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arXiv:0709.3726v1 [hep-ph] 24 Sep 2007

IFT-07-08

Higgs-Boson Mass Limit

within the Randall-Sundrum Model

Bohdan Grzadkowski∗

Institute of Theoretical Physics, University of Warsaw,

Ho˙ za 69, PL-00-681 Warsaw, Poland

John Gunion†

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

valid.

LW−

L→ W+

LW−

Lscattering is discussed within the Randall-Sundrum

PACS numbers: 11.10.Kk, 04.50.+h, 11.15.-q, 11.80.Et

I. INTRODUCTION

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) [1]. 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

?

−? g

?

2M3

Pl5?R + Λ

?

+d4x√−ghid(Lhid− Vhid) +

?

d4x√−gvis(Lvis− Vvis), (1)

where the notation is self-explanatory, see also [2] 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=

∗Electronic address: bohdan.grzadkowski@fuw.edu.pl

†Electronic address: jfgucd@physics.ucdavis.edu

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−Vvis= 24M3

Pl5m0and Λ = −24M3

Pl5m2

0. Then, the following metric is a solution of Einstein’s equations:

?e−2m0b0|y|ηµν |

0 | −b2

? g? µ? ν(x,y) =

0

0

?

. (2)

After an expansion around the background metric we obtain the gravity-matter interactions

Lint= −

1

ΛW

?

n?=0

hn

µνTµν−φ0

ΛφTµ

µ

(3)

where hn

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 [1]. 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 [1]. 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:

10−3<∼m0/mPl<∼10−1.

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 [3].

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)

Pl= 2M3

Pl5/m0 imply

II.THE CUTOFF

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 [4], the associated scale is denoted by ΛNDA.1One

finds

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 [1]. 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

LW−

L→ W+

LW−

LW−

L→ W+

LW−

Lscattering remains

LW−

L→ W+

LW−

Lscattering violates unitarity if ΛNDAis employed

LW−

L→ W+

LW−

L

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).

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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

LW−

L→ W+

LW−

L

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 [3]). 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

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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 [5].

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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)510 2040

Absolute maximum Higgs mass

1435

1.32 × 10−21.8 × 10−32.3 × 10−42.9 × 10−5

associated m1( GeV) 103.2

mmax

h

required m0/mPl

( GeV) 14301430 1430

28.27.2 1.8

m0/mPl= 0.005: Tevatron limit: m1 >??

( GeV)1300

associated m1( GeV)39

mmax

h

930

78

920

156

905

313

m0/mPl= 0.01: Tevatron limit: m1 > 240 GeV

( GeV)1405

associated m1( GeV)78

mmax

h

930

156

910

313

895

626

m0/mPl= 0.05: Tevatron limit: m1 > 700 GeV

( GeV) 930

associated m1( GeV)391

mmax

h

915

782

900

1564

885

3129

m0/mPl= 0.1: Tevatron limit: m1 > 865 GeV

( GeV)920

associated m1( GeV)782

mmax

h

910

1564

893

3128

883

6257

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 [6]) 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

h

, without

LW−

L→ W+

LW−

Lscattering for some√s < Λ, along with the associated m0/mPlvalue and