# Tricking Landau-Yang: How to obtain the diphoton excess from a vector
resonance111This article is registered under preprint number: DESY 15-256,
arXiv:1512.06833

Mikael Chala  Michael Duerr  Felix<EMAIL_ADDRESS>Kai
Schmidt-Hoberg  DESY, Notkestraße 85, D-22607 Hamburg, Germany

###### Abstract

We show that contrary to naive expectations the recently observed diphoton
excess can be explained by a vector resonance, which decays to a photon and a
light scalar $s$, followed by a decay of the scalar into two photons:
$Z^{\prime}\rightarrow\gamma s\rightarrow 3\gamma$. As the two photons from
the scalar decay are highly boosted, the experimental signature is an apparent
diphoton final state. In fact all the necessary ingredients are naturally
present in $Z^{\prime}$ models: Additional fermions with electroweak quantum
numbers are required in order to render the theory anomaly free and naturally
induce the required effective couplings, while the hidden Higgs which gives
mass to the $Z^{\prime}$ can be very light. In particular no new coloured
states are required in this framework. We also show that in such a setup the
width of the resonance can be rather large, while all couplings remain
perturbative.

###### keywords:

Mostly Weak Interactions: Beyond Standard Model

##  1 Introduction

The recently observed excess in the diphoton channel reported by ATLAS [1] and
(with slightly less significance) CMS [2], has triggered a large amount of
interest in the particle physics community [3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48].
Interpretations in terms of particle physics models have mainly assumed a
scalar or tensor resonance at 750 GeV, corresponding to the invariant mass
observed in the diphoton system. The case of a vector resonance seems
problematic due to the Landau-Yang theorem [49, 50], which implies that a
vector resonance cannot decay into two photons. In this letter we show that
the observed excess can nevertheless be due to a new vector resonance, which
decays into three photons (see also [51]). The idea is to have a vector which
decays into a photon and a light singlet scalar which then further decays into
two photons; see figure 1 for the Feynman diagram. If the scalar is light
enough the two photons will be highly boosted and reconstructed as one photon,
leading to the apparent diphoton excess even though there are three photons in
the final state.

A natural candidate for such a vector resonance is the gauge boson of a new
$U(1)^{\prime}$, which gets a mass from a hidden Higgs which is uncharged
under the Standard Model (SM) gauge group. Most $Z^{\prime}$ models need
additional fermions which have SM electroweak charges in order to be anomaly
free and these additional fermions naturally induce the couplings which are
needed in the current context. The Higgs boson can naturally be much lighter
than the $Z^{\prime}$.

As an example we will consider the case of a new $U(1)^{\prime}$ under which
only the third generation quarks are charged. This is motivated by the
observation that no resonance has been observed at the LHC Run 1, implying
that the increase of production cross section needs to be sizeable. In the
case of $b$ quarks in the initial state, the ratio of production cross
sections at 8 and 13 TeV is sufficiently large (5.4) due to the proton PDFs.
As couplings of $\mathcal{O}(1)$ are needed in order to have a large enough
production cross section, the width due to decays into $b\bar{b}$ and
$t\bar{t}$ is sizeable and can easily be as large as $45$ GeV as indicated by
the ATLAS result, while being consistent with searches for $pp\rightarrow
Z^{\prime}\rightarrow\bar{b}b$ and $pp\rightarrow
Z^{\prime}\rightarrow\bar{t}t$.

This letter is organised as follows: In section 2 we introduce the effective
approach and concentrate on the primary decay $Z^{\prime}\rightarrow s\gamma$
while the subsequent decay $s\rightarrow\gamma\gamma$ is discussed in section
3. Details concerning possible ultraviolet (UV) completions are presented in
section 4. Finally, section 5 discusses additional constraints and future
prospects.

