On Standard Model Higgs and Superstring Theories

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Extending the Higgs Boson Reach at the Upgraded Fermilab Tevatron

a r X i v :h e p -p h /9807424v 2 1 D e c 1998Extending the Higgs Boson Reach at the Upgraded Fermilab TevatronTao Han and Ren-Jie ZhangDepartment of Physics,University of Wisconsin,1150University Avenue,Madison,WI 53706,USA(July,1998)We study the observability for a Standard Model-like Higgs boson at an upgraded Tevatron viathe modes p ¯p →gg →h →W ∗W ∗→ℓνjj and ℓ¯ν¯ℓν.We ﬁnd that with c.m.energy of 2TeV and an integrated luminosity of 30fb −1the signal may be observable for the mass range of135GeV <∼m h <∼180GeV at a 3−5σstatistical level.We conclude that the upgraded Tevatronmay have the potential to detect a SM-like Higgs boson in the mass range from the LEP2reach to 180GeV.14.80.Bn,13.85.QkThe Higgs bosons are crucial ingredients in the Stan-dard Model (SM)and its supersymmetric extensions (SUSY).Searching for Higgs bosons has been one of the major motivations in the current and future collider pro-grams since they most faithfully characterize the mecha-nism for the electroweak gauge symmetry breaking.Ex-periments at LEP2will eventually be able to discover a SM-like Higgs boson with a mass about 105GeV [1].The LHC should be able to cover the full range of theoretical interest,up to about 1000GeV [2].It has been discussed extensively how much the Fer-milab Tevatron can do for the Higgs boson search.It appears that the most promising processes continuously going beyond the LEP2reach would be the electroweak gauge boson-Higgs associated production [3–5]p ¯p →W h,Zh.The leptonic decays of W,Z provide a good trigger and h →b ¯b may be reconstructible with ade-quate b -tagging.It is now generally believed that for an upgraded Tevatron with c.m.energy√s =2TeV.Along with the inclusive totalcross section ∗(solid curve),we show the W ∗W ∗(dashes)and Z ∗Z ∗(dots)channels,as well as their various de-cay modes W ∗W ∗→ℓνjj,ℓ¯ν¯ℓνand Z ∗Z ∗→ℓ¯ℓν¯ν,4ℓ.The scale on the right-hand side gives the number of events expected for 30fb −1.We see that for the m h range of current interest,there may be about 1000events produced for W ∗W ∗→ℓνjj and about 100events forW ∗W ∗→ℓ¯ν¯ℓν.This latter channel has been studied at the SSC and LHC energies [10]and at a 4TeV Tevatron [4].We ﬁnd that at√For this mode,we require the ﬁnal state to have an iso-lated charged lepton (ℓ),large missing transverse energy (/E T ),and two hard jets.The leading SM backgrounds arep ¯p →W +2QCD jets ,p ¯p →W W →ℓνjj,(2)p ¯p →W Z (γ∗)→ℓνjj,p ¯p →t ¯t →ℓνjjb ¯b.The background processes are calculated with the fullSM matrix elements at treelevel.FIG.1.The Higgs-boson production cross-section via the gluon-fusion process versus m h at the2TeV Tevatron.The h→W∗W∗(dashes)and Z∗Z∗(dots)channels and various subsequent decay modes are also depicted.To roughly simulate the detector eﬀects,we use the following energy smearing∆E j/E j=0.8/√Eℓ⊕0.01for leptons,(3) where⊕denotes a sum in quadrature.The basic accep-tance cuts used here arep Tℓ>15GeV,|ηℓ|<1.1;p T j>15GeV,|ηj|<3;/ET>15GeV;∆R(ℓj)>0.3,∆R(jj)>0.7.(4) For the sake of illustration,we present our study mostly for m h=140and160GeV and we will generalize the results to the full m h range of interest.The cut eﬃciency for the signal is about35%(60%)for m h≈140(160) GeV.Since there are only two jets naturally appearing in the signal events,the t¯t background can be eﬀectively suppressed by rejecting events with extra hard jets.We therefore imposeJet veto p T j>15GeV in|ηj|<3.(5) The QCD background in(2)has the largest rate.The di-jet in the signal is from a W decay,while that in the QCD background tends to be soft and collinear.We thus impose the cuts on the di-jet:65<m(jj)<95GeV,φ(jj)>140◦;70<m(jj)<90GeV,φ(jj)>160◦,(6) for m h=140and160GeV respectively,where m(jj)is the invariant mass of the di-jet andφ(jj)the opening an-gle of the two jets in the transverse plane.For m h≥160 GeV,the m(jj)distribution has a unique peak because both W bosons are on shell,so the m(jj)cuts in Eq.(6) would not signiﬁcantly harm the signal.On the other hand,for m h≤160GeV,nearly half of the signal willbe cut oﬀby the m(jj)cuts,making this region of theHiggs mass more diﬃcult to explore from this mode.We have also examined other mass variables,such asthe W-boson transverse mass M T(W),di-jet-lepton in-variant mass m(jjℓ),and the cluster transverse massM C,which are deﬁned asM T(W)=p2T(jjℓ)+m2(jjℓ)+/ET.(7)The M T(W)develops a peak near M W for on-shell W decay.An upper cut on this variable below M W can help remove the background from real W decay as long as m his less than160GeV.The cluster transverse mass M Cwould be the most characteristic variable for the signal.It peaks near m h and yields a rather sharp end-pointabove m h.To further improve the signal-to-backgroundratio S/B,weﬁnd the following tighter cuts helpful100<m(jjℓ)<120GeV,/ET<30GeV,120<M C<140GeV, 35<M T(W)<55GeV,2.4<∆R(jj)<3.5;100<m(jjℓ)<130GeV,/ET<50GeV,130<M C<170GeV, 40<M T(W)<90GeV,2.8<∆R(jj)<3.5,(8) for m h=140and160GeV respectively.We show the results progressively at diﬀerent stagesof the kinematical cuts in Table I.We see that for an integrated luminosity of30fb−1,the signal for m h=140 GeV is very weak while that for m h∼160GeV can reacha3σstatistical signiﬁcance.W∗W∗→ℓ¯ν¯ℓν:The W ∗W ∗mass cannot be accurately reconstructed due to the two undetectable neutrinos.However,both the transverse mass M T and the cluster transverse mass M C ,deﬁned asM T =2p 2T (ℓℓ)+m 2(ℓℓ)+/ET ,(12)yield a broad peak near m h .We note that these trans-verse mass variables are very important for the signalidentiﬁcation and for controlling the systematicerror.FIG.2.The cluster transverse mass distributions for ℓ¯ν¯ℓνmode (a)for the signal m h =140,160and 180GeV and the leading SM backgrounds with cuts (10)and (11).With further selective cuts,we show the event rates in (b)for SM background (solid)and background plus signal (dashes)for m h =160GeV.The statistical error bars are also indicated on the background curve in (b).In Fig.2(a),we show the M C distributions for the ℓ¯ν¯ℓνsignal with m h=140,160and 180GeV along with the leading backgrounds after the cuts in (10)and (11).Although the mass peaks in M C are hopeful for signal identiﬁcation,they are rather broad.We may have to rely on the knowledge of the SM background distribu-tion.We hope that with the rather large statistics of the data sample,one may obtain good ﬁt for the normaliza-tion of the background shape outside the signal region in Fig.2(a),so that the deviation from the predicted background can be identiﬁed as signal.Some additional useful cuts arem (ℓℓ)<80GeV (70for m h ≤140GeV),/E T <m h /2,m h /2<M C <m h .(13)The results at diﬀerent stages of kinematical cuts are shown in Table II.Due to the absence of the large QCD background in (2),this pure leptonic mode seems to be statistically more promising than the ℓνjj mode.One may expect a more than 3σ(4σ)eﬀect for m h =140(160)GeV with 30fb −1.It was pointed out in [11]thatsome angular variables implement the information for decay lepton spin correlations and are powerful in dis-criminating against the backgrounds.With some further selective cuts on the angular distributions,we ﬁnd that the S/B can be improved to about 8%and 21%,with the signal rates 2.6and 3.3fb for m h =140and 160GeV,respectively.We show in Fig.2(b)the event rate distri-butions for the SM background (solid)and signal plus background (dashes)for m h =160GeV.The statistical error bars on the background are alsoindicated.FIG.3.The integrated luminosity needed to reach 3σand 5σstatistical signiﬁcance versus m h .The dotted and dashedlines correspond to the ℓνjj and ℓ¯ν¯ℓνmodes respectively.The solid lines are the (quadratically)combined results.In Fig.3,we show the integrated luminosities neededto reach 3σand 5σsigniﬁcance versus m h .The dottedcurves are for the ℓνjj mode and the dashed for ℓ¯ν¯ℓν.We consider that these two modes have rather diﬀer-ent systematic errors,so that we can combine the re-sults for them quadratically.This is shown by the solid curves.We see that with an integrated luminosity of 30fb −1,one may be able to reach at least a 3σsignal for 135GeV <∼m h <∼180GeV.Taking into account the previous studies [3,5,6],we conclude that the upgraded Tevatron with√only for the SM Higgs boson,but also for SM-like ones such as the lightest Higgs boson in SUSY at the decou-pling limit.If there is an enhancement from new physics forΓ(h→gg)×BR(h→W W,ZZ)over the SM ex-pectation,the signal of Eq.(1)would be more viable.If BR(h→b¯b)is suppressed,such as in certain parameter region in SUSY,then the signal under discussion may complement the W h,Zh(h→b¯b)channels at a lower m h region.Our results summarized in Fig.3based on the parton-level simulation are clearly encouraging to signiﬁcantly extend the reach for the Higgs boson search at the up-graded Tevatron.The more comprehensive results with full Monte Carlo simulations in a realistic environment will be reported elsewhere[12].Acknowledgments:We would like to thank V.Barger, E.Berger,R.Demina,M.Drees,T.Kamon,S.Mrenna, J.-M.Qian,S.Willenbrock and J.Womersley for helpful comments.T.H.would like to thank the Aspen Center for Physics for its hospitality during theﬁnal stage of the project.This work was supported in part by a DOE grant No.DE-FG02-95ER40896and in part by the Wisconsin Alumni Research Foundation.Basic Cuts in(4)Cuts in(8)1401601401602349 2.215W jj6.1×1032.1×1031.3×1031755W Z207.0t0.1S/B0.1%0.9%B(30fb−1)0.6 2.4TABLE I.h→W∗W∗→ℓνjj signal and background cross sections(in fb)for m h=140and160GeV,after diﬀerent stages of kinematical cuts.A jet-veto cut in Eq.(5)has been implemented for the t¯t background.Basic Cuts in(10)Cuts in(13)m h[GeV]1401601401607.310 6.39.1backgroundW W2.4×1023216 ZZ(γ∗)180.30.1Z(γ∗)0.00.0t0.20.0S/B 2.7% 3.9%8.0%21%√ 1.5 2.1 3.3 4.2S/TABLE II.h→W∗W∗→ℓ¯ν¯ℓνsignal and background cross sections(in fb)for m h=140and160GeV,after diﬀerent stages of kinematical cuts.The last column corresponds to the reﬁnement of mass cuts and various angular distribution cuts.A jet-veto cut in Eq.(5)has been implemented for the t¯t background.5。

Upper and lower bounds of the lightest CP-even Higgs boson in the two-Higgs-doublet model

M2, where λ ≡ λ3 + λ4 + λ5. The mass of the lighter (heavier) CP-even Higgs boson h
(H) is then given by m2h,H = M121 + M222 ∓ (M121 − M222)2 + 4M142 /2. For the case of v2 ≪ M2, they can be expressed by
RGE below M, and the one-loop 2HDM RGE [7] above M. They are connected at M
by identifying the lightest CP-even Higgs boson in the 2HDM as the SM one in the mass formulas in both the models. We here use this procedure for the case M2 ∼ λiv2 too, because the correction from the SM RGE is numerically very small in this case, although
As the condition of validity of perturbation theory, we here require that the running coupling constants of the Higgs self-couplings and the Yukawa couplings do not blow up below a certain energy scale Λ: this leads the constraints on the coupling constants;

