Searches for the SM Higgs boson by the four LEP experiments have found a 2.3
sigma excess at 98 GeV and a smaller 1.7 sigma at around 115 GeV. We interpret
these excesses as evidence for a Higgs boson coupled to a higher dimensional
singlet scalar. The fit implies a relatively low dimensional mixing scale
mu_{lhd} < 50 GeV, which explains the low confidence level found for the
background fit in the range s^{1/2} > 100 GeV. The data show a slight
preference for a five-dimensional over a six-dimensional field. This Higgs
boson cannot be seen at the LHC, but can be studied at the ILC.Comment: 9 page

Demir

Drees

Ferroglia

Hill

Hooper

J.J. van der Bij

Kane

S. Dilcher

van der Bij

Physics Letters B

English

arXiv

Searches for the SM Higgs boson by the four LEP experiments have found a 2.3 sigma excess at 98 GeV and a smaller 1.7 sigma at around 115 GeV. We interpret these excesses as evidence for a Higgs boson coupled to a higher dimensional singlet scalar. The fit implies a relatively low dimensional mixing scale mu_{lhd}  100 GeV. The data show a slight preference for a five-dimensional over a six-dimensional field. This Higgs boson cannot be seen at the LHC, but can be studied at the ILC

Van der Bij, J J

Dilcher, S

CERN Document Server

arXiv:hep-ph/0605008 v1   30 Apr 2006Freiburg-THEP-06/05Where is Heidi hiding ?Heidi is hiddenin the high-D Higgs Hill !A higher dimensional explanation of theexcess of Higgs-like events at CERN LEPJ. J. van der Bij and S. DilcherInstitut fu¨r Physik, Universita¨t Freiburg,Hermann-Herder-Str. 3, 79104 Freiburg i. Br., GermanyAbstractSearches for the SM Higgs boson by the four LEP experiments have found a2.3 σ excess at 98GeV and a smaller 1.7 σ at around 115GeV. We interpretthese excesses as evidence for a Higgs boson coupled to a higher dimensionalsinglet scalar. The fit implies a relatively low dimensional mixing scale µlhd <50GeV, which explains the low confidence level found for the background fitin the range s1/2 > 100GeV. The data show a slight preference for a five-dimensional over a six-dimensional field. This Higgs boson cannot be seen atthe LHC, but can be studied at the ILC.With the latest developments from high energy colliders like LEP and theTevatron the standard model (SM) has been established up to the loop level.The main missing ingredient is the direct detection of the SM Higgs boson.The four LEP experiments ALEPH, DELPHI, L3 and OPAL have extensivelysearched for the Higgs boson. The final combined result has been publishedin [1]. The absence of a clear signal has led to a lower limit on the Higgs bo-son mass of 114.4 GeV at the 95% confidence level. Although no clear signalwas found the data have some intriguing features, that can be interpreted asevidence for Higgs bosons beyond the standard model. There is a 2.3 σ effectseen by all experiments at around 98 GeV. A somewhat less significant 1.7 σexcess is seen around 115 GeV. Finally over the whole range s1/2 > 100GeVthe confidence level is less than expected from background.Within the minimal supersymmetric standard model (MSSM) and other ex-tensions [2-5], the excesses at 98 GeV and 115 GeV were interpreted as evi-1dence for the presence of two Higgs bosons Hi with couplings reduced by afactor αi to matter g2i = αi g2SM−Higgs. We will call such Higgs bosons frac-tional Higgses in the following. The excess at 98 GeV is well described by a10% fractional Higgs. More precisely [2] gives limits 0.056 < α98 < 0.114 anda mass range 95GeV < mHiggs < 101GeV. The second peak at 115 GeV isthen interpreted as a second Higgs boson with α115 = 0.9. The first peak at98 GeV is rather convincing. The second one at 115 GeV is compatible withthe data, but not really preferred as the data at 115 GeV are also compatiblewith pure background with a similar confidence level. The data correspondroughly to background plus one half of a Higgs boson, however with a largeuncertainty. The MSSM fit is therefore not completely satisfactory and it isnatural to look for other extensions to describe the data.Actually the precision measurements leave only very little space for extensions,as these tend to spoil the agreement with experiment due to a variety of ef-fects, one of the most important of which is the appearance of flavor-changingneutral currents. Even the MSSM has to finely tune a number of parameters.This leaves only one type of extensions that are safe, namely the singlet exten-sions. Experimentally right handed neutrino’s appear to exist. Since these aresinglets a natural extension of the SM is the existence of singlet scalars too.These will only have a very limited effect on