Perturbative and non-perturbative parts of eigenstates and local spectral density of states

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Cabibbo-suppressed decays of the Omega_c^0 - feedback to the Xi_c^+ lifetime

Email addresses: ababic@thphys.irb.hr (A. Babi´ c,), guberina@thphys.irb.hr (B. Guberina,), melic@hippo.irb.hr (B. Meli´ c,), ˇ shrvoje@thphys.irb.hr (H. Stefanˇ ci´ c). 1 It appears astonishig that decay rates of weak and radiative decays are described in terms of few basic quantities, e.g. quark masses, and hadronic expectation values of several leading local operators.
Inclusive hadronic decay rates and lifetimes were calculated a long time ago [3–7] by summing over all possible channels and integrating over some range of energies. A ‘practical’ version of the OPE is used in calculations, i.e. it is assumed that the coeﬃcient functions can be found perturbatively and all nonperturbative eﬀects reside in matrix elements. In real world, however, there are nonperturbative eﬀects even at short distances, and the matrix elements are subject to perturbative corrections too. Suprisingly enough, the theory works rather well, even in the charmed hadron 2 sector, although the expansion parameter µ2 G (D )/mc ≃ 0.5 is not really small (the corresponding parameter in beauty decays is

英语单词词根记忆大全-第十一章

第十一章tributary / 5tribjutEri/ n.adj.支流的,进贡的attribute / 5Atribju:t/ vt.把…归因于 n.属性attribution / Etri5bju:FEn/ n. 归属，归因contribute / kEn5tribju(:)t/ vt.捐献,捐助,投稿contributor / kEn5tribju(:)tE/ n.贡献者,捐助者,投稿者 distribute / dis5tribju:t/ vt.分发,分送,分布distribution / 5distri5bju:FEn/ n.分发,分配,分布retribution / 5retri5bju:FEn/ n. 报应，罚extricable / 5ekstrikEbl/ a.可救出的,可解救的extricate / 5ekstrikeit/ v.拯救,救出extrication / ekstri5keiFEn/ n.救出,解脱inextricable / in5ekstrEkEbl/ a.无法摆脱的,纠缠的intricate / 5intrikit/ a.复杂的,错综的,难懂的intricacy / 5intrikEsi/ n.错综复杂intrigue / in5tri:g/ n.阴谋 vi.密谋,耍诡计contrite / 5kRntrait/ a.悔罪的,痛悔的attrite / E5trait/ adj. 磨损的vt. 将...磨小contrition / kEn5triFEn/ n.悔罪,痛悔detriment / 5detrimEnt/ n.损害,伤害detrimental / 5dZtrE5mZntl/ a.损害的,造成伤害的 detrition / di5triFEn/ n.磨损intrude / in5tru:d/ v.闯入,强行引入intruder / in5tru:dE/ n.入侵者intrusion / in5tru:VEn/ n.私闯,干扰intrusive / in5tru:siv/ a.打扰的,插入的extrude / ik5stru:d/ vt.挤出,逐出extrusion / eks5tru:VEn/ n.挤出,推出extrusive / eks5tru:siv/ a.挤出的,喷出的,突出的protrude / prE5tru:d/ v.突出,伸出protrusion / prou5tru:VEn/ n.伸出,突出protrusive / prou5tru:siv/ a.