arXiv:0709.3726v1 [hep-ph] 24 Sep 2007
Higgs-Boson Mass Limit
within the Randall-Sundrum Model
Institute of Theoretical Physics, University of Warsaw,
Ho˙ za 69, PL-00-681 Warsaw, Poland
Department of Physics, University of California,
Davis CA 95616-8677, USA
Perturbative unitarity for W+
model. It is shown that the exchange of massive 4D Kaluza-Klein gravitons leads to amplitudes
growing linearly with the CM energy squared. Summing over KK gravitons up to a scale Λ and
testing unitarity at√s = Λ, one finds that unitarity is violated for Λ below the ’naive dimensional
analysis’ scale, ΛNDA. It is also shown that the exchange of gravitons can substantially relax the
upper limit from unitarity on the mass of the Standard Model Higgs boson — consistency with
unitarity for all√s below Λ is possible for mh as large as 1.4 TeV, depending on the curvature
of the background metric. Observation of the mass and width (or cross section) of one or more
KK gravitons at the LHC will directly determine the curvature and the scale ΛW specifying the
couplings of matter to the KK gravitons. With this information and a measurement of the Higgs
boson mass it will be possible to determine the precise√s value below which unitarity will remain
Lscattering is discussed within the Randall-Sundrum
PACS numbers: 11.10.Kk, 04.50.+h, 11.15.-q, 11.80.Et
Even though the Standard Model (SM) of electroweak interactions perfectly describes almost all existing experi-
mental data, nevertheless the model suffers from certain theoretical drawbacks. The hierarchy problem is probably the
most fundamental of these: namely, quantum loop corrections in the SM destabilize the weak energy scale O(1 TeV) if
the theory is assumed to remain valid to a much higher scale such as the Planck mass scale O(1019GeV). Therefore,
it is believed that the SM is only an effective theory embedded in some more fundamental high-scale theory that
presumably could contain gravitational interactions. Models that involve extra spatial dimensions could provide a
solution to the hierarchy problem in which gravity plays the major role. The most attractive proposal was formulated
by Randall and Sundrum (RS) . They postulate a 5D universe with two 4D surfaces (“3-branes”). All the SM
particles and forces with the exception of gravity are assumed to be confined to one of those 3-branes called the visible
or TeV brane. Gravity lives on the visible brane, on the second brane (the “hidden brane”) and in the bulk. All
mass scales in the 5D theory are of order of the Planck mass. By placing the SM fields on the visible brane, all the
order Planck mass terms are rescaled by an exponential suppression factor (the “warp factor”) Ω0≡ e−m0b0/2, which
reduces them down to the weak scale O(1 TeV) on the visible brane without any severe fine tuning. To achieve the
necessary suppression, one needs m0b0/2 ∼ 35. This is a great improvement compared to the original problem of
accommodating both the weak and the Planck scale within a single theory.
The RS model is specified by the 5-D action:
S = −d4xdy
Pl5?R + Λ
+d4x√−ghid(Lhid− Vhid) +
where the notation is self-explanatory, see also  for details. In order to obtain a consistent solution to Einstein’s
equations corresponding to a low-energy effective 4D theory that is flat, certain conditions must be satisfied: Vhid=
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†Electronic address: firstname.lastname@example.org
Pl5m0and Λ = −24M3
0. Then, the following metric is a solution of Einstein’s equations:
0 | −b2
? g? µ? ν(x,y) =
After an expansion around the background metric we obtain the gravity-matter interactions
field (the scalar quantum degree of freedom associated with fluctuations of the distance between the branes), ΛW ≃
√2mPlΩ0, where Ω0= e−m0b0/2, and Λφ=√3ΛW. To solve the hierarchy problem, ΛWshould be of order 1−10 TeV,
or perhaps higher . In addition to the radion, the model contains a conventional Higgs boson, h. The RS model
solves the hierarchy problem by virtue of the fact that the 4D electro-weak scale is given in terms of the O(mPl) 5D
Higgs vev, ? v, by:
v0= Ω0? v = e−m0b0/2? v ∼ 1 TeV
However, the RS model is trustworthy in its own right only if the 5D curvature m0is small compared to the 5D
Planck mass, MPl5 . The m0 < MPl 5 requirement and the fundamental RS relation m2
that m0/mPl = 2−1/2(m0/MPl5)3/2should be significantly smaller than 1. Hereafter, we will focus on the range:
The goal of this analysis is to determine the cutoff (defined as some maximum energy up to which the 4D RS
theory is well behaved) and to discuss the unitarity limits on the Higgs boson mass taking into account KK graviton
exchange; for a detailed discussion see .