Supernovae as Probes of Extra Dimensions

a rXiv:076.3551v2[he p-ph]7J ul27Supernovae as Probes of Extra Dimensions V .H.Satheesh Kumar ∗,†,P.K.Suresh ∗and P.K.Das ∗∗∗School of Physics,University of Hyderabad,Hyderabad 500046,India.†Department of Physics,Jain International Residential School,Bangalore 562112,India.∗∗The Institute of Mathematical Sciences,CIT Campus,Taramani,Chennai 600113,India.Abstract.Since the dawn of the new millennium,there has been a revived interest in the concept of extra dimensions.In this scenario all the standard model matter and gauge ﬁelds are conﬁned to the 4dimensions and only gravity can escape to higher dimensions of the universe.This idea can be tested using table-top experiments,collider experiments,astrophysical or cosmological observations.The main astrophysical constraints come from the cooling rate of supernovae,neutron stars,red giants and the sun.In this article,we consider the energy loss mechanism of SN1987A and study the constraints it places on the number and size of extra dimensions and the higher dimensional Planck scale.Keywords:Large Extra Dimensions,Kaluza-Klein Gravitons,Supernovae PACS:11.25.-w,11.25.Wx INTRODUCTION It has been recently noted that the scale of quantum gravity M Pl ≈1019GeV can be brought down to a few TeV in certain class of extra dimensional models [1].The only experimentally veriﬁed scale M EW of Standard Model(SM)interactions in four dimen-sions lies within the TeV scale which allows tolerable quantum corrections.Therefore,the assumption that the 4+n dimensional gravity becomes strong at TeV scale,while the standard gauge interactions remain conﬁned to the four-dimensional spacetime does not conﬂict with today’s data available from low energy gravitational experiments [2].Such a notion of TeV scale gravity solves the hierarchy problem between M EW and M Pl with-out relying on supersymmetry or technicolour.According to this model,the observed weakness of gravity at long distances is due to the presence of n new spatial dimensions large compared to the electroweak scale.This can be inferred from the relation betweenthe Planck scales of the D =4+n dimensional theory MD and the four dimensional theory M Pl ,which is given by M 2Pl ∼R n M n +2D ,(1)where R is the size of the extra dimensions.Putting M D ∼1TeV one ﬁndsR ∼1030these extra dimensions,that is they are conﬁned to only‘3+1-brane’,in the higher di-mensional spacetime called‘bulk’.In this framework the universe is4+n dimensional with the fundamental Planck scale M D near the weak scale,with n≥2new sub-mm sized dimensions where gravity can freely propagate everywhere in the bulk,but the SM particles are localised on the3-brane embedded in this bulk.This theory predicts a variety of novel signals which can be tested using table-top experiments,collider ex-periments,astrophysical or cosmological observations.It has been pointed out that one of the strongest constraints on this physics comes from SN1987A[3].Various authors have done calculations to place such constraints on M D and n[4,5,6,7,8,9,10].In this article,we summarise all the results which have appeared in the literature so far.SUPERNOV A EXPLOSION AND COOLING Supernovae come in two main observational varieties[11].Those whose optical spectra exhibit hydrogen lines are classiﬁed as Type II,while hydrogen-deﬁcient SNe are desig-nated Type I.Physically,there are two fundamental types of supernovae,based on what mechanism powers them:the thermonuclear SNe and the core-collapse ones.Only SNe Ia are thermonuclear type and the rest are formed by core-collapse of a massive star. The core-collapse supernovae are the class of explosions which mark the evolutionary end of massive stars(M≥8M⊙).The collapse can not ignite nuclear fusion because iron is the most tightly bound nucleus.Therefore,the collapse continues until the equation of state stiffens by nucleon degeneracy pressure at about nuclear density(3×1014g cm−3). At this“bounce”a shock wave forms,moving outward and expelling the stellar mantle and envelope.The kinetic energy of the explosion carries about1%of the liberated gravitational binding energy of about3×1053erg and the remaining99%going into neutrinos.This powerful and detectable neutrino burst is the main astro-particle interest of core-collapse SNe.In the case of SN1987A,about1053ergs of gravitational binding energy was re-leased in few seconds and the neutrinoﬂuxes were measured by Kamiokande[12]and IMB[13]collaborations.Numerical neutrino light curves can be compared with the SN1987A data where the measured energies are found to be“too low”.For exam-ple,the numerical simulation in[14]yields time-integrated values Eνe ≈13MeV,E¯νe ≈16MeV,and Eνx≈23MeV.On the other hand,the data imply E¯νe=7.5MeV at Kamiokande and11.1MeV at IMB[15].Even the95%conﬁdence range forKamiokande implies E¯νe <12MeV.Flavor oscillations would increase the expectedenergies and thus enhance the discrepancy[15].It has remained unclear if these and other anomalies of the SN1987A neutrino signal should be blamed on small-number statistics,or point to a serious problem with the SN models or the detectors,or is there a new physics happening in SNe?Since we have these measurements already at our disposal,now if we propose some novel channel through which the core of the supernova can lose energy,the luminosity in this channel should be low enough to preserve the agreement of neutrino observa-tions with theory.That is,L newchannel≤1053ergss−1.This idea was earlier used to put the strongest experimental upper bounds on the axion mass[16].Here,we will con-sider emission of higher-dimensional gravitons from the core.Once these particles areproduced,they escape into extra dimensions,carrying energy away with them.The con-straint on luminosity of this process can be converted into a bound on the M D.The argument is very similar to that used to bound the axion-nucleon coupling strength [16,17,18,19].The“standard model"of supernovae does an exceptionally good job of predicting the duration and shape of the neutrino pulse from SN1987A.Any mecha-nism which leads to signiﬁcant energy-loss from the core of the supernova immediately after bounce will produce a very different neutrino-pulse shape,and so will destroy this agreement[18].Raffelt has proposed a simple analytic criterion based on detailed su-pernova simulations[19]:if any energy-loss mechanism has an emissivity greater than 1019ergs g−1s−1then it will remove sufﬁcient energy from the explosion to invalidate the current understanding of Type-II supernovae’s neutrino signal.CONSTRAINTS ON EXTRA DIMENSIONSThe most restrictive limits on M D come from SN1987A energy-loss argument.If large extra dimensions exist,the usual four dimensional graviton is complemented by a tower of Kaluza-Klein(KK)states,corresponding to new phase space in the bulk.The KK gravitons interact with the strength of ordinary gravitons and thus are not trapped in the SN core.Each KK graviton state couples to the SMﬁeld with the usual gravitational strength according to[20]κL=−and the summation is over all KK states labeled by the levelM2Pln.Tµνis the energy-momentum tensor of the SMﬁelds residing on the brane and hµν, n the KK state.Since for large R the KK gravitons are very light,they may be copiously produced in high energy processes.For real emission of the KK gravitons from a SM ﬁeld,the total cross-section can be written asσ( n),(4)σtot=κ2∑nwhere the dependence on the gravitational coupling is factored out.Because the mass separation of adjacent KK states,O(1/R),is usually much smaller than typical energies in a physical process,we can approximate the summation by an integration.Since we are concerned with the energy loss to gravitons escaping into the extra dimensions,it is convenient and standard to deﬁne the quantities˙εa+b→c which are the rate at which energy is lost to gravitons via the process a+b→c,per unit time per unit mass of the stellar object.In terms of the cross-sectionσa+b→c the number densities n a,b for a,b and the mass densityρ,˙εis given byn a n bσ(a+b→c)v rel E c˙εa+b→c.=TABLE1.Bounds from Nucleon-Nucleon Brehmstrahlung process in SN1987A M D n=2505123.4318420.1 (TeV)n=34 3.6 1.51 2.757 1.26During theﬁrst few seconds after collapse,the core contains neutrons,protons,elec-trons,neutrinos and thermal photons.There are a number of processes in which KK gravitons can be produced.For the conditions that pertain in the core at this time (T∼30−70MeV,ρ∼(3−10)×1014g cm−3),the relevant processes are •Nucleon-Nucleon Brehmstrahlung:NN→NNG KK•Graviton production in photon fusion:γγ→G KK,and•Electron-positron anhilation process:e−e+→G KK.In the SNe,nucleon and photon abundances are comparable(actually nucleons are somewhat more abundant).Nucleon-nucleon bremhmstrahlung is the dominant process relevant for the SN1987A where the temperature is comparable to mπand so the strong interaction between N’s is unsuppressed.In the following we present the bounds derived by various authors based on this process.CONCLUSIONSIn summary,it has been found that KK graviton emission from SN1987A puts very strong constraints on models with large extra dimensions in the case n=2.In this case, for a conservative choice of the core parameters we arrive at a bound on the M D≥30 TeV.We have done similar calculations in the case of plasmons which will be reported elsewhere.Even though taking into account various uncertainties encountered in the calculation can weaken this bound,it is unlikely that it can be pushed down to the phenomenologically interesting range of a few TeV.Therefore this case is still viable for solving the hierarchy problem and accessible to being tested at the LHC.AcknowledgmentsOne of the authors(VHS)would like to thank the organisers of IWTHEP-2007 for their hospitality and for providing an excellent intellectual atmosphere.He thanks Professor R K Kaul for fruitful discussions and hosting at IMSc where part of this article was written under the Summer Research Fellowship of IASc,INSA and NASI.He also thanks Dr.R Chenraj Jain,Mr.K L Ganesh Sharma and Mr.V Venkatachalam for their encouragement.REFERENCES1.N.Arkani-Hamed,S.Dimopoulos and G.R.Dvali,Phys.Lett.B429,263(1998)[arXiv:hep-ph/9803315].2.See J.C.Long et.al.,Nature421,922(2003)and the references therein.3.N.Arkani-Hamed,S.Dimopoulos and G.R.Dvali,Phys.Rev.D59,086004(1999)[arXiv:hep-ph/9807344].I.Antoniadis,N.Arkani-Hamed,S.Dimopoulos and G.Dvali,Phys.Lett.B463,257(1998).4.S.Cullen and M.Perelstein,Phys.Rev.Lett.83,268(1999)[arXiv:hep-ph/9903422].5.V. D.Barger,T.Han, C.Kao and R.J.Zhang,Phys.Lett.B461,34(1999)[arXiv:hep-ph/9905474].6. C.Hanhart,D.R.Phillips,S.Reddy and M.J.Savage,Nucl.Phys.B595,335(2001)[arXiv:nucl-th/0007016].7. C.Hanhart,J. A.Pons, D.R.Phillips and S.Reddy,Phys.Lett.B509,1(2001)[arXiv:astro-ph/0102063].8.S.Hannestad and G.Raffelt,Phys.Rev.Lett.87,051301(2001)[arXiv:hep-ph/0103201].9.S.Hannestad and G.G.Raffelt,Phys.Rev.D67,125008(2003)[Erratum-ibid.D69(2004)029901][arXiv:hep-ph/0304029].10.M.Fairbairn,L.M.Grifﬁths,JHEP0202,024(2002).11.See for a short review on supernovae P K Suresh and V H Satheesh Kumar,Sci.Rep.40,20(2001)[arXiv:astro-ph/0504597].12.K.Hirata et.al.,Phys.Rev.Lett.58,1490(1987).13.R.M.Bionta et.al.,Phys.Rev.Lett.58,1494(1987).14.T.Totani,K.Sato,H. E.Dalhed and J.R.Wilson,Astrophys.J.496216(1998)[arXiv:astro-ph/9710203].15. B.Jegerlehner, F.Neubig and G.Raffelt,Phys.Rev.D541194(1996)[arXiv:astro-ph/9601111].16.R.Mayle et.al.,Phys.Lett.B203188(1988);G.G.Raffelt and D.Seckel,Phys.Rev.Lett.60,1793(1988);M.S.Turner,Phys.Rev.Lett.60,1797(1988).17.R.P.Brinkmann and M.S.Turner,Phys.Rev.D38,2338(1988);H.-T.Janka,W.Keil,G.Raffelt,and D.Seckel,Phys.Rev.Lett.76,2621(1996)[arXiv:];W.Keil,H.-T.Janka,D.N.Schramm,G.Sigl,M.S.Turner,and J.Ellis,Phys.Rev.D56,2419(1997)[arXiv:].18. A.Burrows,R.P.Brinkmann,and M.S.Turner,Phys.Rev.D39,1020(1989).19.G.G.Raffelt,Stars as Laboratories for Fundamental Physics,(Chicago University Press)(1996).20.T.Han,J.D.Lykken and R.Zhang:Phys.Rev.D59,105006(1999).[arXiv:hep-ph/9811350];G. F.Giudice,R.Rattazzi and J. D.Wells,Nucl.Phys.B544,3(1999)[arXiv:hep-ph/9811291].。

