Diphoton resonance from a warped extra dimension
Revised: May
Diphoton resonance from a warped extra dimension
Martin Bauer 0 1 2 5 6
Clara Horner 0 1 2 3 6
Matthias Neubert 0 1 2 3 4 6
0 Ithaca , NY 14853 , U.S.A
1 Johannes Gutenberg University , 55099 Mainz , Germany
2 Philosophenweg 16 , 69120 Heidelberg , Germany
3 PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics
4 Department of Physics & LEPP, Cornell University
5 Institut fur Theoretische Physik, Universitat Heidelberg
6 , W W , ZZ, Z , tt and dijet decay rates. We nd that
We argue that extensions of the Standard Model (SM) with a warped extra dimension, which successfully address the hierarchy and particle physics, can provide an elegant explanation of the 750 GeV diphoton excess recently reported by ATLAS and CMS. A gaugesinglet bulk scalar with O(1) couplings to fermions is identi ed as the new resonance S, and the vectorlike KaluzaKlein excitations of the SM quarks and leptons mediate its loopinduced couplings to photons and gluons. The electroweak gauge symmetry almost unambiguously dictates the bulk matter content and hence the hierarchies of the S ! decay mode is strongly suppressed, such that Br(S ! Z )=Br(S !
Phenomenology of Field Theories in Higher Dimensions

the S ! Z
) < 0:1.
The hierarchy problem for the new scalar boson is solved in analogy with the Higgs boson
by localizing it near the infrared brane. The in nite sums over the KaluzaKlein towers
of fermion states converge and can be calculated in closed form with a remarkably simple
result. Reproducing the observed pp ! S !
signal requires KaluzaKlein masses in
the multiTeV range, consistent with bounds from avor physics and electroweak precision
observables.
Useful side products of our analysis, which can be adapted to almost any model for
the diphoton resonance, are the calculation of the gluonfusion production cross section
(pp ! S) at NNLO in QCD, an exact expression for the inclusive S ! gg decay rate at
N3LO, a study of the S ! tth threebody decay and a phenomenological analysis of portal
couplings connecting S with the Higgs eld.
1 Introduction 2 3 4
6
7
RS model with a bulk scalar eld
Diboson signals from warped space
3.1
3.2
4.1
4.2
Phenomenology of the diphoton resonance
General discussion
Sboson phenomenology in RS models
Diboson couplings induced by KK fermion exchange
Coupling to top quarks
5 Impact of Higgs portal couplings
Threebody decay S ! tth
Conclusions
A RG evolution of the Wilson coe cients cgg and ctt
B Inclusive S ! gg decay rate at N3LO in QCD
new particles with a large multiplicity or sizable couplings to S have to enter the S !
loop for both gluonfusion or bbinitiated production processes [5]. Producing the resonance
from other quarkinitiated states results in a tension with 8 TeV data, while photoninduced
production would require nonperturbatively large couplings [
6, 7
]. Supersymmetric UV
completions of the SM, which motivate such additional degrees of freedom, lack a neutral
scalar candidate with appropriate couplings, and the full parameter space of the Minimal
{ 1 {
Supersymmetric SM is excluded as a consequence [
8
]. One thus has to resort to models
with a low supersymmetrybreaking scale, which allow for a sgoldstino explanation [9], or
Rparity violating scenarios, in which the sneutrino can have large enough couplings to
account for the excess [10, 11]. Composite Higgs models predict several composite
resonances that can facilitate a large diphoton branching ratio [12{15]. Neutral composite
scalars, which appear in nonminimal composite Higgs models with larger coset structure,
as well as the dilaton/radion have been considered as possible candidates for S. While
theoretically motivated, the latter implies a small radius of the extra dimension in order
to enhance the couplings to diphotons [16], unless the Higgsradion mixing is tuned to a
particular value [17, 18]. In this regard, the sgoldstino and radion explanations have similar
anomalous magnetic moment of the muon [8, 19{25].
In this paper we argue that RandallSundrum (RS) models featuring a warped extra
dimension [26], with all SM
elds (with the possible exception of the Higgs boson)
propagating in the bulk, can explain the observed excess in a natural way. We introduce a bulk
scalar singlet, whose only renormalizable interactions  with the exception of a possible
Higgs portal  are couplings to bilinears of vectorlike bulk fermions. Remarkably, for
O(1) couplings of this new scalar the diphoton excess is explained for KaluzaKlein (KK)
masses in the multiTeV range without any additional model building. This mass scale is
su ciently large to avoid constraints from electroweak precision tests, avor physics and
Higgs phenomenology. Our results are largely insensitive to the parameters of the RS
model, such as the
vedimensional (5D) masses of the fermions and their Yukawa
couplings to the Higgs eld. To good approximation the loopinduced couplings of the new
resonance S to diboson states just count the number of degrees of freedom propagating
in the loop (times grouptheory factors). We consider three implementations of the RS
model with di erent fermion contents and present detailed predictions for the gluonfusion
production cross section
(pp ! S) and the rates for the decays S !
, W W , ZZ, Z ,
gg, tt and tth, all of which are found within current experimental bounds. We note in
passing that our scenario is particularly well motivated if one assumes the new scalar to
take on a vacuum expectation value, which generates the fermion bulk mass terms, thus
providing a mechanism for the
avorspeci c localization of fermions along the extra
dimension [27, 28]. In this case, the bulk scalar would assume the role of the localizer eld
rst introduced in the context of split fermion models [29]. We shall explore this intriguing
possibility in future work.
We are aware of only a few papers in which the possibility of an extradimensional
origin of the diphoton signal has been explored. The authors of [
30
] considered a model
with a
at extra dimension.
While such a framework does not address the hierarchy
problem of the Higgs boson and the new scalar resonance, this work shares several technical
similarities with our approach. However, the warped background of RS models makes our
{ 2 {
This paper is organized as follows: in section 2 we brie y introduce the basic
concalculations more demanding. In [
31
] it was assumed that the new resonance couples to
the SM only via loops involving heavy vectorlike leptons. In order to obtain the very
large couplings required in this case [
6, 7
], the construction relies on more than one at
extra dimension, and only SM lepton elds are placed in the bulk. This treatment gives up
the attractive possibility of understanding the avor hierarchies from an extradimensional
perspective. The authors found that the overlap integrals in their calculation required a
cuto , which was introduced by hand and motivated based on stringy arguments. In [32{
34] the new resonance was identi ed with the lowest spin2 KK graviton in warped
extradimension models.
rate at N3LO, and the S !
struction of warped extradimensional models and derive expressions for the mass and the
wavefunction of the bulk scalar S. In section 3 we compute the e ective Wilson coe cients
parameterizing the couplings of S to SM gauge bosons and top quarks by integrating out
the heavy fermionic KK modes, considering both the minimal RS model as well as two
di erent extensions with a custodial symmetry. Section 4 deals with the phenomenology of
the resonance S. In the context of an e ective Lagrangian with local interactions of S with
SM
elds, we rst calculate the gluonfusion production cross section
(pp ! S) at
nexttonexttoleading order (NNLO) in QCD perturbation theory, the inclusive S ! gg decay
, W W , ZZ, Z , tt decay rates at leading order. We then
perform
ts to the diphoton signal in the parameter space of the RS models, taking into
account existing constraints from LHC Run 1 resonance searches. In section 5 we study
the impact of possible Higgs portal interactions of S on the various branching fractions,
including the S ! hh signal. The threebody decay mode S ! tth is studied in section 6,
before we conclude in section 7. Some technical details are relegated to two appendices.
