Table Of ContentarXiv: 1601.02570
Interpreting the 750 GeV diphoton excess in the Minimal Dilaton Model
Junjie Cao,1,2,∗ Liangliang Shang,2,† Wei Su,3,‡ Yang Zhang,3,§ and Jinya Zhu1,¶
1Center for Theoretical Physics, School of Physics and Technology,
Wuhan University, Wuhan 430072, China
2Department of Physics, Henan Normal University, Xinxiang 453007, China
3State Key Laboratory of Theoretical Physics,
Institute of Theoretical Physics, Academia Sinica, Beijing 100190, China
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We try to interpret the 750 GeV diphoton excess in the Minimal Dilaton Model, which
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extendstheSMbyaddingonelinearizeddilatonfieldandvector-likefermions. Wefirstshow
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p by analytic formulae in this framework that the production rates of the γγ, gg, Zγ, ZZ,
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WW∗, tt¯and hh signals at the 750GeV resonance are only sensitive to the dilaton-Higgs
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1 mixing angle θS and the parameter η ≡vNX/f, where f is the dilaton decay constant and
N denotes the number of the fermions. Then we scan the two parameters by considering
] X
h
varioustheoreticalandexperimentalconstraintstofindthesolutionstothediphotonexcess.
p
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p We conclude that the model can predict the central value of the diphoton rate without
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conflicting with any constraints. The signatures of our explanation at the LHC Run II and
h
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the vacuum stability at high energy scale are also discussed.
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PACS numbers: 12.60.Fr, 14.80.Ec, 14.65.Jk, 14.70.Bh
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∗ [email protected]
† [email protected]
‡ [email protected]
§ [email protected]
¶ [email protected]
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I. INTRODUCTION
About four years ago, the hint of a 125 GeV Higgs boson was reported in the diphoton channel
by both the ATLAS and CMS collaborations based on about 5 fb−1 data for each collaboration at
the 7-TeV LHC [1, 2], and this led to the great discovery of the Higgs boson in July 2012 [3, 4].
Recently another excess in the diphoton channel was reported by the first 3.2 fb−1 data at the
13-TeV LHC [5, 6]. This time the invariant mass of the signal locates around 750GeV, and its
local and global significances are about 3.6σ and 2.3σ respectively for the ATLAS analysis, and
2.6σ and 2σ for the CMS analysis. Interestingly, although there exists a ostensible inconsistence
in the width of the resonance 1, both the analyses favored the diphoton production rate at about
4fb in the narrow width approximation. Such a rate is about 104 times larger than the prediction
of the Standard Model (SM) with a 750GeV Higgs boson [8]. Obviously, if this excess is confirmed
in near future, it points undoubtedly to the existence of new physics.
So far more than one hundred theoretical papers have appeared to interpret the excess in new
physics models [9–19], and most of them employed the process gg → S → γγ with S denoting a
scalar particle with mass around 750 GeV to fit the data. From these studies, one can infer two
essential ingredients of the explanations. One is that there must exist other charged and colored
particles to generate by loop effects sufficiently large Sγγ and Sgg interactions. The other is, given
the fact that no excess was observed in the channels such as ZZ, WW∗ and tt¯at the LHC Run I,
the particle S is preferred to be gauge singlet dominated so that the branching ratios of S → ZZ,
WW∗, tt¯are not much larger than that of S → γγ. These requirements guide us in seeking for the
explanations of the excess.