standard model标准模型

CPT violation and the standard modelDon Colladay and V.Alan Kostelecky´Department of Physics,Indiana University,Bloomington,Indiana47405͑Received22January1997͒Spontaneous CPT breaking arising in string theory has been suggested as a possible observable experimen-tal signature in neutral-meson systems.We provide a theoretical framework for the treatment of low-energy effects of spontaneous CPT violation and the attendant partial Lorentz breaking.The analysis is within the context of conventional relativistic quantum mechanics and quantumﬁeld theory in four dimensions.We use the framework to develop a CPT-violating extension to the minimal standard model that could serve as a basis for establishing quantitative CPT bounds.͓S0556-2821͑97͒05211-9͔PACS number͑s͒:11.30.Er,11.25.Ϫw,12.60.ϪiI.INTRODUCTIONAmong the symmetries of the minimal standard model is invariance under CPT.Indeed,CPT invariance holds under mild technical assumptions for any local relativistic point-particleﬁeld theory͓1–5͔.Numerous experiments have con-ﬁrmed this result͓6͔,including in particular high-precision tests using neutral-kaon interferometry͓7,8͔.The simulta-neous existence of a general theoretical proof of CPT invari-ance in particle physics and accurate experimental tests makes CPT violation an attractive candidate signature for nonparticle physics such as string theory͓9,10͔.The assumptions needed to prove the CPT theorem are invalid for strings,which are extended objects.Moreover, since the critical string dimensionality is larger than four,it is plausible that higher-dimensional Lorentz breaking would be incorporated in a realistic model.In fact,a mechanism is known in string theory that can cause spontaneous CPT vio-lation͓9͔with accompanying partial Lorentz-symmetry breaking͓11͔.The effect can be traced to string interactions that are absent in conventional four-dimensional renormaliz-able gauge theory.Under suitable circumstances,these inter-actions can cause instabilities in Lorentz-tensor potentials, thereby inducing spontaneous CPT and Lorentz breaking.If in a realistic theory the spontaneous CPT and partial Lorentz violation extend to the four-dimensional spacetime,detect-able effects might occur in interferometric experiments with neutral kaons͓9,10͔,neutral B d or B s mesons͓10,12͔,or neutral D mesons͓10,13͔.For example,the quantities pa-rametrizing indirect CPT violation in these systems could be nonzero.There may also be implications for baryogenesis ͓14͔.In the present paper,our goal is to develop within an effective-theory approach a plausible CPT-violating exten-sion of the minimal standard model that provides a theoreti-cal basis for establishing quantitative bounds on CPT invari-ance.The idea is to incorporate notions of spontaneous CPT and Lorentz breaking while maintaining the usual gauge structure and properties like renormalizability.To achieve this,weﬁrst establish a conceptual framework and a procedure for treating spontaneous CPT and Lorentz viola-tion in the context of conventional quantum theory.We seek a general methodology that is compatible with desirable fea-tures like microscopic causality while being sufﬁciently de-tailed to permit explicit calculations.We suppose that underlying the effective four-dimensional action is a complete fundamental theory that is based on conventional quantum physics͓15͔and is dynami-cally CPT and Poincare´invariant.The fundamental theory is assumed to undergo spontaneous CPT and Lorentz breaking. In a Poincare´-observer frame in the low-energy effective ac-tion,this process is taken toﬁx the form of any CPT-and Lorentz-violating terms.Since interferometric tests of CPT violation are so sensi-tive,we focus speciﬁcally on CPT violation and the associ-ated Lorentz-breaking issues in a low-energy effective theory without gravity͓21͔.For the most part,effects from deriva-tive couplings and possible CPT-preserving but Lorentz-breaking terms in the action are disregarded,and any CPT-violating terms are taken to be small enough to avoid issues with standard experimental tests of Lorentz symmetry.A partial justiﬁcation for the latter assumption is that the absence of signals for CPT violation in the neutral-kaonsystem provides one of the best bounds on Lorentz invari-ance.Our focus on the low-energy effective model bypassesvarious important theoretical issues regarding the structure ofthe underlying fundamental theory and its behavior at scalesabove electroweak uniﬁcation,including the origin and ͑renormalization-group͒stability of the suppression of CPT breaking and the issue of modeﬂuctuations around Lorentz-tensor expectation values.Since these topics involve the Lor-entz structure of the fundamental theory,they are likely to berelated to the difﬁcult hierarchy problems associated withcompactiﬁcation and the cosmological constant.The ideas underlying our theoretical framework are de-scribed in Sec.II.A simple model is used to illustrate con-cepts associated with CPT and Lorentz breaking,including the possibility of eliminating some CPT-violating effects throughﬁeld redeﬁnitions.The associated relativistic quan-tum mechanics is discussed in Sec.III.Section IV contains a treatment of some issues in quantumﬁeld theory.A CPT-violating extension of the minimal standard model is provided in Sec.V,and the physically observable subset ofPHYSICAL REVIEW D1JUNE1997VOLUME55,NUMBER11550556-2821/97/55͑11͒/6760͑15͒/$10.006760©1997The American Physical SocietyCPT-breaking terms is established.We summarize in Sec. VI.Some of the more technical results are presented in the appendixes.II.BASICSA.Effective model for spontaneous CPT violationWe begin our considerations with a simple model within which many of the basic features of spontaneous CPT vio-lation can be examined.The model involves a single massive Diracﬁeld␺(x)in four dimensions with Lagrangian densityLϭL0ϪLЈ,͑1͒where L0is the usual free-ﬁeld Dirac Lagrangian for a fer-mion␺of mass m,and where LЈcontains extra CPT-violating terms to be described below.For the present discussion,we follow an approach in which the C,P,T and Lorentz properties of␺are assumed to be conventionally determined by the free-ﬁeld theory L0and are used to estab-lish the corresponding properties of LЈ͓22͔.This method is intrinsically perturbative,which is particularly appropriate here since any CPT-violating effects must be small.In Sec. II C,we consider the possibility of alternative deﬁnitions of C,P,T and Lorentz properties that could encompass the full structure of L.We are interested in possible forms of LЈthat could arise as effective contributions from spontaneous CPT violation in a more complete theory.To our knowledge,string theory forms the only class of͑gauge͒theories in four or more dimensions that are quantum consistent,dynamically Poin-care´invariant,and known to admit an explicit mechanism ͓9͔for spontaneous CPT violation triggered by interactions in the Lagrangian.However,to keep the treatment as general as possible we assume only that the spontaneous CPT vio-lation arises from nonzero expectation values acquired by one or more Lorentz tensors T,so LЈis taken to be an effective four-dimensional Lagrangian obtained from an un-derlying theory involving Poincare´-invariant interactions of ␺with T.The discussion that follows is independent of any speciﬁcs of string theory and should therefore be relevant to a nonstring model with spontaneous CPT violation,if such a model is eventually formulated.Even applying the stringent requirement of dynamical Poincare´invariance,an unbroken realistic theory can in prin-ciple include terms with derivatives,powers of tensorﬁelds, and powers of various terms quadratic in fermionﬁelds. However,any CPT-breaking term that is to be part of a four-dimensional effective theory must have mass dimension four.In the effective Lagrangian,each combination ofﬁelds and derivatives of dimension greater than four therefore must have a corresponding weighting factor of a negative power Ϫk of at least one mass scale M that is large compared to the scale m of the effective theory.In a realistic theory with the string scenario,M might be the Planck mass or perhaps a smaller mass scale associated with compactiﬁcation and uni-ﬁcation.Moreover,since the expectations͗T͘of the tensors T are assumed to be Lorentz and possibly CPT violating, any terms that survive in LЈafter the spontaneous symmetry breaking must on physical grounds be suppressed,presum-ably by at least one power of m/M relative to the scale of the effective theory.A hierarchy of possible terms in LЈthus emerges,labeled by kϭ0,1,2,....Omitting Lorentz indices for simplicity,the leading terms with kр2have the schematic formLЈʛ␭M k͗T͘•␺¯⌫͑iץ͒k␺ϩH.c.͑2͒In this expression,the parameter␭is a dimensionless cou-pling constant,(iץ)k represents k four-derivatives acting in some combination on the fermionﬁelds,and⌫represents some gamma-matrix structure.Terms with kу3and with more quadratic fermion factors also appear,but these are further suppressed.Note that contributions of the form͑2͒arise in string theory͓10͔.Note also that naive power count-ing indicates the dominant terms with kр1are renormaliz-able.For kϭ0,the above considerations indicate that the domi-nant terms of the form͑2͒must have expectations͗T͘ϳm2/M.In the present work,we focus primarily on this relatively simple case.Most of the general features arising from CPT and Lorentz violation together with some of our more speciﬁc results remain valid when terms with other values of k are considered,but it remains an open issue to investigate the detailed properties of terms with kϭ1and expectations͗T͘ϳm or those with kϭ2and expectations ͗T͘ϳM.Both these could in principle contribute leading effects in the low-energy effective action.Each contribution to LЈfrom an expression of the form ͑2͒is a fermion bilinear involving a4ϫ4spinor matrix⌫. Regardless of the complexity and number of the tensors T inducing the breaking,⌫can be decomposed as a linear com-bination of the usual16basis elements of the gamma-matrix algebra.Only the subset of these that produce CPT-violating bilinears are of interest for our present pur-poses,and they permit us to provide explicit and relatively simple expressions for the possible CPT-violating contribu-tions to LЈ.For the case kϭ0of interest here,weﬁnd two possible types of CPT-violating term:L aЈϵa␮␺¯␥␮␺,L bЈϵb␮␺¯␥5␥␮␺.͑3͒For completeness,we provide here also the terms appearing for the case kϭ1,where weﬁnd three types of relevant contribution:L cЈϵ12ic␣␺¯ץJ␣␺,L dЈϵ12d␣␺¯␥5ץJ␣␺,L eЈϵ12ie␮␯␣␺¯␴␮␯ץJ␣␺,͑4͒where AץJ␮BϵAץ␮BϪ(ץ␮A)B.In all these expressions,the quantities a␮,b␮,c␣,d␣,and e␮␯␣must be real as conse-quences of their origins in spontaneous symmetry breaking and of the presumed hermiticity of the underlying theory. They are combinations of coupling constants,tensor expec-tations,mass parameters,and coefﬁcients arising from the decomposition of⌫.In keeping with their interpretation as effective coupling constants arising from a scenario with spontaneous symme-try breaking,a␮,b␮,c␣,d␣,and e␮␯␣are invariant under CPT transformations.Together with the standard556761CPT VIOLATION AND THE STANDARD MODELCPT-transformation properties ascribed to␺,this invariance causes the terms in Eqs.͑3͒and͑4͒to break CPT͓23͔.As discussed above,in the remainder of this work we restrict ourselves largely to the expressions in Eq.͑3͒.Allowing both kinds of term in Eq.͑3͒to appear in LЈproduces a model Lagrangian of the formLϭ12i␺¯␥␮ץJ␮␺Ϫa␮␺¯␥␮␺Ϫb␮␺¯␥5␥␮␺Ϫm␺¯␺.͑5͒The variational procedure generates a modiﬁed Dirac equa-tion:͑i␥␮ץ␮Ϫa␮␥␮Ϫb␮␥5␥␮Ϫm͒␺ϭ0.͑6͒Associated with this Dirac-type equation is a modiﬁed Klein-Gordon equation.Proceeding with the usual squaring procedure,in which the Dirac-equation operator with oppo-site mass sign is applied to the Dirac equation from the left, leads to the Klein-Gordon-type expression͓͑iץϪa͒2Ϫb2Ϫm2ϩ2i␥5␴␮␯b␮͑iץ␯Ϫa␯͔͒␺͑x͒ϭ0.͑7͒This equation is second order in derivatives,but unlike the usual Klein-Gordon case it contains off-diagonal terms in the spinor space.These may be eliminated by repeating the squaring procedure,this time applying the operator in Eq.͑7͒with opposite sign for the off-diagonal piece.The result is a fourth-order equation satisﬁed by each spinor component of any solution to the modiﬁed Dirac equation:͕͓͑iץϪa͒2Ϫb2Ϫm2͔2ϩ4b2͑iץϪa͒2Ϫ4͓b␮͑iץ␮Ϫa␮͔͒2͖␺͑x͒ϭ0.͑8͒B.Continuous symmetriesConsider next the continuous symmetries of the model with Lagrangian͑5͒.For deﬁniteness,we begin with an analysis in a given oriented inertial frame in which values of the quantities a␮and b␮are assumed to have been speciﬁed. The effects of rotations and boosts are considered later.The CPT-violating terms in Eq.