2
RS model with a bulk scalar eld
We consider extensions of the SM with a warped extra dimension, described by an S1=Z2
orbifold parameterized by a coordinate
orbifold
xedpoints: the UV brane at
; ], with two branes localized on the
= 0, and the infrared (IR) brane at j j =
.
The curvature k and radius r of the extra dimension are assumed to be of Planck size,
k
1=r
MPl, and the metric reads [26]
ds2 = e 2 ( )
dx dx
r2d 2 =
dx dx
dt2 ;
(2.1)
t
2
2
1
MK2 K
where ( ) = krj j is referred to as the warp factor. The quantity L = ( ) = kr
measures the size of the extra dimension and is chosen so as to explain the hierarchy
34
between the Planck scale and the TeV scale [35].
With the help of the curvature and
the warp factor evaluated on the IR brane,
= e L
10 15, one de nes the KK scale
MKK
k . It sets the mass scale for the lowlying KK excitations of the model and
controls the mass splitting between the KK modes. On the righthand side of (2.1) we
have introduced the dimensionless coordinate t de ned by t = e ( )
2 [ ; 1], which will be
{ 3 {
used throughout this work. It is related to the frequently used conformal coordinate z by
the rescaling z = t=MKK.
The hierarchy problem is solved by localizing the SM Higgs eld on or near the IR
brane, e ectively cutting o
UVdivergent contributions to the Higgs mass at the scale
IR =
MPl
TeV [26]. If gauge bosons and fermions are promoted to 5D bulk
elds,
the avor problem can be addressed in a natural way by means of di erent localizations
of the fermion zero modes along the extra dimension [27, 28]. The large hierarchies in the
spectrum of fermion masses and mixing angles can then be reproduced by small variations
of parameters in the underlying 5D Lagrangian [36]. The minimal RS model with bulk
elds, which has the same gauge symmetry and matter content as the SM, is strongly
constrained by electroweak precision observables [37, 38]. A recent treelevel analysis of
the S and T parameters yields the lower bound [39]
(minimal RS model) :
(2.2)
Since the masses of the lowlying KK excitations are typically several times heavier than
MKK (for example, the lightest KK gluon and photon have a mass of 2:45 MKK [40]), this
puts them out of the reach for discovery at the LHC. Bounds from electroweak precision
observables are considerably relaxed if the electroweak sector respects the custodial
symmetry present in the SM. This implies an enhanced bulk gauge group SU(3)c
and its avorchanging counterparts [44] from receiving too large corrections. As a result,
the bound on the KK scale is lowered to [39]
(custodial RS model) :
(2.3)
Thorough discussions of this model containing many technical details can be found in [45,
46]. In the following, we will consider two di erent versions of the custodial RS model:
one with a symmetric implementation of the quark and lepton sectors (custodial model I),
and one in which the lepton sector is more minimal than the quark sector (custodial
model II) [47].
Besides electroweak precision tests, RS models are constrained by
avor
observables [44, 45, 48{50] and Higgs phenomenology [46, 47, 51{53]. The most severe
avor
constraint comes from K
K mixing [49]. In the minimal model the KK scale is so high
that this bound can be satis ed with a modest 25%
netuning. For the lower values of the
KK mass scale allowed in the custodial model, the avor constraints can either be solved
by means of a 5 10 %
netuning or by enlarging the stronginteraction gauge group in
the bulk [54]. Additional constraints arising from the phenomenology of the Higgs
boson, such as its production cross section and decay rates into
, ZZ and W W , are more
model dependent and can readily be made consistent with present data by adjusting some
model parameters.
{ 4 {
We identify the diphoton resonance with the lightest excitation of a new bulk scalar
eld S(x; ), which is a singlet of the full bulk gauge group. In order to allow for a coupling
of this eld to the scalar density of the vectorlike 5D fermion elds we need to implement
S(x; ) as an odd eld on the S1=Z2 orbifold, such that S(x;
) =
S(x; ). The relevant terms in the action read
Z
2
2
S
2
X
f
sgn( ) f Mf f + S f Gf f
;
where the sum extends over all 5D fermion multiplets f . Even in the minimal RS model
there exists a 4component vectorlike 5D fermion eld for every Weyl fermion of the SM.
The SM fermions correspond to the zero modes of these
elds, which become massive
after electroweak symmetry breaking. Consequently, for each SM fermion there exist two
towers of KK excitations [55]. In extensions of the RS model with a custodial symmetry
additional exotic matter elds are introduced, which have no zero modes but give rise to
additional towers of KK excitations, thereby increasing the number of vectorlike fermions
of the model [41, 42]. The bulk masses Mf and couplings Gf are hermitian matrices in
generation space. By means of eld rede nitions one can arrange that Mf are real, diagonal
matrices. From now on we will always work in this socalled bulk mass basis. The values of
the bulk masses determine the pro les of the SM fermions along the extra dimension, which
generically turn out to be localized near one of the two branes [27, 28]. Note that there is
the intriguing possibility that the bulk masses could be generated dynamically in models
where the scalar eld S acquires a vacuum expectation value w, such that Mf = wGf .
While we leave the detailed construction of such models to future work, we shall assume
that the structure of the couplings Gf follows the structure of Mf .
In (2.4) we have not considered the possibility of a portal coupling
S j j2 connecting
the eld S with the Higgs doublet. We will investigate the phenomenological impact of
such a coupling on the various decay rates of the resonance S in section 4, nding rather
strong constraints. An extradimensional setup, in which the Higgs sector is localized
on the IR brane, where the Z2odd scalar eld S vanishes, might provide a dynamical
explanation for the suppression of the portal interaction. We emphasize, however, that
even with such sequestering a Shh coupling is inevitably induced at oneloop order, since
the 5D bulk fermions can mediate between the IR brane, where the Higgs eld is localized,
and the bulk, where the eld S lives. In our phenomenological analysis in section 4 we will
therefore allow for the presence of a loopsuppressed portal interaction.
The solution of the eld equations satis ed by the KK modes of the scalar eld S
is obtained in complete analogy to the case of a bulk scalar eld studied in [39, 56, 57].
