Hadron Collider Physics symposium 2012

11institutetext: Institute of Modern Physics and Center for High Energy
Physics, Tsinghua University, Beijing 100084, China 22institutetext:
Department of Physics, Tohoku University, Sendai 980-8578, Japan
33institutetext: Department of Physics, Saitama University, Saitama 355-8570,
Japan 44institutetext: Department of Physics, Kyoto University, Kyoto
606-8502, Japan 55institutetext: Institute for Cosmic Ray Research (ICRR),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan

# Minimal dilaton model

Tomohiro Abe 11<EMAIL_ADDRESS>Ryuichiro Kitano 22
<EMAIL_ADDRESS>Yasufumi Konishi 33
<EMAIL_ADDRESS>Kin-ya Oda 44
<EMAIL_ADDRESS>Joe Sato 33<EMAIL_ADDRESS>Shohei
Sugiyama 55<EMAIL_ADDRESS>
###### Abstract

Both the ATLAS and CMS experiments at the LHC have reported the observation of
the particle of mass around 125 GeV which is consistent to the Standard Model
(SM) Higgs boson, but with an excess of events beyond the SM expectation in
the diphoton decay channel at each of them. There still remains room for a
logical possibility that we are not seeing the SM Higgs but something else.
Here we introduce the minimal dilaton model in which the LHC signals are
explained by an extra singlet scalar of the mass around 125 GeV that slightly
mixes with the SM Higgs heavier than 600 GeV. When this scalar has a vacuum
expectation value well beyond the electroweak scale, it can be identified as a
linearly realized version of a dilaton field. Though the current experimental
constraints from the Higgs search disfavors such a region, the singlet scalar
model itself still provides a viable alternative to the SM Higgs in
interpreting its search results.

##  1 Introduction

The ATLAS :2012gk  and CMS :2012gu  experiments have reported the observation
of the particle of mass 125 GeV that is consistent with the Standard Model
(SM) Higgs boson, but with an excess of events in the diphoton decay channel
with the signal strength $1.80\pm 0.30\pm 0.7$ at ATLAS ATLAS diphoton  and
$1.6_{-0.6}^{+0.7}$ at CMS CMS combined . There still remain a logical
possibility that the observed particle is not the SM Higgs but an extra boson
beyond the SM.

Here we report the minimal dilaton model Abe:2012eu  in which the observed
particle is identified as the SM singlet scalar field $S$ of mass around 125
GeV that slightly mixes with the SM Higgs $H$ heavier than 600 GeV. A vector-
like top partner $T$ is introduced so that $S$ can couple to a pair of gluons
or photons through the $T$ loop, just as in the case of the SM Higgs coupling
to them via the top quark loop. The top partner $T$ at the same time cures the
constraints from the electroweak precision data by its loop driving the
Peskin-Takeuchi T-parameter positive and hence canceling the heavy SM Higgs
contributions.

If the vacuum expectation value (vev) of the singlet $f:=\left\langle
S\right\rangle$ is much larger than the electroweak scale $v\simeq 246\,$GeV,
then $S$ can be identified as a linear realization of a dilaton field (hence
the model’s name) associated with an almost scale invariant sector whose
strongly coupled interactions, broken at $f$, give rise to the top quark and
the SM Higgs as composite particles, and the top partner $T$ represents a
fermion in the composite sector Abe:2012eu .

In literature conventional dilaton , the terminology “dilaton model” has been
used for the one in which all the SM particles couple to the dilaton through
the trace of the energy momentum tensor of the SM. This type of model
implicitly assumes that all the SM particles are composite under the strong
dynamics in the ultraviolet region. On the other hand, we take more
conservative approach that the SM except for the top/Higgs sector is a
spectator of the dynamics, and thus the dilaton $S$ couples to the $W$, $Z$
bosons and to light fermions only through the mixing with the Higgs fields
$H$, while the couplings to the gluons and photons are generated only through
the loops of the top quark and its partner. Due to these different origin of
the couplings between two models, the production and decay properties are
quite different. Indeed, we see that our model can give better fit to the LHC
data compared to the SM Higgs boson, while the authors of Ref. conventional
dilaton  have reported that the above mentioned dilaton scenario is rather
disfavored.

