Is the standard Higgs scalar elementary
小学上册第十三次英语第4单元真题试卷(有答案)
小学上册英语第4单元真题试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The _____ (car/bike) is parked outside.2.The process of a solid turning into a gas is called _______.3. A sunflower turns towards the _____.4.I like to watch ______ shows on television.5.The capital of Armenia is _____.6.My brother, ______ (我弟弟), loves to play chess.7.The ________ can be found in many colors.8. A reaction that occurs spontaneously is labeled as a ______ reaction.9. A _______ is a type of reaction that occurs spontaneously.10.The arrangement of atoms in a molecule is known as its _____ structure.11.The Earth's atmosphere is made up of different ______ (gases).12.The skunk's spray is a powerful ________________ (防御).13.The _______ (The Boston Tea Party) protested British taxation without representation.14.My ________ (玩具名称) is a great way to express myself.15.The capital of Kazakhstan is ________ (哈萨克斯坦的首都是________).16.I want to ___ an adventure. (have)17.__________ are used in cleaning products for stain removal.18.I love spending time outdoors, especially at the __________.19.She is wearing ________ shoes.20.The process of converting a solid to a gas is called _______.21.The fish is swimming _____ the water. (in)22.My favorite food is _______ (披萨), and I can eat it every _______ (天).23.We engage in ________ (activities) regularly.24.I enjoy riding my ______ (滑板) at the skate park. It is a lot of ______ (乐趣).25.The ______ helps us learn about media literacy.26.I like to go ______ in the summer.27.My sister loves to __________ (绘画) in her sketchbook.28.The parrot is _______ (colorful).29. A _______ is a substance that can increase the reaction rate.30.The process of extracting oil from plants is called _______.31. A __________ is formed by the interaction of wind and water on rock.32.n Wall fell in ______ (1989年), marking the end of the Cold War. The Berl33. A ______ (青蛙) can jump very far.34.I can enjoy playful activities with my ________ (玩具类型).35.What is the name of the popular card game played with a standard deck of cards?A. PokerB. BridgeC. RummyD. Solitaire答案: A36.The puppy loves to chase its ______ (尾巴). It looks very ______ (搞笑).37.My favorite color is ________.38. A _____ (小马) can be very gentle and friendly.39.The _______ is important for pollination and growth.40.The _______ (小鳗鱼) slithers through the water.41.The ______ (青蛙) catches flies with its tongue.42.They like to ______ music together. (play)43.What is the term for a word that has the opposite meaning?A. SynonymB. AntonymC. HomonymD. Pronoun答案:B44.air quality) affects health and environment. The ____45.What do you call the act of taking care of plants?A. GardeningB. LandscapingC. FarmingD. Horticulture答案: A46.What do we use to eat food?A. KnifeB. ForkC. SpoonD. Plate答案:B Fork47.The study of ancient geological formations is known as ______.48.She is _______ (baking) cookies for the party.49.The Earth’s rotation causes day and ______.50.I want to be a ______ in the future.51.The chemical formula for barium sulfate is ______.52.小猫) curls up in my lap. The ___53.The flowers in the garden attract _______ and buzzing bees.54.My mom, ______ (我妈妈), enjoys gardening and caring for plants.55.How many weeks are in a year?A. 48B. 50C. 52D. 54答案: C56.During lunch, I like to eat _______ (食物) with my friends. We talk about our _______ (事情).57. A thermometer measures ______.58. A neutron has no electrical ______.59.The chemical process of breaking down substances is called _______.60.Nebulas are giant clouds of ______.61.My favorite season is ________.62.The playground is _______ with laughter.63.We watched a documentary about the ________ (极地).64. A reduction reaction involves the gain of ______.65.My dad loves __________ (健身).66.I play board games with my ________ (玩具名称).67.My _____ (父亲) teaches me many things.68.The train station is _______ (在城市中心).69.The beetle has a shiny _______ (壳).70.The ancient Romans built _______ to connect their empire. (道路)71.I have a toy ________ that looks like a dragon.72.What is 4 + 3?A. 5B. 6C. 7D. 8答案:C 773.The chemical formula for methane is __________.74.What is the value of 10 2 + 4?A. 10B. 11C. 12D. 13答案:B75.The smallest unit of a substance is called an ______.76.What is the main ingredient in salad?A. BreadB. LettuceC. RiceD. Meat答案: B77.My sister loves to play with her _____.78.The process of turning liquid into vapor is called ______.79.The __________ (冷战) was a period of political tension between the USA and the USSR.80.I like to go ________ (出租车) with my friends.81.I enjoy ___ (traveling) by train.82.The ________ is a small, quiet creature.83.What is the name of the famous temple in India?A. Taj MahalB. Angkor WatC. Temple of HeavenD. Borobudur答案:A.Taj Mahal84.My sister enjoys singing in the ____ (choir).85.The process of sublimation is when a solid changes to a gas without becoming ______.86.The tree has green ________.87. A squirrel collects ____.88.I have a ______ (小) house.89.The soup is _______ (cooking) on the stove.90.public engagement) encourages participation in decision-making. The ____91.The __________ (历史的潮流) shapes our present.92.I enjoy drawing pictures of my ________ (玩具名称).93.My friend is very __________ (善解人意) and compassionate.94. A _______ is a tool that can help to measure the pressure of gases.95.My ___ (小狗) loves to cuddle.96.What is the capital of New Zealand?A. AucklandB. WellingtonC. ChristchurchD. Hamilton答案: B97.The ____ has a long body and can swim very well.98.The skunk is known for its strong ________________ (气味).99.The _____ (多肉植物) is easy to care for.100.What do we call a fabric made from cotton?A. SilkB. WoolC. DenimD. Canvas答案: C. Denim。
9ElementaryParticlePhysics
1Hale Waihona Puke Overview• In particle physics, an elementary particle or fundamental particle is a particle whose substructure is unknown, thus it is unknown whether it is composed of other particles. Known elementary particles include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle. • Everyday matter is composed of atoms, once presumed to be matter's elementary particles—atom meaning "indivisible" in Greek. Soon, subatomic constituents of the atom were identified. As the 1930s opened, the electron and the proton had been observed, along with the photon, the particle of electromagnetic radiation. At that time, the recent advent of quantum mechanics was radically altering the conception of particles, as a single particle could seemingly span a field as would a wave, a paradox still eluding satisfactory explanation.
Neutron Electric Dipole Moment in Two Higgs Doublet Model
2. CP violation parameter in THDM The simple extension of SM is the one with the two Higgs doublets[6]. This model has the possibility of the soft CP violation in the neutral Higgs sector, which does not contribute to the flavor changing neutral current in the B , D and K meson decays. Weinberg[9] has given the unitarity bounds for the dimensionless parameters of the CPon of the strong-interaction hadronic effects. Weinberg employed the ”naive dimensional analyse”(NDA) as developed by Georgi and Manohar[7] in comput˜ operator on the neutron. However, this method admittedly ing the effect of the GGG provides at best the order-of-magnitude estimation. Moreover, when gluon fields are present, there occurs an indeterminable factor of 4π , which depends on whether one associates a factor gs or 4πgs with each gluon field factor in the interaction Lagrangian[3]. Recently, Chemtob[8] proposed a systematic approach which gives the hadronic matrix elements of the higher-dimension operators involving the gluon fields by using the large Nc current-algebra. In his model, the hadronic matrix elements of the operators are approximated by the intermediate states with the single nucleon pole and the nucleon plus one pion. So, this approach may be a realistic one. We employ his model to get the hadronic matrix elements of the relevant operators in this work. The comparision between results by this approach and NDA will be discussed briefly in the last section. In section 2, the neutral Higgs mass matrix is analyzed and then the magnitudes of the CP violation factors ImZi are estimated. In section 3, the formulation of the neutron EDM with the hadronic matrix elements of the CP violating operators are discussed. Section 4 is devoted to the numerical results of the neutron EDM and some remarks and conclusion are given in section 5.
internal standard of each element -回复
internal standard of each element -回复"Internal Standard of Each Element: A Step-by-Step Understanding"Introduction:The internal standard is an essential concept in analytical chemistry. It refers to a substance added to a sample, which is used as a reference for quantification purposes. By understanding the internal standard of each element, scientists can accurately measure and compare the concentrations of different elements in a sample. This article aims to provide a step-by-step understanding of the internal standard and its significance.1. Define the internal standard and its purpose:The internal standard is a known substance added to a sample before analysis. It must be chemically similar but different from the analyte of interest. The purpose of the internal standard is to compensate for variations in sample preparation, instrument response, and other uncontrollable factors that may affect the analysis. It acts as a reference or calibration compound, allowing for the accurate quantification of the analyte.2. Selection of the appropriate internal standard:The internal standard should possess specific characteristics to be effective. Firstly, it should not be present in the original sample. This ensures that the internal standard does not interfere with the quantification of the analyte. Secondly, it should have similar chemical properties and behavior to the analyte, facilitating consistent response during analysis. Thirdly, it should be easily detectable and distinguishable from the analyte to avoid any confusion in differentiation.3. Preparation and addition of the internal standard:Once the appropriate internal standard is chosen, it must be prepared and added to the sample. The addition should be done carefully, considering the concentration of the analyte in the sample. The internal standard should be added in a known amount to enable quantitative comparison. The addition can be performed manually or using automated systems, depending on the analytical technique and instrument employed.4. Measurement and calibration:After the internal standard is added, the sample is subjected to the chosen analytical technique, such as spectroscopy orchromatography. The internal standard and the analyte are simultaneously detected, providing a basis for calibration. The response of the internal standard is used for normalization and calibration of analyte response. This calibration allows for accurate determination and quantification of the analyte concentration in the sample.5. Calculation of results:Using the calibration data obtained, the concentration of the analyte in the sample can be calculated. The internal standard provides a known reference point, enabling the determination of the analyte's concentration relative to it. By comparing the response parameters of the internal standard and analyte, such as peak area or height, a calibration curve can be constructed. This curve relates the analyte concentration to its corresponding response, facilitating the accurate calculation of the sample's analyte concentration.6. Evaluation of results:The calculated concentration of the analyte is vitally important, but it must also be evaluated based on the chosen method's limit of detection, precision, and accuracy. A thorough evaluation allows forthe determination of any potential sources of error and identifies areas where improvement may be required. By ensuring the accuracy and precision of the calculated results, the internal standard provides confidence in the analytical measurements.Conclusion:The internal standard of each element plays a crucial role in accurate quantification of analytes in samples. By carefully selecting and using the internal standard, consistency and reliability in analytical results can be achieved. Understanding the step-by-step process of employing the internal standard allows scientists to overcome various sources of error, leading to precise and reliable analytical measurements.。
Scalar Glueball, Scalar Quarkonia, and their Mixing
Hai-Yang Cheng1 , Chun-Khiang Chua1† and Keh-Fei Liu1,2
1
Institute of Physics, Academia Sinica Taipei, Taiwan 115, Republic of China
†
Address after August, 2006 : Department of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 320, R.O.C.