Imposing the KK decomposition (with t = e ( ))
(2.4)
the pro le functions nS(t) are obtained from the equation of motion
e ( )
p
r
X
n
S(x; ) =
Sn(x) nS(t) ;
2 + t2x2n
nS(t)
t
= 0 ;
{ 5 {
(2.5)
(2.6)
where Nn is a normalization constant. In order to obtain a relatively light mass mS
m1S
750 GeV for the lightest scalar resonance, we impose the mixed boundary condition
nS(1) =
nS0(1) on the IR brane, which can be engineered by adding branelocalized
terms to the action. In the limits
Dirichlet boundary condition
! 0 and
! 1 one recovers the special cases of the
nS(1) = 0 and the Neumann boundary condition
nS0(1) = 0,
respectively. In the general case, we obtain
and due to the smallness of rn /
approximation. It follows that the mass of the lightest resonance is given by
2 the righthand side can be set to zero to excellent
(2.7)
1(xn) ;
o
(2.9)
(2.10)
(2.11)
= O(1)
(2.12)
x
2
1
750 GeV can be achieved with a moderate tuning of parameters. For
example, with MKK = 2 TeV we need
0:69 for
= 0:5,
0:51 for
= 1,
0:17
for
= 5 and
0:09 for
= 10. The properly normalized pro le function of the lightest
resonance is given by (dropping irrelevant terms vanishing for
1S(t) =
r L(1 + ) t1+
1
x
2
1
4
t
2
1 +
1
2 +
! 0)
+ O(x14) :
where xn = mnS=MKK and 2 = 4 + 2=k2. To obtain canonically normalized kinetic terms
for the KK modes, we must impose the normalization condition
L
t
2 Z 1 dt Sm(t) nS(t) = mn :
Requiring the Dirichlet boundary condition
general solution
nS( ) = 0 on the UV brane, one nds the
nS(t) = Nn t [J (xnt)
rn J
(xnt)] ;
rn =
J ( xn)
J
( xn)
(1
(1 + )
)
xn 2
2
;
(2.8)
controls the localization of the bulk scalar, and in analogy to the case of
a bulk Higgs boson we will assume that
> 0 (i.e., 2 >
4k2) [58]. For values
the scalar has a wide pro le along the extra dimension, while for
1 it is localized near
the IR brane; in fact, we have
1S(t) !=1
r L(1 + )
1
2 +
(t
1) :
While there is no particular reason why the bulk scalar should be localized near the IR
brane, we will nd that our results take a particularly simple form in this limit.
{ 6 {
In the models we consider, the masses of the KK excitations of gauge bosons and fermions
are bound by constraints from electroweak precision and
avor observables to lie in the
multiTeV range. The 750 GeV resonance is considerably lighter, and it is thus justi ed to
integrate out the tower of fermion KK modes in computing the decays of S to diboson or
fermionic nal states. Below the KK mass scale we de ne the e ective Lagrangian
Le = cgg 4
s S G
a G ;a + cW W 4 s2 S W a W
w
;a + cBB 4 c2 S B
w
B
S QLY^u ~ uR + S QLY^d
dR + S LLY^e
eR + h.c. ;
(3.1)
is the scalar Higgs doublet, and sw = sin w and cw = cos w are functions
of the weak mixing angle. Since the mass of the new resonance is much larger than the
electroweak scale, it is appropriate to write the e ective Lagrangian in the electroweak
symmetric phase. Upon electroweak symmetry breaking the second and third operator in
the rst line generate the couplings of S to pairs of electroweak gauge bosons. In particular,
the resulting diphoton coupling is
Le 3 c
4
S F
F
;
with c
= cW W + cBB :
The terms in the second line in (3.1) describe the couplings of S to fermion pairs (with or
without a Higgs boson). In our model these couplings have a hierarchical structure, and
the dominant e ect by far is the coupling to the top quark. Rewriting Re[(Y^u)33] = ctt yt
(after transformation to the mass basis), where yt =
2mt=v is the topquark Yukawa
p
coupling, we can express the corresponding term as
Le 3
ctt mt 1 +
S tt + : : : :
h
v
The Wilson coe cients in the e ective Lagrangian are suppressed by the mass scale of the
heavy KK particles, cii / 1=MKK. In the remainder of this section we will calculate these
coe cients at the matching scale
KK = few
MKK corresponding to the masses of the
lowlying KK modes, which give the dominant contributions.
It is well known that the twogluon operator has a nontrivial QCD evolution [59,
60] and mixes with the operator in (3.3) under renormalization [61]. These e ects are
discussed in detail in appendix A. When the strong coupling
s and the Yukawa coupling
yt are factored out from the de nitions of cgg and ctt as we have done above, evolution
e ects from the high matching scale
KK to the scale
= mS only arise at NLO in
renormalizationgroup (RG) improved perturbation theory. At this order they give rise to
the simple relations
cgg( ) = 1 +
1
matching scale, it is inevitably induced through RG evolution; however, this is a very small
e ect. For
and cgg( )
KK = 5 TeV and
= 750 GeV we nd ctt( )
ctt( KK) + 0:0028 cgg( KK)
1:0045 cgg( KK). Higherorder QCD corrections to the Wilson coe cients at
the high matching scale are likely to have a more important impact. For instance, they
enhance the topquark contribution to the Higgsboson production cross section in the SM
by about 20% [62, 63]. To be conservative we will not include such enhancement factors in
our analysis.
3.1
Diboson couplings induced by KK fermion exchange
Since the scalar eld S is a gauge singlet, its couplings to gauge bosons are induced by
fermion loop diagrams, such as those shown in
gure 1. The relevant couplings in (2.4)
are parameterized by the matrices Gf , while the pro les of the fermions along the extra
dimension depend on the (diagonal) bulk mass matrices Mf . It is conventional to de ne
dimensionless bulk mass parameters by cf =
Mf =k, where the plus (minus) sign holds
for fermion elds whose lefthanded (righthanded) components have even pro le functions
under the Z2 symmetry. In the minimal RS model the SU(2)L fermion doublets have
even lefthanded components, while the SU(2)L fermion singlets have even righthanded
components. In extensions of the RS model elds transforming as SU(2)L triplets also have
even righthanded components. Using the same sign conventions, we de ne dimensionless
couplings gf of S to fermions via
gf =
pk(1 + )
2 +
Gf :
(3.5)
This de nition is analogous to the de nition of the dimensionless Yukawa couplings in
RS models with a bulk Higgs
eld studied in [39, 52, 64]. The
dependent terms
ensure that the dimensionless couplings remain wellbehaved in the limit
! 1 of an IR
branelocalized scalar eld. These matrices are hermitian but, in general, not diagonal in
generation space. Since with the exception of the top quark all SM fermions have masses
much below the electroweak scale, the values of most of the bulk mass parameters cf
cluster near or below the critical value
1=2, below which the zeromode fermion pro le is
localized near the UV brane. For example, a typical set of bulk mass parameters adopted
in [55] ranges from
0:74 for cu1 to
0:47 for cQ3 . The only exception is the parameter
c
t
cu3
+0:34 of the righthanded top quark, which is positive so as to realize a
localization near the IR brane. In our phenomenological analysis below we will for simplicity
assume that the diagonal elements of the matrices gf all have the same sign and magnitude,
with the possible exception of gt
(gu)33.
In close analogy with the case of the induced hgg and h
couplings of the Higgs
boson in models where all SM
eld propagate in the bulk, we nd that the sums over the
in nite towers of KK fermion states in
gure 1 converge and can be calculated in closed
form using 5D fermion propagators [65{67]. In the unbroken phase of the electroweak
gauge symmetry (i.e. for v = 0), there is no mixing between fermion states belonging to
{ 8 {
HJEP07(216)94
S
fn
fn
MK2 K ! m2S
S
di erent multiplets of the gauge group and the fermion propagators are diagonal matrices
in generation space. The Wilson coe cients are then given by sums over the contributions
from the di erent fermion multiplets. Mixing e ects induced by electroweak symmetry
breaking yield corrections of order (mf =mS)2 relative to the leading terms we will compute.
Even for the top quark these corrections are at most a few percent and can safely be
neglected.