伸出来的,突出的detrude / di5tru:d/ v.推倒,扔掉abstruse / Ab5stru:s/ a.难懂的,深奥的obtrude / Eb5tru:d/ v.突出,强加obtrusive / Eb5tru:siv/ a.突出的,炫耀的disturb / dis5tE:b/ vt.打扰,扰乱,弄乱disturbance / dis5tE:bEns/ n.动乱,干扰,侵犯undisturbed / 5Qndis5tE:bd/ a.安静的,镇定的turbid / 5tE:bid/ a.混浊的,紊乱的turbine / 5tE:bin/ n.叶轮机,汽轮机turbulent / 5tE:bjulEnt/ a.骚动的,骚乱的turbulence / 5tE:bjulEns/ n.骚乱,动荡perturb / pE5tE:b/ v.感到不安perturbative / 5pE:tEbeitiv, pE:5tE:bEtiv/ a.扰乱的,使混乱的perturbation / 5pE:tE5beiFn/ n.烦乱,扰乱turmoil / 5tE:mRil/ n.混乱,骚乱turgescence / tE:5dVesns/ n.膨压,肿胀,夸张turgescent / tE:5dVesnt/ a.肿的,浮夸的turgid / 5tE:dVid/ a.浮肿的,浮夸的turgidly ad.肿胀地,夸张地turgidness n.肿胀turgor / 5tE:gE/ n.膨胀,肿胀,浮夸,夸张tutor / 5tju:tE/ vt.教,指导tutoress / 5tju:tEris/ n.女教师tutorial / tju:5tR:riEl/ n.指南tutorship / 5tju:tEFip/ n.家庭教师,辅导untutored / Qn5tju:tEd/ a.未受教育的,粗野的,幼稚的 tuition / tju:5iFEn/ n.教,教诲,学费tuitional a.讲授的,学费的umbrella / Qm5brelE/ n.伞,雨伞umbra / 5QmbrE/ n.暗影umbrage / 5QmbridV/ n.不快,愤怒,树荫umbriferous / Qm5brifErEs/ a.成荫的,投影的 penumbra / pi5nQmbrE/ n.半明半暗之处adumbral / E5dQmbrEl/ a.遮阳的,成荫的adumbrate / 5AdQmbreit/ v.预示abadumbration / AdQm5breiFEn/ n.预兆unite / ju5nait/ vi.联合 vt.使联合 united / ju:5naitid/ a.联合的,团结的,一致的 unitive / 5ju:nitiv/ a.统一的unity / 5ju:niti/ n.单一,统一,团结 disunity / dis5ju:niti/ n.不统一union / 5ju:njEn/ n.联合,团结,协会 reunion / 5ri:5ju:njEn/ n.团圆,重聚unit / 5ju:nit/ n.单位,单元,部件unique / ju:5ni:k/ a.唯一的,独一无二的unify / 5ju:nifai/ vt.统一,使成一体unification / ju:nifi5keiFEn/ n.统一,合一,一致 unison / 5ju:nizn/ n.和谐,一致,齐唱unilateral / ju:ni5lArErEl/ a.单方面,单边的unicycle / 5ju:nisaikl/ n.单轮脚踏车reunite / 5ri:ju:5nait/ v.(使)再结合suburb / 5sQbE:b/ n.郊区,郊外,近郊suburban / sE5bE:bEn/ a.郊外的,偏远的suburbanite / sE5bE:bEnait/ n.郊区居民suburbanize / sE5bE:bEnaiz/ vt.使成市郊,使市郊化 urban / 5E:bEn/ a.都市的,住在都市的urbanite / 5E:bEnait/ n.都市人urbane / E:5bein/ a.温文尔雅的,懂礼的urbanity / E:5bAniti/ n.有礼貌,文雅inurbane / inE5bein/ a.粗野的,粗鄙的,无理的inurbanity / inE5bAniti/ n.粗鄙,无礼粗野,urbanize / 5E:bEnaiz/ vt.使都市化,使文雅urbanization / E:bEnai5zeiFEn/ n.都市化,文雅化urbanology / 5E:bE5nRlEdVi/ n.城市学,都市学exurb / 5egzE:b/ n.远郊地区exurban / ek5sE:bEn, eg5z-/ a.城市远郊的exurbanite / ek5sE:bEnait, eg5zE:-/ n.住在城市远郊的居民 interurban / intE5E:bEn/ a.都市间的use / ju:z, ju:s/ vt.n.用,耗费usage / 5ju:zidV/ n.使用,对待,惯用法used / ju:zd, ju:st/ a.使用过的,二手的,旧的useful / 5ju:zful/ a.有用的,有益的usefulness n.有用,有效性useless / 5ju:zlis/ a.无用的,无价值的user / 5ju:zE/ n.用户,使用者abuse / E5bju:z, E5bju:s/ vt.滥用,虐待 n.滥用 abusive / E5bju:siv/ a.辱骂的,滥用的utility / ju:5tiliti/ n.