Inthiswork,weconsiderinterpretingthediphotonexcessintheMinimalDilatonModel(MDM),
which extends the SM by one gauge singlet field called dilaton [20–22]. Just like the traditional
dilaton theories [23], the dilaton in this model arises from a strong interaction theory with ap-
proximate scale invariance at a certain high energy scale. The breakdown of the invariance then
triggers the electroweak symmetry breaking, and during this process, the dilaton as the pseudo
Nambu-Goldstone particle of the broken invariance can be naturally light in comparison with the
high energy scale. Furthermore, this model assumes that all SM particles except for the Higgs
field do not interact with the dynamics sector, and consequently the dilaton does not couple di-
rectly to the fermions and W, Z bosons in the SM. In this sense, the dilaton is equivalent to an
1 Currentlywithinsufficientexperimentaldata,theATLASanalysisslightlypreferredawidewidthoftheresonance
(about45GeV)toanarrowwidth[5],andbycontrasttheCMSanalysisfavoredanarrowwidth[6]. Veryrecently,
an analysis by combining both the ATLAS data and the CMS data was carried out, and it indicated that the
narrow width was preferred [7].
3
electroweak gauge singlet field. The model also consists of massive vector-like fermions acting as
the lightest particles in the dynamical sector, to which the dilaton naturally couples in order to
recover the scale invariance: M → Me−φ/f. As a result, the interactions between the dilaton and
the photons/gluons are induced through loop diagrams of these fermions. These characters enable
the MDM as a hopeful theory to explain the diphoton excess through the dilaton production.
Discussing the capability of the MDM in explaining the excess is the aim of this work.
Thispaperisorganizedasfollows. WefirstintroducebrieflytheMDMinSectionII,andpresent
in Section III some analytical formulae which are used to calculate the diphoton rate. In Section
IV, we discuss the constraints on the model, its capability in explaining the excess, and also the
related phenomenology at the LHC Run II. For completeness, in section V we turn to discuss the
vacuum stability at high energy scale. Finally, we draw our conclusions in Section VI.
II. THE MINIMAL DILATON MODEL
As introduced in last section, the MDM extends the SM by adding one gauge singlet field
S, which represents a linearized dilaton field, and also vector-like fermions X . The low energy
i
effective Lagrangian is then written as [20, 21]
L = L + 1∂ S∂µS +(cid:88)NX X¯ (cid:18)iD/ − MiS(cid:19)X −V(S,H˜), (1)
SM µ i i
2 f
i=1
where L is the SM Lagrangian without Higgs potential, f is the decay constant of the dilaton
SM
S, M is the mass of the fermion X , and N is the number of the vector-like fermions. The scalar
i i X
potential V(S,H˜) contains terms with explicit breaking of the scale invariance, and its general
form is given by
m2 λ (cid:12) (cid:12)2 (cid:12) (cid:12)4 λ (cid:12) (cid:12)2
V(S,H˜) = SS2+ SS4+m2 (cid:12)H˜(cid:12) +λ (cid:12)H˜(cid:12) + HSS2(cid:12)H˜(cid:12) , (2)
2 4 H(cid:12) (cid:12) H(cid:12) (cid:12) 2 (cid:12) (cid:12)
where m , λ , m , λ and λ are all free real parameters.
S S H H HS
About the Lagrangian in Eq.(1), one should note following points:
• The MDM is actually a low energy theoretical framework describing the breakdown of a
UV strong dynamics with approximate scale invariance, and the dilaton in this theory is
distinguished from the usual one. Explicitly speaking, in the traditional dilaton models
the whole SM sector is usually assumed to be a part of the strong dynamics, and all the
fermions and gauge bosons of the SM are composite particles at the weak scale [23]. Under
4
these theoretical assumptions, the couplings of the linearized dilaton S to the SM fields take
following form [23]
S
L = Tµ, (3)
f µ
where Tµ represents the trace of the energy-momentum tensor of the SM. Through the
µ
interactions in Eq.(3), the dilaton couples directly to the fermions and W, Z bosons in the
SM with the strengthes proportional to the mass of the involved particle. In this way, the
dilaton mimics the properties of the SM Higgs boson. By contrast, in the MDM all SM
particles except for the Higgs field are assumed to be the spectators of the strong dynamics,
and they are all elementary particles. As a result, the dilaton does not couple directly to
these particles.