µν(x) are the Kaluza-Klein (KK) modes (with mass mn) of the graviton field hµν(x,y), φ0(x) is the radion
form0b0/2 ∼ 35. (4)
The cutoff can be estimated in a number of ways. One estimate of the maximum allowed energy scale is that
obtained using the ’naive dimensional analysis’ (NDA) approach , the associated scale is denoted by ΛNDA.1One
ΛNDA= 27/6π(m0/mPl)1/3ΛW, (5)
where ΛW was defined in Eq. (3); its inverse sets the strength of the coupling between matter and gravitons. We
emphasize that ΛNDAis obtained when the exchange of the whole tower of KK modes up to ΛNDAis taken into account.
Physically, ΛNDAis the energy scale at which the theory starts to become strongly coupled and string/M-theoretic
excitations appear from a 4D observer’s point of view . In this presentation, we show that unitarity in the J = 0
partial wave of W+
Lscattering is always violated in the RS model for energies below the ΛNDAscale.
We will define Λ as the largest√s value such that if we sum over graviton resonances with mass below Λ (but do
not include diagrams containing the Higgs boson or radion of the model) then W+
unitary in the J = 0 partial wave. Unitarity of the S-matrix implies that the partial wave amplitudes aJ(s) must
satisfy |Re aJ| < 1/2. As we see from Fig. 1, W+
as the cutoff. A more appropriate cutoff is determined numerically by requiring |Re a0,1,2| < 1/2 for√s = Λ after
summing over KK resonances with mass below Λ. It is important to realize that in the presence of KK gravitons
and the radion, the SM cancellation (between Higgs and gauge boson contributions) of terms ∝ s in the asymptotic
behavior of aJ(s) is spoiled, that is why graviton contributions turn out to be so relevant. In the left-hand plot of
Fig. 2, we display the ratio Λ/ΛNDAas a function of m0/mPl, where Λ is the largest√s for which W+
scattering is unitary when computed including only the KK graviton exchanges. Results are shown for J = 0, 1, and
2. As a function of√s, the J = 0 partial wave is always the first to violate unitarity and gives the lowest value of
Lscattering violates unitarity if ΛNDAis employed
1The 4D condition for the cutoff ΛNDA (which corresponds to the scale at which the theory becomes strongly coupled) is
(ΛNDA/ΛW)2N/(4π)2∼ 1, where N is the number of KK-gravitons lighter than ΛNDA(implying that they should be included in the
low-energy effective theory). For the RS model the graviton mass spectrum for large n is mn ≃ m0πnΩ0, implying N ∼ ΛNDA/(m0πΩ0)
which leads to Eq. (5).
FIG. 1: We plot Re a0,1,2 as functions of m0/mPl as computed at√s = ΛNDA and summing over all KK graviton resonances
with mass below ΛNDA, but without including Higgs or radion exchanges.
FIG. 2: In the left hand plot, we give Λ/ΛNDA as a function of m0/mPl, where Λ is the largest√s for which W+
scattering is unitary after including KK graviton exchanges with mass up to Λ, but before including Higgs and radion exchanges.
Results are shown for the J = 0, 1 and 2 partial waves. With increasing√s unitarity is always violated earliest in the J = 0
partial wave, implying that J = 0 yields the lowest Λ. The right hand plot shows the individual absolute values of Λ(J = 0)
and ΛNDA for the case of Λφ= 5 TeV; Λ/ΛNDA is independent of Λφ
Λ. We will cut off our sums over KK exchanges when the KK mass reaches Λ as determined by the J = 0 amplitude.
