqrt{2}}&0\\\
&&0&-\mu_{\mathrm{eff}}&-\lambda v_{u}\\\ &&&0&-\lambda v_{d}\\\ &&&&2\kappa
s+\mu^{\prime}\end{array}\right),$ (18)

Here $M_{1},~{}M_{2}$ are the soft mass parameters of the gauginos. After
diagonalization, we can have five mass eigenstates $\chi_{1,\cdots,5}$. If the
neutralino ($\chi_{1}$) is the lightest supersymmetric pariticle (LSP), it
will be a good candidate of DM. In the GNMSSM, the self-interaction dark
matter scenario can be realized if the LSP is singlino-like and $h_{1}$ is
light enough and serve as be the mediator between the DM scattering
Wang:2014kja . In the following section we will revisit such a scenario under
the current LHC constraints and investigate the the Higgs exotic decays in the
visible parameter space of explaining the small structure anomalies.

##  IV Self-interacting dark matter in the GNMSSM

From the neutralino mass matrix in Eq.(18), we can see that the GNMSSM
singlino mass is mainly determined by the parameters $\mu^{\prime}$ and
$\kappa$. By tuning these two parameters, the very light singlino-like LSPs
can dominantly annihilate to the singlet-like Higgs bosons and satisfy the
requirements of relic abundance and direct detections Wang:2014kja . Here it
should be mentioned that the correlation between the singlino sector and the
singlet Higgs bosons sector are relaxed by the extra parameter $\mu^{\prime}$
in the GNMSSM, which is different from the situation in ${\cal Z}_{3}$-NMSSM
***In the ${\cal Z}_{3}$-NMSSM, the masses of the singlino and the singlet
Higgs bosons are strongly related by the same parameter $\kappa$. Thus, when
$\kappa\ll\lambda$, Draper:2010ew  the correct relic abundance of the very
light singlino dark matter is obtained by annihilating to the SM fermions via
the very light (several MeV) singlet CP-odd Higgs boson. However, such a light
singlet scalar will greatly enhance the DM-nucleon SI cross section and is
disfavored by the current dark matter direct detection. So, it hardly realizes
the self-scattering dark matter for the cosmological small structure in the
${\cal Z}_{3}$ NMSSM.

In our numerical calculations, we use the newest version of package NMSSMTools
nmssmtools  to perform the scan of the parameter space. All the mass input
parameters are defined at the electroweak scale. We choose the input
parameters of the Higgs sector as

$\displaystyle\lambda,\ \kappa,\ \tan\beta,\ \mu_{\mathrm{eff}},\
A_{\lambda},\ A_{\kappa}.\ m_{3}^{2},\ \mu^{\prime},\ m_{S}^{\prime 2},\
\xi_{F},\ \xi_{S}.$ (19)

Then, we scan the dimensionless parameters in the range

$0.005<|\lambda,\ \kappa|<0.5,~{}~{}1<\tan\beta<50$

and all the other mass dimension parameters in the range

$(-2{\rm TeV},~{}2{\rm TeV}).$

To obtain the very light singlino-like DM and the singlet-like Higgs bosons,
we require

$\displaystyle\xi_{S}$ $\displaystyle\sim$
$\displaystyle\lambda(A_{\lambda}+\mu^{\prime})v_{u}v_{d}-\kappa
s^{2}(A_{\kappa}+4\kappa s+3\mu^{\prime})-\xi_{F}\mu^{\prime},$ (20)
$\displaystyle m_{S}^{\prime 2}$ $\displaystyle\sim$
$\displaystyle\frac{1}{2}\left[\lambda(B_{\mathrm{eff}}+3\kappa
s+\mu^{\prime})\frac{v_{u}v_{d}}{s}-3\kappa
A_{\kappa}s-\kappa\mu^{\prime}s-\xi_{F}\left(4\kappa+\frac{\mu^{\prime}}{s}\right)-\frac{\xi_{S}}{s}\right],$
(21) $\displaystyle\mu^{\prime}$ $\displaystyle\sim$ $\displaystyle-2\kappa
s.$ (22)

Also, we require

$\displaystyle A_{\lambda}=2\kappa s+\mu^{\prime},$ (23)

to forbid the mixing between the CP-odd doublet and the singlet Higgs bosons.
The gaugino input parameters $M_{1},~{}M_{2},~{}M_{3}$ are scanned in the
range of $(-1{\rm TeV},~{}1{\rm TeV})$. In order to give a 125 GeV SM Higgs
boson, we scan the third generation squark mass parameters $m_{Q_{3L}}$,
$m_{U_{3R}}$ and $m_{D_{3R}}$ in the range of $(-5{\rm TeV},~{}5{\rm TeV})$.
The first two generation squark and slepton mass parameters are fixed at 5
TeV. Finally, we tune the parameter $A_{\kappa}$ slightly to give a light
$h_{1}$ at order of 10 MeV. In our scan, we impose the following constraints:

  * •

The SM-like Higgs mass in the range of 123-127 GeV;

  * •

The thermal relic density of the lightest neutralino in the 2$\sigma$ range of
the Planck value planck ;

  * •

The LEP-I bound on the invisible $Z$-decay,
$\Gamma(Z\to\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1})<1.76$ MeV, and the LEP-
II upper bound on
$\sigma(e^{+}e^{-}\to\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{i})<5\times
10^{-2}~{}{\rm pb}$ for $i>1$, as well as the lower mass bounds on the
sparticles from the direct searches at LEP and the Tevatron;

  * •

The constraints from the LHEP-II direct search for the Higgs boson exotic
decays, including the decay modes $h\to h_{1}h_{1},a_{1}a_{1}\to 4f$;

  * •

The constraints from $B$-physics: $B\to X_{s}\gamma$, $B^{+}\to\tau^{+}\nu$,
$\Upsilon\to\gamma a_{1}$, the $a_{1}$–$\eta_{b}$ mixing and the mass
difference $\Delta M_{d}$ and $\Delta M_{s}$.

Figure 2: The survived points under the constraints of relic density and the
scattering cross section in case of $\lambda=0$. The blue points are the
points satisfy the simulation in the Dwarf scale ($\sigma/m_{\chi}\sim
0.1-10\,\,{\rm cm^{2}/g}$, the characteristic velocity is 10 km/s.) The red
points are the points satisfy the simulation in the Milky Way
($\sigma/m_{\chi}\sim 0.1-1\,\,{\rm cm^{2}/g}$, the characteristic velocity is
200 km/s.) The green points are the points satisfy the simulation in the Milky
Way ($\sigma/m_{\chi}\sim 0.1-1\,\,{\rm cm^{2}/g}$, the characteristic
velocity is 1000 km/s.)

In Fig. 2, we show the survived points under the constraints of relic density
and scattering cross section in case of $\lambda=0$. The left panel shows the
results for one mediator in the simplified model Kang:2016xrm , while the
right panel shows the results of the GNMSSM. We can see that the simulation of
small structure gives a stringent constraint on the parameter space for the
self-interacting DM, in particular for one light mediator case. The survival
parameter space in the GNMSSM is larger than the simple model of one mediator.
The main reason is that in the DM self-interaction model Tulin:2013teo  DM can
only annihilate into $hh$ via $t$-channel and $u$-channel while in the GNMSSM
DM can annihilate into $hh$, $ha$ and $aa$ via $t$-channel, $u$-channel and
$s$-channel.

Figure 3: Scatter plots of the samples which satisfy the current experimental
constraints except the direct detection limits of dark matter. The $\Box$
(green) and $\triangle$ (red) samples can and cannot explain the small-scale
cosmic structure, respectively.

Figure 4: Same as Fig. 3, but showing the spin-independent cross section of
dark matter-nucleon scattering. The curves are the current limits of dark
matter direct detections, which can exclude most samples but have no
sensitivity to ultra-light dark matter.

In Fig. 3, we present the scatter plots of the survived samples which can
satisfy the current experimental constraints and can explain the small-scale
cosmic structure. From Fig. 3, it can be seen that the mass of the light CP-
even singlet-like Higgs boson has to be in the range of (0.6, 3) MeV in order
to explain the small structure simulations. The mass of the singlino-like DM
is required to be in the range of (5, 11) GeV. Besides, the couplings
$\lambda$ and $\kappa$ are bounded below 0.02 and $\kappa$ can be much larger
than $\lambda$. We also note that the chargino mass $m_{\chi^{\pm}_{1}}$ is
strongly constrained and should be less than 210 GeV.

In Fig. 4, we display the scatter plots by showing the spin-independent cross
section of dark matter-nucleon scattering. We can see that the cross section
is greatly enhanced by the light mediator $h_{1}$ with mass about 10 MeV. The
current direct detection experiments, such as XENON 2012, have already
excluded most of the samples when the dark matter is not so light. However,
the ultra-light dark matter can still be allowed because the direct detections
have no sensitivity to ultra-light dark matter.