• IntheoriginalversionoftheMDM,theauthorssetN = 1andchosethequantumnumbers
X
of the fermion X same as those of the right-handed top quark. This setting was motivated
i
by topcolor theory [24], which intended to present a reasonable explanation of the relatively
large top quark mass within a minimal framework. However, as we will show below, such a
settingistightlylimitedbythevacuumstabilityofthetheoryatm scaleininterpretingthe
Xi
diphoton excess. Considering that a strong dynamical theory usually involves rich fermion
fields and the assignment on their quantum numbers is somewhat arbitrary, we therefore
consider a more general but also simple case, which assumes that all the vector-like fermions
are identical, and each of them transforms in the (3,1,Y = 2Q ) representation of the
X
(cid:78) (cid:78)
SM gauge group SU(3) SU(2) U(1) . In the following, we vary the number of the
c L Y
fermions N , their common mass m , and also their electric charge Q to discuss the
X X X
diphoton excess.
IfonewritestheHiggsfieldinunitarygaugeviaH˜ = √1 U(0,H)T,thescalarpotentialinEq.(2)
2
can be rewritten as
m2 λ m2 λ λ
V˜(S,H) = SS2+ SS4+ HH2+ HH4+ HSS2H2. (4)
2 4 2 4 4
In the following, we consider the most general situation in which both H and S take vacuum
expectation values (VEV), (cid:104)H(cid:105) = v and (cid:104)S(cid:105) = f, and they mix to form mass eigenstates h and s:
h = cosθ H +sinθ S,
S S
s = −sinθ H +cosθ S. (5)
S S
5
In our scheme for the diphoton excess, h corresponds to the 125 GeV Higgs boson discovered at
the LHC, and s is responsible for the 750GeV diphoton excess by the process gg → s → γγ. So in
the following, we set m = 125GeV, m = 750GeV and v = 246GeV, and for the convenience of
h s
our discussion, we choose η ≡ vN , sinθ , Q , N and m as the input parameters of the MDM
f X S X X X
model. In this case, we have following relations
2η(m2 −m2)sinθ cosθ
λ = h s S S,
HS v2N
X
m2cos2θ +m2sin2θ
λ = h S s S,
H 2v2
η2(m2sin2θ +m2cos2θ )
λ = h S s S . (6)
S 2v2N2
X
With the assumption that the dilaton is fully responsible for the fermion masses, the Yukawa
coupling of X is given by y ≡ mX = ηmX. Obviously y is inversely proportional to N for
i X f vNX X X
fixed η and m . As we will show below, the diphoton rate is only sensitive to the parameters η,
X
sinθ and Q , and does not depend on y directly.
S X X
III. USEFUL FORMULAE IN GETTING THE DIPHOTON EXCESS
In the MDM, the particle s may decay into gg, γγ, Zγ, ZZ, WW∗, ff¯and hh. In this section,
we list the formulae for the widths of these decays, which are needed to get the diphoton rate. As
we will show below, these formulae are helpful to understand our results.
• The widths of φ → γγ,gg,Zγ with φ = h,s:
Γ = Gµα√2m3φ (cid:12)(cid:12)Iφ(cid:12)(cid:12)2, (7)
φ→γγ 128 2π3 (cid:12) γ(cid:12)
Γ = Gµ√αs2m3φ (cid:12)(cid:12)Iφ(cid:12)(cid:12)2, (8)
φ→gg 16 2π3 (cid:12) g(cid:12)
Γ = G2µm2Wαm3φ (cid:32)1− m2Z(cid:33)3(cid:12)(cid:12)Iφ (cid:12)(cid:12)2, (9)
φ→Zγ 64π4 m2 (cid:12) Zγ(cid:12)
φ
where the Iφ, Iφ and Iφ are given by
g γ Zγ
4
Ih = cosθ ×(A (τ )+ A (τ ))+sinθ N ηQ2 A (τ ), (10)
γ S 1 W 3 12 t S c X 12 X
4
Is = −sinθ ×(A (τ )+ A (τ ))+cosθ N ηQ2 A (τ ), (11)
γ S 1 W 3 12 t S c X 12 X
cosθ ηsinθ
Ih = S ×A (τ )+ SA (τ ), (12)
g 2 12 t 2 12 X
sinθ ηcosθ
Is = − S ×A (τ )+ SA (τ ), (13)
g 2 12 t 2 12 X
6
2(1− 8 sin2θ )
Ih = cosθ ×(cosθ C (τ−1,η−1)+ 3 W C (τ−1,η−1))
Zγ S W 1 W W cosθW 12 t t
sin2θ
+4sinθ N ηQ2 WC (τ−1,η−1), (14)
S c X cosθW 12 X X
2(1− 8 sin2θ )
Is = −sinθ ×(cosθ C (τ−1,η−1)+ 3 W C (τ−1,η−1))
Zγ S W 1 W W cosθW 12 t t
sin2θ
+4cosθ N ηQ2 WC (τ−1,η−1). (15)
S c X cosθW 12 X X
In above expressions, A , A , C , C are the loop functions defined in [25] with τ =
1 1 1 1 β
2 2
m2/(4m2) and η = m2/(4m2) for β = W,t,X .