͑5͒leave unaffected the usual global U͑1͒gauge invariance,which has conserved current j␮ϭ␺¯␥␮␺.Charge is therefore conserved in the model.These terms also leave unaffected the usual breaking of the chiral U͑1͒current j5␮ϭ␺¯␥5␥␮␺due to the mass term. In what follows,we denote the volume integrals of the cur-rent densities j␮and j5␮by J␮and J5␮,respectively.The model is also invariant under translations provided the tensor expectations are assumed constant,i.e.,provided the possibility of CPT-breaking soliton-type solutions in the underlying theory is disregarded.This leads to a conserved canonical energy-momentum tensor⌰␮␯given by⌰␮␯ϭ12i␺¯␥␮ץJ␯␺,ץ␮⌰␮␯ϭ0,͑9͒and a corresponding conserved four-momentum P␮.These expressions have the same form as in the free theory.Note, however,that constancy of the energy and momentum does not necessarily imply conventional behavior under boosts or rotations.Note also that the presence of the CPT-violating terms in the Dirac equation destroys the usual symmetriz-ability property of⌰␮␯.The antisymmetric part⌰͓␮␯͔is⌰͓␮␯͔ϵ⌰␮␯Ϫ⌰␯␮ϭϪ14ץ␣͓␺¯͕␥␣,␴␮␯͖␺͔Ϫa[␮j␯]Ϫb[␮j5␯],͑10͒which is no longer a total divergence.The conventional con-struction of a symmetric energy-momentum tensor,involv-ing a subtraction of this antisymmetric part from the canoni-cal energy-momentum tensor,would affect the conserved energy and momentum and is therefore presumably inappli-cable in the present case.The implications of this for a more complete low-energy effective theory that includes gravity remain to be explored.Next,consider the effect of Lorentz transformations,i.e., rotations and boosts.Conventional Lorentz transformations in special relativity relate observations made in two inertial frames with differing orientations and velocities.These transformations can be implemented as coordinate changes, and we call them observer Lorentz transformations.It is also possible to consider transformations that relate the properties of two particles with differing spin orientation or momentum within a speciﬁc oriented inertial frame.We call these par-ticle Lorentz transformations.For free particles under usual circumstances,the two kinds of transformation are͑in-versely͒related.However,this equivalence fails for particles under the action of a backgroundﬁeld.The reader is warned to avoid confusing observer Lorentz transformations͑which involve coordinate changes͒or par-ticle Lorentz transformations͑which involve boosts on par-ticles or localizedﬁelds but not on backgroundﬁelds͒with a third type of Lorentz transformation that within a speciﬁed inertial frame boosts all particles andﬁelds simultaneously, including background ones.The latter are sometimes called ͑inverse͒active Lorentz transformations.For the case of con-ventional free particles,they coincide with particle Lorentz transformations.We have chosen to avoid applying the terms active and passive here because they are insufﬁcient to dis-tinguish the three kinds of transformation and because in any case their interpretation varies in the literature.The distinction between observer and particle transforma-tions is relevant for the present model,where the CPT-violating terms can be regarded as arising from con-stant backgroundﬁelds a␮and b␮.The point is that these eight quantities transform as two four-vectors under observer Lorentz transformations and as eight scalars under particle Lorentz transformations,whereas they are coupled to cur-rents that transform as four-vectors under both types of trans-formation.This means that observer Lorentz symmetry is still an invariance of the model,but the particle Lorentz group is͑partly͒broken.Physical situations with features like this can readily be identiﬁed.For example,an electron with momentum perpen-dicular to a uniform background magneticﬁeld moves in a circle.Suppose in the same observer frame we instanta-neously increase the magnitude of the electron momentum without changing its direction,causing the electron to move in a circle of larger radius.This͑instantaneous͒particle boost676255DON COLLADAY AND V.ALAN KOSTELECKY´leaves the backgroundﬁeld unaffected.However,if insteadan observer boost perpendicular to the magneticﬁeld is ap-plied,the electron no longer moves in a circle.This is viewed in the new inertial frame as an EϫB drift caused bythe presence of an electricﬁeld.In this example,the back-ground magneticﬁeld transforms into a different electromag-neticﬁeld under observer boosts but͑by deﬁnition͒is un-changed by particle boosts,in analogy to the transformation of a␮and b␮in the CPT-violating model.From the viewpoint of this example,the unconventional aspect of the CPT-violating model is merely that the con-stantﬁelds a␮and b␮are a global feature of the model.Theycannot be regarded as arising from localized experimentalconditions,which would cause them to transform under par-ticle Lorentz transformations as four-vectors rather than as scalars.The behavior of a␮and b␮as backgroundﬁelds andhence as scalars under particle Lorentz transformations is aconsequence of their origin as nonzero expectation values ofLorentz tensors in the underlying theory.These Lorentz-tensor expectations break those parts of the particle Lorentzgroup that cannot be implemented as unitary transformationson the vacuum.This is in parallel with other situations in-volving spontaneous symmetry breaking,such as ones com-monly encountered in the treatment of internal symmetries.The preservation of observer Lorentz symmetry is an im-portant feature of the model.It is a consequence of observerLorentz invariance of the underlying fundamental theory.This symmetry is unaffected by the appearance of tensorexpectation values by virtue of its implementation via coor-dinate transformations.As an illustration of its use in theeffective model,we show that it permits a further classiﬁca-tion of types of CPT-violating term according to the ob-server Lorentz properties of a␮and b␮.Thus,for example, if b␮is future timelike in one inertial frame,it must be futuretimelike in all frames.This implies that a class of inertial frames can be found in which b␮ϭb(1,0,0,0),where calcu-lations are potentially simpliﬁed.A similar argument for the lightlike or spacelike cases shows that the CPT-violating physics of the four components of b␮can in each case bereduced to knowledge of its Lorentz type and a single num-ber specifying its magnitude.Inertial frames within this ideal class are determined by the little group of b␮,which can in turn be used to simplify͑partially͒the form of a␮.The reader is cautioned that the class of inertial framesselected in this way may be distinct from experimentallyrelevant inertial frames such as,for example,those deﬁnedusing the microwave background radiation and interpretingthe dipole component in terms of the motion of the Earth.The point is that,given an inertial frame,the process ofspontaneous Lorentz violation in the underlying theory is assumed to produce some values of a␮and b␮.In this spe-ciﬁc inertial frame,there is no reason a priori why these values should take the ideal form described above.One is merely assured of the existence of some frame in which the ideal form can be attained.The current J␭␮␯for particle Lorentz transformationstakes the usual form when expressed in terms of the energy-momentum tensor:J␭␮␯ϭx[␮⌰␭␯]ϩ14␺¯͕␥␭,␴␮␯͖␺.͑11͒This current is conserved at the level of the underlying theory with spontaneous symmetry breaking,but in the ef-fective low-energy theory where the spontaneous breaking appears as an explicit symmetry violation the conservation property is destroyed.In the latter case,the corresponding Lorentz charges M␮␯obeydM␮␯dtϭϪa[␮J␯]Ϫb[␮J5␯].͑12͒Given explicit values of a␮and b␮in some inertial frame, Eq.͑12͒can be used directly to determine which Lorentz symmetries are violated.Note that if either a␮or b␮van-ishes,the Lorentz group is broken to the little group of the nonzero four-vector.This means that the largest Lorentz-symmetry subgroup that can remain as an invariance of the model Lagrangian͑5͒is SO͑3͒,E͑2͒,or SO͑2,1͒.Since a␮and b␮represent two four-vectors in four-dimensional space-time,they can deﬁne a two-dimensional plane.Transforma-tions involving the two orthogonal dimensions have no effect on this plane.This means that the smallest Lorentz-symmetry subgroup that can remain is a compact or noncom-pact U͑1͒.In a realistic low-energy effective theory,CPT-violating terms would break the particle Lorentz group in a manner related to the breaking given by Eq.͑12͒.Since no zeroth-order CPT violation has been observed in experiments, CPT-violating effects in the string scenario are expected to be suppressed by at least one power of the Planck mass rela-tive to the scale of the effective theory.However,the inter-esting and involved issue of exactly how small the magni-tudes of a␮and b␮͑or their equivalents in a realistic model͒must be to satisfy current experimental constraints lies be-yond the scope of the present work.We conﬁne our remarks here to noting that the partial breaking of particle Lorentz invariance discussed above generates an effective boost de-pendence in the CPT-breaking parameters.This could pro-vide a deﬁnite experimental signature for our framework if CPT violation were detected at some future date.C.Field redeﬁnitionsFor the discussions in the previous subsections,we adopted a practical approach to the deﬁnition of CPT and Lorentz transformations.It involves treating C,P,T and Lorentz properties of␺as being deﬁned via the free-ﬁeld theory L0and subsequently using them to establish the sym-metry properties of LЈ.This approach requires caution,how-ever,because in principle alternative deﬁnitions of the sym-metry transformations could exist that would leave the full theory L invariant.Considerﬁrst an apparently CPT-and Lorentz-violating model formed with a␮only,deﬁned in a given inertial frame by the LagrangianL͓␺͔ϭL0͓␺͔ϪL aЈ͓␺͔.͑13͒Introducing in this frame aﬁeld redeﬁnition of␺by a spacetime-dependent phase,␹ϭexp͑ia•x͒␺,͑14͒556763CPT VIOLATION AND THE STANDARD MODELthe Lagrangian expressed in terms of the newﬁeld is L͓␺ϭexp(Ϫia•x)␹͔ϵL0͓␹͔.This shows that the model is equivalent to a conventional free Dirac theory,in which there is no CPT or Lorentz breaking,and thereby provides an example of redeﬁning symmetry transformations to main-tain invariance͓24͔.The connection between the Poincare´generators in the two forms of the theory can be found explicitly by substitut-ing␺ϭ␺͓␹͔in the Poincare´generators for L͓␺͔and ex-tracting the combinations needed to reproduce the usual Poincare´generators for L0͓␹͔.Weﬁnd that the charge and chiral currents j␮and j5␮take the same functional forms in both theories but that the form of the canonical energy-momentum tensor changes,⌰␮␯ϭ12i␹¯␥␮ץJ␯␹ϩa␯␹¯␥␮␹,͑15͒producing a corresponding change in the Lorentz current J␭␮␯.This means that in the original theory L͓␺͔we could introduce modiﬁed Poincare´currents⌰˜␮␯and J˜␭␮␯that have corresponding conserved charges generating an unbro-ken Poincare´algebra.These currents are given as functionals of␺by⌰˜␮␯ϭ⌰␮␯Ϫa␯j␮,J˜␭␮␯ϭJ␭␮␯Ϫx[␮a␯]j␭.͑16͒The existence of this connection between the two theories depends critically on the existence of the conserved current j␮.In the model͑5͒with both a␮and b␮terms,the compo-nent L aЈcan be eliminated by aﬁeld redeﬁnition as before but there is no similar transformation removing L bЈbecauseconservation of the chiral current j5␮is violated by the mass. In the massless limit of this model the chiral current is con-served,and we can eliminate both a␮and b␮via theﬁeld redeﬁnition␹ϭexp͑ia•xϪib•xy5͒␺.͑17͒For the situation with m 0,however,this redeﬁnition would introduce spacetime-dependent mass parameters.The term L aЈin Eq.͑3͒is reminiscent of a local U͑1͒coupling,although there is no local U͑1͒invariance in the theory͑5͒.It is natural and relevant to our later consider-ations of the standard model to ask how the above discussion ofﬁeld redeﬁnitions is affected if the U͑1͒invariance of the original theory is gauged.Then,the term L aЈhas the same form as a coupling to a constant background electromagnetic potential.At the classical level,this would be expected to have no effect since it is pure gauge.However,a conven-tional quantum-ﬁeld gauge transformation involving both␺and the electromagnetic potential A␮cannot eliminate a␮, since the theory is invariant under such transformations.In-stead,the electromagneticﬁeld can be taken as the sum of a classical c-number backgroundﬁeld A␮and a quantumﬁeld A␮,whereupon a␮can be regarded as contributing to an effective A␮.Conventional classical gauge transformations can be performed on the c-number potential A␮,while leav-ing the quantumﬁelds␺and A␮unaffected.This changes the Lagrangian but should not change the physics.In fact, the resulting gauge-transformed Lagrangian is unitarily equivalent to the original one under aﬁeld redeﬁnition on␺of the form discussed above for the ungauged model.To summarize,in the gauged theory the CPT-breaking term L aЈcan be interpreted as a background gauge choice and eliminated via aﬁeld redeﬁnition as in the ungauged case. We note in passing that related issues arise for certain non-linear gauge choices͓25͔and in the context of efforts to interpret the photon as a Nambu-Goldstone boson arising from͑unphysical͒spontaneous Lorentz breaking͓26–32͔.In typical models of the latter type,a four-vector bilinear con-densate͗␺¯␥␮␺͘plays a role having some similarities to that of a␮.The model͑5͒involves only a single fermionﬁeld.All CPT-violating effects can also be removed from certain theories describing more than one fermionﬁeld in which each fermion has a term of the form L aЈ.For example,this is possible if there is no fermion mixing and each such CPT-violating term involves the same value of a␮,or if the fermions have no interactions or mixings that acquire spacetime-dependence upon performing theﬁeld redeﬁni-tions.However,in generic multifermion theories with CPT violation involving fermion-bilinear terms,it is impossible to eliminate all CPT-breaking effects throughﬁeld redeﬁni-tions.Nonetheless,since Lagrangian terms that spontane-ously break CPT necessarily involve paired fermionﬁelds, at least one of the quantities a␮can be removed.This means that only differences between values of a␮are observable. Examples appear in the context of the CPT-violating exten-sion of the standard model discussed in Sec.V.III.RELATIVISTIC QUANTUM MECHANICS In this section,we discuss some aspects of relativistic quantum mechanics based on Eq.͑6͒,with␺regarded as a four-component wave function.The results obtained provide further insight into the nature of the CPT-violating terms and are precursors to the quantumﬁeld theory.The analo-gous treatment in the context of the standard model involves several fermionﬁelds,for which CPT-violating terms of the form L aЈcannot be altogether eliminated.We therefore ex-plicitly include the quantity a␮in the following analysis, even though it could be eliminated by aﬁeld redeﬁnition for the simple one-fermion case.In fact,the reinterpretation of negative-energy solutions causes the explicit effects of a␮to be more involved than might otherwise be expected.The modiﬁed Dirac equation͑6͒can be solved by assum-ing the usual plane-wave dependence,␺͑x͒ϭeϪi␭␮x␮w͑␭ជ͒.͑18͒In this equation,w(␭ជ)is a four-component spinor satisfying ͑␭␮␥␮Ϫa␮␥␮Ϫb␮␥5␥␮Ϫm͒w͑␭ជ͒ϭ0.͑19͒For a nontrivial solution to exist,the determinant of the ma-trix acting on w(␭ជ)in this equation must vanish.This means that␭␮ϵ(␭0,␭ជ),where͓33͔␭0ϭ␭0(␭ជ),must satisfy the requirement͓͑␭Ϫa͒2Ϫb2Ϫm2͔2ϩ4b2͑␭Ϫa͒2Ϫ4͓b␮͑␭␮Ϫa␮͔͒2ϭ0.͑20͒676455DON COLLADAY AND V.ALAN KOSTELECKY´。