The fermion representations of the custodial RS models have been discussed in detail
in [43, 45{47]. We begin with a brief description of the quark sector. As a consequence of the
discrete PLR symmetry, the lefthanded bottom quark needs to be embedded in an SU(2)L
SU(2)R bidoublet with isospin quantum numbers TL3 =
TR3 =
1=2. This assignment
xes the quantum numbers of the remaining quark elds uniquely. In particular, the
righthanded downtype quarks have to be embedded in an SU(2)R triplet in order to obtain a
U(1)X invariant Yukawa coupling. We choose the same SU(2)L
SU(2)R quantum numbers
for all three quark generations, which is necessary to consistently incorporate quark mixing
in the anarchic approach to avor in warped extra dimensions. Altogether, there are fteen
di erent quark states in the up sector and nine in the down sector (for three generations).
The boundary conditions give rise to three light modes in each sector, which are identi ed
with the SM quarks. These are accompanied by KK towers consisting of groups of fteen
and nine modes of similar masses in the up and down sectors, respectively. In addition,
there is a KK tower of exotic fermion states with electric charge Q
= 5=3, which exhibits
nine excitations in each KK level. In order to compute the Wilson coe cients cgg, cW W
and cBB in (3.1) it is most convenient to decompose these multiplets into multiplets under
U(1)Y . There are two SU(2)L doublets and one triplet
cQ :
0 u(+) 1
L
L
L
0 ( ) A ;
7
6
c 1 :
R
BB U 0 ( ) CCC ;
B
R
DR0( ) A2
as well as four singlets
cu :
uc (+)
R
2
3
;
cd :
DR(+)
1
3
;
U
( )
R
2
3
;
We only show the chiral components with even Z2 parity; the other chiral components are
odd under the Z2 symmetry. The subscript denotes the hypercharge of each multiplet.
{ 9 {
3
( )
R
5
3
:
(3.6)
(3.7)
The superscripts on the elds specify the type of boundary conditions they obey on the
UV brane. Fields with superscript (+) obey the usual mixed boundary conditions allowing
for a light zero mode, meaning that we impose a Dirichlet boundary condition on the
pro le functions of the corresponding Z2odd elds. These zero modes correspond to the
SM quarks. Fields with superscripts ( ) correspond to heavy, exotic fermions with no
counterparts in the SM. For these states, the Dirichlet boundary condition is imposed
on the Z2even
elds so as to avoid the presence of a zero mode. The UV boundary
conditions for the elds of opposite Z2 parity are of mixed type and follow from the eld
equations. Above we have indicated the bulk mass parameters associated with the various
multiplets.1 The three parameters contained in the matrix c 1 can be related to the other
ones by extending the PLR symmetry to the part of the quark sector that mixes with the
lefthanded downtype zero modes [46]. It then follows that c 1 = cd. Whether or not this
equation holds turns out to be irrelevant to our discussion.
In the custodial model I the lepton sector is constructed in analogy with the quark
sector [45]. It consists of two SU(2)L doublets and one triplet
(3.8)
(3.9)
B
BB N R0( ) CC ;
C
ER0( ) A
0
as well as four singlets
c :
c (+)
R
0
;
ce :
ER(+)
1
;
NR
( )
0
;
( )
R
1
:
Again we only show the chiral components with even Z2 parity. There are fteen di erent
lepton states in the neutrino sector and nine in the chargedlepton sector. The boundary
conditions give rise to three light modes in each sector, which are identi ed with the SM
neutrinos and charged leptons. These are accompanied by KK towers consisting of groups
of fteen and nine modes in the two sectors, respectively. In addition, there is a KK tower
of exotic lepton states with electric charge Q
= +1, which exhibits nine excitations in each
KK level. The three parameters contained in the matrix c 3 can be related to the other
ones by requiring an extended PLR symmetry, in which case c 3 = ce. In the custodial
model II the lepton sector is more minimal [47]. It consists of one SU(2)L doublet and
two singlets
cL :
L
(+) !
e(+)
L
1
2
;
ce :
ec (+)
R
1
;
N R0( )
0
:
(3.10)
The choice of the boundary conditions is such that the zero modes correspond to the light
leptons of the SM, without a righthanded neutrino. Note that the minimal RS model is
1Fields belonging to the same SU(2)R multiplet have equal bulk mass parameters. The two doublets
associated with cQ form a bidoublet under SU(2)L
SU(2)R, while the three singlets associated with cd
form a triplet under SU(2)R.
obtained by simply omitting all multiplets containing elds carrying a superscript \( )"
from the above list.
The calculation of the oneloop diagrams in
gure 1 proceeds in complete analogy
with the corresponding calculation for a bulk Higgs eld performed in [39]. One evaluates
the amplitude in terms of an integral over 5D propagator functions, employs the KK
representation of these functions in terms of in nite sums, simpli es the resulting expression
and recasts it in the form of an integral over a single 5D fermion propagator. Adapting
these steps to the present case, we obtain the expressions (for v = 0)
cgg =
which only di er in grouptheory factors. The sum in the rst line runs over quark states
only. Here df is the dimension of the SU(2)L multiplet, Tf is the Dynkin index of SU(2)
(Tf = 1=2 for doublets, Tf = 2 for triplets, and Tf = 0 for singlets), Yf is the hypercharge of
the multiplet, and the color factor Ncf equals 3 for quarks and 1 for leptons. The variables
x and y (with y
1
y) are Feynman parameters. The quantity Tf ( p2) denotes an
integral over the product of mixedchirality components of the 5D fermion propagator
with momentum p2 and 5D coordinates t = t0 with the pro le of the scalar resonance S.
Explicitly, we nd in the Euclidean region2 p
2E =
p2 > 0
Tf (p2E) =
r
2 +
L p
1 +
Z 1
dt 1S(t) Tr ( gf )
3 matrices in generation space. The KK representation of the
propagator functions reads
fAB(t; t0; p2E) =
X
n
mn
p2E + m2n F A(n)(t) FB
(n)y(t0) ;
where the normalization of the fermion pro les F A(n)(t) with A = L; R is such that [69]
Z 1
dt F A(m)y(t) F A(n)(t) = mn :
(3.13)
(3.14)
The zero modes are massless in the limit where v = 0 and hence give no contribution to
the result at leading order.
Note that the sum over KK modes in (3.13) is logarithmically divergent by naive power
counting, since the masses of the KK modes have approximately equal spacing.
Nevertheless an explicit calculation of the in nite sum leads to a nite and wellbehaved answer,
2This relation holds under the assumptions that Tf (p2E) vanishes for pE !