φ β β Z β i
About these formulae, one should note that the terms proportional to cosθ in the expres-
S
sions of Is are contributed by the dilaton component of s, while those proportional to sinθ
i S
come from the H-component of s. One should also note that in the case of sinθ ∼ 0, which
S
is required by the null excess in the channels such as ZZ and hh at the 750GeV invariant
mass(seebelow)andalsobythe125GeV Higgsdata, Is, Is andIs arealldominatedbythe
γ g Zγ
contribution from the vector-like fermions, and consequently they are correlated. Explicitly
speaking, we have Is : Is : Is = N Q2 : 1 : NcQ2X sin2θW in the limit m ,m (cid:29) m . This
γ g Zγ c X 2 2 cosθW s X Z
correlation may sever as a test of the model at future LHC experiments.
• The widths of the decays s → VV∗ with V = W,Z.
If one parameterizes the effective sVV∗ interaction as
A = g m (As gµν +Bspµpν)(cid:15) (p )(cid:15) (p ),
sVV∗ V V V V 2 1 µ 1 ν 2
then the decay width of s → VV∗ is given by[13]
G m3 4m4 (cid:113)
Γ = δ F√s V λ(m2 ,m2 ;m2)×
s→VV∗ V 16π 2 m4 V V s
s
(cid:20) (cid:18) (p ·p )2(cid:19) (cid:18)(p ·p )3 (cid:19)
As As∗× 2+ 1 2 +(As Bs∗+As∗Bs)× 1 2 −p ·p
V V m4 V V V V m4 1 2
V V
(cid:18) (p ·p )4 (cid:19)(cid:21)
+ BsBs∗× m4 + 1 2 −2(p ·p )2 , (16)
V V V m4 1 2
V
where δ = 2(1) for V = W(Z) respectively and λ(x,y,z) = ((z−x−y)2−4xy)/z2.
V
In the MDM, we have
As (cid:39) −sinθ , Bs (cid:39) 0,
W S W
α
As (cid:39) −sinθ + cosθ N ηQ2 tan2θ p ·p A (τ ),
Z S 4πm2 S c X W 1 2 12 X
Z
α
Bs (cid:39) − cosθ N ηQ2 tan2θ A (τ ).
Z 4πm2 S c X W 12 X
Z
7
Note that in the expressions of As and Bs, we have included the one-loop corrections. This
Z Z
is because in case of sinθ ∼ 0, the corrections are not always smaller than the tree level
S
contributions. Also note that in getting As and Bs, to a good approximation we have
Z Z
neglected the Z boson mass appeared in the loop functions, and that is why we can express
the corrections in term of the simple function A (τ ).
1 X
2
• The width of the tree-level decay s → ff¯with f denoting any of the fermions in the SM:
Γ = sin2θ 3Gµ√m2fms(cid:16)1− 4m2f(cid:17)32. (17)
s→ff¯ S 4 2π m2
s
Note that for this kind of decays, the widths are proportional to sin2θ .