Figure 1: Feynman diagram for the process $Z^{\prime}\rightarrow 3\gamma$

##  2 Model-independent parametrisation and $Z^{\prime}$ decays

In the following we will study the case of a new $U(1)^{\prime}$ gauge group
under which only the third generation quarks are charged. We assume the
$U(1)^{\prime}$ factor to be spontaneously broken by a complex scalar $S$
which is charged under the $U(1)^{\prime}$ but singlet with respect to the SM
gauge group. The Lagrangian we consider is

$\displaystyle\mathcal{L}=\quad$
$\displaystyle\mathcal{L}_{\text{SM}}+\sum_{f=b,t}g^{\prime}q_{f}Z^{\prime\mu}\,\bar{f}\gamma_{\mu}f-\frac{1}{4}F^{\prime\mu\nu}F^{\prime}_{\mu\nu}$
$\displaystyle+(D^{\mu}S)^{\dagger}D_{\mu}S+\mu_{s}^{2}\,S^{\dagger}S-\lambda_{s}\left(S^{\dagger}S\right)^{2},$
(1)

where $D^{\mu}=\partial^{\mu}+i\,g^{\prime}q_{S}\,Z^{\prime\mu}$,
$F^{\prime\mu\nu}=\partial^{\mu}Z^{\prime\nu}-\partial^{\nu}Z^{\prime\mu}$,
and we have assumed vectorial couplings to quarks only333See the discussion in
[52] for why axial couplings can be problematic.. Once the hidden Higgs
acquires a vev $w$ we define

$S=\frac{1}{\sqrt{2}}(s+w)$ (2)

so that the Lagrangian in the unitary gauge can be written as

$\displaystyle\mathcal{L}=\quad$
$\displaystyle\mathcal{L}_{\text{SM}}+\sum_{f=b,t}g^{\prime}q_{f}Z^{\prime\mu}\,\bar{f}\gamma_{\mu}f-\frac{1}{4}F^{\prime\mu\nu}F^{\prime}_{\mu\nu}$
$\displaystyle+\frac{1}{2}m_{Z^{\prime}}^{2}\,Z^{\prime\mu}Z^{\prime}_{\mu}+\frac{1}{2}g^{\prime
2}q_{S}^{2}\,Z^{\prime\mu}Z^{\prime}_{\mu}(s^{2}+2\,s\,w)$
$\displaystyle+\frac{1}{2}\partial^{\mu}s\partial_{\mu}s+\frac{\mu_{s}^{2}}{2}(s+w)^{2}-\frac{\lambda_{s}}{4}(s+w)^{4}\;,$
(3)

with $m_{Z^{\prime}}^{2}=g^{\prime 2}q_{S}^{2}w^{2}$. In addition, there can
be mixing terms in the extended Higgs sector which we assume to be small for
simplicity. The model as written down above is anomalous and therefore
additional fermions charged under the electroweak and $U(1)^{\prime}$ gauge
groups are required to render the theory anomaly free. Interestingly, these
additional fermions will generically induce effective couplings between the
singlet $s$ and the two abelian gauge bosons, that we parametrise as

$\mathcal{L}\supset
c_{BZ^{\prime}}sB^{\mu\nu}F_{\mu\nu}^{\prime}+c_{BB}sB^{\mu\nu}B_{\mu\nu}\;.$
(4)

As the anomaly cancellation conditions do not fix the additional field content
uniquely, these coefficients are a priori unknown. In the equation above,
$B^{\mu\nu}$ stands for the $U(1)_{Y}$ field strength (note that there are
hidden insertions of the vev of the hidden Higgs $S$ which is why these terms
are gauge invariant). In section 4 we will provide an estimate of the
magnitude of these operators for a particular UV completion as an example.