1
I.
INTRODUCTION
Despite of the fact that the q q ¯ and glueball contents of the iso-singlet scalar mesons f0 (1710), f0 (1500) and f0 (1370) have been studied extensively, it has been controversial as to which of these is the dominant glueball. Partly due to the fact that f0 (1500), discovered in pp ¯ annihilation at LEAR, ′ ¯ , ηη and ηη modes which are not compatible with a simple q has decays to ππ, K K ¯q picture [1] and that the earlier quenched lattice calculations [2] predict the scalar glueball mass to be ∼ 1550 MeV, it has been suggested that f0 (1500) is primarily a scalar glueball [3]. Furthermore, because ¯ , it has been thought that of the small production of ππ in f0 (1710) decay compared to that of K K ¯ relative f0 (1710) is primarily ss ¯ dominated. On the other hand, the smaller production rate of K K to ππ in f0 (1370) decay leads to the conjecture that f0 (1370) is governed by the non-strange light quark content. Based on the above observations, a flavor-mixing scheme is proposed [3] to consider the glueball and q q ¯ mixing in the neutral scalar mesons f0 (1710), f0 (1500) and f0 (1370). χ2 fits to the measured scalar meson masses and their branching ratios of strong decays have been performed in several references by Amsler, Close and Kirk [3], Close and Zhao [4], and He et al. [5]. A common feature of these analyses is that, before mixing, the ss ¯ mass MS is larger than the pure glueball mass √ ¯)/ 2) mass MN , with MG close to 1500 MeV MG which, in turn, is larger than the N (≡ (uu ¯ + dd and MS − MN of the order of 200 ∼ 300 MeV. However, there are several serious problems with this scenario. First, the isovector scalar meson a0 (1450) is confirmed to be a q q ¯ meson in lattice calculations [6, 7, 8, 9, 10] which will be discussed later. As such, the degeneracy of a0 (1450) and ∗ (1430), which has a strange quark, cannot be explained if M is larger than M by ∼ 250 MeV. K0 S N Second, the most recent quenched lattice calculation with improved action and lattice spacings extrapolated to the continuum favors a larger scalar glueball mass close to 1700 MeV [11, 12] (see below for discussion). Third, if f0 (1710) is dominated by the ss ¯ content, the decay J/ψ → φf0 (1710) is expected to have a rate larger than that of J/ψ → ωf0 (1710). Experimentally, it is other way around: the rate for ωf0 (1710) production is about 6 times that of J/ψ → φf0 (1710). Fourth, it is well known that the radiative decay J/ψ → γf0 is an ideal place to test the glueball content of f0 . If f0 (1500) has the largest scalar glueball component, one expects the Γ(J/ψ → γf0 (1500)) decay rate to be substantially larger than that of Γ(J/ψ → γf0 (1710)). Again, experimentally, the opposite is true. Other scenarios have been proposed. Based on their lattice calculations of the quenched scalar glueball mass at 1625(94) MeV at the infinite volume and continuum limits [13] and the ss ¯ meson mass in the connected insertion (no annihilation) at ∼ 1500 MeV, Lee and Weingarten [14, 15] considered a mixing scheme where f0 (1500) is an almost pure ss ¯ meson and f0 (1710) and f0 (1370) ¯ meson respectively, but with substantial mixing between are primarily the glueball and uu ¯ + dd the two (∼ 25% for the small component). With the effective chiral Lagrangian approach, Giacosa et al. [16] performed a fit to the experimental masses and decay widths of f0 (1710), f0 (1500) and f0 (1370) and found four possible solutions, depending on whether the direct decay of the glueball component is considered. One of the solutions (see Appendix A) gives f0 (1710) as the pure glueball, while f0 (1370) and f0 (1500) are dominated by the quarkonia components, but with strong mixing
Natural Mass Hierarchy of Z Boson and Scalar Top in No-Scale Supergravity
a rXiv:h ep-ph/5231v26J un2TIT-HEP-448KEK-TH-696hep-ph/0005231May,2000Natural Mass Hierarchy of Z Boson and Scalar Top in No-Scale Supergravity Yuichi Chikira ∗Department of Physics,Tokyo Institute of Technology Oh-okayama,Meguro,Tokyo 152-0033,Japan and Yukihiro Mimura †Theory Group,KEK,Oho 1-1,Tsukuba,Ibaraki 305-0801,Japan Abstract A study has shown that a ‘no-scale’model makes a hierarchy between the scalar top mass and the Z boson mass naturally.The supersymmetry breaking parameters are constrained by flavor changing neutral currents in the minimal supersymmetric standard model.One solution of the problem is that the gaugino mass is the only source of supersymmetry breaking parameters at thePlanck scale.However,in such a scenario,we need a cancellation between the Higgs mass parameters under the minimization condition of the Higgs potential.We insist that there is no such cancellation in the no-scale model,and that the no-scale model provides a prediction of the scalar top mass and the lightest Higgs mass.The lightest Higgs mass is predicted to be m H =110±5GeV.1.IntroductionSupersymmetric theories now stand as the most promising candidates for a uni-fied theory beyond the standard model[1].Accurate data remarkably favor the supersymmetric grand unified theory(GUT)over any non-supersymmetric the-ory[2].Supersymmetry helps to resolve the gauge hierarchy problem[3].In non-supersymmetric standard models,because the squared Higgs mass receives a quadratic divergent correction radiatively,we cannot explain the hierarchy between the weak scale and the grand unified scale naturally.Supersymmetry removes the quadratic divergences and provides a framework for naturally explaining the widely separated hierarchy.Under those contexts,the idea of radiative breaking of the electroweak symme-try[4]is very popular.It is very attractive to explain the breaking of electroweak symmetry through large logarithms between the Planck(or GUT)scale and the weak scale.The radiative corrections drive an up-type Higgs mass-squared param-eter negative for a large top Yukawa coupling,and thus the electroweak symmetry breaks down.The radiative symmetry breaking mechanism has consequences for the supersymmetric particle spectrum,and provides important constraints on the particle spectrum.These constraints also provides a slight puzzle.In the radiative breaking mech-anism,the Z boson mass is related to the supersymmetry breaking parameters.We thus believe that supersymmetric particles are not very heavy compared with the Z boson.However,the experimental lower bounds for the supersymmetric particle masses are becoming larger day by day,and it seems that we must require afine tuning between the parameters in the Higgs potential[5].It is well known thatflavor changing neutral currents(FCNC)make important constraints on the supersymmetry breaking scalar masses[1,6].We require that the scalar quark eigenmasses have degeneracy∗of a few percent when the scalar masses are on the order of O(100)GeV.One solution concerning the scalar quark mass degeneracy is to consider the type of minimal gaugino mediation[7].Namely, the gaugino mass is almost the only source for supersymmetry breaking at the Planck scale.The supersymmetry breaking parameters which haveflavor indicesare sufficiently small compared to the gaugino mass at the Planck scale,and become sufficiently large due to renormalization groupflow at the low energy scale.Though this scenario is very attractive,such large gaugino masses cause thefine tuning described above to the Higgs potential.Are there any mechanisms in whichfine tuning is not required,even if the gaugino mass is large?In this paper,we insist that the‘no-scale’supergravity model[12]does not re-quire anyfine tuning of the Higgs potential.The no-scale models are very suitable for the scenario of minimal gaugino mediation.We consider the supersymmetric particle spectrum in no-scale models,where the magnitude of the supersymmetry breaking parameters is also determined radiatively.We can investigate the theo-retical upper bounds in no-scale models,and can judge the bounds at near future colliders.Especially,we insist that the no-scale models suggest a natural mass hierarchy between the Z boson mass and the supersymmetry breaking masses.The organization of this paper is as follows.In Section2,we review unnatural tuning in the Higgs potential in Minimal Supersymmetric Standard Model(MSSM). In Section3,we review no-scale supergravity models.In Section4,we explain how to calculate the particle spectrum in our framework.In Section5,we show the results for the particle spectrum and discuss its bounds.Finally,we conclude in Section6with a summary of our results.2.Unnatural Tuning in Z Boson MassThe tree level neutral Higgs potential in MSSM is given byV(0)=m21|H0d|2+m22|H0u|2−(m23H0d H0u+c.c.)+g2+g′22=−µ2+m2Hd−m2H u tan2β2m23sin2β=∗Since we would not like to consider the Physics beyond GUT,those parameters are given at the GUT scale later.†To plot thefigure,we consider the1-loop corrected scalar potential(3.2).‡We have a problem which is so called theµ-problem;why is the supersymmetric parameterµof the same order as the supersymmetry breaking parameters.0.81 1.21.4 1.6µ0 /M 1/20100200300400500M Z [G e V ]M 1/2 = 250 GeV M 1/2 = 200 GeV M 1/2 = 150 GeV M 1/2 = 100 GeV Figure 1:We show the Z boson mass as a function of µ0/M 1/2for various gaugino masses.In this figure,we set m 0and A 0to be zero.We choose the B 0parameter so that tan β=10at the point M Z =91GeV.On the left side of this figure,tan βbecomes close to 1,and the Higgs potential is destabilized.Since parameters µand m 2H u depend on the scale Q ,we should know the scale wherewe require tuning between µ2and m 2H u .The scale is the one where 1-loop correctedpotential becomes small.We denote the scale as Q ˜t ,since it is nearly equal to the mass of scalar top quarks.Then,the physical Z boson mass is approximated byM 2Z ∼M 2Z (Q ˜t ),(2.7)whereM 2Z (Q )≡−2m 22(Q ).