1, and that pE dTf =dpE
vanishes for pE = 0 and pE ! 1. We have checked that these conditions are satis ed in our models.
hinting at a nontrivial interplay of the pro le functions for the various KK fermions. The
calculation of the propagator functions has been discussed in detail in the literature [39, 66{
68]. It requires solving a secondorder di erential equation subject to appropriate boundary
conditions. We obtain
fRL(t; t; p2E) =
2
1
2MKK
d( )(cf ; pE; t) ;
(3.15)
where the overall sign is the same as that in (3.5), and the superscript \( )" refers to the
boundary conditions (normal or twisted) obeyed by the fermion multiplet f . The functions
d( ) are diagonal matrices, whose entries depend on the bulk mass parameters. Explicitly
they are given by (omitting the matrix notation for simplicity)
D1(cf ; a; t; p^E) = I cf 12 (ap^E) Icf 12 (p^Et) Icf + 12 (ap^E) I cf + 12 (p^Et) ;
D2(cf ; a; t; p^E) = I cf 12 (ap^E) Icf + 12 (p^Et) Icf + 12 (ap^E) I cf 12 (p^Et) ;
with p^E
pE=MKK, are given in terms of modi ed Bessel functions. In our case these
functions are evaluated (by analytic continuation to the timelike region) at momenta of
order p2
m2S
MK2 K, so that it is possible to expand these complicated expressions in a
power series. This yields
d( )(cf ; pE; t) = k0( )(cf ; t) + p^2E k2( )(cf ; t) + O(p^4E) ;
k0(+)(cf ; t) = 1 +
k2(+)(cf ; t) =
1 + 2cf
2F 2(cf ) t1+2cf
2t2 1 t 1 2cf
1 ;
k0( )(cf ; t) = 1 ;
+ 2(1
2) F 4(cf ) (1
t1+2cf
1
4cf2 )(3 + 2cf )
1
1
4cf2
4cf2
is the wellknown zeromode pro le [27, 28], which is exponentially small for all fermions
with the exception of the righthanded top quark. We note the exact boundary values
2(1 + cf ) t3+2cf
(1+2cf )2(3+2cf )
1 + 2(t1+2cf
(1 2cf )(3+2cf )
1
t
1 2cf :
F 2(cf ) =
1
1 + 2cf
1+2cf
where (dropping irrelevant terms in )
(+)(cf ; ) =
Analogous expressions hold for the Wilson coe cients cW W and cBB, as is evident
from (3.11). In
gure 2 we show the exact numerical results for
( )(cf ; ) as functions
of the bulk mass parameter cf for various values of . Even for MKK as low as 2 TeV
we nd that the corrections of O(m2S=MK2 K) are very small and can safely be neglected.
Moreover, for all fermions other than the righthanded top quark it is an excellent
approximation to neglect the exponentially small quantity F 2(cf ), while for the righthanded top
quark we can replace F 2(ct)
1 + 2ct. Note that in the limit
a scalar resonance localized on the IR brane, we obtain the exact result
! 1, corresponding to
( )(cf ; ) ! 0,
and hence the Wilson coe cients in this limit are simply given in terms of sums over the
diagonal elements of the matrices gf , meaning that they essentially count the number of
5D fermionic degrees of freedom. For simplicity, we will adopt this approximation in
displaying the following results. In our numerical work we will use the correct expressions,
which are obtained by replacing gt
(gu)33 ! 3+ +22cctt gt.
1+
We now collect our results for the Wilson coe cients in the three versions of the RS
model, adopting these approximations. For the custodial model I we nd
Tr 2gQ +
1
3
2 gd +
3
2
g 1
Tr 3gQ + 6g 1 + gL + 2g 3
16ge
3MKK
gt
6MKK
;
12ge ;
MKK
Tr
25
3
gQ +
3 gu + 10gd + 4g 1 + gL + 2ge
236ge
9MKK
4gt ;
9MKK
cgg =
For the custodial model II we nd instead
1, from which it follows that kn( )(cf ; 1) =
! 0 where possible)
m2S
4MK2 K
t
2
Using these results, it follows that (taking
cgg =
dt (2 + ) t1+
1
1
1
3MKK f=q 2
7m2S
120MK2 K k2( )(cf ; t) + : : :
X df Tr gf 1 +
( )(cf ; ) ;
4
4
Tr 3gQ + 6g 1 +
Tr
25
3
1
2 gL
19ge ;
while cgg is unchanged. In the minimal RS model, the corresponding expressions read
cgg =
1
2 gd
Tr gQ +
Tr
Tr
3
2
1
6
gQ +
gQ +
1
2
4
gL
3 gu +
2ge ;
MKK
1
3 gd +
1
2
gL + ge
11ge
6MKK
gt
6MKK
;
26ge
9MKK
4gt
9MKK
:
(3.25)
In the last step in each line we have assumed, for simplicity, that all diagonal entries of
the matrices gf are equal to a universal value ge . While there is no particular reason why
this should be true, the near equality of all cf parameters other than ct suggests that such
an approximation might be reasonable. Note that any reference to the parameter
has
disappeared, except for the small correction term multiplying gt.
In our phenomenological discussion of diboson decays in section 4 the ratio of the
will play an important role.
Neglecting the small
2:19 + 0:04 gt=ge in the custodial model I,
Wilson coe cients cBB and cW W
correction terms, we
nd cBB=cW W
in the minimal RS model.
3.2
Coupling to top quarks
cBB=cW W
2:60+0:05 gt=ge in the custodial model II, and cBB=cW W
1:44+0:22 gt=ge
The resonance S has treelevel couplings to the SM fermions, which are induced after
electroweak symmetry breaking. These interactions are very similar to the couplings of a
bulk Higgs to fermions studied in [39]. The largest e ects arise in the upquark sector. We
will discuss them for the case of the minimal RS model, but the
nal result is the same
in the custodial models. Using the zeromode pro le functions derived in [55], we nd
that the corresponding terms in the e ective Lagrangian read (neglecting terms of order
m2h=MK2 K)
Lferm =
X S(x) u(Lm)(x) u(Rn)(x) (2 + )
m;n
xn a^(mU)y F (cQ) tcQ gQ F (cQ)
Z 1
0
t1+cQ
+ xm a^(mu)y F (cu)
t1+cu
1+2cu t cu
1 + 2cu
dt t1+
1 + 2cQ
1+2cQ t cQ a^(U)
n
gu F (cu) tcu a^(u) + h.c. ;
n
(3.26)
HJEP07(216)94
where xn = mn=MKK, and n = 1; 2; 3 label the three lowestlying states u, c and t. The
integrand involves a product of a Z2even fermion pro le with a Z2odd one, and for
the SM fermions the latter one arises from the mixing of the zero modes with their KK
excitations induced by electroweak symmetry breaking. As a consequence, the overlap
integrals scale with the masses of the fermions involved. The 3dimensional vectors ^a(nU)
and a^(nu) describe the mixings in avor space and are normalized to unity. Their entries are
strongly hierarchical, with the largest entry at position n. The most important interaction
involves the coupling of S to a pair of top quarks. For the Wilson coe cient ctt in (3.3),
we nd to a good approximation
ctt
1
1
MKK
MKK
gQ
1
(gQ)33 +
F 2(cQ)
3 +
+ 2cQ
2 +
3 +
+ 2ct
+ gu 1
gt :
F 2(cu)
3 +
+ 2cu
33
(3.27)
4
Phenomenology of the diphoton resonance
In this section we express the production cross section and the rates for the decays of
the resonance S into SM particles in terms of the Wilson coe cients in the e ective
Lagrangians (3.1) and (3.3). These results are general and can be applied to any model in
which the couplings of S to SM particles are induced by the exchange of some heavy new
particles. We will then apply these general results to the case of the RS models studied in
At Born level, the cross section for the production of the resonance S in gluon fusion at
the previous section.