The main production channel for the $Z^{\prime}$ at the LHC is Drell-Yan
production with $b$ quarks in the initial state. As there is a large
suppression due to the proton PDFs, this production cross section is typically
rather small. Nevertheless, for large couplings there are relevant constraints
from resonance searches in $b\bar{b}$ and $t\bar{t}$ final states, if the
branching ratios are sizeable (which is naturally the case for the model under
consideration). Generally the partial widths of the $Z^{\prime}$ into SM
fermions can be written as

$\Gamma(Z^{\prime}\rightarrow
f\bar{f})=\frac{m_{Z^{\prime}}N_{c}}{12\pi}\sqrt{1-4\,x_{f}}\,(g^{\prime}q_{f})^{2}\left[1+2x_{f}\right]\;,$
(5)

where $x_{f}=m_{f}^{2}/m_{Z^{\prime}}^{2}$ and $N_{c}=3$ for quarks. The decay
modes into $b\bar{b}$ and $t\bar{t}$ will dominate the overall width. For
$q_{b}=q_{t}$ the branching into each of the two channels is about 0.5. The
partial widths of the $Z^{\prime}$ to $s\gamma$ and $sZ$ are typically smaller
and are given by

$\displaystyle\Gamma(Z^{\prime}\rightarrow s\gamma)=$
$\displaystyle\frac{(c_{BZ^{\prime}})^{2}c_{W}^{2}}{24\pi}m_{Z^{\prime}}^{3}\;,$
(6) $\displaystyle\Gamma(Z^{\prime}\rightarrow sZ)=$
$\displaystyle\frac{(c_{BZ^{\prime}})^{2}s_{W}^{2}}{24\pi}m_{Z^{\prime}}^{3}\;,$
(7)

where $c_{W}$ ($s_{W}$) stands for the cosine (sine) of the Weinberg angle.
Clearly the decay $Z^{\prime}\rightarrow sZ$ is suppressed by a factor of
$c_{W}^{2}/s_{W}^{2}\sim 3.3$ compared to $Z^{\prime}\rightarrow s\gamma$.
Finally the decay width of $s$ to $\gamma\gamma$ is given by

$\Gamma(s\rightarrow\gamma\gamma)=\frac{(c_{BB})^{2}c_{W}^{4}}{4\pi}m_{s}^{3}\;.$
(8)

Several constraints on this model can be set in light of the results from the
LHC Run 1. The upper bounds on the production cross section times branching
ratio into $b\bar{b}$ and $t\bar{t}$ final states are about 1 pb [53] and 0.55
pb [54] at the 95% C.L., respectively. The diphoton channel is also
constrained by Run 1 data to be below 2 fb at the 95% C.L. [55]. Given that
the cross section of the observed excess at 13 TeV can be estimated to be
around 5–10 fb, it is important to have a significant increase of the
production cross section with the centre of mass energy. For $b$ quarks in the
initial state, this is indeed the case with the ratio of cross sections at 8
and 13 TeV being slightly larger than 5. Another constraint is that the total
width of the resonance should not be too large (the best fit width measured by
ATLAS is about 45 GeV, although there is currently a large uncertainty on this
number and CMS prefers a narrow width). Potential constraints coming from
electroweak precision data, in particular those due to the renormalization of
the $Z$ vertices [56], set negligible bounds as long as left-handed and right-
handed quarks carry the same charge under the $U(1)^{\prime}$ group.

In order to study the LHC phenomenology we have implemented this generic
parametrisation including the five-dimensional operators in MadGraph 5 [57] by
means of Feynrules v2 [58]. We take $q_{b}=q_{t}\equiv q_{f}$ and fix the
scalar mass $m_{s}=1$ GeV.444We show in the next section that the resulting
two photons will be reconstructed as one. In figure 2 we show the region in
the $g^{\prime}q_{f}-c_{BZ^{\prime}}$ plane that can account for the observed
excess at 13 TeV without conflicting with previous searches. Indeed, the blue
region is excluded by diphoton searches at 8 TeV, while in the red region the
cross section of the diphoton signal at 13 TeV is smaller than 5 fb. Very
large couplings to quarks produce too many $t\bar{t}$ final states and are
hence excluded. Searches for $b\bar{b}$ are less constraining and not shown in
the plot. Finally, a line indicating a $Z^{\prime}$ width of 45 GeV is also
shown. Significantly larger widths are disfavoured as an explanation of the
observed excess.