(2.8)We define the scale Q 0where M 2Z (Q )vanish.The Q 0is the scale where electroweaksymmetry breaks down at the tree level.Expanding M 2Z (Q )by ln Q around scaleQ 0,we obtain M 2Z ∼2ln(Q 0d ln Q(Q 0).(2.9)From this point of view,the fine tuning in the Z boson mass is translated into the tuning between the scalar quark mass scale Q ˜t and the scale Q 0.As stated in Ref.[9],electroweak symmetry breaks down only when the scale Q ˜t is less than Q 0.This fact means the same thing as saying that the parameter µshould be the right edge in Fig.1when the gaugino mass becomes greater.There are many implications about naturalness in the literature.In Ref.[10], it is pointed out that a heavy scalar mass m0(which masses are of the order of1 TeV)relaxes thefine tuning.Ref.[8]suggests that a lessfine-tuned model should be selected as a scenario candidate for supersymmetry breaking.It is preferred that the gaugino mass is not unified at the GUT scale(e.g.D-brane model)in the reference. In this literature,we feel that thefine tuning in the Higgs potential is not dispelled. Are there any models in which cancellation occurs naturally?In this paper,we suggest that we have already had a model which can explain the heavy gluino mass naturally without anyfine tuning.This model is no-scale supergravity.In folklore,it is said that more severefine tuning is required in the no-scale supergravity model rather than in ordinary models.We believe,however, that this interpretation is not correct.To see this,we give a brief review of no-scale supergravity in the next section.3.No-Scale SupergravityIn this section,we briefly review the no-scale supergravity,and we consider the natural mass hierarchy between the Z boson and the scalar top in the no-scale model.In the hidden sector model[11],we separatefields into two sectors,which are a visible sector and a hidden sector.The observablefields(quarks,leptons and Higgsfields)are involved in the visible sector.The hiddenfields,which break supersymmetry,exist in the hidden sector,and couple with the observablefields through only gravitational interaction.The F terms of the hiddenfields have VEVs due to the dynamics in only the hidden sector in ordinary hidden sector models.In other words,the scale of supersymmetry breaking is determined with no relation to our visible sector.However,it is possible that a scalar potential for the hidden sectorfields isflat at the tree level,and that the VEVs of the hiddenfields determine radiatively accompanied with visible sector dynamics.Such theories are called no-scale supergravity[12].Let us see how the gravitino mass is determined in the no-scale ing the minimization conditions(2.3)and(2.4),we obtain the tree level MSSM scalar potential at the minimal point,V(0) mim =−1Since the Z boson mass is proportional to the gravitino mass∗,the potential involving the hidden sector is unbounded from below.However,there exists a1-loop corrected scalar potential,V(1)=1Q2−3DR scheme[13].As a result,the scalar potential is stabilized if Str M4>0[14], and the gravitino mass is determined dynamically.We emphasize here that the gravitino mass is not independent on the visible sector parameter,namelyµ,in the no-scale models.The naturalness argument in the no-scale model differs from arguments in the ordinary ones due to such a dependence.To confirm the natural hierarchy between the Z boson mass and the supersym-metry breaking masses,we will overview the minimization with respect to gravitino mass[12].Since the total scalar potential does not depend on the renormalization point,we can evaluate the potential at a scale where the1-loop corrected potential vanishes,V(1)(v u,v d;Q)=0.(3.3) This scale is approximately the mass scale of the scalar top quarks(Q˜t),Q˜t≡(m˜t1m˜t2)1/2.(3.4) We can thenfind the minimal value of the effective scalar potential[12],V min∼−CQ4˜t ln Q2˜tQ2˜t=1.(3.6) It is important that Q˜t is very close to Q0,Q˜t=Q0/e1/2.(3.7)Substituting it into Eq.(2.9),wefind the Z boson mass formula in the no-scale model as follows for large tanβ:M2Z∼dd ln Q m2Hu=1d ln Qµ2=1010*******400500M 1/2 [GeV]−1−0.8−0.6−0.4−0.200.20.4V e f f /(100 G e V )4Figure 2:Effective potential Eq.(4.1)minimized by the Higgs vacuum expectation values,v u and v d .At first,we show an effective potential which is minimized by Higgs VEVs v d and v u (Fig.2).This figure is drawn in the minimal case,where m 20=0and A 0=B 0=0.The horizontal axis is for the gaugino mass.We can see that there is a minimum with respect to the gaugino mass.Since the gaugino mass is the most sensitive parameter for the supersymmetry breaking Higgs mass squared,we normalize the following dimensionful parameters of MSSM:{m 20,M 1/2,A 0,B 0,µ0},(4.2)divided by the gaugino mass M 1/2,and we adopt the following four dimensionless parameters:{ˆm 20,ˆA 0,ˆB 0,ˆµ0}(4.3)as parameters for no-scale models.The hat denotes that the parameters are normal-ized by the gaugino mass (squared).The gaugino mass is determined in minimizing the potential if we fix the hatted parameters ∗.The freedom for ˆµ0is consumedwhen the Z boson mass is fixed as 91GeV.If we fix tan β,ˆB0is consumed and the remaining free parameters are only ˆm 0and ˆA0.To show our numerical analysis,we evolve the supersymmetry breaking param-eters with the full two-loop RGEs[16].Since the potential does not ideally depend on the renormalization point,we may choose any scale.Nevertheless,we minimize the potential(4.1)near to the scale where the electroweak symmetry breaks down at the tree level tofix our aim. This is because we must consider the threshold effect for supersymmetric particles, for instance,scalar quarks and gluinos.Therefore,we adopt our method as follows. Firstly,we minimize the effective potential with respect to the Higgs VEVs and the gaugino mass at the scale above those supersymmetric particle masses.After minimization,we include one-loop threshold corrections from supersymmetric par-ticles[17],andfix the physical quantities.We take as inputsα−1em(M Z)=127.9, sin2θW(M Z)DR scheme.5.Numerical ResultsIt is convenient that we present the RGE solution by the following parameteri-zation[5].The dimensionful parameters at low energy are written using the GUT scale parameters.First of all,the up-type Higgs mass squared,m22,is written asm22=1.0µ20−0.05m20−1.75M21/2−0.34M1/2A0−0.10A20(5.1) in the case of tanβ=10.The mass of the Z boson is M2Z∼−2m22.It is easy tosee that we requirefine tuning betweenµ0and M1/2,if the gaugino is much heavier than Z boson.It is worth noting that the coefficient of m20is very small in the RGEis solution in the expression of m22.This is because the’focus-point scale’for m2Hu of the order of100GeV[10]∗.In contrast,the RGE solution for dm22/d ln Q isdm22∗In the reference[10],it seems that the coefficient of m2is of opposite sign to ours.In ourcalculation,the sign is reversed when we take the top quark pole mass as M t=172GeV.m 0 [GeV]-2000-10001000A 0 [G e V ]Figure 3:Contour plot for the gaugino mass as a function of m 0and A 0in the case of tan β=10.the m 0and A 0are tuned in the lower right region in the figure.We prefer a small m 0because of FCNC constraints.Therefore,we do not regard the large m 0region.In Fig.4,we show m 0-M 1/2plot at A 0=0.We can qualitatively understand its elliptic shape from Eq.(5.2).Fig.5shows the chargino masses as a function of m 0for various A 0.We remark that the dots are plotted every 0.2interval for ˆm 0(not m 0),and 0.5interval for ˆA0,thus the density of the dots is not related to the probability of the parameters.This remark is also applied in the figure below.In Fig.6,we show the gluino mass,lightest chargino mass and lightest neutralino mass as a function of m 0for various A 0in the same way as in Fig.5.The important prediction for the no-scale model is that the scalar top masses are almost determined independently of the scalar mass,m 0.The scalar top masses are plotted in Fig.7.In supersymmetric models,the lightest Higgs mass is bounded by M Z at the tree level.However,this upper bound is corrected by the 1-loop potential [19].Since the scalar top masses are almost determined,the lightest Higgs mass is also predictable in the no-scale model.We plot the lightest Higgs mass for tan β=5,10,30in Fig.8.It is important that the lightest Higgs mass is 110±5GeV for small m 0.The small m 0is favored for FCNC constraints.In calculating the lightest Higgs mass,we adopt the 2-loop approximate formula for the mass in Ref.[20].0200400600m 0 [GeV]50100150200250M 1/2 [G e V ]Figure 4:m 0-M 1/2plot at A 0=0in the case of tan β=10.0200400600800m 0 [GeV]0200400600[G e V]Figure 5:Chargino masses.The heavier and lighter chargino masses are approxi-mately µand the wino mass respectively.We cut the lighter chargino mass,which is smaller than 85GeV.m 0 [GeV]0200400600[G e V]Figure 6:Gluino mass,lightest chargino mass and lightest neutralino mass.The 1-loop correction for gluino mass is included.The lighter chargino mass,which is smaller than 85GeV,has been cut.m 0 [GeV]0200400600800[G e V ]Figure 7:Scalar top masses.The scalar top masses are determined up to left-right mixing.0200400600800m 0 [GeV]100105110115120M H [G e V ]Figure 8:Lightest Higgs mass for tan β=5,10,30.The dots are plotted every 0.2interval for ˆm 0(not m 0),and 0.5interval for ˆA0.The density of the dots is not related to the probability of the parameters.We line the recent LEPII bound on the non-observation of e +e −→ZH [21]for one’s information.6.DiscussionIn order to see our insist visibly,we show figure (Fig.9)in the corresponding plot to Fig.1.Again,we set parameter m 20and A 0to be zero.We choose the B 0parameter so as to be tan β=10at the point M Z =91GeV ∗.We plot the Z boson mass as a function of µ0/M 1/2(=ˆµ0).There is no weird constraint for the parameter ˆµ0for electroweak symmetry breaking,contrary to the case in Fig.1.Therefore,the model-building God can create the MSSM parameters without considering whether electroweak symmetry can break down at low energy.Our Z boson mass (91GeV)does not lie on a special point,contrary to the ordinary case.The following quantity [22]is usually used for measuring the sensitivity of the Z boson mass for variations in parameter a ,∆a =∂ln M 2Z ∗Incidentally,tan βis just 10when the B 0parameter is equal to zero.0.9 1.1 1.31.5 1.7µ0 /M 1/2110100100010000M Z [G e V ]GluinoZ BosonZ Boson (tree)Figure 9:Z boson mass and gluino mass as a function of µ0/M 1/2in the case of the no-scale model.Their mass ratio is approximately constant to µ0/M 1/2.We also plot the tree level Z boson mass formula,Eq.