4.1
General discussion
the LHC is given by
where
Z 1 dx
y x
(pp ! S) =
64 s
2( ) m2S cg2g( ) ffgg(m2S=s; ) ;
s
ffgg(y; ) =
fg=p(x; ) fg=p(y=x; )
(4.1)
(4.2)
is the gluongluon luminosity function. The factorization and renormalization scales should
be chosen of order
mS. It is well known from the analogous Higgs production cross
p
s
8 TeV
13 TeV
MSTW2008 [73] NNPDF30 [74] PDF4LHC15 [75]
(cgg=TeV)2, for di erent sets of parton distribution functions. The quoted errors are estimated
from scale variations.
section that higherorder QCD corrections have an enormous impact on the cross section.
We have calculated these corrections up to NNLO and including resummation e ects using
an adaption of the public code CuTe [70] developed in [71, 72]. Table 1 shows our results for
the ratio (pp ! S)=cg2g( ) for the default scale choice
= mS and di erent sets of parton
distribution functions (PDFs). Taking the MSTW2008 PDFs as a reference, we obtain
N8TNeLVO(pp ! S) = (44:9 +12::64 +12::87) fb
N13NTLeOV(pp ! S) = (203 +160+56) fb
cgg(mS) 2
TeV
cgg(mS) 2
TeV
;
;
where the errors refer to scale variations and the variations of the PDFs. The program
CuTe also predicts the pT distribution of the produced S bosons, and we
nd that this
distribution peaks around 22 GeV. The higherorder corrections enhance the cross section
by more than a factor 2 compared with the Born cross section in (4.1).
The Wilson coe cients in (3.1) also contribute to possible decays of the new resonance
into the electroweak diboson
nal states
, W W , ZZ and Z , and into hadronic nal
states such as gg and tt. The partial decay rates for the former channels are
(4.3)
(4.4)
(S !
) =
624m33S (cW W + cBB)2 ;
where xW;Z = m2W;Z =m2S. While the Wilson coe cients are evaluated at the scale
the gauge couplings and Weinberg angle are evaluated at the scale appropriate for the
nalstate bosons. We use (mZ ) = 1=127:94 for Z and W bosons,
= 1=137:04 for the
photon, and s2
W = 0:2313 for the weak mixing angle. Apart from known quantities, the
two Wilson coe cients cW W and cBB determine these four rates entirely. It follows that
any ratio of two rates is a function of the ratio cBB=cW W , which in turn is characteristic
for the model under investigation. This is illustrated in
gure 3. Note that the S ! Z
decay rate in particular strongly depends on the value of cBB=cW W . In the RS models
considered in the previous section this decay mode turns out to be strongly suppressed.
the S !
rate as functions of cBB=cW W . Assuming
2:11 < cBB=cW W < 2:23 in custodial model I, 2:50 < cBB=cW W < 2:65 in custodial model II, and
1:00 < cBB=cW W < 1:66 in the minimal RS model.
The partial rates for decays into hadronic nal states are
where all running quantities should be evaluated at
mS. In the second case mt( ) is the
running topquark mass (we use mt(mS) = 146:8 GeV), whereas the mass ratio xt = mt2=m2S
entering the phasespace factor involves the topquark pole mass mt = 173:34 GeV. In
many scenarios the dijet decay mode S ! gg is the dominant decay channel and hence
enters in the calculation of the branching fractions for all other decay modes. It is therefore
important to calculate this partial rate as accurately as possible. Using existing calculations
of the Higgsboson decay rate (h ! gg) up to O( s5) in the heavy topquark limit [78, 79]
it is possible to derive an exact expression for the S ! gg decay rate to the same accuracy.
This is discussed in detail in appendix B. We nd that the impact of radiative corrections
is signi cantly smaller than in the Higgs case, and that the perturbative series at
exhibits very good convergence. We obtain KgNg3LO(mS)
1:348.
= mS
4.2
Sboson phenomenology in RS models
We are now ready to explore the phenomenological consequences of our calculations. The
challenge is to reproduce the observed diphoton rate in (1.1), while at the same time
respecting existing bounds on dijet, diboson and tt resonance searches3 from Run 1 of the
3Note that the reported bound for the S ! tt channel might in fact be considerably weaker due to
interference e ects not considered in the experimental analyses [80]. The potential impact of these e ects
jj
< 2:5 pb [85]
diboson and tt resonance searches performed in Run 1 of the LHC ( s = 8 TeV).
p
HJEP07(216)94
reproduced at 1 (light blue) and 2 (dark blue). The black dashed line corresponds to the central
value shown in (1.1). The two upper panels refer to the custodial models I and II, while the lower
left panel refers to the minimal RS model. Regions excluded by bounds from resonance searches in
Run 1 data (at 95% CL) are shaded gray with boundaries drawn in red (dijets), purple (tt), blue
(W W ), orange (ZZ) and green (Z ). We use
= 1 and ct = 0:4. The lower right panel shows the
variation of the central t values with the localization parameter
of the scalar pro le, for
= 1
ing bounds at p
LHC, which are collected in table 2. Assuming that the new resonance S is predominantly
produced via gluon fusion, as is the case in the RS models we study, the
corresponds = 13 TeV are obtained by multiplying these numbers with the boost
factor 4.52 corresponding to the ratio of the production cross sections in (4.3).
In gure 4, we present plots in the MKK=ge
gt=ge
plane showing the 1 and 2
t regions to the diphoton excess in light and dark blue. The central t values are shown
by the dashed black line. Regions excluded by bounds from resonance searches in data
collected during the 8 TeV run of the LHC are shaded gray with a boundary drawn in red
(dijet searches), purple (tt searches), blue (W W searches), orange (ZZ searches) and green
(Z
searches). Throughout we use
= 1 and ct = 0:4 for the parameters entering the
contribution from the SU(2)Lsinglet top quark. The lower right plot shows the variation
of the central t result for di erent values of the localization parameter , namely
= 1 (red). It is apparent that there is
only a minor dependence on this parameter. Changing ct does not substantially alter the
t either.
We observe that in the two versions of the RS model with a custodial symmetry the
diphoton signal can be reproduced over a wide range of parameters without any ne tuning
and without violating any of the bounds from other searches. Depending on the choice of
gt=ge one obtains values for MKK=ge in the range between 2 and 8 TeV. If the KK scale
MKK is close to the lower bound (2.3) allowed by electroweak precision tests, this requires
couplings ge in the range 0.25 to 1, which are well inside the perturbative region. In this
scenario some of the lowlying KK excitations could have masses around 4 TeV, in which
case they might be discovered in Run 2 of the LHC. If the KK mass scale is signi cantly
higher a direct discovery of KK excitations will not be possible at the LHC. Nevertheless,
even for MKK
5 TeV (implying KK resonance masses near 10 TeV) the diphoton signal
can be explained with a modest coupling ge
1. In the minimal RS model the parameter
space in which the diphoton signal can be explained is more constrained. We nd values in
the range MKK=ge
0:4 1, which for a KK scale as high as the bound (2.2) enforced by
electroweak precision tests requires large couplings ge
5 12, close to the perturbativity
limit. One also needs to require that the ratio gt=ge is negative so as to avoid the strong
constraint from tt resonance searches (see, however, footnote 3).