效用,有用,实用utilitarian / 5ju:tili5tZEriEn/ a.功利的,实利的utilize / 5ju:tilaiz/ vt.利用utilization / 5ju:tilai5zeiFEn/ n.利用utilizable / 5ju:tilaizEbl/ a.可以利用的inutile / in5ju:tail/ a.无益的,无用的inutility / inju5tiliti/ n.没有益处,无用之物vacation / vE5keiFEn/ n.假期,休假vacant / 5veikEnt/ a.空的,未被占用的vacancy / 5veikEnsi/ n.空白,空缺vacate / vE5keit/ v.空出evacuate / i5vAkjueit/ v.撤退,撤离evacuation / ivAkju5eiFEn/ n.撤退,走开 evacuee / ivAkju5i:/ n.撤离者,被疏散者vacuous / 5vAkjuEs/ a.发呆的,无意义的vacuity / vA5kju:iti/ n.贫乏,无聊,空白vacuum / 5vAkjuEm/ n.真空,真空吸尘器vacuumize / vAkjuEmaiz/ vt.真空包装,抽成真空 void / vRid/ a.空的,无效的avoid / E5vRid/ vt.避免,躲开,撤消 devoid / di5vRid/ a.空的,缺少的invade / in5veid/ vt.入侵,侵略,侵袭invader / in5veidE/ n.侵略者invasion / in5veiVEn/ n.入侵,侵略,侵犯invasive / in5veisiv/ a.入侵的evade / i5veid/ v.逃避,规避evasion / i5veVEn/ n.逃避evasive / i5veisiv/ a.逃避的pervade / pE:5veid/ v.弥漫,普及pervasion / pE5veiVEn/ n.扩散,渗透pervasive / pE5veisiv/ a.遍及的,弥漫的vagary / 5veigEri/ n.奇想,异想天开vague / veig/ a.模糊的,含糊的extravagant / iks5trAvigEnt/ a.奢侈的,过度的extravagance / iks5trAvigEns/ n.奢侈,过度,放纵的言行 vagabond / 5vAgEbEnd/ n.浪荡子,流浪者vagabondage / 5vAgEbRndidV/ n.流浪,流浪生活vagrant / 5vAgrEnt/ n.游民a.漂泊的vagile / 5vAdVil/ a.漫游的divagate / 5daivEgeit/ v.离题,漂泊divagation / 5daivE5geiFn/ n.流浪,误入歧途invalid / 5invEli:d, in5vAlid/ n.病人 a.有病的,无效的 invalidity / invE5liditi/ n.无效力invalidism / 5invEli:dizm/ n.病弱,病身valiant / 5vAljEnt/ a.英勇的valour / 5vAlE/ n.英勇valorous / 5vAlErEs/ a.勇敢的,无畏的convalesce / kRnvE5les/ v.（病）康复,复原convalescence / kRnvE5lesns/ n.病后康复期convalescent a.康复期的,渐愈的valid / 5vAlid/ a.有效的,正当的validity / vE5liditi/ n.有效,效力,正确validate / 5vAlideit/ v.使…生效valor / 5vAlE/ n.勇气invalidate / in5vAlideit/ v.使无效力,证明无效vanish / 5vAniF/ vi.突然不见,消失vanity / 5vAniti/ n.虚荣,自负evanesce / i:vE5nes/ vi.逐渐消失,消散evanescence / i:vE5nesns/ n.逐渐消失,幻灭,瞬间 evanescent / i:vE5nesnt/ a.短暂的,瞬息的vantage / 5va:ntidV/ n.优势,有利情况advance / Ed5va:ns/ vi.前进,提高 n.进展advantage / Ed5va:ntidV/ n.优点,优势,好处advantageous / 5AdvEn5teidVEs/ a.有利的disadvantage / 5disEd5va:ntidV/ n.不利,不利地位 evaporable / i5vApErEbl/ a.易蒸发性的,蒸发性的 evaporate / i5vApEreit/ vt.使蒸发 vi.蒸发vapor / 5veipE/ n.汽,蒸气vaporize / 5vApEraiz/ v.(使)蒸发variability / vZEriE5biliti/ n.变化性,变化无常 variable / 5vZEriEbl/ a.易变的 n.变量variance / 5vZEriEns/ n.矛盾,不同variant / 5vZEriEnt/ a.变体variation / 5vZEri5eiFEn/ n.变化,变动,变异varied / 5vZErid/ a.杂色的,各式各样的variegate / 5vZErigeit/ vt.使成斑驳,使多样化variform / 5vZErifR:m/ a.多种形态的various / 5vZEriEs/ a.各种各样的,不同的vary / 5vZEri/ vt.改变,使多样化invariable / in5vZEriEbl/ n.