##  V Higgs exotic decays in the GNMSSM with self-scattering dark matter

Now we display in Fig. 5 the Higgs exotic decays in the GNMSSM parameter space
which can satisfy the current experimental constraints (including the direct
detection limits of dark matter) and can explain the small-scale cosmic
structure. From Fig. 5 we can see that the branching ratios of these channels
become larger with the increase of $\kappa$. This is because that the
branching ratios of these decay channels are dominantly determined by the
Higgs self-coupling parameter $\kappa$ for the fixed masses of $h_{1}$,
$a_{1}$ and $\chi_{1}$. However, it should be noted that the small structure
simulations require a small $\kappa$ so that the branching ratios of the decay
channels $h_{SM}\to h_{1}h_{1},a_{1}a_{1}$ can only reach about 3%, while the
invisible decay can only reach 0.5%.

Figure 5: Same as Fig. 3, but showing the branching ratios of the exotic
decays of the 125 GeV SM-like Higgs boson.

Figure 6: Same as Fig. 3, but showing the cross sections of
$\sigma(e^{+}e^{-}\to Zh_{SM})\times Br(h_{SM}\to XX)$ for $\sqrt{s}=250$ GeV,
where $X=h_{1},a_{1},\chi$. The vertical lines (from left to right) correspond
to the mass thresholds of $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$
and $b\bar{b}$ final states, respectively.

In Fig. 6, we show the cross sections of $\sigma(e^{+}e^{-}\to Zh_{SM})\times
Br(h_{SM}\to XX)$ ( $X=h_{1},a_{1},\chi$) for $\sqrt{s}=250$ GeV. From Fig. 6,
we can see that the cross section of $e^{+}e^{-}\to Zh_{SM}\to Zh_{1}h_{1}$
can maximally reach 6 fb, which corresponds to $3\times 10^{4}$ events for the
highest luminosity ${\cal L}=5000$ fb-1 at an $e^{+}e^{-}$ collider. When
$2m_{e}<m_{h_{1}}$, the CP-even $h_{1}$ will dominantly decay to the
$e^{+}e^{-}$ final states. Then the most promising channel of searching for
such a light CP-even Higgs boson will be $e^{+}e^{-}\to Zh_{SM}\to
Zh_{1}h_{1}\to 2j+4\ell$ at an $e^{+}e^{-}$ collider. While the cross section
of $e^{+}e^{-}\to Zh_{SM}\to Za_{1}a_{1}$ can maximally reach 3 fb and the
light CP-odd $a_{1}$ will mainly decay to $b\bar{b}$, which leads to a
distinctive signature of $2\ell+4b$ at an $e^{+}e^{-}$ collider. ††† The
observability of light $a_{1}$ through the process $pp\to Vh_{SM}\to 2\ell+4b$
has been investigated in Cheung:2007sva ; han ; zhang  and is found to reach
$3\sigma$ if $Br(h_{SM}\to a_{1}a_{1})>15\%$ at LHC with a luminosity ${\cal
L}=300$ fb-1 zhang .. Due to the clean environment, with the mass window cut
$|m_{4b}-m_{h_{SM}}|<10$ GeV, such a signature may be observed at a future
$e^{+}e^{-}$ collider.

##  VI Conclusions

We revisited the general NMSSM which can have self-interacting dark matter to
explain the cosmological small structure. In this scenario, the dark matter is
the ultra-light singlino-like neutralino ($\chi$) which self-interacts through
exchanging the ultra-light singlet-like scalars ($h_{1},a_{1}$). Since these
light scalars and neutralinos inevitably interact with the 125 GeV SM-like
Higgs boson ($h_{SM}$), the Higgs will have exotic decays $h_{SM}\to
h_{1}h_{1},a_{1}a_{1},\chi\chi$. We first showed the parameter space required
by the explanation of the cosmological small structure and then displayed the
Higgs exotic decays. We found that in such a parameter space the Higgs exotic
decays can have branching ratios of a few percent, which could be accessible
in the future $e^{+}e^{-}$ colliders.

## Acknowledgments

We thank Lei Wu and Jin Min Yang for discussions and comments. This work was
supported by the Natural Science Foundation of China under grant number
11375001, and by the talent foundation of education department of Beijing.