Figure 2: Parameter space in the plane of the $Z^{\prime}$ couplings to third
generation quarks and to $s\gamma$ with bounds from LHC Run 1 diphotons
(blue), signal cross section below 5 fb (red) as well as bounds from
$t\bar{t}$ resonance searches (green). The white region can explain the
observed excess and is consistent with all bounds.

##  3 Mimicking one photon with two

So far we have concentrated on the production cross section of the
$Z^{\prime}$ and its decay into $s\gamma$. Now we turn to the subsequent
decays of the scalar $s$ in order to determine whether the observed excess can
be reproduced in our model. The central observation is that the effective
Lagrangian introduced in eq. (4) does not only account for the production of
$s$ from decays of the $Z^{\prime}$, but it also leads to the decay processes
$s\rightarrow\gamma\gamma$, $s\rightarrow ZZ$ and $s\rightarrow Z\gamma$.
Provided the scalar is sufficiently light, only the first decay channel will
be kinematically allowed,555We emphasise that any additional fermions should
be heavier than about 100 GeV given their electroweak quantum numbers and that
mixing in the Higgs sector is negligible. Consequently the branching fraction
into fermions of the scalar $s$ is expected to be small. so that we can assume
$\text{BR}(s\rightarrow\gamma\gamma)=1$.

Our central observation is that the decay chain $Z^{\prime}\rightarrow
s\gamma\rightarrow 3\gamma$ may be interpreted as a diphoton resonance,
provided the two photons produced in the scalar decay are highly collimated
(see for instance [59] for a similar analysis in the context of LEP). If the
scalar $s$ is produced with a boost factor $\gamma_{s}$ and velocity
$\beta_{s}$ in the laboratory frame, the distribution of opening angles
$\alpha$ is given by

$\frac{\mathrm{d}N}{\mathrm{d}\alpha}=\frac{1}{4\,\gamma_{s}\,\beta_{s}}\frac{\cos\alpha/2}{\sin^{2}\alpha/2}\frac{1}{\sqrt{\gamma_{s}^{2}\,\sin^{2}\alpha/2-1}}\;,$
(9)

which (assuming $\gamma_{s}\gg 1$) is strongly peaked towards the minimum
opening angle $\alpha_{\text{min}}\simeq 2/\gamma_{s}$. Integrating the above
distribution, we find that in 90% of the scalar decays the opening angle
between the two photons is smaller than $4.6/\gamma_{s}$.

The typical angular resolution of the electromagnetic calorimeter in ATLAS and
CMS is of order $20\>\text{mrad}$ [60, 61]. Hence, for $\gamma_{s}\gtrsim
200$, the vast majority of the scalar decays will appear in the detector as a
single photon rather than two individual photons. This number agrees with an
ATLAS study on the process $h\rightarrow aa\rightarrow 4\gamma$ for a light
pseudoscalar $a$ [62]. It was found that for $m_{a}<400\>\text{MeV}$, the two
photons from the pseudoscalar decay are indistinguishable and the decay chain
therefore mimics $h\rightarrow\gamma\gamma$ in the detector. Since
$m_{h}=125\>\text{GeV}$, this upper bound on the mass corresponds to the
requirement $\gamma_{a}>150$.666We thank Kerstin Tackmann and James Beacham
for insightful discussions on this point.

The consideration above leads us to consider a very light scalar with mass
$m_{s}<2\,\text{GeV}$. On the other hand, since we require the scalar to decay
within the detector, it cannot be arbitrarily light. The decay length is given
by

$l=\beta_{s}\,\gamma_{s}/\Gamma\approx\frac{4\pi\,E_{s}}{c_{BB}^{2}\,c_{W}^{4}\,m_{s}^{4}}\;.$
(10)

Clearly, we must require $l\ll 1\>\text{m}$ in order for the majority of the
scalars to decay within the detector, but it is certainly possible to have
$l\sim 10\>\text{cm}$.777Note that it is not easily possible to reconstruct
the tracks of the two photons and hence there is no problem with the scalar
decaying from a displaced vertex (see also [8]).