不变的,永恒的 intervene / 5intE5vi:n/ vi.干涉,干预convene / kEn5vi:n/ vt.集合intervention / 5intE5venFEn/ n.干涉convention / kEn5venFEn/ n.习俗,惯例,公约 prevent / pri5vent/ vt.预防,阻止prevention / pri5venFEn/ n.预防,防止preventive / pri5ventiv/ a.预防性的event / i5vent/ n.事件,大事,事变eventful / i5ventful/ a.变故多的,重要的,多事的supervene / 5su:pE5vi:n/ v.接着发生,被附加advent / 5AdvEnt/ n.来到,来临adventitious / Adven5tiFEs/ a.偶然的,外来的adventure / Ed5ventFE/ n.冒险,惊险活动adventurer / Ed5ventFErE/ n.冒险家invent / in5vent/ vt.发明,创造,捏造invention / in5venFEn/ n.发明,创造,捏造avenge / E5vendV/ v.为…复仇,为…报仇avenger / E5vendVE/ n.复仇者revenge / ri5vendV/ vt.替…报仇 n.报仇revengeful / ri5vendVful/ a.报复的,深藏仇恨的 vengeful / 5vendVful/ a.复仇心重的,报复的vengeance / 5vendVEns/ n.报仇,报复very / 5veri/ ad.很,完全 a.真的verily / 5verili/ ad.实在,真veracious / vE5reiFEs/ a.诚实的,说真话的veracity / vE5rAsEti/ n.真实性,诚实verism / 5viErizm/ n.真实主义verity / 5veriti/ n.真实,真理veritable / 5veritEbl/ a.真正的verify / 5verifai/ vt.证实,查证,证明verifiable / 5verifaiEbl/ a.能作证的,能证实的 verification / verifi5keiFEn/ n.验证verisimilar / veri5similE/ a.好象真实的verisimilitude / verisi5militju:d/ n.逼真,可能性verbal / 5vE:bEl/ a.言辞的,文字的,动词的verbalize / 5vE:bElaiz/ v.描述verbatim / vE:5beitim/ a.逐字的,照字面的verbiage / 5vE:biidV/ n.罗嗦,冗长verbose / vE:5bEus/ a.冗长的,罗嗦的verb / vE:b/ n.动词verbify / 5vE:bifai/ vt.使动词化adverb / 5AdvE:b/ n.副词adverbial / Ed5vE:biEl/ a.副词的,状语的n.状语subvert / sEb5vE:t/ v.颠覆,推翻subversion / sEb5vE:FEn/ n.颠覆subversive / sEb5vE:siv/ a.颠覆性的,破坏性的 divert / dai5vE:t/ vt.使转向 vi.转移diversion / dai5vE:FEn/ n.转移,改道diverse / dai5vE:s/ a.不一样的,相异的diversify / dE5vE:sE5fai/ vt.使多样化diversification / daivE:sifi5keiFEn/ n.变化,多样化diversity / dai5vE:siti/ n.多样,变化万千averse / E5vE:s/ a.厌恶的,反对的aversion / E5vE:FEn/ n.嫌恶,憎恨avert / E5vE:t/ vt.转开,避免,防止adverse / 5AdvE:s/ a.不利的,敌对的,相反的adversity / Ed5vE:siti/ n.不幸,灾难adversary / 5AdvEsEri/ n.对手,敌手reverse / ri5vE:s/ vt.颠倒,翻转 n.背面reversion / ri5vE:FEn/ n.返回,逆转reversible / ri5vE:sEbl/ a.可逆的revert / ri5vE:t/ v.回复reversal / ri5vE:sl/ n.颠倒,逆转,撤销irreversible / 5iri5vE:sEbl/ a.不能撤回的,不能取消的 introvert / 5intrEvE:t/ v.内向的人introversion / intrE5vE:FEn/ n.内向性,内省性extrovert / 5ekstrouvE:t/ n.性格外向者convert / kEn5vE:t, 5kRnvE:t/ vt.使转变,使改变conversion / kEn5vE:FEn/ n.转变,转化,改变versatile / 5vE:sEtail/ a.多方面的,通用的converse / kEn5vE:s, 5kRnvE:s/ n.相反的事物 a.相反的 conversely / 5kRn5vE:sli, 5kRnvE:s-/ ad.相反地 convertible / kEn5vE:tEbl/ a.可转换的,n.敞篷车 controversy / 5kRntrEvE:si/ n.争论,辩论,争吵controversial / 5kRntrE5vE:FEl/ a.