Large extra dimension effects in Higgs boson production at linear colliders and Higgs facto

a r X i v :h e p -p h /0303259v 2 3 O c t 2003HIP-2003-19/TH ROME1-1352/2003Large extra dimension eﬀects in Higgs boson production at linear colliders and Higgs factoriesAnindya Datta 1,Emidio Gabrielli 1,and Barbara Mele 21Helsinki Institute of Physics,POB 64,University of Helsinki,FIN 00014,Finland 2Istituto Nazionale di Fisica Nucleare,Sezione di Roma,and Dip.di Fisica,Universit`a La Sapienza,P.le A.Moro 2,I-00185Rome,Italy1IntroductionIn recent years much attention has been paied to theories where the weakness of thegravitational coupling is explained by the presence of large compact extra spatial dimensions,as shown in [1].In such theories,while standard model (SM)ﬁelds are conﬁned in the usual 4-dimensional space,the gravity can propagate in the full high dimensional space,and its intensity is diluted in the large volume of the extra dimensions.1The Newton’s constant G N in the3+1dimensional space is then related to the corresponding Planck scale M D in the D=4+δdimensional space by(1)G−1N=8πRδM2+δDwhere R is the radius of a compact manifold assumed to be on a torus.According to the present limits on Newton’s law[2],one could have M D∼1TeV if the number of extra dimensions isδ≥2.A crucial consequence of this framework is that quantum gravity eﬀects could become strong at the TeV scale and measurable at future high-energy colliders.Af-ter integrating out the compact extra dimensions,the eﬀective Einstein theory in 3+1dimensions reliably describes the interactions of the extra-dimensional gravitons with gauge and matterﬁelds[3,4,5].An essentially continuum spectrum of massive Kaluza-Klein(KK)excitations of the standard gravitonﬁeld arises,forδnot larger than about6.When summing over the allowed spectrum of KK states either in the inclusive production or in the exchange of virtual KK gravitons,the small coupling (E/M P)2associated to a single graviton production/exchange(where E is the typ-ical energy of the process and M P is the Plank mass)is replaced by the quantity (E/M D)2+δ.Then,for M D∼1TeV,processes involving gravitons could well be detected at present and future high-energy colliders.This possibility has been quite thoroughly explored in a number of papers[3]-[8].Regarding processes with virtual KK graviton exchange,it is well known that in general the corresponding amplitude is divergent and not computable in the eﬀective theory[3].In particular,the real part of the amplitude,Re[A],needs an ultraviolet cut-oﬀ.This means that in general the theory is not predictive in the sector of virtual KK graviton exchange.On the other hand,the imaginary part of the amplitude, Im[A],isﬁnite and cutoﬀindependent,being connected to the branch-cut singularity of real graviton emission[3].In a recent paper[8],we stressed that this can have important consequences,when considering standard model(SM)resonant processes interfering with virtual KK graviton exchange graphs.In fact,the corresponding interference,that is dominated by Im[A],turns out to beﬁnite,and predictable in terms of the fundamental Plank scale M D and the number of extra dimensionsδ.In[8],we applied this observation to LEP physics,and computed the eﬀects on the e+e−→f¯f physical observables of the interference of the virtual KK graviton-exchange amplitude with the resonant SM amplitude at the Z boson pole.We found that,although the corresponding impact on total cross-sections vanishes,there are ﬁnite modiﬁcations of diﬀerent asymmetries,whose relative eﬀect amounts,in the most favorable cases,to about10−4.2In the present paper,we want to extend this approach to the case of a heavy Higgsboson(H)production at future linear e+e−colliders andµµcolliders.By heavy,wewill imply m H≥200GeV.Graviton interference eﬀects on cross-sections turns out to be proportional to the ratio of the total width over the mass of the resonance state,dueto the imaginary part of the SM amplitude[8].Then,one expects toﬁnd remarkablymore conspicuous graviton interference eﬀects in a heavy Higgs boson productionthan in the Z0boson pole physics,due to the rapidly growing Higgs width with theHiggs mass.One can compare,e.g.,ΓZ/M Z≃0.027withΓH/m H≃0.072,0.29 for m H=400,700GeV,respectively.Moreover,the imaginary part of the graviton amplitude grows quite rapidly with the process c.o.m.energy.+P_P e+W+W−e−W +e+W −Σn He−P_Pνν−νν−G Sn n,Figure1:Feynman diagrams of the processes in Eq.(2).At linear e+e−colliders[9],we consider the Higgs production via vector boson fusion with the subsequent H decay into pairs of heavy particlese+e−(W W)→ν¯νH→ν¯νW W,ν¯νZZ,ν¯νt¯t.(2)Feynman diagrams for these processes are presented in Figure1.This process is one of the dominant H production mechanism at linear colliders,and becomes the main one at√S≫M W can be reliably treated in the eﬀective-W approximations[11],by convoluting the cross-sections for the subprocessesW W→H→W W,ZZ,t¯t,(4)andn{W W→G∗n,S∗n→W W,ZZ,t¯t}(5)with the appropriate W distributions in the electron beam(same for ZZ initiated processes).On the Higgs boson resonance,the interference of the processes eqs.(4)and(5) will be dominated by the imaginary part of the graviton/graviscalar amplitude,that, as discussed above,isﬁnite and predictable in terms of M D andδ.The aim of the present paper is to determine the amount by which the Higgs production cross-sections and distributions can be aﬀected by the interference with the KK graviton/graviscalar amplitude.We then discuss the possibility to measure such an eﬀect(and,hence,toﬁnd a footprint of a large extra dimension theory)at realistic linear collider machines.In the last part of the paper we also analyze KK graviton/graviscalar interference eﬀects in Higgs production at a possibleµµcollider acting as a Higgs boson factory [12],through the channelsµ+µ−→H→W W,ZZ,t¯t,(6)andn µ+µ−→G∗n,S∗n→W W,ZZ,t¯t .(7)This process would presumably be aﬀected by smaller theoretical uncertainty,and could provide an even more sensitive probe to large extra dimension eﬀects.2The virtual-graviton exchange amplitudeWe are interested in computing the interference of the exchange of a virtual KK spin-2graviton and a virtual KK spin-0graviscalar with a resonant SM scattering amplitude.Hence,we will analyze in particular an s-channel KK exchange amplitude. We will follow in this section the approach of[3].The graviton-mediated scattering amplitude in the momentum space is obtained by summing over all KK modesA=1s−m2n Tαβ+1δ+2 TµµTνν8π),and Tµνis the energy-momentum tensor of the scatteringﬁelds.Theﬁrst and second terms in Eq.(8) corresponds to graviton and graviscalar exchanges respectively(here m n represents4both the graviton and graviscalar masses without loss of any generality).In the unitary gauge,the projector of the graviton propagator,Pµναβ,is given byPµναβ=13ηµνηαβ+ (9)whereηµνis the Minkowski metric.Dots represent terms proportional to the graviton momentum qµ,that,being qµTµν=0,give a vanishing contribution to the amplitude. The trace of Tµνis nonvanishing only for massive initial andﬁnal states.Since the energy-momentum tensors do not depend on KK indices,one can per-form the sum(over n)irrespective of the scattering process,and Eq.(8)becomesA=S(s)T,S(s)=1s−m2n,T=TµνTµν−1M2+δD dδq T 122) −s2−1(11)where we assumed m2n=q2T,with q T the graviton momentum orthogonal to the brane.In the interference with a resonant amplitude,only Im[S(s)]will contribute,withIm[S(s)]=−π2sδ−22),with n integer.Hence,imaginary part of the amplitude isﬁnite and predictable,only depending on the D-dimensional Plank scale M D and on the number of extra dimensionsδ.It also grows quite rapidly with√3Interference eﬀects in the W W partonic cross-sectionsIn this section,we determine the eﬀects on the angular distributions and cross-sections of the W W fusion Higgs production processes in Eq.(2)arising from their interfer-ences with the corresponding KK graviton/graviscalar exchange processes in Eq.(3). We start from the partonic W W initiated amplitude for the processesW W→H→W W,ZZ,t¯t,(13)andn{W W→G∗n,S∗n→W W,ZZ,t¯t},(14)we compute their interferences,and then convolute them(along with the correspond-ing SM cross sections as in Eq.(13)with the eﬀective-W distributions in the initial electron/positron beams.We notice that due to the diﬀerent spin properties of the Higgs(s=0),graviton (s=2)and graviscalar(s=0)intermediate states,only the graviton will have a nontrivial impact on the angular distribution.On the other hand,the latter eﬀect will vanish in the total cross section,since diﬀerent spin amplitudes turn out to be orthogonal.An analogous eﬀect can be observed in the Z boson-graviton interference in[8].The initial W polarizations that are relevant for Higgs production are the ones where both the W’s are either transverse(with opposite polarization projection)or longi-tudinal.We call the two combinations,λ=T andλ=L,respectively.If P stands for one of the possibleﬁnal particles W,Z and t in Eq.(13),the polarization dependent angular distribution for the process W+W−→P¯P via Higgs exchange plus interference eﬀects with the graviton/graviscalar mediated scattering reads,near the Higgs boson pole(i.e.,for|√d cosθ=¯σPλ†Contributions coming from the real part of the amplitudes are suppressed by terms of order |ˆs−m2H|/m2H in this case.6Higgs-exchange total cross section for the process W +W −→P ¯P,¯σPλ=1(ˆs −m 2H )2+m 2H Γ2Hˆs −4m 2WρP λˆs ˆs is thetotal energy of the initial W ’s in their c.o.m.frame,and ξP are numerical coeﬃcients (ξt =ξW =1,and ξZ =14ρt T (x ),ρt T (x )=34ρV T (x ),ρV T (x )=(x 2−4xr V +12r 2V )δ+2,∆P 2,λ=R δf Pλˆs 4√ˆs ˆs(17)where c P are numerical coeﬃcients (c W =43,and c t =42x +43f VL (x )=−1x −2f VT (x ),f VT (x )=2(x 2−4xr V +12r 2V ).(18)We recall that interference eﬀects arising from the real part of the amplitudes (thatwe are neglecting)are suppressed by terms of order |ˆs −m 2H |/m 2H on the Breit-Wigner resonance.When convoluting the partonic W W cross sections with the W ’s eﬀective ﬂuxes in the collider beams,it will be useful to approximate the Breit-Wigner propagator in Eq.(16)by a Dirac delta function1m H ΓH δ(ˆs −m 2H ).(19)7A few basic features of the distribution in Eq.(15)can be discussed even beforemaking the convolution with the W’sﬂuxes.First of all,as anticipated,the spinstructure of the intermediate states determine aﬂat(Higgs-like)angular distributionfor the graviscalar interference contribution,aﬀecting the total cross section by anamount∆P0×σSM.On the other hand,the spin-2gravitons give rise to a(1−3cos2θ) angular distribution in the W W c.o.m frame,that gives a vanishing result on the to-tal cross section.Nevertheless,an angular analysis of theﬁnal state will reﬂect thenontrivial impact of the(1−3cos2θ)distribution in the W W c.o.m system on thelaboratory-frame angular characteristics.For instance,some angular cut on the direc-tions of theﬁnal states P with respect to the electron/positron beams will originatea non null eﬀect(weighted by the coeﬃcients∆P2,L and∆P2,T)in the integrated crosssections.In order to establish the general relevance of the present eﬀects,it is of coursecrucial to analyze the numerical values of the coeﬃcients∆P0and∆P2,λfor interestingcases of the model parameters.In Tables1and2the most favorable(experimentallyallowed)case of M D=1TeV withδ=2is presented for the processes W+W−→W+W−and W+W−→t¯t,respectively.The coeﬃcients∆P0and∆P2,λhave been √evaluated atΓH(GeV)∆W2,L2005.9×10−52.8×10−58.5−5.1×10−44002.8×10−31.1×10−367.−4.4×10−36001.6×10−27.3×10−3200.−1.6×10−28005.2×10−22.4×10−2Table1:Numerical values of∆2,T,∆2,L and∆0for various Higgs masses ifδ=2and M D=1TeV,for the process W+W−→W+W−.The leading dependence on the Higgs mass of the coeﬃcients∆P0and∆P2,λarises from the Rδbehavior as a function of m H andΓH.In general,from Eq.