(3.8).does not cause any fine-tuning problem,contrary to ordinary models.It is just an event that a value of the Z boson mass is selected.Some people may say that it is also just an event in other supersymmetry break-ing scenarios.This opinion is obviously true.However,the predictive ability in the no-scale model is completely different.The subtractive tuning in the ordinary model does not have any predictive power.The supersymmetry breaking mass scale may be of the order of 10TeV in the ordinary model.On the other hand,we do not require any subtractive tuning in the no-scale model,and predict that all of the supersymmetric particles (except gravitino)appear below about 500-600GeV.Especially,we can judge the no-scale model when we search the Higgs boson or gauginos in the near future.This predictive ability is our motivation concerning the no-scale model.For theoretical physicists,it is important to search predictive models.To say more,it is important that we recognize that the fine-tuning in the Higgs potential may impose the no-scale supergravity,and we consider predictions of the no-scale models.This is a process of Physics to access the unknown world.Note added:While completing this paper,we received a paper by R.Barbieri and A.Strumia [23]which also considers that the electroweak breaking scale becomes related to the supersymmetry breaking scale by a loop factor in a similar way to us.AcknowledgmentsY.M.would like to thank to N.Okada for the discussion of the no-scale supergrav-ity.This work was supported by JSPS Research Fellowships for Young Scientists.A Notation and ConventionThe superpotential of minimal supersymmetric standard model(MSSM)is presented asW=Y u Q·H u U c+Y d H d·QD c+Y e H d·LE c+µH d·H u,(A.1) where the SU(2)inner product is defined asH d·H u≡H T dǫH u,ǫ= 01−10 .(A.2) Here,Q,U c,D c,L,E c are matter chiral superfields,and H u and H d are Higgs doublets.We denote the soft supersymmetry breaking terms asV soft=m2Hd|H d|2+m2H u|H u|2+m2˜q˜q˜q†+m2˜u˜u R˜u†R+m2˜d ˜dR˜d†R+m2˜ℓ˜ℓ˜ℓ†+m2˜e˜e R˜e†R+(A u Y u˜q·H u˜u c R+A d Y d H d·˜q˜d c R+A e Y e H d·˜ℓ˜e c R+h.c.)+(BµH d·H u+h.c.)(A.3) To clarify our notation,we present the left-right component in the scalar top quark mass matrix and chargino mass matrix in the following.The left-right mixing is(A t+µcotβ)m t.(A.4) The chargino mass matrix is presented asMχ+= M2√2M W sinβ−µ .(A.5) The supergravity theories are given by the K¨a hler potential K,the superpotential W and the gauge kinetic function f.The scalar potential is given in supergravity asV=e K[g ij∗(D i W)(D j∗W∗)−3W W∗].(A.6)Using the K¨a hler transformation G=K+log W+log W∗,we obtainV=e G[G i G i−3].(A.7) The no-scale K¨a hler potential[12]is written asG=−3ln(T+¯T−h(φ∗i,φi))+ln|W(φi)|2,(A.8) where T is a modulifield andφarefields in the visible sector.The function h is a K¨a hler potential for the visiblefields.The scalar potential is thusV=3|W|2∂φi2.(A.9)If the global supersymmetric conditions,∂W/∂φi=0,are satisfied,the scalar po-tential for T isflat and the gravitino mass,m3/2=e G/2,is not determined.Expanding the K¨a hler potential with respect to visiblefields,Q,we write the K¨a hler potential[24]asK=ˆK(T,T∗)+˜K ij∗(T,T∗)Q i Q j∗+1|ˆW|˜µij+m3/2(H ij−G α∗∂α∗H ij),(A.13)Y ijk=eˆK/2ˆW∗The gaugino mass is given by the gauge kinetic function,f a,as1M a=[8]G.L.Kane and S.F.King,Phys.Lett.B451(1999)113,M.Bastero-Gil,G.L.Kane and S.F.King,Phys.Lett.B474(2000)103.[9]G.Gamberini,G.Ridolfiand F.Zwirner,Nucl.Phys.B331(1990)331.[10]J.L.Feng,K.T.Matchev and T.Moroi,Phys.Rev.D61(2000)075005;hep-ph/0003138.[11]R.Nath,R.Arnowitt and A.Chamseddine,Phys.Rev.Lett.49(1982)970.[12]For a review on no-scale supergravity,hanas and D.V.Nanopoulos,Phys.Rep.145(1987)1.[13]S.Coleman and E.Weinberg,Phys.Rev.D7(1973)1888.[14]C.Kounnas,hanas,D.V.Nanopoulos and M.Quir´o s,Nucl.Phys.B236(1984)438;J.L.Lopez,D.V.Nanopoulos and K.Yuan,Phys.Rev.D50(1994) 4060.[15]J.A.Casas,V.Di Clemente and M.Quir´o s,Nucl.Phys.B553(1999)511;D.V.Gioutsos,hep-ph/9905278.[16]S.P.Martin and M.T.Vaughn,Phys.Rev.D50(1994)291;I.Jack and D.R.T.Jones,Phys.Lett.B333(1994)372;Y.Yamada,Phys.Rev.D50(1994)3537;I.Jack,D.R.T.Jones,S.P.Martin,M.T.Vaughn and Y.Yamada,Phys.Rev.D50(1994)5481.[17]J.Bagger,K.Matchev and D.Pierce,Phys.Lett.B348(1995)443;D.Pierce,J.Bagger,K.Matchev and R.-J.Zhang,Nucl.Phys.B491(1997)3.[18]R.Arnowitt and P.Nath,Phys.Rev.D46(1992)3981.[19]Y.Okada,M.Yamaguchi and T.Yanagida,Prog.Theor.Phys.85(1991)1;H.E.Haber and R.Hempfling,Phys.Rev.Lett.66(1991)1815;J.Ellis,G.Ridolfiand F.Zwirner,Phys.Lett.B257(1991)83;Phys.Lett.B262(1991) 477.[20]S.Heinemeyer,W.Hollik and G.Weiglein,Phys.Lett.B455(1999)179.[21]W.J.Marciano,hep-ph/0003181.[22]R.Barbieri and G.F.Giudice,Nucl.Phys.B306(1993)63.[23]R.Barbieri and A.Strumia,hep-ph/0005203.[24]V.S.Kaplunovsky and J.Louis,Phys.Lett.B306(1993)269.[25]G.F.Giudice and A.Masiero,Phys.Lett.B206(1988)480.21。
基本粒子物理1
“Particle Physics”Leptonic Weak InteractionsNeutrinos and Neutrino OscillationsThe CKM Matrix and CP ViolationThe handouts are fairly complete, however there a number of decent books:Michaelmas 2009based on the language of particle physicsthe unit of action : Units become (i.e. correct dimensions):MomentumLength Energy Momentum Time Length AreaNow all quantities expressed in powers of GeVSimplify algebra by setting:of andNOW:electric chargehas dimensionsUnless otherwise stated, Natural Units are used throughout these handouts, , , etc.Michaelmas 2009Will use 4-vector notation withIn particle physics, usually deal with relativistic particles. calculations to be Lorentz Invariant . L.I.quantities formed from , e.g.withNOTATIONFour vector scalar product :written as either:or or Quantities evaluated in the centre of mass frame orkinematic variables: s , t and u Centre-of-mass energy, s :Since this is a L.I.quantity, can evaluate in any frame. Choose the most convenient, i.e. the centre-of-mass frame:This is a scalar product of two four-vectors Lorentz Invariant is the total energy of collision in the centre-of-mass frameCurrent understanding embodied in theFundamental interaction strength is given by chargecoupling “constant”(both gg Convenient to express couplings in terms ofProf. M.A. Thomson in sense that:All intermediate particles arei.e.Feynman diagrams mainly used toRequires understanding of theory and experimental datais Transition Matrix Elementis density of final statesis number of transitions per unit time from initial stateto final state –not Lorentz Invariant !Rates depend on MATRIX ELEMENT and DENSITY OF STATES the ME contains the fundamental particle physicsHamiltoniancalculations of particle decay rates and cross sections:calculation of interaction Matrix Element:Interaction by particle exchange and Feynman rulestreatment of spin-half particles:Dirac Equationand a few mathematical tricks along, e.g. the Dirac Delta FunctionMichaelmas 20092Want to calculate the decay rate in first orderaswhere N is the normalisation andFor decay rate calculation need to know:Transition matrix element from perturbation theory p yp xp Volume of single state in momentum space:Normalising to one particle/unit volume gives number of states in element:Integrating over an elemental shell in momentum-space givesApply boundary conditions ( ):Therefore density of states in Golden rule:Wave-function vanishing at box boundariesquantised particle momenta: withAny function with the above properties can represent(an infinitesimally narrow Gaussian) In relativistic quantum mechanics delta functions prove extremely usefulfor integrals over phase space, e.g. in the decayandexpress energy and momentum conservationProf. M.A. Thomson Michaelmas 2009Start from the definition of a delta functionNow express in terms of whereand then change variablesFrom properties of the delta function (i.e. here onlynon-zero at )Rearranging and expressing the RHS as a delta functionRewrite the expression for density of states using a delta-functionintegrating over all final state energies but energy conservation nowtaken into account explicitly by delta functionHence the golden rule becomes:the integral is over all “allowed ”final states of any energy2T in a two-body decay, only need to considerone particle : mom. conservation fixes the other However, can include momentum conservation explicitly by integrating overthe momenta of both particles and using another G -fnEnergy cons.Mom. cons.Density of states a/J Conclude that a relativistic invariant wave-function normalisation particles per unit volumeNormalise to 2E particles/unit volumeLorentz Invariant Matrix Element , , in terms of the wave-functions normalised to particles per unit volumeused normalised to 1 particle per unit volume is normalised to per unit volumeProf. M.A. Thomson Note:uses relativistically normalised wave-functions. It is This form of is simply a rearrangement of the original equationbut the integral is now the factor of Now expressing in terms of givesis inversely proportional to exactly what one would expect from time dilation (Energy and momentum conservation in the delta functionsBecause the integral is Lorentz invariant (i.e. frame independent) it can be evaluated in any frame we choose. The C.o.M. frame is most convenient 2Integrating over