S
• The width of the tree level decay s → hh:
Γ = |Cshh|2 (cid:18)m2s −m2(cid:19)12 , (18)
s→hh 16πm2 4 h
s
where
C = −6λ vsinθ cos2θ +6λ fsin2θ cosθ
shh H S S S S S
+λ (−vsin3θ +fcos3θ −2fsin2θ cosθ +2vsinθ cos2θ )
HS S S S S S S
2m2
(cid:39) − s sinθ .
S
v
In getting the final expression of C , we have used the relation m2 (cid:29) m2 and sinθ ∼ 0
shh s h S
to neglect some unimportant terms. Just like the decays s → WW∗ and s → tt¯, Γ is
s→hh
proportional to sin2θ .
S
With these formulae, the total width of the scalar s and the s-induced diphoton rate can be
written as
Γ = Γ +Γ +Γ +Γ +Γ +Γ +Γ +Γ , (19)
tot s→gg s→γγ s→Zγ s→ZZ s→WW∗ s→ff¯ s→hh new
Γ Γ
σ13TeV = φ→gg | ×σ√SM (H)× s→γγ, (20)
γγ ΓSM mH(cid:39)750GeV s=13TeV Γ
H→gg tot
where the Γ in Eq.(19) represents the contribution from the exotic decays of s, which may
new
exist if the MDM is embedded in a more complex theoretical framework, ΓSM denotes the decay
H→gg
width of the SM Higgs H into gg with m = 750GeV, and σ√SM (H) = 735fb is the NNLO
H s=13TeV
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production rate of the H at the 13 TeV LHC [29]. Obviously, if Γ is determined mainly by Γ ,
tot gg
the rate can be approximated by
Γ
σ13TeV (cid:39) φ→γγ| ×σ√SM (H) ∝ η2Q4 , (21)
γγ ΓSM mH(cid:39)750GeV s=13TeV X
H→gg
while if Γ takes a fixed value, we have
tot
(cid:18) (cid:19)
45GeV
σ13TeV = ×σ ×(ηQ )4, (22)
γγ Γ norm X
tot
where the normalized cross section σ is equal to 0.019 fb (0.018 fb) for m = 1TeV (1.5TeV).
norm X
From the discussion in this section, one can get following important conclusions:
• The widths of s → gg,γγ,Zγ or the production rates of the gg, γγ and Zγ signals at the
LHC are correlated by
9α2 9α2
Γ : Γ : Γ (cid:39) 1 : Q4 : tan2θ Q4 (cid:39) 1 : 0.03Q4 : 0.004Q4 . (23)
s→gg s→γγ s→Zγ 2α2 X 4α2 W X X X
s s
• The widths listed from Eq.(7) to Eq.(18) depend on the number of the vector-like fermions
N only through the parameter η ≡ vNX. As a result, explaining the diphoton excess puts
X f
non-trivial requirements on the combination vNX, instead of on the individual parameter
f
N or y = ηmX.
X X vNX
• Since the recent LHC searches for right-handed heavy quarks have required m (cid:38) 900 GeV
X
[26–28] and thus τ ≡ m2/(4m2 ) < 0.2, the loop functions appeared in the widths change
X s X
slightly with the further increase of m . This implies that the widths and also the cross
X
section have a very weak dependence on the value of m . As a result, the results obtained
X
in this work are only sensitive to the parameters η, sinθ and Q .
S X
At this stage, one can infer that the parameter N may also be understood as the total
X
number of the vector-like fermions with the electric charge Q in the strong dynamics
X
because the contributions of the fermions to the diphoton rate are roughly identical. Since
the particle content of a strong dynamics is usually rich, N is naturally larger than 1.
X
We remind that the second and third conclusions depend on the assumption that the dilaton is
fully responsible for the masses of the vector-like fermions, and within our knowledge, they were
not paid attention to in previous literatures.