(17),on the8m H(GeV)∆t2,T∆t029.2.7×10−3500−1.3×10−23.3×10−3 123.1.6×10−2700−5.6×10−21.4×10−2 309.5.0×10−2G F M2D m H m H .(20)Atδ=2,R2∼m HΓH,and this largely explains the increase of∆P0and∆P2,λwith m H observed in Tables1and2.One can note that for m H>500GeV most of the coeﬃcients are quite large,and could have an impact on the measurable cross-sections.At m H=800GeV,all the coeﬃcients amount to a few percent.The most striking one seems to be the graviton-interference case in the top quark channel,that gives∆t2,T≃−0.1at m H≃800GeV.On the other hand,increasing the Planck scale M D can quite aﬀect the coeﬃcient values considered above.From Eq.(20),one has Rδ∼1/M2+δD.Increasing M D by a factor2would imply,for instance,a reduction by a factor about1/16on the coeﬃcients values shown in Tables1and2.4Interference eﬀects in the e+e−cross-sectionsIn the previous section,we have shown that,after integrating over the full range of cosθthe W+W−→P¯P angular distribution,the graviton interference,weighted by the function(1−3cos2θ),vanishes.Only graviscalar-interference eﬀects survive, aﬀecting the total cross sections by a factor(1+∆P0).The latter will modify the cor-responding total cross-sections in e+e−collisions.In order to pin down the graviton-interference coeﬃcients∆P2,λ,one should instead optimize the angular analysis of the process by deﬁning proper strategies(like angular cuts or new asymmetries)that can enhance the graviton contribution(cf.[8]).To this end,it is crucial to consider the laboratory-frame angular distribution for the complete process e+e−(W W)→ν¯νP¯P, that can be obtained by properly boosting the subprocess W+W−→P¯P according to the initial W Wﬂuxes in the electron/positron beams.9In a laboratory frame where the initial W W systems moves with velocityβ,the W+W−→P¯P angular distribution in Eq.(15)becomesdσPλ2 1+∆P0+∆P2,λF(θL,β) J(θL,β),(21) whereF(θL,β)≡1−3 cosθL−β(1−βcosθL)2,(22) andθL is the P scattering angle in laboratory frame.Above,we have neglected terms of order m2W/E2e.We then fold the above partonic cross sections with the probabilities P Wλ(x)of emitting from an e+(e−)beam a real W with polarizationλand fraction of the beam momentum x.The e+e−(W W)→ν¯νP¯P diﬀerential cross-section can be written as dσP ee(S)d cosθL (23) whereˆs=x1x2S and√64π2x2+2(1−x)m2WP W L(x)=g2x .(24)We now can have the laboratory-frame angular distribution for the complete pro-cess e+e−(W W)→ν¯νP¯P,by convoluting Eq.(21)with the W-ﬂuxes.By introduc-ing the variablesτ=m2Hx2+τ,r H=m2H32Sm3HΓhr H−4ρP T(r H),(26)and,by making use of Eq.(19),one has from Eqs.(21)and(23) dσP eexP W T(x)P W T τ-0.006-0.004-0.00200.0020.004-1-0.500.51d σ/d c o s θL (f b )cos θL ∆2∆0m H = 300 GeV√s ee = 0.5 TeV(a)-0.06-0.04-0.0200.020.04-1-0.500.51d σ/d c o s θL (f b )cos θL ∆2∆0m H = 300 GeV m H = 500 GeVm H = 500 GeV↑√s ee = 1 TeV ↓↓↑(b)Figure 2:Graviton (solid line)and graviscalar (dashed line)contributions to the an-gular distribution in the laboratory frame [Eq.(27)]for e +e −center-of-mass energies (a)500GeV and (b)1TeV.I L 0(θL )=(r H −2)2x P W L (x )P W L τxP W T (x )P W T τ4 1τdx x J (θL ,β)F (θL ,β),(31)where the factor 2in the transverse functions I T 0,2(θL )comes from the two diﬀerentinitial W polarization transverse projections that contribute to a spin-0state.For the graviton component,the zeros of the (1−3cos 2θ)distribution in the W W c.o.m.frame are in general shifted by the W W boosts to higher values of |cos θ|.In Figure 2a,we show (by the symbol ∆0)the graviscalar contribution and (by the symbol ∆2)the graviton contribution (including both the longitudinal and the trans-verse part)to the total interference with the SM amplitude in the angular distribution in Eq.(27),at√αǫλ= ǫ−ǫd cosθL Iλ2(θL)S=1TeV case(cf.Figure2b),one has,for m H=300GeV,ǫ≃0.86withαT≃0.18andαL≃0.09.For m H=500GeV,one can selects the central range where the graviton distribution keeps negative values, and setǫ≃0.60.Correspondingly,one hasαT≃0.13andαL≃0.43.Hence,one in general expects that the graviton coeﬃcients in Table1will con-tribute to the measured cross section with reduction factors of a few tens percent according to Eq.(32).At the same time,the graviscalar contribution,having the sameﬂat distribution as the SM signal,will always contribute by the total relative amount∆0to both the total and the cut cross section.5Gravity interference eﬀects atµ+µ−collidersA cleaner framework where to study gravity interference eﬀects on the Higgs boson pole is clearly given by a Higgs boson factory.Although presently challenging from a technological point of view,aµ+µ−collider with c.o.m.energies around m H is the natural place where to realize a Higgs boson factory[12].One then should consider the gravity interference with the Higgs exchange diagram for the processµ+µ−→P¯P,(34) with P=t,W,Z.For unpolarized initial states,the cross section for the latter process,including interference contributions with the graviscalar and graviton exchange graphs,can be expressed near the Higgs pole as:dσP2 1+∆P0+∆P2(1−3cos2θ) ,(35)12where¯σP=d P(s−m2H)2+m2HΓ2hs−4m2µ(s−4m2µ)ρP s2(x2−4x+12),(37)is the SM total cross section for the process in Eq.(34).Here,√2. The coeﬃcients∆P0are the same as in Eq.(17)∆P0=Rδc P δ−13and c Z=23Rδ,∆W,Z2=Rδ2x2−4x+12,(39)with x=s/m2W,Z.Note that∆t,W2=∆t,W2,T,with coeﬃcients∆t,W2,Tdeﬁned as in Eq.(17).As aconsequence of the above identities,a few numerical interesting values of∆t,Wand∆t,W2,for M D=1TeV andδ=2,can be found back in the Tables1and2. Even in the processµ+µ−→P¯P the gravity interference eﬀects can be quite large for high Higgs boson masses.Also in this case,in order to enhance the graviton contribution(that vanishes in the total cross section)it would be suﬃcient to properly exclude in the measured cross-section the forward-backward direction.This can be straightforwardly done in this case by properly cutting theθrange in Eq.(34).6ConclusionsIn this paper,we computed gravity interference eﬀects in Higgs boson production at future colliders in the framework of the models based on large compact extra di-mensions proposed in[1].In particular,we considered the Higgs production channel via W W fusion at linear colliders(that we treat in the eﬀective W approximation) with a subsequent Higgs decay into pairs of heavy particles(W W,ZZ,t¯t).We also analyzed Higgs production and decay channels atµ+µ−Higgs factories.The interfer-ence of graviton/graviscalar exchange diagrams with resonant Higgs production and decay channels has the advantage with respect to usual virtual graviton/graviscalar13exchange channels to lead to a completely predictive determination in terms of the Planck scale M D and number of extra dimensionsδ.The eﬀect on the SM angular distribution in general increases with the Higgs boson mass(forδ=2,the eﬀect is proportional to m HΓH).The graviscalar interference,that does not alter the shape of the distributions,changes its normalization by a few percent for m H>500GeV, if M D≃1TeV andδ=2.On the other hand,due to the diﬀerent spin properties of the graviton and Higgs boson amplitude,the graviton interference alters the angular shape by a universal (1−3cos2θ)distribution(in the W+W−orµ+µ−c.o.m.frame)with a coeﬃcient that is again of the order of a few percent for m H>500GeV.The latter distribution is averaged to zero in the total cross section.Hence,in order to select a graviton eﬀect, we suggest angular-cut strategies that enhance the graviton interference contribution in the measured cross section.In order to detect such indirect graviton eﬀects in Higgs cross section measure-ments,it is crucial that the actual experimental set up will be able to reach the required sensitivity.While assessing theﬁnal precision of muon colliders is prema-ture at the moment,quite a few studies on this subject have been carried out for the linear e+e−colliders[9].In particular,the precision expected on the measurement of the cross section for Higgs boson production via W W fusion has been considered in [14](see also[15])for a light Higgs decaying predominantly into b quark pairs,and is of the order of a few percent.A detailed study for heavier Higgs bosons(that are the relevant ones for our study)is presently missing,to our knowledge.Anyhow,a percent precision in the cross section measurements should allow to detect some eﬀect at least in the most favorable case of M D≃1TeV andδ=2at both linear colliders and Higgs factories.The eﬀect scales as∼1/M2+δwith the Planck mass scale.DA complete treatment(i.e.,beyond the eﬀective W approximation)of the cross-section in the W W fusion process at linear colliders is not expected to alter our conclusions.Note that,by the time experiments at linear colliders should be operating,the LHC will have presumably observed the direct production of gravitons in the range of parameters that could be relevant for our precision measurements.In particular, a direct graviton signal is expected,forδ=2,3,4,for M D up to a few TeV’s[16,17]. The information derived from the direct graviton production and observation at LHC will deﬁnitely help in disentangling the deviations in the Higgs cross sections and distributions analyzed in the present paper.14AcknowledgmentsWe would like to thank M.Giovannini,M.Porrati,and M.Testa for useful discussions.E.G.and A.D.would also like to thank the Physics Department of University of Roma “La Sapienza”,while E.G.thanks also the CERN Theory Division,for their kind of hospitality during the preparation of this work.A.D.and E.G.also thank Academy of Finland(project number48787)forﬁnancial support.References[1]N.Arkani-Hamed,S.Dimopoulos and G.Dvali,Phys.Lett.B429(1998)263;I.Antoniadis,N.Arkani-Hamed,S.Dimopoulos,and G.Dvali,Phys.Lett.B436 (1998)257.[2]S.Dimopoulos and G.F.Giudice,Phys.Lett.B379(1996)105;J.C.Long,H.W.Chan and J.C.Price,Nucl.Phys.B539:23-34,1999.[3]G.F.Giudice,R.Rattazzi and J.D.Wells,Nucl.Phys.B544(1999)3.[4]E.A.Mirabelli,M.Perelstein and M.E.Peskin,Phys.Rev.Lett.82(1999)2236.[5]T.Han,J.D.Lykken and R.Zhang,Phys.Rev.D59(1999)105006.[6]N.Arkani-Hamed,S.Dimopoulos and G.Dvali,Phys.Rev.D59(1999)086004;J.L.Hewett,Phys.Rev.Lett.82(1999)4765;C.Bal´a zs,H.-J.He,W.W.Repko and C.-P.Yuan,Phys.Rev.Lett.83(1999)2112;E.Dudas and J.Mourad,Nucl.Phys.B575(2000)3;E.Accomando,I.Antoniadis and K.Benakli,Nucl.Phys.B579(2000)3;S.Cullen,M.Perelstein and M.E.Peskin,Phys.Rev.D62 (2000)055012;W.D.Goldberger and M.B.Wise,Phys.Lett.B475(2000)275;B.Grzadkowski and J.F.Gunion,Phys.Lett.B473(2000)50;G.F.Giudice,R.Rattazzi and J.D.Wells,Nucl.Phys.B630(2002)293;Nucl.Phys.B595(2001) 250;H.Davoudiasl,J.L.Hewett and T.G.Rizzo,Phys.Rev.Lett.84(2000) 2080;Phys.Lett.B473(2000)43;Phys.Rev.D63(2001)075004;T.G.Rizzo, Phys.Rev.D64(2001)095010;E.Gabrielli and B.Mele,Nucl.Phys.B647 (2002)319;J.Hewett,M.Spiropulu,Ann.Rev.Nucl.Part.Sci.52(2002)397;E.Dvergsnes,P.Osland and N.Ozturk,hep-ph/0207221;T.G.Rizzo,eConf15C010630(2001)P301,hep-ph/0108235;M.Cavagli´a,hep-ph/0210296;T.G.Rizzo,JHEP0302(2003)008;G.C.Nayak,hep-ph/0211395;G.F.Giudice,A.Strumia,hep-ph/0301232;N.G.Deshpande,D.K.Ghosh,hep-ph/0301272.[7]ndsberg,arXiv:hep-ex/0105039,and references therein.[8]A.Datta,E.Gabrielli and B.Mele,Phys.Lett.B552(2003)237.[9]J.A.Aguilar-Saavedra et al.[ECFA/DESY LC Physics Working Group Collab-oration],‘TESLA Technical Design Report Part III:Physics at an e+e-Linear Collider,hep-ph/0106315.[10]See,for instance,M.Spira and P.M.Zerwas,hep-ph/9803257.[11]S.Dawson,Nucl.Phys.B249(1985)42.[12]see,e.g.,HIGGS FACTORY2001SNOWMASS REPORT,/hep/hfactory/index.html[13]K.Hagiwara et al.,Phys.Rev.D6*******(2002),/.[14]K.Desch and N.Meyer,LC-PHSM-2001-025In*2nd ECFA/DESY Study1998-2001*1694-1704.[15]S.Dawson and S.Heinemeyer,Phys.Rev.D66(2002)055002.[16]L.Vacavant and I.Hinchliﬀe,J.Phys.G27(2001)1839.[17]L.Vacavant,“Search for extra dimensions at LHC”,talk given at the Inter-national Europhysics Conference on High Energy Physics EPS,July17th-23rd, 2003,Aachen,Germany.16。