radiative corrections, since theyappear only in two-loop calculations. For a mini-review on singlet extensionssee [7].In this letter we will try to fit the data within the specific model of a SMHiggs boson coupled to a higher dimensional (d = 5, 6) singlet scalar, wherewe assume that the higher dimensions are open and flat. This leads to a Higgspropagator of the form:DHH(q2) =[q2 +M2 − µ8−dlhd (q2 +m2)d−62]−1. (1)The masses M , m and µlhd are the free parameters of the model and are de-fined more precisely later. The scale µlhd stands for low-to-high-dimensionalmixing mass and measures the coupling of the high-d singlet scalar and the 4-ddoublet. The propagator contains a particle peak and a continuum. This modelhas, like the MSSM, three free parameters to fit the data. We will take theexcess at 98 GeV at face value and interpret it as the delta-peak in the prop-agator. The excess at 115 GeV is interpreted as an enhancement due to thecontinuum of the Higgs propagator. Because of the uncertainty of this excesswe will only demand that a Higgs integrated spectral density∫ρ(s)ds > 30%is present in the range 110GeV < s1/2 < 120GeV. We will find that withthese conditions one gets a relatively small highdimension-lowdimension mix-2ing scale µlhd < 50GeV. This result has a non-trivial consequence. It impliesthat the continuum starts very close to the delta-peak. This would thereforeexplain why in the range s1/2 > 100GeV the confidence level of the data isless than expected for background. This cannot be explained by the MSSM.The effects of singlets appear in two forms, one is the mixing with the SMHiggs, the other is invisible decay. In this paper we are interested in puremixing models. It is actually possible to have a Higgs model that has onlyHiggs-mixing. If one starts with an interaction of the form HΦ†Φ, where His the new singlet Higgs field and Φ the SM Higgs field, no interaction of theform H3 or H4 is generated with an infinite coefficient [6]. Therefore one canleave the H3 and H4 interactions out of the Lagrangian without violatingrenormalizability. This type of model can easily be extended to more fieldsHi. The situation becomes quite interesting, when one allows for an infinitenumber of fields, in particular when one assumes the field H to be moving ind = 4+ δ dimensions. Normally speaking this would lead to a nonrenormaliz-able theory. However since the only interaction is of the form HΦ†Φ, which issuperrenormalizable in four dimensions, the theory stays renormalizable. Ananalysis of the power counting of divergences shows that one can associate thecanonical dimension 1 + δ/2 to the H-field. This means that the theory staysrenormalizable as long as δ ≤ 2. When one assumes, that the extra dimen-sions are compact, for instance a torus, with radius R = L/2π, one can simplyexpand the d-dimensional field H(x) in terms of Fourier modes:H(x) =1√2Lδ/2∑~kH~k(xµ) ei 2piL~k~x, H~k = H∗−~k. (2)Here xµ is a four-vector, ~x is δ-dimensional and the δ components of ~k takeonly integer values.One then finds a Lagrangian of the form:L = −12DµΦ†DµΦ−12∑(∂µHk)2+M204Φ†Φ−λ8(Φ†Φ)2−∑ m2k2H2k−g2Φ†Φ∑Hk .(3)The masses of the Hk fields are given by m2k = m20 + m2d~k2. Here ~k is aδ-dimensional vector and m0 is a d-dimensional mass term for the field H .This term is necessary to insure stability of the vacuum. The last term can bewritten as a so-called brane-bulk termS =∫d4+δxδ∏i=1δ(x4+i)H(x)Φ†Φ. (4)This Lagrangian leads to spontaneous symmetry breaking. Both the Φ-field3and the Hk-fields will receive a vacuum expectation value, which we call vrespectively hk. In the unitary gauge we write Φ = v + φ and Hk = hk + xk.The conditions for the minimum of the potential V ′ = 0 read:∂V∂Φ|v,hk =λ2v3 − M202v + gv∑khk = 0, (5)∂V∂Hk|v,hk = m2khk +gv22= 0, (6)⇒ v2 = M20λ− g2∑k 1m2k; hk =−gM202m2k(λ− g2∑k1m2k). (7)From the second derivatives one then finds the mass matrix :Lmass =−M22φ2 −∑km2k2x2k + gvφ∑kxk, (8)with M2 = 32λv2 − M202+ g∑k hk = λv2.One can derive the Higgs propagator without knowing the eigenstates ex-plicitely by inverting the matrix:Dij(q2)−1 =q2 +M2 −gv . . .−gv q2 +m2i 0... 