We nd it useful to de ne a benchmark point for each model and study the individual
branching fractions for the various S decay modes for these points. Speci cally, we choose
the points indicated by the orange stars in gure 4, for which (with
= 1 and ct = 0:4)
Minimal model :
Custodial model I :
Custodial model II :
MKK=ge = 0:7 TeV;
MKK=ge = 4:0 TeV;
MKK=ge = 3:0 TeV;
gt=ge =
gt=ge =
gt=ge = 0 :
1:5 ;
0:5 ;
(4.6)
In the upper portion of table 3 we collect the branching ratios into the various nal states
for these benchmark models. Note that the S ! tt decay rate is only calculated at lowest
order in QCD and hence a icted with some uncertainty. The S ! tt branching ratio
has rst been pointed out in [81] and has recently been reemphasized in [82, 83].
Minimal
Custodial I
Custodial II
Minimal
Custodial I
Custodial II Minimal
gg
and for the benchmark parameter points de ned in (4.6). In the center and lower portions of the
table we show the branching ratios in the presence of a small portal coupling 1 = 0:02 and 0.04,
respectively, see section 5. The small contributions to the S ! hh and S ! tth branching ratios
resulting from the portal coupling 2 in (5.2) and (6.2) have been set to 0.
is rather sensitive to the choice of gt=ge , while the remaining branching fractions only
mildly depend on this parameter. The threebody decay mode S ! tth will be discussed in
section 6. In the last column we show the total decay width of S, which is very small in our
models. Given the existing Run 1 dijet bound shown in table 2, it is impossible to obtain a
total width exceeding a few GeV in any model in which the decay S ! gg has a signi cant
branching ratio. This is below the experimental resolution of approximately 10 GeV on
m . In our framework we can therefore not accommodate the best t value tot
45 GeV
reported by ATLAS [1]. Rather, the numbers shown in the table correspond to values
tot=mS
(1:1 4:3) 10 4. We recall, however, that the large width tot
0:06 mS is only
slightly preferred by the ATLAS analysis, leading to an improvement of the t by 0:3 over
a narrowwidth scenario. An independent analysis in [84] concludes that the largewidth
scenario is disfavored by a combination of the ATLAS and CMS analyses of the 13 TeV
data, and only slightly preferred taking into account the 8 TeV data, because it is easier
to absorb the signal of a broad resonance in the background model (the local signi cance
changes at most by 0:5 between these options).4
We observe that there are rather striking di erences between the three RS models
considered here, even though any of the three benchmark points reproduces the diphoton
signal and is consistent with all other bounds. In particular, the S !
W W , ZZ and
tt branching ratios vary signi cantly from one model to another, indicating that future
measurements of these modes will provide very interesting clues about the underlying
model. Note also that in all cases we nd that the S ! Z
branching fraction is very
4After the submission of our paper, CMS has reported a combined analysis of the 8 TeV and 13 TeV
data using three templates with
tot=mS = 1:4 10 4, 1:2 10 2 and 5:6 10 2 [CMS Collaboration,
CMSPASEXO16018]. The best t is obtained for the narrowwidth assumption with tot=mS = 1:4 10 4. A
value of this order is indeed predicted in our models.
S
S
h
h
t
t¯
S
S
h
S
h
W +, Z
W −, Z
S
h
h
h
small, so that it will be challenging to observe this mode in our scenarios. On the other
hand, not seeing the S ! Z signal would be as important a nding as seeing it.
5
Impact of Higgs portal couplings
The most general renormalizable Lagrangian includes besides the operators in (3.1)
potential Higgs portal interactions (see e.g. [
5, 8, 91, 92
])
Le =
1 mS S j j
2
2
2 S2 j j2 3
2
1 mS S (v + h)2
4
2 S2 (v + h)2 :
(5.1)
In RS models the couplings 1 and 2 can be suppressed at tree level by localizing the Higgs
sector on or near the IR brane, where the Z2odd bulk eld for the resonance S vanishes.
However, starting at oneloop order the portal couplings will be induced through diagrams
analogous to that shown in
gure 1, but with the external gauge elds replaced by Higgs
bosons. Note that this diagram exists even if only the \rightchirality" couplings of the
Higgs bosons are included. Also, below the electroweak scale the e ective Lagrangian (3.1)
gives rise to a contribution to the portal coupling 1 proportional to ctt from topquark
loop graphs. RS models thus provide a rationale for why the portal interactions should be
suppressed (by small overlap integrals or a loop factor), but it would be unjusti ed to omit
them altogether.
1
After electroweak symmetry breaking, the rst portal interaction gives rise to three
interesting (and potentially dangerous) e ects. First, the presence of a tadpole for the eld
S requires that we de ne the physical eld by the shift S ! S
this shift in the Lagrangian (3.1) generates corrections to the SM Yukawa couplings and
wavefunction corrections to the gauge
elds. Otherwise these corrections do not have
observable e ects. Second, there is a treelevel decay S ! hh, generated by the upper two
diagrams shown in gure 5, whose decay rate (at leading order in the portal coupling 1)
( 1v2)=(2mS). Performing
is given by
(S ! hh) =
32
mS 21 1 +
3m2h
m2S
Third, there is a mass mixing between the scalar resonance S and the Higgs boson. At
leading order in the portal coupling
1 this gives rise to treelevel contributions to the
(S ! W W ) =
0:5 for the custodial model I and gt=ge = 0 for the custodial model II as in (4.6),
= 1 and ct = 0:4. The constraints from Run 1 resonance searches refer to W W (blue),
ZZ (orange), Z (green), tt (purple), dijets (red) and hh (brown). See text for further explanations.
S ! tt, W W , ZZ decay amplitudes induced by the lower two diagrams in gure 5. The
corrected expression for the S ! tt decay width is then given by
xh
2
:
The corrected expression for the S ! W W decay rate reads
mS p
16
1
+
1
1
xh
4xW
2
m2S c2W W
2
s
2
W
2
(1
(1 4xW +12x2W )
6mS cW W
4xW + 6x2W )
s
2W 1
1
xh
xW (1 2xW ) ;
(5.3)
(5.4)
and similarly for S ! ZZ. The mixing between the elds S and h also has an impact on
the properties of the Higgs boson. The physical Higgs boson is given by the combination
sin S), where sin 2 = 2 1mSv=(m2S
m2h)
0:67 1, with mS and mh refering
to the physical masses after the eld rede nitions. Existing measurements of the Higgs
branching fractions constrain cos to be larger than 0.86 at 95% CL [93, 94], which implies
the rather weak constraint j 1j < 1:3 in our model.
The S ! hh decay channel and the admixture of a treelevel coupling to W W and
ZZ induced by the mixing with the Higgs boson can have a signi cant impact on our
phenomenological analysis. In gure 6, we present the 1 and 2
t regions to the diphoton
excess in the custodial RS models in the presence of the portal coupling
1
. These plots
are analogous to those shown in gure 4, except that we have xed the values of gt=ge to
those of the benchmark points in (4.6). The meaning of the colors of the various curves
S
h
h
t
h
t¯
t
t¯
S
S
h
t
h
t¯
t
h
t¯
h
h
S
t
t¯
t
t¯
xh
1
xh
:
is the same as before. In all cases the LHC Run 1 bound on the S ! ZZ rate provides
the strongest constraint, excluding portal couplings j 1j & 0:06 (0.07) in the custodial
RS model I (II). The impact of a small portal coupling
1 = 0:02 or 0.04 on the various
branching ratios is shown in the central and lower portions of table 3. Even for such a
small coupling the S ! W W , ZZ branching ratios can be signi cantly enhanced, and a
sizable S ! hh branching fraction can open up.