争论的,争议的controvert / 5kRntrE5vE:t/ v.反驳,驳斥inverse / in5vE:s/ a.相反的,倒转的inversely ad.相反地,倒转地invert / in5vE:t, 5invE:t/ vt.使反转,使颠倒,使转化 perverse / pE5vE:s/ a.刚愎自用的,故意作对的perversion / pE5vE:Fn/ n.堕落,曲解pervert / pE:5vE:t/ v.使堕落,使变坏transverse / 5trAnzvE:s/ a.横向的,横断的traverse / 5trAvE:s/ vt.横越,横切,横断diverge / dai5vE:dV/ v.分歧,分开divergent / dai5vE:dVEnt/ a.离题的,偏差的,分歧的converge / kEn5vE:dV/ vi.聚合,集中于一点advert / 5AdvE:t/ vi.注意,留意n.广告conversant / kRn5vE:sEnt/ a.精通的,熟知的conversation / 5kRnvE5seiFEn/ n.会话,非正式会谈 evert / i5vE:t, i:-/ v.使外翻,翻转inadvertent / inEd5vE:tEnt/ a.不注意的,疏忽的 inversion / in5vE:FEn/ n.倒置deviate / 5di:vieit/ vt.使背离,偏离via / 5vaiE/ prep.经过,通过viaduct / 5vaiEdQkt/ n.高架桥deviation / 5di:vi5eiFEn/ n.背离devious / 5diviEs/ a.不正直的,弯曲的obviate / 5Rbvieit/ v.排除（困难）obvious / 5RbviEs/ a.明显的,显而易见的pervious / 5pE:viEs/ a.能被通过的previous / 5pri:vjEs/ a.先的,前的 ad.在前trivial / 5triviEl/ a.琐碎的,平常的convey / kEn5vei/ vt.传送,运送,传播voyage / vRidV/ n.vi.航海,航空convoy / 5kRnvRi/ v.护航,护送envoy / 5envRi/ n.使者,代表,跋词,后记multilateral / 5mQlti5lAtErEl/ a.多边的,多国的convict / kEn5vikt, 5kRnvikt/ n.囚犯,罪犯 vt.宣告有罪 convince / kEn5vins/ vt.使确信,使信服evict / i5vikt/ v.(依法)驱逐evince / i5vins/ v.表明,表示invincible / in5vinsEbl/ a.不可征服的,难以制服的 victor / 5viktE/ n.胜利者a.胜利者的divide / di5vaid/ vt.分开,分配,除division / di5viVEn/ n.分,分配,除法divisive / di5vaisiv/ a.区分的,不和的,分裂的individual / 5indi5vidjuEl/ a.个别的,独特的subdivide v.再分,细分vigil / 5vidVil/ n.守夜vigilance / 5vidVilEns/ n.警戒,警惕,失眠症vigilant / 5vidVilEnt/ a.机警的,警惕的invigilate / in5vidVileit/ v.监考vincinage n.邻近,附近vicinal / 5visnl/ a.附近的,同地方的vicinity / vi5siniti/ n.邻近,附近地区villa / 5vilE/ n.别墅,<英>城郊小屋 village / 5vilidV/ n.乡村,村庄villain / 5vilEn/ n.坏蛋violate / 5vaiEleit/ vt.违犯,违背,侵犯violation / 5vaiE5leiFEn/ n.违犯violator / 5vaiEleitE/ n.违犯者violative / 5vaiEleitiv/ a.违犯的,侵犯的violence / 5vaiElEns/ n.猛烈,激烈,暴力violent / 5vaiElEnt/ a.猛烈的,狂暴的violable / 5vaiElEbl/ a.易受侵犯的visible / 5vizEbl/ a.可见的,看得见的invisible / in5vizEbl/ a.看不见的,无形的visit / 5vizit/ vt.n.访问,参观visitor / 5vizitE/ n.访问者,游客revisit / 5ri:5vizit/ vt.再访,重游advise / Ed5vaiz/ vt.劝告,建议,通知advisor / Ed5vaizE/ n.顾问,指导老师advisory / Ed5vaizEri/ a.劝告的revise / ri5vaiz/ vt.修订,校订,修改revision / ri5viVEn/ n.修订,修改,修订本previse / pri5vaiz/ vt.预知,预先警告prevision / pri:5viVEn/ n.先见,预感supervisory / sju:pE5vaizEri/ a.管理的,监督的 visual / 5vizjuEl/ a.看的,看得见的visage / 5vizidV/ n.脸,面貌vision / 5viVEn/ n.视觉,想象力,梦幻television / 5teli5viVEn/ n.电视,电视机televise / 5telivaiz/ v.广播,播映visa / 5vi:zE/ n.签证vt.