0. . . . (9)First we define the auxiliary quantity Q:Q = 1−∑kg2v2(q2 +M2)(q2 +m2k). (10)Dij(q2) is found to be:Dij(q2) =1Q1q2+M2gv(q2+M2)(q2+m2j)gv(q2+M2)(q2+m2i)(Qq2+m2iδij +g2v2(q2+M2)(q2+m2i)(q2+m2j)) . (11)In the continuous case the Higgs propagator, which corresponds to the (1,1)4component of D, becomes:DHH(q2) =[q2 +M2 − g2v2Γ(1− δ/2)(4π)δ/2(q2 +m20)δ−22]−1. (12)After redefining variables this becomes:DHH(q2) =[q2 +M2 − µ8−dlhd (q2 +m2)d−62]−1. (13)We will now consider in more detail the two cases d = 5 and d = 6 separatelyand compare these with the data. Other values of d are in principle possible,but have no simple physical interpretation as higher dimensional fields.First we consider the case d = 5. The propagator becomes of the form:DHH(q2) =[q2 +M2 − µ3lhd(q2 +m2)−1/2]−1. (14)This corresponds to a Ka¨lle´n-Lehmann spectral density:ρ(s) = θ(m2 − s) 2(m2 − speak)3/22(m2 − speak)3/2 + µ3lhdδ(s− speak)+θ(s−m2)πµ3lhd (s−m2)1/2(s−m2)(s−M2)2 + µ6lhd, (15)where speak satisfies:speak −M2 + µ3lhd(m2 − speak)−1/2 = 0. (16)The peak always exists at a positive value of speak as long as µ3lhd < mM2,which we assume satisfied, since this corresponds to the condition that weexpand around a minimum of the potential. Actually this form of the spectraldensity is only correct as long as M > m. In the case m > M in principlemore peaks can appear, however this would imply for the residue at the poleα98 > 1/3, which is not allowed by the data.We now consider the case d = 6. This case is special, since it corresponds to thelimiting dimension, where the theory is still renormalizable. In the limit d→ 6one notices that the propagator has a logarithmic singularity. This is becausethe coupling constant becomes dimensionless and is running as a function ofthe renormalization scale. In contrast to normal four-dimensional models, the5running appears already at the tree level of the theory. Specializing to this caseand expanding the Γ-function around d = 6 the propagator can be written as:DHH(q2) =[q2 +M2 + µ2lhd log(q2 +m2µ2lhd)]−1. (17)The spectrum contains a pole term for a positive mass as long as:M2 + µ2lhd log(m2/µ2lhd) > 0. (18)When this condition is not fulfilled the pole becomes a tachyon. The reasonis as before in the d = 5 case, that in this case the potential due to theattractive H-exchange dominates over the repulsive λφ4 term and the vacuumis not stable.The corresponding Ka¨lle´n-Lehmann spectral density is:ρ(s) = θ(m2 − s) m2 − speakm2 + µ2lhd − speakδ(s− speak)+ θ(s−m2) µ2lhd[ s−M2 − µ2lhd log((s−m2)/µ2lhd) ]2 + π2 µ4lhd. (19)With these formulas we now try to describe the LEP data. We start with thecase d = 5. The delta-peak will be assumed to correspond to the peak at 98GeV, with a fixed value of α98. Ultimately we will vary the location of the peakbetween 95GeV < mpeak < 101GeV and 0.056 < α98 < 0.144. After fixing α98and mpeak we have one free variable, which we take to be µlhd. If we also take afixed value for µlhd all parameters and thereby the spectral density is known.We can then numerically integrate the spectral density over selected rangesof s. The allowed range of µlhd is subsequently determined by the data at 115GeV. Since the peak at 115 GeV is not very well constrained, we demandhere only that the integrated spectral density from sdown = (110GeV)2 tosup = (120GeV)2 is larger than 30%. This condition, together with formula(15), which implies:ρ(s) <(s−m2)1/2π µ3lhd, (20)leads to the important analytical result:23π µ3lhd[ (sup −m2peak)3/2 − (sdown −m2peak)3/2 ] > 0.3 (21)This implies µlhd < 53GeV. Using the constraint from the strength of thedelta-peak, it follows that the continuum starts very close to the peak, the6difference being less than 2.5 GeV. This allows for a natural explanation, whythe CL for the fit in the whole range from 100 GeV to 110 GeV is somewhatless than what is expected by pure background. The enhancement can be dueto a slight, spread-out Higgs signal. Actually when fitting the data with theabove conditions, one finds for small values of µlhd, that the integrated spectraldensity in the range 100 GeV to 110 GeV can become rather large, whichwould lead to problems with the 95% CL limits in this range. We thereforeadditionally demand that the integrated spectral density in this range is lessthan 30%. There is no problem fitting the data with these conditions. Asallowed ranges we find:95GeV < m < 101GeV111GeV < M < 121GeV26GeV < µlhd < 49GeV (22)As a final consistency check we notice that the results are in agreementwith the upper limit on the Higgs boson mass from precision measurementsmH < 190GeV. The integrated spectral density above 190GeV is less than1%. This upper limit on the Higgs mass is actually so large, because of themeasurement of the b-asymmetry, which is away from the other measure-ments, that are very close together. One can follow the analysis from [8] withthe latest data and finds at the 95% CL mH < 124GeV from the leptonicdata on sin2θ(lept), mH < 144GeV from mW and mH < 109GeV from the twocombined. Even this limit is not inconsistent with our model, which may betaken as additional confirmation. One can actually allow for the integratedspectral