6
Threebody decay S ! tth
The e ective Lagrangian (3.3) contains a treelevel coupling of the new resonance S to
a tt pair and a Higgs boson. It is interesting to ask if this coupling might explain the
enhanced tth production rates reported by ATLAS and CMS [95, 96]. At tree level, the
threebody decay S ! tth is mediated by the diagrams shown in
gure 7. Note that
both portal couplings introduced in (5.1) contribute here. Introducing the dimensionless
variables z = mt2t=m2S and w = mt2h=m2S, we obtain the Dalitz distribution
d2 (S ! tth)
where
B =
2xt
1
1
xh
1
1
mS ctt z
w
z + xt + xh
w
1 +
1
3xh
+
xh
1
2xt
xt
v
2
m2S 1
1 +
1
w
z + xt + xh
w
xt
mS ctt 1
(1
pxh)2, where
The phase space for the variables w and z is wmin(z)
wmax(z) and 4xt
z
wmax=min(z) =
(1
xh)2
4z
1
4z
4xt)
p(1
xh)2
4zxh
2
:
pz(z
1
(6.1)
(6.2)
(6.3)
expect that the rate for pp ! S ! tth at p
Our results for the S ! tth branching ratio obtained by integrating over the Dalitz plot are
shown in the penultimate column in table 3. This branching ratio is typically two orders of
magnitude smaller than the S ! tt branching fraction. This will be true in any model in
which the decays of the new resonance can be described in terms of local operators. Given
the existing upper bound from tt resonance searches in Run 1 shown in table 2, we would
s = 8 TeV cannot be larger than about 12 fb.
This falls short by far to explain the enhanced Higgs production rate in association with
tt. In our speci c models, the predicted tth production rates do not exceed 8 fb.
7
The recent hint of a 750 GeV diphoton excess, observed in the data from Run 2 of the LHC
by both ATLAS and CMS, could be the rst direct manifestation of physics beyond the
SM. If veri ed by future analyses, this excess will most likely have been created by the
decay of a new, scalar boson with a mass of 750 GeV produced either in gluon fusion or in
bbinitiated scattering. Many possible interpretations of such a boson have been proposed.
Remarkably, the simple addition of a single new scalar to the SM as well as several well
motivated UV completions  including the Minimal Supersymmetric SM  have already
been excluded.
In this paper, we have shown that the diphoton signal can be reproduced from a very
straightforward extension of the popular warped extradimension models introduced by
Randall and Sundrum. In these models, our spacetime is enlarged by a warped extra
dimension in order to give natural explanations of the gauge and avor hierarchy problems
of the SM. We have identi ed the diphoton resonance with the lightest excitation of an
additional bulk scalar
eld. Such a scalar might serve as the localizer
eld providing
a dynamical generation of the bulk mass parameters of the 5D fermions and is thus a
natural ingredient of RS models with bulk matter
elds. In addition to a model with a
minimal particle content, we have discussed two implementations of RS models providing
a custodial protection mechanism for electroweak precision observables. These models
feature a larger number of fundamental 5D fermions compared with the SM. As a result,
the gluonfusion production process and the decay into two photons are enhanced by the
large multiplicity of vectorlike KK fermion states propagating in the loop. By summing
up the contribution from the in nite towers of KK fermions into 5D propagator functions
we were able to derive remarkably simple analytic expressions for the e ective couplings to
gluons or photons, which to very good approximation simply count the number of fermionic
degrees of freedom, weighted by grouptheory factors. For the custodial RS models, we
have found that with O(1) couplings of the resonance S to fermions and KK masses in the
multiTeV range one can explain the diphoton signal without violating any of the Run 1
bounds from resonance searches in various diboson and dijet channels.
Useful side products of our analysis, which can be adapted to any model for the
diphoton resonance that at the scale
= mS can be mapped onto an e ective Lagrangian
with local interactions, are the calculation of the gluonfusion production cross section
(pp ! S) at NNLO in QCD, an expression for the inclusive S ! gg decay rate at N3LO,
a study of the S ! tth threebody decay mode, and a phenomenological analysis of portal
couplings connecting S with the Higgs eld.
We conclude that our simple extension of the established RS models can deliver one
of the most elegant and minimal explanations for the observed diphoton excess. Assuming
that the resonance will survive future veri cations, it could hint at the existence of a
warped extra dimension and open the door to detailed studies of the parameter space of
RS models.
Acknowledgments
HJEP07(216)94
(A.1)
(A.2)
(A.3)
We are grateful to Joachim Kopp for useful discussions which have initiated this project, to
Daniel Wilhelm for adapting the CuTe program to the calculation of the pp ! S cross
section, and to Ulrich Haisch for drawing our attention to the possible impact of interference
e ects in S ! tt resonance searches. M.B. acknowledges the support of the Alexander von
Humboldt Foundation. The work of C.H. and M.N. was supported by the Advanced Grant
EFT4LHC of the European Research Council (ERC), the DFG Cluster of Excellence
Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA EXC 1098),
grant 05H12UME of the German Federal Ministry for Education and Research (BMBF) and
the DFG Graduate School Symmetry Breaking in Fundamental Interactions (GRK 1581).
A
RG evolution of the Wilson coe cients cgg and ctt
The mixing among the Wilson coe cients ctt and cgg in the e ective Lagrangians (3.1)
and (3.3) is described by the RG equations
d
d
cgg( ) = gg( s) cgg( ) ;
ctt( ) =
tg( s) cgg( ) ;
d
d
where s = s( ), and gg is given by the exact expression [59, 60]
gg( s) = s2 dd s
( s)
2
s
in terms of the function ( s) =
d s( )=d . The leading contribution to tg( s) has
been obtained rst in [61]. The exact solutions to the evolution equations are
cgg( ) =
ctt( ) = ctt( 0)
( s( ))= s2( )
( s( 0))= s2( 0) cgg( 0) ;
2
( 0)
s
( s( 0)) cgg( 0)
Z s( )
s( 0)
d
tg( )
2
:
s
2
The perturbative expansions of the anomalous dimension and function read
( s) =
2
s
1
2
where 1 = 24CF [61].
0 + 1
4
s + : : : ;
tg( s) = 1 4
+ : : : ;
(A.4)
! gg decay rate at N3LO in QCD
An analytic expression for the inclusive S ! gg decay rate at N3LO in QCD perturbation
theory can be derived using existing calculations of the Higgsboson decay rate (h ! gg)
up to O( s5) obtained in [78, 79]. Taking the heavy topquark limit and dividing the result
by the threeloop expression for the matching coe cient Ct(mt; ) obtained in [62, 63],
we nd
with
and (for Nc = 3 colors and nf = 6 light avors)
Lh ;
49 2
16
2821025
where Lh = ln( 2=m2S). Numerically, we
turbative expansion KgNg3LO = 1 + 0:32768 + 0:03325
excellent convergence.
nd for the Kfactor at
= mS the
per0:01290
1:348, which exhibits
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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