Figure 3: Constraints and interesting regions in the $m_{s}-c_{BB}$ parameter
plane. The solid (dashed) lines correspond to constant scalar decay length
(photon opening angle). In the green shaded region we expect the acceptance to
be larger than $90\%$, meaning that the large majority of scalars will decay
within the detector and give a signal that mimics a single photon. For
comparison, we also show bounds on the parameter space from beam-dump
experiments and collider searches (see [63]).

In figure 3 we illustrate these considerations in the $m_{s}-c_{BB}$ parameter
plane. Since we expect $c_{BB}$ to be comparable to $c_{BZ^{\prime}}$, i.e.
$c_{BB}\sim 10^{-1}\>\text{TeV}^{-1}$ we conclude that $m_{s}$ should not be
smaller than about $200\>\text{MeV}$. In the example presented below, we find
$c_{BB}\sim 5\times 10^{-3}\>\text{TeV}^{-1}$ (indicated by the dotted
horizontal line), which would limit $m_{s}$ to a relatively narrow range
around $1\text{--}2\>\text{GeV}$. In this mass range it is possible to have a
sufficiently short decay length and at the same time a sufficiently small
opening angle between the two photons produced in the decay.

##  4 A $U(1)^{\prime}$ model example

Field |  $SU(3)_{C}$ |  $SU(2)_{L}$ |  $U(1)_{Y}$ |  $U(1)^{\prime}$
---|---|---|---|---
$\Sigma_{L}$ | $\mathbf{1}$ | $\mathbf{2}$ | $Y_{1}$ | $Q_{1}$
$\Sigma_{R}$ | $\mathbf{1}$ | $\mathbf{2}$ | $Y_{1}$ | $Q_{2}$
$\rho_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $Y_{2}$ | $Q_{1}$
$\rho_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $Y_{2}$ | $Q_{2}$
$\xi_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $Y_{3}$ | $Q_{1}$
$\xi_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $Y_{3}$ | $Q_{2}$
$S$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $Q_{1}-Q_{2}=-3q_{f}$
Table 1: Fermionic and scalar field content of a possible anomaly-free UV
completion, and the corresponding quantum numbers [64].

Having discussed the excess in a model-independent framework up to now, let us
now give an example for a UV complete model that has the required properties.
For our analysis we have assumed that only the third generation quarks are
charged under the new $U(1)^{\prime}$ gauge group, with $q_{t}=q_{b}\equiv
q_{f}$. As commented earlier, new fermionic states charged under the
$U(1)^{\prime}$ and the SM gauge group are required to make this model anomaly
free. It can easily be checked that all relevant anomalies are cancelled with
the fermions given in Table 1, which are vector-like under the SM gauge group
and do not need to carry colour charge. This solution for the cancellation of
anomalies was first found in the context of gauging baryon and lepton numbers
[64]. The only requirements on the hypercharges and the $U(1)^{\prime}$
charges are

$Q_{1}-Q_{2}=-3q_{f}$ (11)

and888Note that the linear condition on the hypercharges is only relevant if
$|Q_{1}|\neq|Q_{2}|$.

$\displaystyle 2Y_{1}^{2}-Y_{2}^{2}-Y_{3}^{2}$ $\displaystyle=-\frac{1}{2},$
(12) $\displaystyle 2Y_{1}-Y_{2}-Y_{3}$ $\displaystyle=0.$ (13)

The latter conditions on the hypercharges are solved by choosing (see e.g.
[48])

$\left(Y_{1},Y_{2},Y_{3}\right)=(a,\ a+\frac{1}{2},\ a-\frac{1}{2})\;,$ (14)

where $a$ is a free parameter. Due to $U(1)^{\prime}$ gauge symmetry the new
fermions cannot have bare mass terms, but they will have Yukawa interactions
with the new scalar $S$ provided the $U(1)^{\prime}$ charge of the scalar $S$
satisfies the condition $q_{S}=Q_{1}-Q_{2}$, which implies $q_{S}=-3q_{f}$
from anomaly cancellation. Once $S$ obtains its vacuum expectation value, it
will generate mass terms for the new fermions, as well as a mass term for the
$Z^{\prime}$ gauge boson.