density in the low mass region to be larger than given in our analysis,if one allows for a fraction of invisible decays of the Higgs boson. Indeed ifthis strong limit is valid, a low mass spread-out Higgs boson appears to bethe only possibility consistent with experiment.We now repeat the analysis for the case d = 6. The analytic argument givesthe result:sup − sdownπ2 µ2lhd> 0.3 (23)which implies µlhd < 28GeV. Because of this low value of µlhd it is difficult toget enough spectral weight arond 115 GeV and one also tends to get too muchdensity below 110 GeV. So the fit was only possible in a restricted range. Wefound the following limits:95GeV < m < 101GeV106GeV < M < 111GeV22GeV < µlhd < 27GeV (24)7As far as the limit from the precision data there is no problem here either.The spectral weight above 190 GeV is a bit larger, because the six-dimensionalspectrum falls off somewhat more slowly at high values of s, however is stillless than 4%. Though not quite ruled out, the six-dimensional case thereforeseems to be somewhat disfavoured compared to the five-dimensional case.From the discussion it is clear that the data can be well described by the as-sumption of a higher dimensional Higgs field. The model is renormalizable andcontains only two parameters beyond the SM. The relevant mass scales are allcomparable to each other and no fine tuning is present. The model obviouslyfits the data better than pure background or the MSSM, due to the fact thatfractional Higgses are possible at 98 GeV and 115 GeV. Also it provides anexplanation for the slight discrepancy in the range 100 GeV to 110 GeV, thatis not explained by another model. What is not clear is how significant theoverall results are. For this an analysis of the confidence level on a bin by binbasis would be necessary. From the discussion given in [1] we cannot reliablyextract this significance. This can only be done by the experimental groups.Assuming independence in the different mass ranges a very rough estimategives a 3.5 σ − 4 σ overall signal.If this model is indeed true, what are the consequences for the Higgs searchat the LHC? The situation appears to be essentially hopeless. Most of thespectral density is tied up in the low Higgs mass region. In this region theHiggs search is performed at the LHC in the rare decay modes H → γγ andH → ZZ∗ → l+l−l+l−, both of which heavily rely on an excellent mass reso-lution in order to extract the Higgs signal. Since now there is no mass peak,these signals do not work. In the range 150 GeV-180 GeV one can try to usethe dominant H → W ∗W ∗ decay mode. However this does not work either,because the integrated spectral density in this region is less than 2% for thed = 5 case and less than 6% for the d = 6 case. Also the golden decay modesabove the Z threshold are too small, since the integrated spectral density above190 GeV is less than 4%. The only possibility to study this type of Higgs bosonappears to be the international linear collider (ILC), where the Higgs signalcan be studied independently of its decay modes. The spectral density canbe reconstructed from the Bjorken process e+e− → ZH , by looking at theinvariant mass of the Higgs boson, that can be extracted, if one knows theZ momentum precisely. Since backgrounds and signal can be calculated andmeasured very precisely, this model presents no particular problem for the ILC.Acknowledgement We thank Dr. D. Schlatter for an illuminating conversa-tion. We thank Prof. M. Veltman, Drs. A. Ferroglia, A. Lorca and B. Tauskfor discussions and careful readings of the manuscript.8References[1] LEP working group for Higgs boson searches, Phys. Lett. B565, 61 (2003).[2] M. Drees, Phys. Rev. D71, 115006 (2005).[3] G.L. Kane, T.T. Wang, B.D. Nelson, L.T. Wang, Phys. Rev. D71, 035006(2005).[4] D. Hooper, T. Plehn, Phys. Rev. D72, 115005 (2005).[5] D.A. Demir, L. Solmaz, S. Solmaz, Phys. Rev. D73, 016001 (2006).[6] A. Hill, J. J. van der Bij, Phys. Rev. D36, 3463 (1987).[7] J.J. van der Bij, Phys. Lett. B636, 56 (2006).[8] A. Ferroglia, G. Ossola, A. Sirlin, Talk at the XVI Ciclo di Incontri, Torinoapril 2004, Freiburg-THEP-04/11, NYU-TH/04-06-28, hep-ph/0406334.9

A higher dimensional explanation of the excess of Higgs-like events at CERN LEP

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A higher-dimensional explanation of the excess of Higgs-like events at CERN LEP

A higher dimensional explanation of the excess of Higgs-like events at
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A higher dimensional explanation of the excess of Higgs-like events at CERN LEP

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