We see that the Λ so defined is typically a significant fraction of ΛNDA, but never as large as ΛNDA. Still, it is quite
interesting that the unitarity consistency limit Λ tracks the ’naive’ ΛNDAestimate fairly well as m0/mPlchanges over
a wide range of values (for a qualitative ’derivation’ see ). The right-hand plot of Fig. 2 shows the actual values of Λ
and ΛNDAas functions of m0/mPlfor the case of Λφ= 5 TeV. Note that for larger m0/mPlthey substantially exceed
the input inverse coupling scale Λφ, whereas for smaller m0/mPl they are both substantially below Λφ. In other
words, using either Λ or ΛNDA, one concludes that Λφ, and equally ΛW, are themselves not appropriate estimators
for the maximum scale of validity of the model. The left-hand plot of Fig. 3 shows Re a0 as a function of√s for
the case of Λφ= 5 TeV for two different mhvalues and with and without radion and/or KK gravitons included. In
the case where we include only the SM contributions for mh= 870 GeV, the figure illustrates unitarity violation as
FIG. 3: For Λφ = 5 TeV — left (Λφ = 10 TeV — right), we plot Re a0 as a function of√s for five cases: 1) solid (black)
mh= 870 GeV, SM contributions only; 2) short dashes (red) mh= 870 GeV, with an unmixed radion of mass mφ= 500 GeV
included, but no KK gravitons (we do not show the very narrow φ resonance); 3) dots (blue) as in 2), but including the
sum over KK gravitons taking m0/mPl = 0.01 (m0/mPl = 0.05) — Re a2 is also shown for this case; 4) long dashes (green)
mh= 1000 GeV (915 GeV), with an unmixed radion of mass mφ= 500 GeV, but no KK gravitons); 5) as in 4), but including the
sum over KK gravitons taking m0/mPl= 0.01 (m0/mPl= 0.05). The Λ and ΛNDA values for m0/mPl= 0.01 (m0/mPl= 0.05)
are indicated by vertical lines.
Re a0asymptotes to a negative value very close to −1/2, implying that mh= 870 GeV is very near the largest value
of mhthat is allowed by unitarity in the SM. If we add in just the radion contributions (for mφ= 500 GeV2– the φ
resonance is very narrow and is not shown), then a sharp-eyed reader will see (red dashes) that Re a0is a bit more
negative at the highest√s plotted, implying earlier violation of unitarity. However, if we now include the full set of
KK gravitons, which enter with an increasingly positive contribution, taking m0/mPl= 0.01 (dotted blue curve) one
is far from violating unitarity due to Re a0< −1/2 for√s values above mh= 870 GeV; instead, the positive KK
graviton contributions, which cure the unitarity problem at negative Re a0for√s above mh, cause unitarity to be
violated at large√s, but above Λ, as Re a0passes through +1/2. In fact, in the case of a heavy Higgs boson we see
that Re a2actually violates unitarity earlier than does Re a0. However, even using Re a2as the criterion, unitarity
is first violated for√s values above the Λ value appropriate to the m0/mPl= 0.01 value being considered, but still
below ΛNDA. In fact, it is very generally the case that unitarity is not violated at√s = Λ (which is typically a sizable
fraction of ΛNDA) no matter how small we take m0/mPl. However, as we shall see, unitarity can be violated in the
vicinity of√s ∼ mhif mhis large and m0/mPlis sufficiently small.
Looking again at the left plot of Fig. 3, we observe that if mhis increased to 1000 GeV, the purely SM plus radion
contributions (long green dashes) show strong unitarity violation at large√s due to Re a0< −1/2. However, if we
include the KK gravitons (long dashes and two shorter dashes in magenta), the negative Re a0 unitarity violation
disappears and unitarity is instead violated at higher√s. Thus, it is the KK gravitons that can easily control whether
or not unitarity is violated for√s < Λ for a given value of mh.
III.THE HIGGS-BOSON MASS LIMIT
As we have already seen, the Higgs plus vector boson exchange contributions have a large affect on the behavior of
Re a0(whereas the radion exchange contributions are typically quite small in comparison). It is particularly interesting
to consider cases with a very heavy Higgs boson, focusing on small values of m0/mPl. For mh = 870 GeV and
Λφ= 10 TeV, the result appears as the left-hand plot of Fig. 4. Note that for the very small value of m0/mPl= 0.0001,
unitarity is only just satisfied for√s ∼ mhand that Re a0exceeds +1/2 near√s ∼ mh. This is a general feature in
the case of a heavy Higgs; there is always a lower bound on m0/mPlcoming purely from unitarity. The right-hand plot
2The radion contribution is always negligible in the scenario discussed here if mφremains in the range mφ∈ [10,1000] GeV and Λφis
above 1 TeV. This is however not true in the context of curvature-Higgs mixing as discussed in .
of Fig. 4 shows how high we can push the mass of the Higgs boson without violating unitarity. For mh= 1430 GeV,
we are just barely consistent with the unitarity limit |Re a0| ≤ 1/2 (until large√s>∼Λ) if m0/mPl= 0.0018 (and
excursion to Re a0< −1/2 at higher√s values (but still below Λ). There are no experimental limits (coming from
direct production of KK gravitons) of which we are aware on the m0/mPl values considered in Fig. 4 . For such
values, the KK gravitons would have very small masses, an experimental analysis in that range of m0/mPlis needed.