using the G -function: since the G -function imposes Writing In the C.o.M. frame andFor convenience, hereis written asProf. M.A. Thomson Michaelmas 2009i Timposes energy conservation. determines the C.o.M momenta ofthe two decay productsi.e. for can be integrated using the property of function derived earlier (eq. where is the value for whichAll that remains is to evaluate•can be obtained fromBut from , i.e. energy conservation:VALID FOR ALL TWO-BODY DECAYS !In the particle ’s rest frameDifferential Cross sectionintegrate over allother particlesa particle of type a traversesv aInteraction probability obtained from effectivecross-sectional area occupied by then b v VRate per particle of type a=total reaction rate ==Rate = Flux x Number of targets x cross sectionMichaelmas 2009•Start from Fermi ’s Golden Rule:whereis the transition matrix for a normalisation of 1/unit volume4Now For 1 target particle per unit volumethe parts are not Lorentz InvariantTo obtain a Lorentz Invariant form use wave-functions normalised to particles Matrix elementThe integral is now written in a Lorentz invariant formThe quantity can be written in terms of a four-vector scalar product and is therefore also Lorentz Invariant Consequently cross section is a Lorentz Invariant quantityTwo special cases of Lorentz Invariant Flux:Centre-of-Mass Frame •Target (particle 2) at restMichaelmas 20094•Start from •HereThe integral is exactly the same integral that appeared in the particle decay calculation but with replaced byIn the case of elastic scattering For calculating the total cross-section (which is Lorentz Invariant) the previous page (eq. (4)) is sufficient. However, it is not so useful for calculating the differential cross section in a rest frame other than the e 1e –e –Start by expressing in terms of Mandelstam t square of the four-momentum transferbecause the angles in refer to the C.o.M frame For the last calculation in this section, we need to find a L.I.expression for Product offour-vectorstherefore L.I.Michaelmas 2009Want to express in terms of Lorentz Invariant12x In C.o.M. frame: givingFinally, integrating over (assuming no dependence of ) gives:thereforehenceLorentz Invariant differential cross sectionAll quantities in the expression for are Lorentz Invariant and any rest frame . It should be noted thatis a constant, fixed by energy/momentum conservationAs an example of how to use the invariant expressionwe will consider elastic scattering in the laboratory frame in the limit where we can neglect the mass of the incoming particlee.g. electron or neutrino scatteringMichaelmas 2009e–e1e.g.TWish to express the cross section in terms of scattering angle of theIntegratingThe rest is some rather tedious algebra…. start from four-momentaso hereBut from (E,p) conservationand, therefore, can also express t in terms of particles 2and 4is a constant (the energy of the incoming particle) soEquating the two expressions for t givesParticle 1 massless usinggivesIn limitProf. M.A. Thomson Michaelmas 2009The calculation of the differential cross section for the case where 1can not be neglected is longer and contains no more “physics ”(see appendix II).Again there is only one independent variable , T , which can be seen from conservation of energyGeneral form for 2 2 Body Scattering in Lab. Frame is a function of Where is a function of particle massesScattering cross section in C.o.M. frame:Invariant differential cross section (valid in all frames):Differential cross section in the lab. frame (m1 0)withSummary of the summary:To show this is Lorentz invariant, first considerMichaelmas 2009TagainBut now the invariant quantity t :Michaelmas 20093/d(cos T ),first differentiateto giveDifferentiate wrt. cos T(AII.2)obtain。
Energy and entropy conservation for dynamical black holes
a r X i v :g r -q c /0408008v 2 22 N o v 2004Energy and entropy conservation for dynamical black holesSean A.HaywardDepartment of Physics,National Central University,Jhongli,Taoyuan 320,Taiwansean hayward@(Dated:revised 28th October 2004)The Ashtekar-Krishnan energy-balance law for dynamical horizons,expressing the increase inmass-energy of a general black hole in terms of the infalling matter and gravitational radiation,isexpressed in terms of general trapping horizons,allowing the inclusion of null (isolated)horizons aswell as spatial (dynamical)horizons.This first law of black-hole dynamics is given in differentialand integral forms,regular in the null limit.An effective gravitational-radiation energy tensor isobtained,providing measures of both ingoing and outgoing,transverse and longitudinal gravitationalradiation on and near a black hole.Corresponding energy-tensor forms of the first law involve apreferred time vector which plays the role for dynamical black holes which the stationary Killingvector plays for stationary black holes.Identifying an energy flux,vanishing if and only if the horizonis null,allows a division into energy-supply and work terms,as in the first law of thermodynamics.The energy supply can be expressed in terms of area increase and a newly defined surface gravity,yielding a Gibbs-like equation,with a similar form to the so-called first law for stationary blackholes.A Clausius-like relation suggests a definition of geometric entropy flux.Taking entropy asarea/4for dynamical black holes,it is shown that geometric entropy is conserved:the entropyof the black hole equals the geometric entropy supplied by the infalling matter and gravitationalradiation.The area or entropy of a dynamical horizon increases by the so-called second law,notbecause entropy is produced,but because black holes classically are perfect absorbers.PACS numbers:04.70.Bw,04.30.Db,04.70.DyI.INTRODUCTION Black holes are perhaps the most exotic and energetic objects in the universe.Their theoretical history is long and winding:the earliest such solution to the field equations of Einstein’s theory of General Relativity [1]was found by Schwarzschild [2]almost immediately,but not understood as such for decades [3].In a few years around 1970,there was rapid theoretical progress,with the introduction of the term black hole by Wheeler [4]and the development of the classical four laws of black-hole mechanics [5,6,7,8],supposedly analogous to the laws of thermodynamics.Since then,astrophysical evidence has increasingly accumulated not only for stellar-mass supernova-remnant black holes,but for supermassive black holes,mysteriously present at the heart of most if not all galaxies and powering active galactic nuclei [9].Cataclysmic events such as binary black-hole mergers are predicted to produce gravitational waves which are observable on or near our home planet,for which a new generation of detectors is being developed [10].Consequently,recent years have seen a great deal of work on numerical simulations to study how black holes evolve according to given initial conditions,and what gravitational radiation they may produce [11].Such progress leaves the textbook theory of black holes seriously out of date.Much is known about stationary black holes,for instance the zeroth and first laws just mentioned,but dynamical black holes are much more complex.Of the classical laws,only Hawking’s area theorem has generality,but it applies to event horizons,which are theoretical constructs which cannot be located by mortals.It is quite timely that Hawking has recently recanted,writing that “a true event horizon never forms,just an apparent horizon”[12].Unfortunately,Hawking’s definition of apparent horizon [6]is also not the most appropriate to define black holes,due to its global nature and slicing dependence;for instance,the Schwarzschild black hole may be globally sliced so that there is no apparent horizon [13].About ten years ago,the author began a program to understand local,dynamical properties of black holes [14,15].The basic idea is that black holes contain trapped surfaces,where both ingoing and outgoing light wavefronts are converging,and that one can locate the surface of the black hole by marginal surfaces ,where outgoing light rays are instantaneously parallel.A trapping horizon is a hypersurface foliated by marginal surfaces.Locally classifying trapping horizons as future or past,and outer or inner,it was proposed that a future outer trapping horizon char-acterizes non-degenerate black holes.Some general results were that:there are future trapped surfaces just inside such a horizon;the horizon is achronal,being null only in the locally stationary case and otherwise spatial,assuming the null energy condition;the marginal surfaces have spherical topology,assuming the dominant energy condition;and the area A of the horizon is non-decreasing,A ′≥0,and increasing if spatial.The last property is analogous to Hawking’s area theorem,but for a practically locatable horizon.Trapping horizons can be numerically located by so-called apparent-horizon finders [16,17],which actually find marginal surfaces;they do not check every surface inthe hypersurface to see whether it is outer trapped,as required by the definition of apparent horizon.Under smooth-ness assumptions[6,18],apparent horizons are marginal surfaces,but not vice versa.Incidentally,the resolution to the supposed black-hole-information paradox is simple,using versions of the above results for matter violating the null energy condition:as a black hole evaporates,the ingoing negative-energy Hawking radiation causes the trapping horizon to shrink and become temporal,so that information can cross it in both directions[19,20].There never was a paradox,just a fundamental misunderstanding,that black holes are usefully defined by event horizons.In terms of trapping horizons,a comprehensive picture of black-hole dynamics wasfirst developed in spherical symmetry.In this case,there are local definitions of active gravitational mass-energy E[21]and surface gravityκ[22]which have many physically expected properties.The gradient dE of energy,expressed in terms of the energy-momentum of the matter,divides naturally into an energy-supply term Aψand a work term wdV(V=4The classical second law,originally due to Clausius,who used it to define entropy,is˙S≥˙Q/ϑ.