## References

  * (1) T. Bringmann, J. Hasenkamp and J. Kersten, arXiv:1312.4947 [hep-ph].
  * (2) A. A. Klypin, A. V. Kravtsov, O. Valenzuela and F. Prada, Astrophys. J. 522, 82 (1999); A. V. Kravtsov, Adv. Astron. 2010, 281913 (2010); J. Zavala, et al., Astrophys. J. 700, 1779 (2009).
  * (3) R. K. de Naray and K. Spekkens, Astrophys. J. 741, L29 (2011); M. G. Walker and J. Penarrubia, Astrophys. J. 742, 20 (2011).
  * (4) M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. 415, L40 (2011); Mon. Not. Roy. Astron. Soc. 422, 1203 (2012).
  * (5) S. Tulin, H. -B. Yu and K. M. Zurek, Phys. Rev. Lett. 110, 111301 (2013).
  * (6) P. Ko and Y. Tang, arXiv:1402.6449 [hep-ph]; arXiv:1404.0236 [hep-ph].
  * (7) R. Laha and E. Braaten, arXiv:1311.6386 [hep-ph]. R. Laha, Phys. Rev. D 92, 083509 (2015) [arXiv:1505.02772 [hep-ph]].
  * (8) R. Foot and S. Vagnozzi, arXiv:1602.02467 [astro-ph.CO]; Y. Tang, arXiv:1603.00165 [astro-ph.CO]; T. Binder, L. Covi, A. Kamada, H. Murayama, T. Takahashi and N. Yoshida, arXiv:1602.07624 [hep-ph]; T. Bringmann, H. T. Ihle, J. Kersten and P. Walia, arXiv:1603.04884 [hep-ph]; M. R. Buckley, J. Zavala, F. Y. Cyr-Racine, K. Sigurdson and M. Vogelsberger, Phys. Rev. D 90, no. 4, 043524 (2014) [arXiv:1405.2075 [astro-ph.CO]].
  * (9) P. Fayet, Nucl. Phys. B 90, 104 (1975); Phys. Lett. B 69, 489 (1977).
  * (10) For pheno studies of NMSSM, see, e.g., J. R. Ellis, et al., Phys. Rev. D 39, 844 (1989); M. Drees, Int. J. Mod. Phys. A 4, 3635 (1989); P. N. Pandita, Phys. Lett. B 318, 338 (1993);Phys. Rev. D 50, 571 (1994); S. F. King, P. L. White, Phys. Rev. D 52, 4183 (1995); B. Ananthanarayan, P.N. Pandita, Phys. Lett. B 353, 70 (1995); Phys. Lett. B 371, 245 (1996); B. A. Dobrescu, K. T. Matchev, JHEP 0009, 031 (2000); V. Barger, P. Langacker, H.-S. Lee, G. Shaughnessy, Phys. Rev. D 73,(2006) 115010; R. Dermisek, J. F. Gunion, Phys. Rev. Lett. 95, 041801 (2005); G. Hiller, Phys. Rev. D 70, 034018 (2004); F. Domingo, U. Ellwanger, JHEP 0712, 090 (2007); Z. Heng, et al., Phys. Rev. D 77, 095012 (2008); R. N. Hodgkinson, A. Pilaftsis, Phys. Rev. D 76, 015007 (2007); Phys. Rev. D 78, 075004 (2008); W. Wang, Z. Xiong, J. M. Yang, Phys. Lett. B 680, 167 (2009); J. Cao, J. M. Yang, Phys. Rev. D 78, 115001 (2008); JHEP 0812, 006 (2008); U. Ellwanger, C. Hugonie and A. M. Teixeira, Phys. Rept. 496, 1 (2010); M. Maniatis, Int. J. Mod. Phys. A25 (2010) 3505; U. Ellwanger, Eur. Phys. J. C 71, 1782 (2011); J. Cao, et al., JHEP 1311, 018 (2013); JHEP 1304, 134 (2013); JHEP 1309, 043 (2013); JHEP 1206, 145 (2012); JHEP 1210, 079 (2012); JHEP 1203, 086 (2012); Phys. Lett. B 703, 462 (2011); JHEP 1011, 110 (2010); JHEP 1510, 178 (2015); JHEP 1510, 030 (2015); JHEP 1506, 152 (2015) [arXiv:1504.07869 [hep-ph]]; C. Han, et al., JHEP 1404, 003 (2014); Z. Kang, et al., arXiv:1102.5644 [hep-ph]; JCAP 1101, 028 (2011); J. Kozaczuk and S. Profumo, arXiv:1308.5705 [hep-ph]. S. F. King, M. Mühlleitner, R. Nevzorov and K. Walz, Nucl. Phys. B 901, 526 (2015) [arXiv:1508.03255 [hep-ph]]; Phys. Rev. D 90, no. 9, 095014 (2014) [arXiv:1408.1120 [hep-ph]]; Nucl. Phys. B 870, 323 (2013) [arXiv:1211.5074 [hep-ph]]; S. F. King, M. Mühlleitner and R. Nevzorov, Nucl. Phys. B 860, 207 (2012) [arXiv:1201.2671 [hep-ph]].