##  2 Minimal dilaton model

Our starting Lagrangian Abe:2012eu  can be written in terms of the singlet
scalar $S$, the SM Higgs $H$, the left handed quark doublet of the third
generation $q_{3L}$, and the top partner $T$ as

$\displaystyle\mathcal{L}$ $\displaystyle=\mathcal{L}_{\text{SM}}-{1\over
2}\partial_{\mu}S\partial^{\mu}S-\tilde{V}(S,H)$
$\displaystyle\quad-\overline{T}\left(\not{D}+yS\right)T-\left[y^{\prime}\overline{T}(q_{3L}\cdot
H)+\text{h.c.}\right],$ (1)

where $\mathcal{L}_{\text{SM}}$ is the SM Lagrangian except for the Higgs
potential and $\tilde{V}(S,H)$ is the potential for the scalars

$\displaystyle\tilde{V}(S,H)$ $\displaystyle={m_{S}^{2}\over
2}S^{2}+{\lambda_{S}\over 4!}S^{4}+{\kappa\over
2}S^{2}\left|H\right|^{2}+m_{H}^{2}\left|H\right|^{2}+{\lambda_{H}\over
4}\left|H\right|^{4}.$ (2)

The detailed potential shape (2) is unimportant, as its couplings can be
translated into the following two parameters relevant to the Higgs searches:
the Higgs-singlet mixing $\theta_{H}$ and the ratio of the vevs $\eta:=v/f$.
The relation is shown in Appendix B in Ref. Abe:2012eu . We identify the
lighter mass eigenstate to be the observed 125 GeV boson and the heavier to be
almost the SM Higgs with mass $\gtrsim 600$ GeV. The vector-like top partner
$T$ is set to be an $SU(2)_{L}$ singlet and to have a hypercharge $Y=2/3$ so
that it can decay into the top quark through a small mixing $\theta_{L}\sim
y^{\prime}v/\sqrt{2}yf$.

When the SM Higgs mass is heavy, the Peskin-Takeuchi S,T parameters severely
constrain the model. In the SM, the Higgs mass heavier than 185 GeV is ruled
out at the 95% CL ALEPH:2010aa . In the minimal dilaton model, the $T$ loop
gives positive contribution to the Peskin-Takeuchi T-parameter and cures the
model from the heavy Higgs effects. In Fig. 1, we show the constraints from
the S,T parameters on the top-partner mass $m_{t^{\prime}}$ and the mixing
$\theta_{L}$ between the top and top partner, at the 95% CL for the SM Higgs
mass $m_{h}=600$ GeV and the Higgs-singlet mixing angle $\theta_{H}=0$,
$\pi/6$, and $\pi/3$ Abe:2012eu .

Figure 1: Allowed region plot by the Peskin-Takeuchi S,T parameters at the 95%
CL in the plane of top-partner mass $m_{t^{\prime}}$ vs top-top partner mixing
$\theta_{L}$; The SM Higgs mass is taken to be $m_{h}=600$ GeV and the Higgs-
singlet mixing angle is chosen as $\theta_{H}=0$, $\pi/6$, and $\pi/3$
Abe:2012eu .

##  3 Higgs signal at LHC

Let us show our prediction for the Higgs searches at the LHC. In Ref.
conventional dilaton , the parameter $c_{X}$ is defined as the ratio of the
coupling of $S$ to $XX$ (or $X\bar{X}$), in the amplitude, to that of the SM
Higgs at 125 GeV. The minimal dilaton model gives Abe:2012eu

$\displaystyle c_{V}=c_{F}$ $\displaystyle=\sin\theta_{H},$ $\displaystyle
c_{t}$
$\displaystyle=\cos^{2}\theta_{L}\sin\theta_{H}+\eta\sin^{2}\theta_{L}\cos\theta_{H},$
$\displaystyle c_{g}$ $\displaystyle=\eta\cos\theta_{H}+\sin\theta_{H},$
$\displaystyle c_{\gamma}$ $\displaystyle=\eta
A_{t^{\prime}}\cos\theta_{H}+A_{\text{SM}}\sin\theta_{H},$ $\displaystyle
c_{\text{inv}}$ $\displaystyle=0,$ (3)

where $A_{t^{\prime}}\simeq 16/9$, $A_{\text{SM}}\simeq-6.5$, the subscript
“$F$” stands for all the SM fermions except the top quark, and “$V$” for $W$
or $Z$.