NON-ISOTROPIC GEOGRAPHIC MODELING

THREE PRESENTATIONS ON GEOGRAPHICALANALYSIS AND MODELING1) NON-ISOTROPIC MODELING2) SPECULATIONS ON THE GEOMETRY OF GEOGRAPHY3) GLOBAL SPATIAL ANALYSISbyWaldo Tobler:University of CaliforniaSanta Barbara CA93106-4060tobler@NATIONAL CENTER FOR GEOGRAPHIC INFORMATION AND ANALYSISTECHNICAL REPORT 93-1February 1993NON-ISOTROPIC GEOGRAPHIC MODELINGWaldo ToblerFive of the most important and useful theoretical models in the study of geography are, in historical order: the Von Thunen model of agricultural land use; Weber’s model of industrial location; Walter Christaller’s central place formulation; the gravity model of spatial interaction; and Hagerstrand’s model of the geographical spread of innovation. All of these models are of fundamental importance in understanding the world around us. They also have in common the fact that they simplify reality by invoking an isotropic geographic space. Of course they do this to varying degree, and some then relax this assumption. Most of them are also computationally explicit models, again to varying degrees. Both of these aspects of the models are the subject of my discussion.Current Geographic Information Systems have the potential of overcoming the isotropic plane assumption, even though the elegant pedagogic simplicity of models using this assumption will always obtain. Consider first the Von Thunen model. Here agricultural land use patterns are derived as a function of market price, transport cost to market, and agricultural production costs. Under the isotropic assumption, and a single market, the geographic arrangment of crops and land uses takes the concentric form. Years ago, for an Economic Geography class near Detroit, I wrote a simple computer program to demonstrate this, with five commodities, including strawberries, wheat, other foodstuffs, and Cadillacs. Given the other inputs I could demonstrate computer printouts of circularly arranged optimal land uses, and could even run the program with multiple market places to get mixtures of partially concentric patterns, and could include a climatic gradient to get non-circular patterns. All this was done off-line, in a batch environment (circa 1965). My computer program could not incorporate positional variations in transportation cost, even though Von Thunen himself gave such a demonstration introducing a river into the geographic space (Figure I).Several GISs can now provide more realistic calculations. One of our graduate students, R. Dodson, has implemented the Von Thunen model within the IDRISI system, and has done some work on an implementation of the Weber model. This system allows variable transportation costs to be included. However he still uses synthetic landscapes. But this need not be the case. To see how this can be improved upon, I would like to contrast the situation with a popular computer pastime. What I have in mind is the "Flight Simulator" in which one can pretend to be an airplane pilot. The geographically interesting part is that one can purchase a variety of landscapes, and "fly" around these. Each of these "landscapes" or airport vicinities, can be purchased separately. Imagine having the same thing for the USA, or the world, and running Von Thunen, Weber, Christaller, or Hagerstand models on real landscapes. For, say $20, you might purchase "The Los Angeles Region" as a backdrop for your "spread of AIDS" model, or whatever model excites you. I envision the day when we can purchase alternate geographic landscapes for such geographical simulations. These would contain real geographic data (soils, transportation, population, etc., depending on the class of model), for a particular place, region, nation, or continent (or even time period). Indeed, I propose that we consider the development of such a library of modules. The trend is clearly in this direction, and a few prototypes already exist, the Digital Elevation Models and Digital Line Graphs of the US Geological Survey, and the TIGER files of the US Bureau of the Census being two primitive (but useful) examples of proto-modules.A common element in many geographic models is the assumption of Euclidean distance; this is generally the basic postulate of the isotropic plane. How can a model such as that of Christaller relating central places be adapted to fit the highlands of some country in a pre-automobile time? Can it be modified to take into account todays of complex transportation systems which distort topological and metrical realtions? Current Geographic Information Systems indeed enable one, given a transportation disutility on a particular class of land, to calculate more realistic time- or cost-distances between places. I find this particularly exciting and have worked out a simple example for walking in variable terrain using the hiking function (Figure II) estimated from empirical data given by Imhof (1950, pp 217-220).In order to use this function one simply calculates the slope of the terrain, and then converts this to a walking velocity. It is easiest to do if the terrain is given in the form of a geographic "matrix", with elevations at equally spaced increments in two directions (Figure III).Then one can compute, from any inital point, the minimum time path to all other places (Figure IVa). Connecting places at equal time-distances yields isochronic lines, or "geographic circles" (Figure IVb). Putting in the gradients to this contour map then yields two sets of orthogonal lines (Figure IVc), isomorphic withGauss’ geodesic polar coordinates for which the metric takes on a particularly simple form. Now it would make sense to reformulate Christaller’s central place model in terms of a general Gaussian metric (of the form). Unfortunately transportation facilities with modern railroads, highways, and airplanes introduce topological complexities which are not easily treated within the framework of continuous Gaussian manifolds. This is because real geographic circles have holes and disjoint pieces (Figure V; Tobler 1961). GeographicInformation Systems with these capabilities generally are forced to use network models.The computablity of the theoretical models also raises a pedagogic issue. Current Geographic Information Systems seem to appear in three contexts. One use is in applied fields; the bureaucratic inventorying of natural resources or facilities management. A second use is in research, where specific problems are under investigation. Here attempts are now being made to incorporate various sorts of additional spatial analysis capabilities into GIS’s. The third appearance of GIS’s is in courses on GIS. What is missing is their appearance in substantive geography courses such as Historical Geography, Political Geography, Economic Geography, Cultural Geography, Physical Geography, Regional Geography of Country X or area Y. Now imagine some future date when all of these courses are lab courses, or have a laboratory associated with them, as most physical geography courses (and some others) already have. As I see it this lab might be a computational GIS lab. The point is to bring the GIS into the substantive courses (not only in geography), and not to leave it in an isolated GIS techniques course. I recently taught a course on human migration and made one of the hours into a set of computational exercises. Professor Golledge has been using SimCity in a similarmanner in his course on Urban Geography. Going somewhat further, imagine that all social science courses mightcome with a computer lab some day in the future. Not just for word processing, but real Computational Social Science. There is a clear tendency in that direction which needs to be encouraged.REFERENCESChristaller,W (1935) Die zentralen Orte in Sueddeutschland, J. Fischer, Jena.Dodson,R (1991): VT/GIS The Von Thunen GIS Package, Technical Paper 91-27, Santa Barbara, NCGIA. Dussart,F (1959): "Les courbes isochrones de la ville de Liege pour 1958-1959", Bull. Soc. Belge d’Etudes Geogr. XXVIII, 1:59-68.Hagerstrand,T (1968) "A Monte Carlo Approach to Diffusion", pp 368-384 of Berry, B; Marble, D: Spatial Analysis, Prentice Hall, Englewood Cliffs.Imhof,E (1950): Gelaende und Karte, Rentsch, Zurich.Riedel,J (1911): "Neue Studien ueber Isochronenkarten", Pet. Geogr. Mitt., LVII,1:281-284.Tobler, W (1961): Map Transformations of Geographic Space, PhD dissertation, Seattle, University of Washington Von Thunen,J (1826): Der Isolierte Staat in Beziehung auf Landwirschaft und Nationaloekonomie, G. Fischer, Jena; 678 pp.Weber,A (1909): Ueber den Standort der Industrien, Tuebingen.Presented at the NCGIA sponsored conference on Geographic Information Systems in the Social Sciences, Santa Barbara, Feb. 1991SPECULATIONS ON THE GEOMETRY OF GEOGRAPHYWaldo ToblerConsider the set of all places on the surface of the earth which you could reach within one hour of where you are now. The outer edge of this set forms a geographical ’circle’ of one hour radius. Francis Galton introduced the term isochrone for such a circle in the 1880’s, but the word (in a somewhat different context) was probably already used by Snellius or Bernoulli.What a curious circle it is! Its circumference is hardly 2*pi*r, its area, in square hours, is not pi*r^2. The circle most probably has holes in it, and probably consists of disjoint pieces when shown on an ordinary geographical map. The shape of this circle depends on the place and time of day at which you start your journey. Our daily environment, as a geometry, seems more complicated than the Riemannian geometry envisaged by Einstein, at least as far as map images on the earth are concerned. A resident of Paris will have a geographical circle of one hour radius with a different circumference, shape, and area from that of a person in Los Angeles. This is a bit like the geometry on the surface of a cucumber; here a circle at the narrow end will differ from one near the middle. A familiarity with non Euclidean geometry renders this quite understandable. Next, try drawing concentric circles on a potato from some starting point. Now put in the orthogonal trajectories to these circles. Then note the similarity to Gaussian polar geodesic coordinates. This is usually obscured because the radial geodesics are usually not drawn on geographical isochronic maps. The map of travel times from central Dallas is typical.The analogy to non Euclidean spaces clearly suggests that one try to model at least a part of our travel time geometry as a two manifold of variable curvature.To this end I have assembled several tables, of which the following is a short selection:a. Road distances between cities in Switzerland.b. Automobile travel times between these places.c. Similar data for Austriad. Travel times between addresses in New York City.e. Airline travel times between cities in the United States.f. Airline travel costs, in dollars, amongst the foregoing.g. Bus travel times and costs between the above cities.h. Parcel shipment costs between cities in the U.S.i. Distances between US cities estimated by a sample of individuals.j. Ocean route lengths between world ports.In each instance the location names (i.e., latitude and longitude) of the places are available.I would now like to model these relationships with an equation of the form (summation convention):from which the Gaussian curvature, geodesic curvature, etc., might be calculated. A difficulty is that this metric tensor is derived assuming that Euclidean geometry holds in the small, that is, for very short distances. In geographic space this assumption is often not satisfied, as noted again below.A cartographic application might be that one could make polar geodesic maps centered on any place. More practical usefulness would be obtained if one could directly compute the travel costs (times) between places knowing only their latitude and longitude, the mode of transportation, and the time of day. This would be like the computation of spheroidal distances from geodetic coordinates - the sort of thing one does on a hand calculator nowadays. This would require the storage of thethe distances, in a least squares manner. These methods can be applied no matter what the units are by ajudicious scaling. A travel time topography can thus be created The Gaussian weights might now beestimated, because we know that the distance stretching is approximated, as a function of direction, by theratio2..2. Differential geometry needs to be married with the method of least squares to estimate thecan therefore be expected to show only minor deviations from their values as computed from the spherical or spheroidal values. Can we then simply introduce a small "fudge" factor into the formula, but retaining the basic form of the metric tensor? Oceans complicate this.4. Transport costs usually increase at a decreasing rate with distance, i.e. are concave down. This is easily translated into an azimuthal map projection,There are many examples. A short list and illustrations of the earth mapped using these functions are shown. It is clear that any monotonic (concave down) transportation cost can be approximated by some such function, to produce maps scaled in cost (or time) from some point. Less obvious is the fact that this can also be done in an area preserving manner. Lambert’s (1772) azimuthal equal area map projection already has this property. Another equal area map of the globe can be obtained usingThis yields a curious map, as shown in the figure. Generally the travel disutilities are different in all directions so that even more complicated maps need be drawn. But the equations for Gaussian curvature are relatively simple in polar geodesic coordinates, i.e., knowing that the metric is given by thenand alsoTransportation costs which are non-monotonic with respect to geographic distance, as in the airline cost examples cited later, render these things more difficult.5. Many of the tables are not symmetric, . Will this alone introduce enough asymmetry into the results? What happens to the rest of conventional differential geometry under thesecircumstances? Perhaps one must add a term6. Several items in the table exhibit discontinuous rates. In (h), for example, the diagonal of the table is non zero. Shipment within a city costs a finite amount. Often the rates form a step function - several cities at differentdistances fall into the same rate class. The parcel post map shows the effect.7. Within cities we might consider Minkowski metrics, sometimes called city block, Manhattan, or taxicabgeometries. Circles satisfying are hyperbolae, and might represent transport relations in the vicinity of arterial road intersections. The so-called Karlsruhe metric also yields realistic isochrones for certain simiplesituations. A few empirical studies have already been done on this. Can one introduceThe movement of people, of ideas, of technology, of disease, or of money, use complex and intricate transportation systems. These distort our globe in profound ways. One can get from Los Angeles to Chicago, to New York City, or to Washington, D.C., quicker than one can get to Arcata in Northern California. The airfare from Santa Barbara to Sacramento is 107% of that from Los Angeles to New York City. By this measure Columbus, Ohio, today is 151% as far from Los Angeles as is New York, and Madison, Wisconsin, is 219% further from San Francisco than is Milwaukee. Only a few minutes with an airline fare schedule will yield many such examples. Relations on our earth are certainly shrinking, but they are also becoming more warped, turned inside out, and distorted almost beyond imagination. Because of this Bunge has proposed that we use more realistic globes, with maps printed on balloons in which well connected places are constrained by interior strings before inflation. The resulting indentations and bulges after inflation come closer to the realities of current geography than do commercially available globes. In this presentation I have tried to capture aspects of this new geography in terms of geometry, and it is almost possible to believe that Euclidean geometry could not be invented by a careful observer today. Of course the world was complicated in terms of transportation even in Euclid’s day. The major modern changes, aside from speed, being limited access air travel, railroads, highways, and electronic communication networks. These distort the simpletopology of our sphere-like earth.Some literature:Angel, S, and Hyman, G., 1976, Urban Fields, Pion, London, 179 pp.Blumenthal, L., 1953, Theory and Applications of Distance Geometry, Clarendon, Oxford.Bunge, W., 1966, Theoretical Geography, 2nd ed., Lund Studies in Geography, Gleerup, Lund.Dorigo, G., and Tobler, W., 1983, "Push Pull Migration Laws", Annals, Assn. Am. Geographers, 73(1):1-17. Eisenhart, L., 1960, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York.Finsler, P., 1951, Ueber Kurven and Flaechen in allgemeinen Raumen, Birkhauser, Basel.Gatrell, A., 1983, Distance and Space: A Geographic Perspective, Clarendon Press, Oxford, 195pp.Gauss, C., 1902, General Investigations on Curved Surfaces (1827), Princeton University, Princeton, 127 pp. Gauss, C., 1910, "Untersuchungen ueber Gegenstaende der Hoehren Geodasie, 1844", reprinted in Oswald’s Klassiker der exacten Wissenshaften, Nr. 177, Wagrain. Leipzig.Krause, E., 1975, Taxicab Geometry, Addison Wesley, BostonKreyszig, E., 1959, Differential Geometry, University of Toronto, Toronto.Kumler, M., 1989, "Directional Bias in the Ratio of Highway Distances to Great Circle Distances Between Major Cities in the US", Seminar paper, Geography Department, University of California, Santa Barbara, 8 pp.Love, R., and Morris, J., 1972, "Mathematical Models of Road Travel Distances", Management Science, 25:130-139 Love, R., and Morris, J. 1979, "Modeling Inter-City Road Distances by Mathematical Functions", Operational Research Quarterly, 23:61-71.Medvedkov, Y., 1965, Ekonomgeograficheskaya Uzychennosty Raionov Kapitalisticheskogo Mira, Prilojeniya Matematiki v Ekonomucheskoi Geografii, 2, Institute for Scientific Information, Moscow; pp. 110-120. Meyhew, L., 1986, Urban Hospital Location, Allen & Unwin, London, 166 pp.Misner, C., Thorne, K., and Wheeler, J., 1973, Gravitation, Freeman, San Francisco, pp. 305-309.Muller, J-C., 1978, "The Mapping of Travel Time in Edmonton, Alberta", The Canadian Geographer, XXII(3):195-210 Muller, J-C., 1979, "La Cartographie d’une Metrique Non-Euclidean: Les Distances Temps", L’Espace Geographique, 3:116-128.Muller, J-C., 1983, "Die Nichteuklidische Darstellung Funktionaler Raume", Kartographische Nachrichten, 33(1):10-19.Muller J-C. 1984, "Canada’s Elastic Space: A Portrayal of Route and Cost Distances", The Canadian Geographer,18:46-62.Nordbeck, S., 1964, Framstaellning av Kartor med Hjaelp av Siffermaskiner, (The production of Maps with the Help of Digital Computers), Meddelanden Fran Lunds Universitets Geografiska Institution, Avhandlingear 40, pp. 56-60. Nordbeck, S., 1963, "Comparing Distances in Road Nets", Papers, Regional Science Assn., 12:207-220.Okabe, A., Boots, B., & Sugihara, K., 1992, Spatial Tesselations, Wiley, New York, pp. 188-191.Pieszko, H., 1970, Multidimensional Scaling in Riemann Space, PhD Thesis, University of Illinois, Urbana.Puu, T., 1979, The Allocation of Road Capital in Two Dimensional Space, North Holland, Amsterdam. Reichenbach, H., 1958, The Philosophy of Space and Time, Dover, New York, 295 pp.Riemann, B., 1854, Ueber die Hypothesen welche der Geometrie zugrunde liegen, Dissertation, Goettingen (reprinted in Weyl, 1923, op. cit.; also in Weber, ed., 1953, Collected works of B. Riemann, Dover, New York).Rund, H., 1957, Differential Geometry of Finsler Space, Springer, Berlin.Schilling, F. 1928, "Konstruction kuerzester Wege in einem Gelaende", Zeitschrift fuer angewandte Mathematik und Mechanik, 8:45-68.Shreider, Y., 1974, What is Distance?, University of Chicago Press, Chicago, 71 pp.Tissot, M., 1881, Memoire sur la Representation des Surfaces et les Projections des Cartes Geographiques, Gautier-Villars, Paris.Tobler, W., 1962, Map Transformations of Geographic Space, PhD thesis, University of Washington, Seattle. Tobler, W., 1962, Studies in the Geometry of Transportation, DA-44-177-TC-685, US Army Transportation Research Command, Northwestern University Transportation Research Center, Evanston.Trunin, Y., & Serbenyuk, S., 1968, "Maps of Accessibility in the Analysis of Economic-Geographic Space: Cartographic Transformation of a Surface of Negative Curvature", Voprosii Geographii, pp. 179-186 (in Russian). Vaughan, R., 1987, Urban Spatial Traffic Patterns, Pion, London.Werner, C., 1968, "The Law of Refraction in Transportation Geography: its Multivariate Extension", The Canadian Geographer, XII, 1:28-40.Werner, C., 1985, Spatial Transportation Modeling, Sage, Beverly Hills.Weyl, H., 1923, Mathematische Analyse des Raumproblems, Barcelona Lectures, Springer, Berlin (reprinted inH.Weyl, 1958, Das Kontinum und andere Monographien, Chelsea, New York).Zaustinsky, E., 1959, Spaces with Non-Symmetric Distance, Memoir 34, American Mathematical Society, 91 pp.GLOBAL SPATIAL ANALYSISWaldo ToblerWe now have available several books on spatial analysis {Anselin 1988; Arbia 1989; Cressie 1991; Haining 1990; Gaile & Willmott 1981; Griffith 1988, 1990; Ripley 1981, 1990; Unwin 1981, Upton & Fingleton 1985}. This ismost encouraging, but is it not strange that almost none of these works considers that the earth is homeomorphic to a sphere? The term sphere does not even occur in the index in most of these books. This is also true of older books like {Berry & Marble 1968, and Bennett 1979}. A similar criticism applies to all the books published in the last two decades under titles similar to "Statistics for Geographers", or "Statistical Geography", etc. This seems a most curious omission. The same comment seems to apply to the (literally) hundreds of "Geographical Information Systems" (GIS’s). The one exception, explicitly designed to consider the spheroidal earth, is the "Hipparchus" system developed by Hrvoje Lukatela of Calgary, Alberta {Lukatela 1987}. One other system, GIS-Plus, uses latitude and longitude for its coordinate system. Most GISs can convert from latitude and longitude to map projections but do not treat the spherical units as their primary referencing system. And, although there has been a continued plea for more analysis capability, most of the current GIS’s do not claim to be "Geographical Analysis Systems" (GAS’s). Facilitation of analysis is the thrust of the current NCGIA initiative and I argue that this should be extended to include analysis of global distributions.The inference that I make from my observation is that the greatest demand is for parochial, local studies and not for global analysis. This is inconsistent with the increased interest in global problems, especially as they relate to the global environment, terrestrial warming, ozone depletion, and so on, and even to global economic relations. How can this "flat earth" syndrome be overcome and round earth thinking be brought into the textbooks, mainstream research papers and monographs, and GIS’s or GAS’s? Much blame must be put on the teaching of Euclidean geometry in the elementary schools instead of the more natural earth oriented Riemann (elliptical) geometry. It is probably hopeless to attempt to change this. Should everyone know that the circumference of a circle increases as two pi times the sine of the radius, (which means it eventually goes to zero), and that the area of a circle increases in proportion to the square of the sine of one half of the circular radius? Or that the circumference of a circle on the surface of an ellipsoid depends not only on the circle’s radius but also on where one puts the center {Blaschke 1949}? No one seems to teach analytic geometry on the sphere; the most recent book which I have found is nearly 100 years old {Heger 1908}. Knowing these sorts of things would make it much easier to understand the complexities of "geographical circles", the set of all places attainable within a given number of hours (or dollars). These circles have a circumference bounded by isochrones (isotims), the radii are time geodesics, and the circles’ shape depends on where, and when, one begins to travel. These circles often have disjoint pieces, or holes, on the surface of the earth due to air travel. Thus transportation systems induce a geometry even more complicated than that of a sphere or ellipsoid.The only fields in which global analysis routinely appears are geodesy, meteorology, and oceanography, collectively sometimes known as geophysics {Moritz 1980}. Geologists today increasingly invoke spherical ideas, especially since the development of the theory of continental drift, and a statistican occasionally wanders into this domain {Mardia 1972, Watson 1983}. Thus I can cite recent books such as that of {Fischer, et al. 1987, Washington & Parkinson 1986}, and {Daley 1991}, and older ones {e.g., Chapman & Bartels 1940} that consider the geometric nature of the earth. Generally these works are subject specific and specialized. Still they provide a starting place.If one were to develop a course of studies introducing analysis on the earth, treated spherical, what should the class of problems to be treated cover? One approach is to take the standard statistics book and redo the problems on a sphere. The surface of the earth is topologically spherical, even with bumps like the Matterhorn; it is a two dimensional manifold lacking an edge, but with some discontinuities and unsmooth derivatives, and is perhaps even fractal-like if we don’t look too closely. The boundaries above and below are of no concern so that one is immediately into two dimensional spatial statistics (which, as Bill Bunge would remind us, neglects the danger from intercontinental missiles). First, of course, come the two dimensional descriptive statistics, frequency histograms, density functions, means, variances, binomials, normals, transformations, etc., all on the spherical surface. About the only idea from this set that one routinely encounters is the population center of the United States and how it moves from census to census. Even the few papers on centrography generally only consider planar values. Then it perhaps gets more interesting. Scatter diagrams and correlation between observations given by latitude and longitude. We need to know how to rotate to principal axes, a pair of orthogonal great circles (factor analysis on a sphere would need this). Or fit a "straight line", which now becomes an arc of a great circle through a set of point locations; that is, find the pole of this great circle on the sphere. Or does a spherical quadratic (small circle, parabola, or hyperbola), cubic, quartic, or quintic fit better? Or a loxodrome, or some other transcendental curve? Or a nice spline, or a weighted nonparametric smooth curve? Here we have some types of questions addressed in {Fisher et al. 1987}. Is this interesting? Do we need spherical markerboards for the classroom demonstrations? Now move on to the analysis of spherical point patterns in the style of planar analysis {Diggle 1983, Getis and Boots 1978}. A fleeting reference to Voronoi polygons on the sphere is found in (Okabe, et al. 1992). Quadrat analysis must obviously be modified to become quadrilateral analysis. Or one can map。