Since the additional fermions are charged under both $U(1)_{Y}$ and
$U(1)^{\prime}$, they will naturally generate the couplings $c_{BZ^{\prime}}$
and $c_{BB}$ introduced in eq. (4). As long as the additional fermions are
heavy compared to $m_{Z^{\prime}}$ and $m_{s}$, we can estimate their
contribution to be

$\displaystyle c_{BZ^{\prime}}$
$\displaystyle=\mathop{\sum_{\text{heavy}}}_{\text{fermions}}\frac{g_{Y}\,g^{\prime}\,Q_{Y}\,Q^{\prime}}{12\,\pi^{2}\,w}\;,$
$\displaystyle c_{BB}$
$\displaystyle=\mathop{\sum_{\text{heavy}}}_{\text{fermions}}\frac{g_{Y}^{2}\,Q_{Y}^{2}}{24\,\pi^{2}\,w}\;,$
(15)

where $g_{Y}$ and $g^{\prime}$ denote the gauge couplings of $U(1)_{Y}$ and
$U(1)^{\prime}$, respectively, $Q_{Y}$ and $Q^{\prime}$ are the corresponding
vector charges of the new fermions (i.e.
$Q^{\prime}=(Q^{\prime}_{L}+Q^{\prime}_{R})/2$) and $w$ is the vev of the
second Higgs as defined in eq. (2).

Substituting the $Z^{\prime}$ mass for $w$, these expressions can be rewritten
as

$\displaystyle c_{BZ^{\prime}}$
$\displaystyle=\mathop{\sum_{\text{heavy}}}_{\text{fermions}}\frac{g_{Y}\,g^{\prime
2}\,Q_{Y}\,Q^{\prime}\,q_{f}}{4\,\pi^{2}\,m_{Z^{\prime}}}\;,$ $\displaystyle
c_{BB}$
$\displaystyle=\mathop{\sum_{\text{heavy}}}_{\text{fermions}}\frac{g_{Y}^{2}\,Q_{Y}^{2}\,g^{\prime}\,q_{f}}{8\,\pi^{2}\,m_{Z^{\prime}}}$
(16)

leading to

$\displaystyle c_{BZ^{\prime}}$ $\displaystyle\sim
0.012\>\text{TeV}^{-1}\times\mathop{\sum_{\text{heavy}}}_{\text{fermions}}g^{\prime
2}\,Q_{Y}\,Q^{\prime}\,q_{f}\;,$ $\displaystyle c_{BB}$ $\displaystyle\sim
0.002\>\text{TeV}^{-1}\times\mathop{\sum_{\text{heavy}}}_{\text{fermions}}Q_{Y}^{2}\,g^{\prime}\,q_{f}$
(17)

for $m_{Z^{\prime}}=750$ GeV. Since several fermions are expected to give a
sizeable contribution to this expression, it is perfectly conceivable to
obtain large enough effective couplings to account for the observed excess for
gauge couplings and charges of order unity. For example, for $q_{f}=1/3$ the
bound on the total width of the $Z^{\prime}$ implies $g^{\prime}\leq 1.8$ (see
figure 2). Assuming that this bound is saturated and substituting the charges
from table 1, we obtain

$\displaystyle c_{BZ^{\prime}}$ $\displaystyle\sim
0.025\>\text{TeV}^{-1}\times a(Q_{1}+Q_{2})\;,$ $\displaystyle c_{BB}$
$\displaystyle\sim
0.0012\>\text{TeV}^{-1}\times\left(4a^{2}+\frac{1}{2}\right)\;.$ (18)

Setting for example $a=1$, $Q_{1}=4$ and $Q_{2}=Q_{1}+3\,q_{f}=5$ gives
$c_{BZ^{\prime}}=0.23\>\text{TeV}^{-1}$ and $c_{BB}=0.005\>\text{TeV}^{-1}$,
which are both consistent with the constraints shown in figures 2 and 3. We
note that rather large fermion charges are required in order to obtain
sufficiently large couplings. However, the contributions of the new fermions
to $c_{BZ^{\prime}}$ and $c_{BB}$ will be larger if their masses are
comparable to $m_{Z^{\prime}}/2$, which is easily possible given that the new
fermions do not carry colour charge. In such a set-up, smaller charges may be
sufficient in order to explain the diphoton excess. A detailed study of the
implication of such a UV completion is left to future work.