In Fig. 5, we plot the resultant signal strength, which is the ratio of the
model cross section to the corresponding SM Higgs one at 125 GeV, for the
singlet vev $f=246$ GeV ($\eta=1$). We see that the diphoton signal can be
enhanced in the dilatonic region $\theta_{H}\sim 0$, whereas other processes
are suppressed.

Figure 2: Upper: Signal strength for $f=246$ GeV ($\eta=1$). “GF” denotes the
case that $S$ is produced by the gluon fusion process, while “VBF” by the
vector boson fusion, Higgs-strahlung, or $ttH$ process. Lower: The same for
the Higgs to digluon process; Shown for comparison though hardly observable at
the LHC.

The minimal dilaton model predicts different production cross sections between
GF and VBF/VH/ttH processes. In $H\to\gamma\gamma$ search, composition of
these production channels differs category by category and are summarized in
Table 2 in Ref. :2012gu  for CMS and in Table 6 in Ref. ATLAS_diphoton  for
ATLAS. We define $\varepsilon^{i}_{X}$ as the proportion of the production
process $X$ within a category $i$. Note that $\sum_{X}\varepsilon^{i}_{X}=1$
by definition for each category $i$, where a summation over $X$ is always
understood as for all the relevant production channels: GF, VBF, VH, and ttH.
GF is the dominant production process and satisfies
$\varepsilon^{i}_{\text{GF}}\lesssim 90\%$ in production processes other than
dijet category. In the dijet category, the dominant production process is VBF,
and $\varepsilon_{\text{VBF}}\lesssim 70\%$.

When acceptance of a production channel $X$ for a category $i$ is $a^{i}_{X}$,
the estimated value of a signal fraction under the given set of cuts $i$
becomes

$\displaystyle\varepsilon^{i}_{X}$
$\displaystyle={a^{i}_{X}\sigma^{\text{SM}}_{X}\over\sum_{Y}a^{i}_{Y}\sigma^{\text{SM}}_{Y}},$
(4)

where $\sigma^{\text{SM}}_{X}$ is the Higgs production cross section in the SM
through the channel $X$. Given $\\{\varepsilon^{i}_{X}\\}$, we can compute the
signal strength under the imposed cuts for each category $i$

$\displaystyle\hat{\mu}^{i}(h\to\gamma\gamma)$
$\displaystyle={\sum_{X}a^{i}_{X}\,\sigma_{X}\over\sum_{Y}a^{i}_{Y}\,\sigma_{Y}^{\text{SM}}}\,{\operatorname{BR}(s\to\gamma\gamma)\over\operatorname{BR}(h\to\gamma\gamma)_{\text{SM}}}$
$\displaystyle=\sum_{X}\varepsilon^{i}_{X}\left|c_{X}\right|^{2}\,{\left|c_{\gamma}\right|^{2}\over
R(s\to\text{all})},$ (5)

where
$R(s\to\text{all}):=0.913\left|c_{V}\right|^{2}+0.085\left|c_{g}\right|^{2}+0.002\left|c_{\gamma}\right|^{2}$.
We have assumed that the acceptance $a^{i}_{X}$ under the category $i$ does
not change from that of the SM for each production channel $X$.

For the $ZZ\to 4l$ and $WW\to l\nu l\nu$ decay channels, we assume that all
the signals are coming from GF and, hence, we approximate

$\displaystyle\hat{\mu}(s\to VV)$
$\displaystyle=\left|c_{g}\right|^{2}\,{\left|c_{V}\right|^{2}\over
R(s\to\text{all})}$ (6)

for $VV=WW$ and $ZZ$.

As all the signal strengths are obtained, we perform a chi-square test with
the Gaussian approximation for all the errors

$\displaystyle\chi^{2}$
$\displaystyle=\sum_{i}\left(\hat{\mu}_{i}-\mu_{i}\over\sigma_{i}\right)^{2},$
(7)

where summation over $i$ is for all the diphoton categories as well as the
$WW$ and $ZZ$ channels. Of course this is a naive fit without taking into
account the off-diagonal elements of the correlation matrix for various
categories. For example, this type of naive weighting does not reproduce the
central value nor the error of the diphoton signal strength. Although we are
aware of this fact, this is the best we can do with the current data made
public. The result should be regarded as an illustration at best.