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TABLE I. Upper limits on various 750GeV resonant signals at 8-TeV LHC set by either ATLAS or CMS
collaboration [13].
Channel jj [35, 36] hh [37–40] WW∗ [41, 42] ZZ [41, 43] Zγ [44] tt¯[45, 46]
95% C.L. limits 1800 fb 35 fb 37 fb 12 fb 3.6 fb 450 fb
IV. NUMERICAL RESULTS AND DISCUSSIONS
Inthissection,wediscussthediphotonexcessintheMDM.Inordertogetthefavoredparameter
space for the excess, we fix Q = 2, 5 and m = 1TeV, 1.5TeV at each time, and scan following
X 3 3 X
parameter space
0 < η ≤ 2, |tanθ | ≤ 0.1. (24)
S
During the scan, we consider following theoretical and experimental constraints:
• Thevacuumstabilityatthescaleofm = 750GeVforthescalarpotential,whichcorresponds
s
to the requirement 4λ λ −λ2 > 0 [20].
H S HS
• Constraintsfromtheperturbativityatthescaleofm = 750GeV,whichrequiresλ ,λ ,λ (cid:46)
s S H HS
√
4π, and y (cid:46) 4π/ N [18].
X c
• Constraints from the electroweak precision data. We calculate the Peskin-Takeuchi S and T
parameters [30] with the formulae presented in [20], and construct χ2 by following experi-
ST
mental fit results with m = 125 GeV and m = 173 GeV [31]:
h,ref t,ref
S = 0.06±0.09, T = 0.10±0.07, ρ = 0.91. (25)
ST
In our calculation, we require that the samples satisfy χ2 ≤ 6.18.
ST
• Experimental constraints from the 125 GeV Higgs data, which include the updated exclusive
signal rates for γγ, ZZ∗, WW∗, b¯b and ττ¯ channels [32, 33]. We perform the fits like our
previous paper [22, 34], and require the samples to coincide with the combined data at 2σ
level.
• Experimental constraints from the null results in the search for the 750 GeV resonance
through other channels such as s → ZZ,hh at Run I, just like what we did in [13]. The
upper bounds on these channels at 95% C.L. are listed in Table I.
10
FIG. 1. The fit results of the MDM to the 750GeV diphoton data together with the LHC Run I constraints
listed in Table I, which are projected on the σ13TeV −Γ planes for Q =2/3 (left panel) and Q =5/3
γγ tot X X
(right panel) respectively. The regions filled by the colors from gray to deep blue represent the parameter
spaces that can fit the diphoton data within 3σ, 2σ and 1σ level respectively, and by contrast the regions
covered by straw color are excluded by the constraints. The boundaries for the hh, ZZ and WW∗ channels
are also plotted, which correspond to blue lines, red lines and brown lines respectively, and the other
constraintslistedinTableIaretooweaktobedrawnonthepanels. Ineachpanel,thegreenlinerepresents
the best-fit samples. In getting this panel, we have set Γ = 0 and m = 1TeV, and we checked that
new X
m =1.5TeV predicts roughly same results, which reflects that our results are insensitive to m .
X X
For each sample surviving the constraints, we perform a fit to the 750GeV diphoton data
collected at the 8 TeV and the 13 TeV LHC. In doing this, we use the method introduced in [9],
where the data were given by
√
0.63±0.25 fb CMS at s = 8 TeV,
√
0.46±0.85 fb ATLAS at s = 8 TeV,
µexp = σ(pp → γγ) = (26)
i √
5.6±2.4 fb CMS at s = 13 TeV,
√
6.2+2.4 fb ATLAS at s = 13 TeV,
−2.0
and the χ2 function was given by [9, 13]
γγ
4
(cid:88)
χ2 = χ2,
i
i=1
2[µexp−µ +µ ln µi ] for the 13 TeV ATLAS data,
i i i µexp
χ2 = i (27)
i (µeixp−µi)2 for the other three sets of data,
σ2exp
µ
i
with µ denoting the theoretical prediction of the diphoton rate.
i