(3)These integral forms of the laws respectively require the pressure and temperature to be spatially constant.The entropy may be divided into entropy supply S◦,given by˙S=˙Q/ϑ(4)◦and entropy production S−S◦.Then the second law may be written as˙S≥˙S(5)◦which expresses entropy production.In words:S is the entropy of the system,where system means a comoving volume of material,and S◦is the entropy supplied to the system.Thus the second law implies that the total entropy of an isolated system,such as the whole universe,cannot decrease.Here it should be stressed that dynamical black holes are not isolated systems,since they absorb energy and entropy.Then the property that black holes have non-decreasing area,A′≥0,normally called the second law of black-hole mechanics,is actually not analogous to the second law of thermodynamics.Entropy production and entropy increase have entirely different meanings for non-isolated systems. Equality in the second law holds in thermostatics,traditionally called equilibrium thermodynamics or reversible thermodynamics.In the thermostatic case,thefirst and second laws for an inviscidfluid imply˙H=ϑ˙S−p˙V(6)which is the Gibbs equation,or rather its material or comoving form.Note that what is normally called thefirst law of black-hole mechanics for stationary black holes[5],involving area A=4S and surface gravityκ=2πϑ,is actually analogous to the Gibbs equation,rather than thefirst law of thermodynamics.The latter does not involve temperature or entropy,but simply expresses energy balance.Thermodynamics can be formulated as a localfield theory,with H and S replaced by thermal energy density and entropy density respectively,Q replaced by a thermalflux vector q such that˙Q=− ∗n·q(7) and S◦replaced by an entropyflux vectorϕ=q/ϑsuch that˙S=− ∗n·ϕ(8)◦where the integrals are over a surface bounding the system,with vector area element∗n.In terms of these and other fields,e.g.density,velocity and stress for afluid,thefirst and second laws and the Gibbs equation can be localized [42,43,44,45].These three localized equations or inequalities can then be used to derive dissipative relations,in the simplest case the Fourier equation for q and the Newtonian-fluid equation for the viscous stress,leading to the Navier-Stokes equation.Thus it should be stressed that thefirst law and comoving Gibbs equation are still assumed fundamentally and fruitfully in true(non-equilibrium)thermodynamics as well as in thermostatics.Widespread folklore to the contrary is sometimes used to argue that the so-calledfirst law of black-hole mechanics,obtained as a property of stationary black holes,should not be expected to generalize to dynamical black holes.Again,this does not constitute a correct analogy with true thermodynamics.III.DUAL-NULL DYNAMICSTrapping horizons are generally defined as hypersurfaces which may have any causal nature,foliated by marginal surfaces.To study them,it is useful to employ the formalism of dual-null dynamics[46,47],describing two families of null hypersurfaces,intersecting in a two-parameter family of transverse spatial surfaces,as summarized in this section.There are various reasons:marginal surfaces are defined as extremal surfaces of null hypersurfaces;a spatial trapping horizon locally determines a unique dual-null foliation,generated from the marginal surfaces in the null normal directions;and the null limit,where a dynamical horizon reduces to an isolated horizon,is naturally included in the formalism,whereas more conventional treatments of spatial hypersurfaces become degenerate in the null limit, basically because normal vectors become tangent.For a null trapping horizon,the dual-null foliation is not unique, so subtleties remain in describing partial spatial,partially null trapping horizons.Denoting the space-time metric by g and labelling the null hypersurfaces by coordinates x±which increase to the future,the normal1-formsn±=−dx±(9)therefore satisfyg−1(n±,n±)=0.(10)The relative normalization of the null normals may be encoded in a function f defined bye f=−g−1(n+,n−)(11)where the metric sign convention is that spatial metrics are positive definite.Some readers may prefer to write g+−=g−1(n+,n−)for more manifest invariance and remember that g+−<0.The induced metric on the transverse surfaces,the spatial surfaces of intersection,is found to beh=g+2e−f n+⊗n−(12) where⊗denotes the symmetric tensor product.The dynamics are generated by two commuting evolution vectors u±:[u+,u−]=0(13)where the brackets denote the Lie bracket or commutator.Thus there is an integrable evolution space spanned by (u+,u−).There are two shift vectorss±=⊥u±(14)where⊥indicates projection by h.The null normal vectorsl±=u±−s±=e−f g−1(n∓)(15) are future-null and satisfyg(l±,l±)=0(16)g(l+,l−)=−e−f(17)l±·dx±=1(18)l±·dx∓=0(19)⊥l±=0(20) where a dot denotes symmetric contraction.In a coordinate basis(u+,u−,e a)such that u±=∂/∂x±,where e a=∂/∂x a is a basis for the transverse surfaces,the metric takes the formg=h ab(dx a+s a+dx++s a−dx−)⊗(dx b+s b+dx++s b−dx−)−2e−f dx+⊗dx−.(21) Then(h,f,s±)are configurationfields and the independent momentumfields are found to be linear combinations of the following transverse tensors:θ±=∗L±∗1(22)σ±=⊥L±h−θ±h(23)ν±=L±f(24)ω=1FIG.1:A dual-null foliation:the commuting evolution vectors u±=∂/∂x±generate the transverse surfaces S,while theirnull normal projections l±generally do not commute.where∇and D are the covariant derivatives of g and h respectively,and∇∧n±=0has been used.Here(±) indicates a label,not an index;ζ(±)are two generally distinct transverse1-forms.One can compose them as a2-form βin the normal space,defined byβ(µ,ν)=⊥((µ♯·∇)ν)for normal1-forms(µ,ν),with componentsβ±∓=e fζ(∓),β±±=0,but such notation becomes cumbersome.Likewise,one can compose the expansions and shears into a secondfundamental form,but it is more convenient to separate them.One subtle point concerns the evolution vectors u±versus the the null normal vectors l±,differing by the shiftvectors s±.In a numerical evolution,one would be evolving using thefield equations with⊥L uderivatives on the±left-hand side,since such Lie propagation of a point takes it to other points with the same angular coordinates.Inparticular,evolving along u+then u−takes one to the same point as evolving along u−then u+,since they commute, as depicted in Fig.1.However,l±generally do not commute(as measured byω),so that evolving along l+then l−takes one to a generally different point,though in the same transverse surface,as evolving along l−then l+.On the other hand,for analytical purposes it is easier to use l±than u±,writing thefield equations with⊥L±derivatives on the left-hand side.The same issue exists in the3+1formalism,where the evolution vector differs by the lapse function and shift vector from the unit normal vector.Another subtle point is that l±are not general tetrad vectors, since they are defined in terms of n∓,which must be closed.In particular,this means that f cannot befixed to zero for a general dual-null foliation.Another way to see this is that its derivatives L±L∓f are determined by the Einstein system in terms of the free initial data[47],even in spherical symmetry.The dual-null Hamilton equations and integrability conditions for vacuum Einstein gravity were derived previously[47],with matter terms added subsequently[27].Denoting projections of the energy tensor T by T±±=T(l±,l±) and T+−=T(l+,l−),the relevant components of thefield equations are justL±θ±=−ν±θ±−θ2±/2−||σ±||2/4−8πT±±(27)L∓θ±=−θ+θ−−e−f ℜ/2−|ζ(±)|2+D·ζ♯(±) +8πT+−(28)whereℜis the Ricci scalar of h(conventionally positive for spheres),a sharp(♯)denotes the contravariant dual with respect to h−1=h♯(index raising),|ζ|2=ζ·ζ♯and||σ||2=σ:σ♯,where the colon denotes double symmetric contraction.Units are such that Newton’s gravitational constant is unity.Thefirst equation is the well-known null focusing equation and the second has been called the cross-focusing equation[14,15].The null energy condition impliesT±±≥0(29) and the dominant energy condition additionally impliesT+−≥0.(30)IV.TRAPPING HORIZONSThe dual-null formalism may be applied to one-parameter families of transverse surfaces generated by a vectorξ=ξ+l ++ξ−l −Dξ±=0.(31)This means that ξ=∂/∂x is normal to the constant-x transverse surfaces,so that ξ±can be taken outside transversesurface integrals .The area of the transverse surfaces isA = ∗1(32)and the area radiusR =16π∗ ℜ+e f θ+θ− (34)will be used as a measure of the active gravitational mass-energy on a transverse surface.On a stationary black-hole horizon,it is also known as the irreducible mass:the mass which must remain even if rotational or electrical energy is extracted.Consider a trapping horizon generated by a vector ξ=∂/∂x ,so that the constant-x surfaces are marginal surfaces,where one of the null expansions θ±vanishes.This leaves the freedom to relabel the marginal surfaces,x →ˆx (x ),under which all the key equations will be manifestly invariant.Equations holding on a trapping horizon will be denoted by the weak equality symbol ∼=.Initially,the case θ+∼=0will be considered in detail,with the case θ−∼=0included subsequently.The fundamental equation describing the evolution of a trapping horizon is0∼=L ξθ+=ξ+L +θ++ξ−L −θ+(35)where L ξdenotes the Lie derivative along ξ.This will be used together with the Einstein equation to derive the first law for any trapping horizon.Even without the Einstein equation,one can use it to deduce relationships between the signs of ξ±and L ±θ+,which determine the causal nature of the trapping horizon and whether its area increases or decreases,viaL ξA = ∗(ξ+θ++ξ−θ−).