Keeping the above caution in mind, we show the constraints from the data from
the LHC Higgs searches in Fig 3. For comparison, we also show in Fig. 4 the
same plot including more recent data presented at HCP2012 and after HCP and
after . We see that the dilaton like region $\eta^{-1}\gg 1$,
$\left|\theta_{H}\right|\ll 1$ is disfavored by the Higgs data alone, but
there still remains phenomenologically viable region at
$\left|\theta_{H}\right|\lesssim\pi/6$ and $\eta^{-1}\lesssim 1.5$.

Figure 3: Favored regions within 90, 95 and 99% confidence intervals, enclosed
by solid, dashed, and dotted lines, respectively. Density (area) of favored
region decreases (increases) in according order Abe:2012eu .

Figure 4: The same as Fig. 3, including the data presented at HCP2012 and
after.

##  4 Summary

We have examined the possibility that the 125 GeV boson observed at the LHC is
not the SM Higgs but an extra singlet scalar $S$. If the vev well exceeds the
electroweak scale, $f=\left\langle S\right\rangle\gg 246$ GeV, then $S$ can be
regarded as a linearly realized dilaton that is associated with the
quasiconformal dynamics making up the top and Higgs particles as composite
ones. However, such a parameter region is disfavored by the current Higgs
data. Though such a linear dilaton interpretation is marginally excluded, this
model with a singlet scalar and a vector-like top partner still provides a
phenomenologically viable alternative to the SM Higgs to fit the current LHC
data.

## Acknowledgments

This work is supported in part by the Grants-in-Aid for Scientific Research
No. 23740165 (R. K.), No. 23104009, No. 20244028, No. 23740192 (K. O.), and
No. 24340044 (J. S.) of JSPS.

## References

  * (1) G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235 [hep-ex]].
  * (3) The ATLAS Collaboration, “Observation and study of the Higgs boson candidate in the two photon decay channel with the ATLAS detector at the LHC,” ATLAS-CONF-2012-128.
  * (4) The CMS Collaboration, “Combination of standard model Higgs boson searches and measurements of the properties of the new boson with a mass near 125 GeV,” CMS PAS HIG-12-045.
  * (5) T. Abe, R. Kitano, Y. Konishi, K. -y. Oda, J. Sato and S. Sugiyama, “Minimal Dilaton Model,” Phys. Rev. D 86 (2012) 115016 [arXiv:1209.4544 [hep-ph]].
  * (6) [ALEPH and CDF and D0 and DELPHI and L3 and OPAL and SLD and LEP Electroweak Working Group and Tevatron Electroweak Working Group and SLD Electroweak and Heavy Flavour Groups Collaborations], “Precision Electroweak Measurements and Constraints on the Standard Model,” arXiv:1012.2367 [hep-ex].
  * (7) P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, “Reconstructing Higgs boson properties from the LHC and Tevatron data,” JHEP 1206 (2012) 117 [arXiv:1203.4254 [hep-ph]];
P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, “Is the resonance at 125
GeV the Higgs boson?,” arXiv:1207.1347 [hep-ph];
D. Carmi, A. Falkowski, E. Kuflik, T. Volansky and J. Zupan, “Higgs After the
Discovery: A Status Report,” arXiv:1207.1718 [hep-ph].

  * (8) The ATLAS Collaboration, “Observation of an excess of events in the search for the Standard Model Higgs boson in the $\gamma\gamma$ channel with the ATLAS detector,” ATLAS-CONF-2012-091.
  * (9) The ATLAS Collaboration, “Observation and study of the Higgs boson candidate in the two photon decay channel with the ATLAS detector at the LHC,” ATLAS-CONF-2012-168; “An update of combined measurements of the new Higgs-like boson with high mass resolution channels” ATLAS-CONF-2012-170;
The CMS Collaboration, “Combination of standard model Higgs boson searches and
measurements of the properties of the new boson with a mass near 125 GeV,” CMS
PAS HIG-12-045.