Λφ= 10 TeV). Any lower value of m0/mPl leads to Re a0> +1/2 at√s ∼ mhand any higher value leads to an
FIG. 4: We plot Re a0,1,2 as functions of√s for mh= 870 GeV and mh= 1430 GeV, taking mφ= 500 GeV and Λφ= 10 TeV,
and for the m0/mPl values indicated on the plot. Curves of a given type become higher as one moves to lower m0/mPl values.
We have included all KK resonances with mn < Λ (at all√s values). Each curve terminates at√s = ΛNDA, where ΛNDA at
a given m0/mPl is as plotted earlier in Fig. 2. The value of√s at which a given curve crosses above Re a0 = +1/2 is always
slightly above the Λ (plotted in Fig. 2) value for the given m0/mPl.
Λφ( TeV)5 10 2040
Absolute maximum Higgs mass
1.32 × 10−21.8 × 10−32.3 × 10−42.9 × 10−5
associated m1( GeV)103.2
m0/mPl= 0.005: Tevatron limit: m1 >??
associated m1( GeV) 39
m0/mPl= 0.01: Tevatron limit: m1 > 240 GeV
associated m1( GeV)78
m0/mPl= 0.05: Tevatron limit: m1 > 700 GeV
associated m1( GeV) 391
m0/mPl= 0.1: Tevatron limit: m1 > 865 GeV
associated m1( GeV)782
TABLE I: Unitarity limits on mh for various Λφ and m0/mPl values.
In Table I, we summarize the primary implications of our results by showing a number of limits on mh for the
choices of Λφ= 5, 10, 20 and 40 TeV. The first block gives the very largest mhthat can be achieved, mmax
violating unitarity in W+
mass m1of the lightest KK graviton. Unfortunately, no Tevatron limits (see ) have been given for the associated
very small m0/mPlvalues. Even if they end up being experimentally excluded, it is still interesting from a theoretical
Lscattering for some√s < Λ, along with the associated m0/mPlvalue and
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perspective that in the RS model unitarity can be satisfied for all√s values below the Λ cutoff of the theory for
a Higgs boson mass substantially higher than the usual 870 GeV value applicable in the SM context. One finds
is typically of order 1.4 TeV if one chooses the optimal value for m0/mPl(for Λφin a reasonable range:
5 TeV ≤ Λφ≤ 40 TeV). It is also noteworthy that the required values of m0/mPlare quite consistent with model
Table I also gives the mmax
value achievable for the four Λφcases listed above for various fixed m0/mPl. Also given
are the associated m1values and the Tevatron direct production limit when available. For some of the cases that are
clearly consistent with Tevatron limits, unitarity is satisfied for mhvalues as high as ∼ 915 GeV.
We have discussed perturbative unitarity for W+
3-branes and shown that the exchange of massive 4D Kaluza-Klein gravitons leads to amplitudes growing linearly
with the CM energy squared. We have found that the gravitational contributions cause a violation of unitarity for
√s below the natural cutoff of the theory, ΛNDA, as estimated using naive dimensional analysis.
In practice, to determine the cutoff the two basic RS model parameters ΛW and m0/mPlmust be extracted from
experiment, as should be possible at the LHC. If the Higgs mass has also been measured, then the maximum√s for
Lscattering obeys unitarity in the RS model can be found from the results of this paper.
The most important result obtained here is the determination of the maximal Higgs boson mass allowed by requiring
and always close to ΛNDA): one finds mmax
≤ 1.4 TeV — to achieve the upper limit, a particular ΛW-dependent
m0/mPlvalue is necessary.
We should emphasize here that we do not need to consider the effects of the scalar field(s) that are responsible for
stabilizing the inter-brane separation at the classical level. These fields are normally chosen to be singlets under the
SM gauge groups (sample models include those of Refs. [7, 8]), and will thus have no direct couplings to the WLWL
channel. For the purpose of this work, the only effect of the inter-brane stabilization is to determine the radion mass.
Lwithin the Randall-Sundrum theory with two
Lscattering be consistent with unitarity for all√s values below the scale Λ (defined earlier
This work is supported in part by the Ministry of Science and Higher Education (Poland) in years 2006-8 as research
project N202 176 31/3844, by EU Marie Curie Research Training Network HEPTOOLS, under contract MRTN-
CT-2006-035505, by the U.S. Department of Energy grant No. DE-FG03-91ER40674, and by NSF International
Collaboration Grant No. 0218130. B.G. acknowledges the support of the European Community under MTKD-CT-
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