(36)For clarity,all such inequalities are collected in the next section,so that the remainder of the article applies to any trapping horizon.V.AREA AND SIGNATURE LA WSTrapping horizons were previously classified[14]into one of four non-degenerate types:future (respectively past)if θ−<0(respectively θ−>0)and outer (respectively inner)if L −θ+<0(respectively L −θ+>0)on the trapping horizon θ+∼=0.For each type,there are trapped surfaces (θ+θ−>0)to one side of the horizon and untrapped surfaces (θ+θ−<0)to the other side,which is not guaranteed if the inequalities are relaxed even to non-strict inequalities.The causal type of the horizon is determined pointwise by the relative signs of ξ±:spatial if they have opposite signs,null if one vanishes and the other does not,and temporal if they have the same (non-zero)sign.Since the null energy condition (29)and focusing equation (27)imply L +θ+≤0on the trapping horizon,it follows from the fundamental equation (35)that outer trapping horizons are achronal (spatial or null),while inner trapping horizons are causal (temporal or null);the signature law [14].Furthermore,fixing the orientation of ξby ξ+>0,it follows from (36)that the area of a future outer or past inner trapping horizon is non-decreasing,L ξA ≥0,while the area of a past outer or future inner trapping horizon is non-increasing,L ξA ≤0;the area law [14].As corollaries,the horizon is null and has instantaneously constant area if and only if the ingoing energy density T +++Θ++vanishes,where Θ++=||σ+||2/32π(68)can be understood subsequently as the effective energy density of ingoing gravitational radiation.As mentioned in the Introduction,non-degenerate black holes may be characterized by future outer trapping horizons.Ashtekar &Krishnan instead defined dynamical horizons as spatial future trapping horizons.Then θ−<0on the horizon andξ±have opposite signs.Choosing the orientation of l±such that l+is outward and l−inward,ξ+>0,ξ−<0and it follows directly from(36)that the area is increasing,LξA>0,a strict version of the above area law.Actually,for black holes,one is normally interested in future trapping horizons which are either spatial(dynamical)or future-null(isolated),or partially spatial and partially null.In such casesξ+>0andξ−≤0,which immediately gives the non-strict area law LξA≥0.Note also from the fundamental equation(35)that a dynamicalhorizon satisfies L−θ+≤0under the null energy condition,so it is either a future outer trapping horizon or degenerate. The degenerate cases allow dynamical horizons in space-times without trapped surfaces[49],reflecting the need for something like the outer condition to characterize a black hole.In practice,the outer horizon of a black hole is likely to satisfy both definitions,except when it becomes stationary or instantaneously stationary.An evaporating black hole may also be described using trapping horizons.The only difference with the above discussion is that the null energy condition is violated by Hawking radiation,for which the ingoing radiation has negative energy density,T++<0.Assuming that this dominates the positive energy density of ingoing gravitational radiation,T+++Θ++<0,the focusing equation(27)implies L+θ+>0on the trapping horizon.For an outer horizon,the fundamental equation(35)implies thatξ±have the same sign,so that the horizon is temporal,while (36)shows that the area is decreasing for a future horizon,LξA<0.Thus the black-hole horizon is shrinking and two-way traversable.Clearly matter can escape from an evaporating black hole.The strange belief that information cannot escape from an evaporating black hole seems to be based on the impractical event-horizon definition of black hole as a region of no escape.VI.FIRST LA W:ENERGY FLUX AND WORKHenceforth completely general trapping horizons will be considered,so that all the following displayed equations will apply not only to outer black-hole horizons under the usual energy conditions,but to inner black-hole horizons,white holes,cosmological horizons,wormhole mouths and evaporating black holes.Expanding the fundamental relation (35)using the focusing equations(27–28)yields0∼=Lξθ+∼=−ξ+ 8πT+++||σ+||2/4 +ξ− 8πT+−−e−f ℜ/2−|ζ|2+D·ζ♯ (37) whereζ=ζ(+)temporarily simplifies the notation.Multiplying by e f/8πand integrating over the transverse surfaces, using the Gauss-Bonnet theorem ∗ℜ=8π,the Gauss divergence theorem ∗D·α=0and rearranging yieldsξ+ ∗e f T+++||σ+||28π ∼=−ξ−/2.(38)Sinceξ−=0in the null case,this shows that both T++andσ+must vanish on a null horizon[14],assuming the null energy condition.Here spherical topology has been assumed;otherwise,for compact orientable transverse surfaces, the right-hand side of(38)is multiplied by1−γ,whereγis the genus or number of handles.Then the dominant energy condition impliesγ≤1,leading to the topology law[14]:the transverse surfaces are either spherical or toroidal,the latter case requiring very special conditions(including vanishing Gaussian curvature)and anyway being excluded for (non-degenerate)outer trapping horizons,L−θ+<0.The Hawking mass-energy(34)satisfiesE∼=R/2(39) on a trapping horizon,which can be regarded as a generalization of irreducible mass from stationary to non-stationary black holes,since the area law ensures its irreducibility under the null energy condition.Then the identity(38)yields LξE∼=LξRξ− ∗e f T+++||σ+||28π LξR.(40) This is a dual-null differential version of the energy-balance law found by Ashtekar&Krishnan[29,30],as compared more explicitly in the next section.One mayfix the normalization f∼=0and,for a spatial horizon,one mayfix the scaling of the null normals such thatξ+/ξ−∼=−1and the generating vectorξsuch that LξR∼=1.In the following, all gauge freedom will be retained for generality,but readers may wish on afirst reading to mentally set f∼=0and, if interested only in spatial trapping horizons,ξ+/ξ−∼=−1.The four terms are all geometrical invariants of the dual-null foliation,as shown explicitly below,and therefore of the horizon(as an embedded hypersurface)unless it becomes null.Since the formalism is manifestly covariant on the transverse surfaces,checking invariance reduces to writing e f=−g+−and matching±indices.The four terms in parentheses in(40)are all manifestly positive,assuming the dominant energy condition.The T++ term gives the energyflux of the matter propagating in the null direction into the horizon.Consequently it is natural to interpret theσ+shear term as giving the energyflux of the transverse gravitational radiation propagating in the null direction into the horizon.This term has the same form as that of the Bondiflux of gravitational radiation at null infinity[50,51],the same form as a localized energyflux of gravitational radiation in a quasi-spherical approximation [27],and a similar form to the energyflux of linearized gravitational radiation in the high-frequency approximation [7],so its physical interpretation seems sound.The T+−term gives a matter energy density,so theζterm can be interpreted as giving a corresponding gravitational energy density.Ashtekar&Krishnan interpreted it as also due to gravitational radiation,and here it will be interpreted as an energy density of longitudinal gravitational radiation. This is much less familiar than transverse gravitational radiation and is absent in all the above approximations,but the interpretation can be understood in a spin-coefficient formulation,to be presented elsewhere[52].It should be mentioned that there is a widespread belief that longitudinal gravitational radiation does not exist in Einstein gravity, apparently due to an argument in linearized theory that the longitudinal modes are purely gauge-dependent,and the fact that only the transverse mode contributes to the Bondiflux.On thefirst point,Szekeres characterized ingoing and outgoing,transverse and longitudinal gravitational radiation by their effect via the geodesic equation on a“gravitational compass”of test particles[53,54].On the second point,it can be shown that the energy densities of the outgoing transverse and longitudinal modes fall offnear future null infinity as1/R2and1/R4respectively[51,52]. The expression(40)separates thefirst term in parentheses,which vanishes for null horizons(assuming the null energy condition),from the second term in parentheses,which is generally non-zero for horizons of any causal nature. This separation need not appear for spatial trapping horizons,but it will be stressed in the following,since the null case is a physically important limit,where dynamical horizons reduce to isolated horizons,or more prosaically,where a growing black hole ceases to grow.The next task is to write the new law in a more manifestly invariant form.The spherically symmetric case is a useful guide;there the unifiedfirst law was found as dE=Aψ+wAdR for certain invariantsψand w of the matter energy tensor[22].The corresponding formulae readw m=−trace T/2(41)ψm=T·(dR)♯+w m dR(42) where the trace is in the normal space and the subscript m is introduced to refer to the matter.In the current generalized context,these quantities are invariants of the dual-null foliation.Explicitly,w m=e f T+−(43)(ψm)±=−e f T±±L∓R(44) whereψ=ψ+dx++ψ−dx−.Comparing with thefirst law(40),one can define corresponding quantities for the gravitational radiation byw g=|ζ|2.(46)32πAs in the spherically symmetric case[22],one may callw=w m+w g(47) the work density andψ=ψm+ψg(48) the energyflux1-form.On a trapping horizon withθ+∼=0,it follows that L+R∼=0,ψ−∼=0,ξ·ψ∼=ξ+ψ+and LξR∼=ξ−L−R,yieldingLξR.(49)ξ·ψ∼=−ξ+32πThus thefirst law(40)becomesLξE∼= ∗ξ·ψ+ ∗wLξR.(50)This is the first law of black-hole dynamics ,in the desired geometrically invariant form.