  * (11) A. Kaminska, G. G. Ross, K. Schmidt-Hoberg and F. Staub, arXiv:1401.1816 [hep-ph]; H. M. Lee, et al., Nucl. Phys. B 850, 1 (2011) [arXiv:1102.3595 [hep-ph]]; G. G. Ross and K. Schmidt-Hoberg, Nucl. Phys. B 862, 710 (2012) [arXiv:1108.1284 [hep-ph]]; A. Kaminska, G. G. Ross and K. Schmidt-Hoberg, JHEP 1311, 209 (2013) [arXiv:1308.4168 [hep-ph]]; G. G. Ross, K. Schmidt-Hoberg and F. Staub, JHEP 1208, 074 (2012) [arXiv:1205.1509 [hep-ph]]; K. Schmidt-Hoberg, F. Staub and M. W. Winkler, JHEP 1301, 124 (2013) [arXiv:1211.2835 [hep-ph]]; K. Schmidt-Hoberg and F. Staub, JHEP 1210, 195 (2012) [arXiv:1208.1683 [hep-ph]].
  * (12) J. E. Kim and H. P. Nilles, Phys. Lett. B 138, 150 (1984).
  * (13) J. Cao, et al., JHEP 1405,056 (2014); Phys. Lett. B 703, 292 (2011); JHEP 1007, 044 (2010).
  * (14) W. Wang, Z. Xiong, J. M. Yang and L. -X. Yu, JHEP 0911, 053 (2009).
  * (15) D. Hooper and T. M. P. Tait, arXiv:0906.0362 [hep-ph].
  * (16) F. Wang, W. Wang, J. M. Yang and S. Zhou, Phys. Rev. D 90, 035028 (2014) [arXiv:1404.6705 [hep-ph]].
  * (17) D. T. Nhung, M. Mühlleitner, J. Streicher and K. Walz, JHEP 1311, 181 (2013) [arXiv:1306.3926 [hep-ph]].
  * (18) O. Adriani, et al., Phys. Rev. Lett. 102, 051101 (2009).
  * (19) M. Aguilar et al. [AMS Collaboration], Phys. Rev. Lett. 110, 141102 (2013).
  * (20) G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267, 195 (1996).
  * (21) Z. Ahmed et al. [CDMS-II Collaboration], Science 327, 1619 (2010).
  * (22) E. Aprile et al. [XENON100 Collaboration], Phys. Rev. Lett. 107, 131302 (2011).
  * (23) D. S. Akerib et al. [LUX Collaboration], arXiv:1310.8214 [astro-ph.CO].
  * (24) N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. D 79, 015014 (2009).
  * (25) P. S. Krstić and D. R. Schultz, Phys. Rev. A 60, 3 (1999).
  * (26) S. Tulin, H. -B. Yu and K. M. Zurek, Phys. Rev. D 87, 115007 (2013).
  * (27) U. Ellwanger, C. Hugonie and A. M. Teixeira, Phys. Rept. 496, 1 (2010) doi:10.1016/j.physrep.2010.07.001 [arXiv:0910.1785 [hep-ph]].
  * (28) U. Ellwanger, J. F. Gunion and C. Hugonie, JHEP 0502, 066 (2005).
  * (29) P. Draper, et al., Phys. Rev. Lett. 106 (2011) 121805.
  * (30) http://www.sciops.esa.int/SA/PLANCK/docs/Planck 2013 results.
  * (31) H. J. Kang and W. Wang, arXiv:1601.00373 [hep-ph].
  * (32) R. Bernabei et al. [DAMA and LIBRA Collaborations], Eur. Phys. J. C 67, 39 (2010) [arXiv:1002.1028 [astro-ph.GA]].
  * (33) K. Cheung, J. Song and Q. S. Yan, Phys. Rev. Lett. 99 (2007) 031801 [hep-ph/0703149];
  * (34) M. Carena, T. Han, G. -Y. Huang and C. E. M. Wagner, JHEP 0804, 092 (2008).
  * (35) J. Cao, et al., Phys. Rev. D 82, 051701 (2010); JHEP 1311, 018 (2013).