信用卡visionary / 5viVEnEri/ a.幻影的,幻想的,梦想的visionphone n.电视电话vista / 5vistE/ n.远景,展望visualize / 5vizjuElaiz/ vt.使看得见,使具体化,设想supervise / 5sju:pEvaiz/ vt.vi.监督,监视provisional / prE5viVEnl/ a.暂时的,临时的improvise / 5imprEvaiz/ v.即席而作envisage / in5vizidV/ vt.面对,正视,想象envision / en5viVEn/ vt.想象,预想advisable / Ed5vaizEbl/ n.明智的,可取的evident / 5evidEnt/ a.明显的,明白的evidence / 5evidEns/ n.根据,证据,证人provide / prE5vaid/ vt.提供,装备,供给provision / prE5viVEn/ n.供应,预备,存粮provident / 5prRvidEnt/ a.深谋远虑的,节俭的improvident / im5prRvidEnt/ a.不顾将来的,不节俭的video / 5vidiEu/ n.录像 a.录像的evidently / 5evidEntli/ ad.明显地,显然provided / prE5vaidid/ conj.倘若providence / 5prRvidEns/ n.深谋远虑,远见providential / prRvi5denFEl/ a.幸运的vital / 5vaitl/ a.生命的,有生命力的vitality / vai5tAliti/ n.活力,生命力,生动性vitalize / 5vaitlaiz/ v.激发revitalize / ri:5vaitElaiz/ v.使复活,使重新充满活力 devitalize / di:5vaitElaiz/ vt.夺去生命,使衰弱vitamin / 5vitEmin/ n.维生素,维他命vivacious / vi5veiFEs/ a.活泼的,快活的vivid / 5vivid/ a.鲜艳的,生动的vivify / 5vivifai/ vt.给与生气,使生动,使活跃vivisect / vivE5sekt/ vi.活体解剖vt.解剖vivisection / vivi5sekFEn/ n.活体解剖revive / ri5vaiv/ vt.vi.苏醒,复兴survive / sE5vaiv/ vt.幸免于 vi.活下来survival / sE5vaivEl/ n.幸存,残存,幸存者convivial / kEn5viviEl/ a.欢乐的,狂欢的conviviality / kEnvivi5Aliti/ n.欢宴,高兴,欢乐survivor / sE5vaivE/ n.生还者,残存物revival / ri5vaivEl/ n.苏醒,复兴,复活vivacity / vai5vAsEti/ n.快活vivarium / vai5vZEriEm/ n.动物园,植物园revivify / ri:5vivifai/ v.(使)再生,(使)复活vocal / 5vEukEl/ a.直言不讳的,嗓音的,有声的 vocalist / 5voukElist/ n.流行歌手,声乐家convocation / kRnvE5keiFEn/ n.召集,会议evocation / evou5keiFEn/ n.唤出,唤起,招魂provocation / 5prRvE5keiFEn/ n.激怒provocative / prE5vRkEtiv/ a.挑衅的,刺激的,挑逗的 revocation / revE5keiFEn/ n.废弃,取消revocable / 5revEkEbl/ a.可废除的abvocabulary / vE5kAbjulEri/ n.词汇表,词汇,语汇vocation / vEu5keiFEn/ n.职业,行业avocation / Avou5keiFEn/ n.副业,嗜好vociferous / vou5sifErEs/ a.喧哗的,大叫大嚷的evocative / i5vRkitiv/ a.唤起的,激起的advocate / 5AdvEkit/ n.拥护者 vt.提倡,拥护advocacy / 5AdvEkEsi/ n.拥护,鼓吹,辩护,辩护术evoke / i5vEuk/ vt.唤起,引起convoke / kEn5vEuk/ v.召集会议provoke / prE5vEuk/ vt.激怒,惹起,驱使revoke / ri5vEuk/ v.废除,取消volition / vou5liFEn/ n.自决,自主volitional / vou5liFEnl/ a.意志的,凭意志的volitive / 5vRlitiv/ a.意志的,表示意志的benevolent / bi5nevElEnt/ a.慈善的,好意的benevolence / bi5nevElEns/ n.善心,仁心malevolence / mE5levElEns/ n.恶意,狠毒malevolent / mE5levElEnt/ a.有恶意的,恶毒的volunteer / 5vRlEn5tiE/ n.志愿者 vt.志愿voluntary / 5vRlEntEri/ a.自愿的,志愿的involuntary / in5vRlEntEri/ a.自然而然的,无意识的 revolution / 5revE5lu:FEn/ n.革命,旋转,绕转revolutionary / 5revE5lu:FEnEri/ a.革命的 n.革命者 evolve / i5vRlv/ vt.