纹理物体缺陷的视觉检测算法研究--优秀毕业论文

摘要
在竞争激烈的工业自动化生产过程中，机器视觉对产品质量的把关起着举足轻重的作用，机器视觉在缺陷检测技术方面的应用也逐渐普遍起来。与常规的检测技术相比，自动化的视觉检测系统更加经济、快捷、高效与安全。纹理物体在工业生产中广泛存在，像用于半导体装配和封装底板和发光二极管，现代化电子系统中的印制电路板，以及纺织行业中的布匹和织物等都可认为是含有纹理特征的物体。本论文主要致力于纹理物体的缺陷检测技术研究，为纹理物体的自动化检测提供高效而可靠的检测算法。纹理是描述图像内容的重要特征，纹理分析也已经被成功的应用与纹理分割和纹理分类当中。本研究提出了一种基于纹理分析技术和参考比较方式的缺陷检测算法。这种算法能容忍物体变形引起的图像配准误差，对纹理的影响也具有鲁棒性。本算法旨在为检测出的缺陷区域提供丰富而重要的物理意义，如缺陷区域的大小、形状、亮度对比度及空间分布等。同时，在参考图像可行的情况下，本算法可用于同质纹理物体和非同质纹理物体的检测，对非纹理物体的检测也可取得不错的效果。在整个检测过程中，我们采用了可调控金字塔的纹理分析和重构技术。与传统的小波纹理分析技术不同，我们在小波域中加入处理物体变形和纹理影响的容忍度控制算法，来实现容忍物体变形和对纹理影响鲁棒的目的。最后可调控金字塔的重构保证了缺陷区域物理意义恢复的准确性。实验阶段，我们检测了一系列具有实际应用价值的图像。实验结果表明本文提出的纹理物体缺陷检测算法具有高效性和易于实现性。关键字: 缺陷检测；纹理；物体变形；可调控金字塔；重构
Keywords: defect detection, texture, object distortion, steerable pyramid, reconstruction
II

MaxDEA

Detailed Contents
Chapter 1: Main Features of MaxDEA ..................................................8
1.1 Main Features ............................................................................................... 8 1.2 Models in MaxDEA...................................................................................... 9 1.3 What’s NEW ............................................................................................... 12 1.4 Compare MaxDEA Editions ..................................................................... 17
3.1 Import Data ................................................................................................ 19 3.2 Define Data ................................................................................................. 24 3.3 Set and Run Model..................................................................................... 25 3.4 Export Results ............................................................................................ 77

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a rX iv:physics /0649v1[physics.ge n-ph]2J un2ON STANDARD MODEL HIGGS AND SUPERSTRING THEORIES AFSAR ABBAS Institute of Physics,Bhubaneswar-751005,India (e-mail :afsar@iopb.res.in)Abstract It is shown that in the Standard Model,the property of charge quantiza-tion holds for a Higgs with arbitrary isospin and hypercharge.These deﬁning quantum numbers of the Higgs remain unconstrained while the whole basic and fundamental structure of the Standard Model remains intact.Hence it is shown that the Higgs cannot be a physical particle.Higgs is the underlying ‘vacuum’over which the whole ediﬁce of the Standard Model stands.Also on most general grounds it is established here that as per the Standard Model there is no electric charge above the electro-weak phase transition tempera-ture.Hence there was no electric charge present in the Early Universe.The Superstring Theories are ﬂawed in as much as they are incompatible with this requirement.11Charge Quantization in the Standard Model The SM assumes a repetitive structure for each generation of quarks and leptons[1].Let us start by looking at theﬁrst generation[2]of quarks and leptons(u,d,e,ν)and assign them to SU(N C)⊗SU(2)L⊗U(1)Y(where N C=3)representation.To keep things as general as possible this brings inﬁve unknown hypercharges[3].Let us now deﬁne the electric charge in terms of the diagonal generators of SU(2)L⊗U(1)Y asQ=T3+bY(1) In the SM SU(N C)⊗SU(2)L⊗U(1)Y is spontaneously broken through the Higgs mechanism to the group SU(3)c⊗U(1)em.In SM the Higgs is assumed to be a doublet[4].However we do not use this restriction either and assume the Higgsφto have any isospin T and arbitrary hypercharge Yφ.The isospin Tφ3component of the Higgs develops a nonzero vacuum expectation value<φ>o.Since we want the U(1)em generator Q to be unbroken we require Q<φ>o=0.This right awayﬁxes b in(3)and we getQ=T3−(Tφ32(1+12(−1+1For N C=3these are the correct charges in the SM.Note that this charge quantization in the SM holds for Higgs for arbitrary T and arbitrary hypercharge.Hence as far as charge quantization is concerned,the values of T and Yφremain unconstrained.This point,for the special case of the Higgs doublet was already noted by the author earlier[4].2Higgs particle-a ghost!Let us continue with the rest of the structure of the SM and see how our general Higgs with unconstrained and unspeciﬁed isospin T and hypercharge Yφﬁts into it.We can write the covariant derivative of the SM asDµ=∂µ+ig1Tφ3YφY l)W0µg22+(g12Tφ3YφY L)Bµ+g2W0µg22+(g12Tφ3YφY l Bµ)q L+¯e R iγµ(ig1Tφ3Weﬁnd that from rho parameter also[5]the solutions for the isospin of the Higgs are inﬁnite in number.Again,nothing in the theory demands that oneﬁxes the isospin to a particular value.The point is that the full structure of the SM stands intact without con-straining the quantum numbers isospin and/or the hypercharge of the Higgs to any speciﬁc value.All the particles that have been isolated in the labo-ratory or have been studied by any other means,besides having a speciﬁc mass,have deﬁnite quantum numbers which identify them.In the case of Higgs here,no one knows of its mass and more importantly its quantum numbers like isospin and hypercharge,as shown above,are not speciﬁed. The hypercharge of all the other particles are speciﬁed as being proportional to the Higgs hypercharge which itself remains unconstrained.That is,all the hypercharges of particles are rooted on to the Higgs hypercharge which itself remains free and unspeciﬁed.Hence Higgs is very diﬀerent from all known particles.Because of the above reasons Higgs cannot be a physical particle which may be isolated and studied.It must be just the‘vacuum’which sets up the structure of the whole thing.So Higgs is a ghost which shall not manifest itself as a genuine particle in the laboratory[5].3Superstring Theories-intrinsicallyﬂawed! Now we ask the question,with this generalized picture what happens to the electric charge when the full Standard Model symmetry is restored.Note that the expression for Q in(2)arose due to spontaneous symmetry breaking of SU(N C)⊗SU(2)L×U(1)Y(for N C=3)to SU(N C)×U(1)em through the medium of a Higgs with arbitrary isospin T and hypercharge Yφ. What happens when at higher temperature,as for example found in the early universe,the SU(N C)⊗SU(2)L⊗U(1)Y symmetry is restored?Then the pa-rameter‘b’in the electric charge deﬁnition remains undetermined.Note that ‘b’wasﬁxed above due to spontaneous symmetry breaking through Higgs. Without it‘b’remains unknown.Hence when the electroweak symmetry is restored,irrespective of the Higgs isospin and hypercharge the electric charge disappears as a physical quantity.Therefore weﬁnd that there was no electric charge in the early universe.Here attention is drawn to the fact that all putative extensions of the Standard Model should reduce smoothly and consistently to the Standard4Model at low energies.Not only that,all these extensions should be consis-tent with the predictions of the Standard Model at very high temperatures. Contrary to naive expectations,the SM does make speciﬁc predictions at very high temperatures too.For example one clear-cut prediction of the Standard Model as shown here and also shown earlier,is that at high enough temper-atures(as in the early universe)when the unbroken SU(3)⊗SU(2)⊗U(1) symmetry was restored,there was no electric charge.GUTs and other stan-dard extensions of the SM are incompatible with this requirement[6].What about Superstring Theories?Quite clearly,generically in Super-string Theories electric charge exists right up to the Planck Scale[7].Hence as per this theory the electric charge,as an inherent property of matter,has existed right from the beginning[8].This is not correct in the SM.As shown here and earlier,the electric charge came into existence at a later stage in the evolution of the Universe when the SU(2)L⊗U(1)Y group was sponta-neously broken to U(1)em.It was never there all the time.This is because electric charge is a derived quantity.Hence weﬁnd that in this regard the Superstring Theories are inconsistent with the SM and henceﬂawed[9].5References[1]K.S.Babu and R.N.Mohapatra,Phys.Rev.Letts.63,938(1989).[2]X.-G.He,G.C.Joshi,H.Lew,B.H.J.McKellar and R.R.Volkas,Phys.Rev.D40,3140(1989).[3]5.A.Abbas,Phys.Lett.238B,344(1990).[4]A.Abbas,J.Phys.G.16,L163(1990).[5]A.Abbas,‘What is the Standard Model Higgs?’,hep-ph/9912243.[6]A.Abbas,Physics Today,July1999,p.81-82.[7]X.G.Wen and E.Witten,Nucl.Phys.B261,651(1985).[8]A.N.Schellekens,Phys.Letts.B237,363(1990)[9]A.Abbas,‘Electric charge,Early Universe and the Superstring Theo-ries’,hep-ph/9912244.6。