(The Lie derivative L ξacting on integral scalars like E and R is just the partial derivative ∂/∂x ).The energy flux ξ·ψvanishes if the horizon is null,while the work density w is generally non-zero for horizons of any causal nature.The two terms ∗ξ·ψand ∗wLξR may be called respectively the energy-supply and work terms,by analogy with the first law of thermodynamics (1).The above derivation of the first law applied to a trapping horizon with θ+∼=0.For a trapping horizon with θ−∼=0,one obtains the same formula with a different w .For completeness one can definew (±)=e f T +−+|ζ(±)|2L ξR +w (53)is the combined energy density ,where the division into energy-supply and work terms can be ignored in this section.It can be independently divided into matter and gravitational-radiation terms,ǫ=ǫm +ǫg ,in the obvious way,yielding the explicit expressionsǫm ∼=e f T +−−ξ±32π4|ζ(±)|2−e f ξ±g xx ∧dx(57)where s is arc length along the horizon-generating vector ξ=∂/∂x and√g (ξ,ξ)=∂s/∂x(58)is the corresponding scale factor,then the integral first law can be written as[E ]∼= ˜∗˜ǫ(59)where˜ǫ=ǫL ηR =η·ψ+wL ηR (60)is the proper energy density andη=ξ/√2l ±,but it is now straightforward to check theirexpressions against the explicit expressions (54–55),which reduce to ǫm ∼=T +++T +−and ǫg ∼=(||σ+||2+4|ζ(+)|2)/32πfor θ+∼=0,with ˜ǫreducing to ǫ/√g xx here,their coordinate r is x here,their σis σ+/√2/g xx ,rescaling the outward null normal l +ratherthan the horizon-generating vector ξ(to η=ξ/√2/g xx ,ˆτ)in their gauge choice,where ˆτis the unit normal vector to a spatial trapping horizon.However,unit ˆτdoes not exist for null trapping horizons.Nevertheless a natural normal does exist for any trapping horizon,namely the vector τdual to the generating vector ξin the normal space:τ=ˆ∗ξ(63)where ˆ∗is the vectorial Hodge operator of the normal space,with orientation chosen so that τ=ξ+l +−ξ−l −,meaning that τis future-pointing for outward-pointing ξ.Then τis normal to the horizon,g (ξ,τ)=0,⊥τ=0,has equal and opposite normalization g (τ,τ)=−g (ξ,ξ)and is regular in the null limit,becoming null itself,τ→ξ.In spherical symmetry,the Kodama vector χprovides a preferred flow of time,reducing to the stationary Killing vector for Schwarzschild and Reissner-Nordstr¨o m black holes.It has a dual relation to the energy E which can be written as L ξE =AT (χ,τ),for any normal vector ξand its orthogonal dual τ.This vector can be generalized byχ=ˆ∗(dR )♯(64)or the curl of R in the normal space,with components χ=e f (L +R l −−L −R l +).Then χis orthogonal to R and the transverse surfaces,χ·dR =0,⊥χ=0,has normalization g (χ,χ)=−g −1(dR,dR )and becomes null on a trapping horizon,g (χ,χ)∼=0,with χ∼=±(dR )♯for θ±∼=0.Flow lines of χand (dR )♯are sketched in Fig.2for typical gravitational collapse to a black hole,assuming cosmic censorship;for a comprehensively analyzed case,see Christodoulou [55]for the massless Klein-Gordon field in spherical symmetry.For a future outer trapping horizon[14],the area-radius vector (dR )♯is spatial and χis temporal just outside the horizon,and (dR )♯is temporal and χspatial just inside the horizon.In terms of these vectors,there is a remarkably simple and manifestly invariant expression for the matter energy density:ǫm L ξR =T (χ,τ).(65)This holds for any foliation of spatial surfaces in any space-time,generated by a normal vector ξwith orthogonal dual τ,without any gauge conditions.Then the integrated matter flux is[E ]m ∼= ∗T (χ,τ)∧dx.(66)。
2024年高三英语统计学分析单选题30题
2024年高三英语统计学分析单选题30题1.The average height of a group of people is calculated by adding up all the heights and then dividing by the _____.A.number of peopleB.sum of heightsC.difference in heightsD.product of heights答案：A。
本题考查平均数的计算方法。
平均数是所有数据之和除以数据的个数，这里就是把所有人的身高加起来然后除以人数。
选项B“sum of heights”是身高总和，不是计算平均数的除数。
选项C“difference in heights”是身高差，与平均数计算无关。
选项D“product of heights”是身高乘积，也与平均数计算无关。
2.In a statistical survey, the mode is the value that _____.A.appears most frequentlyB.has the highest sumC.is the averageD.is the middle value答案：A。
本题考查众数的概念。
众数是一组数据中出现次数最多的数值。
选项B“has the highest sum”是和最大，与众数无关。
选项C“is the average”是平均数，与众数不同。
选项D“is the middle value”是中位数，不是众数。
3.The median of a set of data is found by arranging the data in orderand then finding the _____.rgest valueB.smallest valueC.middle valueD.average value答案：C。
The scalar sector of the Randall-Sundrum model
only the Standard Model vacuum solution of ∂V (h)/∂h = 0 (h is the Higgs field) is allowed. We then turn to our main focus: the consequences of the curvaturescalar mixing ξ R H † H (where H is a Higgs doublet field on the visible brane), which causes the physical mass eigenstates h and φ to be mixtures of the original Higgs and radion fields. First, we discuss the theoretical constraints on the allowed parameter space. Next, we give precise procedures for computing the h and φ couplings given the physical eigenstate masses, mh and mφ , ξ and the new physics scales of the model. Relations among these new-physics scales are discussed and a set of values not far above the smallest values required by precision electroweak constraints and RunI data is chosen. A simple result for the sum of the ZZh and ZZφ squared couplings relative to the ZZhSM squared coupling is derived. We demonstrate that this sum rule in combination with LEP/LEP2 data implies that
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
(3)
with gσ , an effective coupling, and mσ , an effective mass. Implications of this possible non-perturbative effect are most easily described in terms of the iso-doublet auxiliary field Σ≡ such that the Lagrangian gσ ¯ tR ψL , − m2 σ (4)
UCL-IPT-95-22
Is the standard Higgs scalar elementary?
D. Delepine∗, J.-M. G´ erard and R. Gonzalez Felipe
arXiv:hep-ph/9512339v1 18 Dec 1995
Institut de Physique Th´ eorique Universit´ e catholique de Louvain B-1348 Louvain-la-Neuve, Belgium
after resummation of the multiple scalar exchanges in the heavy quark-antiquark channels. The 4-fermion form-factor is expected [6] to depend on the q 2 transfermomentum in the following way G (q 2 ) =
2. The scalar Lagrangian.
Let us assume that the Yukawa coupling gt of the top quark alone is indeed responsible for the electroweak symmetry breaking below the cutoff scale Λ. If such is the case, the relevant Lagrangian for the fundamental iso-doublet scalar field H simply becomes
+ ¯ LH = ∂µ H + ∂ µ H − m2 H H H + gt (ψL tR H + h.c.) ,
(1)
where H =
H0 tL and ψL = . − H bL 1
The crucial non-perturbative effect possibly induced by a large gt Yukawa coupling is the appearance of a 4-fermion interaction ¯L tR t ¯R ψL , LN JL = Gψ (2)
∗
Research assistant of the National Fund for the Scientific Research
1. Introduction.
Well before the advent of QCD, Nambu and Jona-Lasinio (NJL) [1] introduced a four-fermion interaction to break the chiral symmetry of strong interactions. In modern language, this NJL effective Lagrangian is expected to be induced by multiple gluon exchanges in the light quark-antiquark channels. The electroweak gauge-couplings of the Standard Model are not strong enough to trigger a similar breaking of the flavour SU (2)L × U (1) gauge symmetry. But new gauge interactions beyond the Standard Model might generate effective four-fermion interactions [2]-[4]. In that approach, the scalar field h responsible for the symmetry ¯ composite state with mh = 2mt , if QCD effects are ignored. breaking is a pure tt It is however quite remarkable that the strongest force in the electroweak sector of the Standard Model is due to the Yukawa coupling of the recently observed top quark [5] to the fundamental Higgs field . It is therefore quite legitimate to investigate ¯ condensation without having to invoke new physics beyond the the possibility of a tt Standard Model [6]. Indeed, the Yukawa coupling itself might generate a four-fermion interaction at some scale Λ, in a way similar to what is happening in QCD. In this letter, we analyze the implications of this minimal scenario on the scalar mass spectrum. For that purpose, we assume that the SU (2)L × U (1) gauge couplings and the scalar self-coupling do not participate at all in the symmetry breaking. In that case, the electroweak symmetry breaking is also triggered by top quark loops ¯ composite [7] such that the standard Higgs boson is a linear combination of the tt state and of the fundamental scalar. In particular, if the Λ scale is around 1 TeV, the elusive Higgs scalar is mainly a composite state with a mass of about 80 GeV.
Abstract In the standard electroweak model, the measured top quark mass requires a sizeable Yukawa coupling to the fundamental scalar. This large coupling alone might induce a dynamical breaking of the electroweak symmetry as well as non-perturbative effects. If such is the case, even a standard Higgs scalar as ¯ component induced by the top light as 80 GeV should have a non-negligible tt condensate.
q2
+ ¯ LN JL = ∂µ Σ+ ∂ µ Σ − m2 σ Σ Σ + gσ (ψL tR Σ + h.c.) .
(5)
is equivalent to Eq.(2). The scalar Lagrangian LH + LN JL at the source of the SU (2)L × U (1) breaking below the scale Λ is then defined by Eqs.(1) and (5). Notice that the absence of a kinetic term for the Σ field at the scale Λ would imply the compositeness boundary condition [4] gσ (Λ) = ∞ on the physically normalized coupling gσ . But the coupled renormalization group equations for gt and gσ require the ratio gσ /gt to be independent of the scale. This would obviously be in contradiction with the Lagrangian given in Eq.(1) where gt (µ) is supposed to have a fixed and finite value at the Λ scale. This kinetic term is important since it allows the Σ field to contribute also to the W gauge-boson mass through the covariant derivatห้องสมุดไป่ตู้ve, once the (small) SU (2) gauge coupling g2 is switched on. Therefore, both the loop-induced vacuum expectation values of the elementary field H 0 and of the composite field Σ0 contribute to the W gauge-boson mass 1 2 0 m2 W = g2 ( H 2 and to the top quark mass