##  5 Discussion

We have shown that extending the SM with a new $U(1)^{\prime}$ gauge group can
provide an explanation for the recently observed diphoton excess around $\sim
750$ GeV by both ATLAS and CMS. The central idea is that a very light scalar
decaying into two photons can mimic a single photon if it is produced with
sufficiently large boost factor. This observation allows us to evade the
Landau-Yang theorem and to obtain the diphoton excess from a vector resonance.

As a specific example we have discussed the addition of a new complex scalar
degree of freedom and a set of new fermions in order for the new gauge
$Z^{\prime}$ boson to have a mass and the full theory to remain anomaly free.
It turns out that loops of new extra fermions give rise to trilinear vertices
allowing the $Z^{\prime}$ to decay into $s\gamma$ and the light scalar $s$ to
decay into $\gamma\gamma$ respectively. The corresponding couplings are
expected to be of the same order, namely $c_{BZ^{\prime}}\sim c_{BB}\sim
0.1\>\text{TeV}^{-1}$, if the charges carried by the new fermions are large
enough. Interestingly, these values can make the cross section into diphotons
large enough to account for the observed excess and, at the same time, make
the light scalar mimic a one-photon signature while still decaying within the
detector.

In fact, the idea of reconstructing two photons as a single one may also be of
interest to other models. One interesting example is provided by a scalar
resonance $\Phi$ with mass $m_{\Phi}\sim 750$ GeV that decays into two very
light scalars $s$ that subsequently decay into two photons. The relevant
interactions can be written as

$\mathcal{L}\supset
c_{GG}\,\Phi\,G_{\mu\nu}^{a}G^{\mu\nu\,a}+\kappa_{\Phi}\Phi s^{2}.$ (19)

Figure 4: Allowed and excluded parameter space regions in the plane of the
$\Phi$ couplings to gluons and to $s$.

In figure 4 we show the parameter space region in the $c_{GG}-\kappa_{\Phi}$
plane allowed by the diphoton searches at both 8 and 13 TeV as well as the
best fit width. Searches for dijets are not sensitive to this region of the
parameter space.

The two scenarios can be easily distinguished with additional data. If the
diphoton excess is due to the process $Z^{\prime}\rightarrow s\gamma$, a
corresponding signal in $Z^{\prime}\rightarrow sZ\rightarrow\gamma\gamma Z$
should be discovered in the near future. On the other hand, no such additional
decay channel is expected if the diphoton excess is due to a scalar decaying
into two very light scalars.

Finally, we emphasize that the gauge boson of a new $U(1)^{\prime}$ gauge
group can mix with the neutral gauge bosons of the SM. Indeed, given that we
include fermions charged under both $U(1)_{Y}$ and $U(1)^{\prime}$, loop
processes will in general lead to kinetic mixing of the form
$-\frac{1}{2}\sin\epsilon F^{\prime\mu\nu}B_{\mu\nu}$ [65, 66]. The presence
of such a kinetic mixing term can modify electroweak precision observables and
lead to sizeable decays of the $Z^{\prime}$ into SM leptons, providing
promising ways to constrain this scenario. However, the precise magnitude of
these effects depends on the specific UV completion. We leave a detailed
investigation of these effects to future studies.

## Acknowledgments

We thank James Beacham and Kerstin Tackmann for very helpful discussions and
Joerg Jaeckel for valuable comments on the draft. This work is supported by
the German Science Foundation (DFG) under the Collaborative Research Center
(SFB) 676 Particles, Strings and the Early Universe as well as the ERC
Starting Grant ‘NewAve’ (638528).

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