使进化,使发展evolution / 5i:vE5lju:FEn/ n.进化,演化,发展evolutionism / evE5lu:FEnizm/ n.进化论a.进化论的evolutionist / evE5lu:FEnist/ n.进化论者a.进化论的 voluble / 5vRljubl/ a.流利的,健谈的volume / 5vRljum/ n.卷,册,容积,音量involve / in5vRlv/ vt.使卷入,牵涉involvement / -mEnt/ n.连累,包含intervolve / intE5vRlv/ v.互卷,互相牵连revolve / ri5vRlv/ v.(使)旋转,考虑,循环出现revolver / ri5vRlvE/ n.左轮手枪voluminous / vE5lju:minEs/ a.长篇的,多产的,庞大的 revolutionize / revE5lu:FEnaiz/ vt.宣传革命,大事改革 convolution / kRnvE5lu:FEn/ a.旋绕的,费解的devolve / di5vRlv/ vt.转移,传下,退化,衰落voracious / vE5reiFEs/ a.狼吞虎咽的,贪婪的voracity / vE5rAsiti/ n.贪食,贪婪devour / di5vauE/ v.吞食,（一口气）读完omnivorous / Rm5nivErEs/ a.杂食的,兴趣广泛的granivorous / grE5nivErEs/ a.食谷类的carnivorous / kB:5nivErEs/ a.食肉的abinsectivorous / insek5tivErEs/ a.食虫的,食虫动植物的 frugivorous / fru:5dVivErEs/ a.常食果实的carnivore / 5ka:nivR:/ n.食肉动物herbivorous / hE5bivErEs/ a.食草的divulge / dai5vQldV/ v.泄露,透露vulgar / 5vQlgE/ a.粗俗的,庸俗的vulgarian / vQl5gZEriEn/ n.俗物,俗人vulgarism / 5vQlgErizm/ n. 粗俗vulgarity / vQl5gArEti/ n. 粗俗，低级vulgarization / vQlgErai5zeiFEn/ n. 俗化, 卑俗化vulgarize / 5vQlgEraiz/ vt. 使通俗化, 使庸俗化vulnerable / 5vQlnErEbl/ a.易受伤害的,脆弱的vulnerability / vQlnErE5biliti/ n.弱点,攻击 vulnerary / 5vQlnErEri/ a.医治创伤的vulnerate / vQlnEreit/ vt.使受伤害invulnerable / in5vQlnErEbl/ a.无法伤害的 outwit / aut5wit/ v.以机智胜过wisdom / 5wizdEm/ n.智慧,才智,名言wise / waiz/ a.有智慧的,聪明的wiseacre / 5waizeikE/ n.自以为万事通的人wisecrack / 5waizkrAk/ n.俏皮话wit / wit/ n.智力,才智,智能witenagemot / 5witinEgi5mout/ n.国会,民会 zeal / zi:l/ n.热心,热情,热忱zealless / 5zi:llis/ a.无热情的,冷淡的zealot / 5zZlEt/ n.狂热份子,热心者zealotic / zi5lRtik/ adj.狂热的zealously ad.热心地,狂热地zoology / zou5RlEdVi/ n.动物学zoologist / zEu5RlEdVist/ n.动物学家zoo / zu:/ n.动物园zootomy / zou5RtEmi/ n.动物解剖zoolite / 5zouElait/ n.动物化石zoophilist / zou5Rfilist/ n.爱护动物者。

Two lectures on heavy quark production in hadronic collisions

Abstract These lectures present a pedagogical introduction to the physics of heavy-ﬂavour production in hadronic collisions. The ﬁrst lecture gives the theoretical background, with a discussion of leading-order calculations and of the eﬀects of next-to-leading-order corrections. The origin and implications of the large logarithmic corrections appearing at this order are presented in an elementary way. The second lecture provides a survey of current experimental data on charm and bottom production, and describes their comparison with theoretical predictions. We emphasize the role played by some non-perturbative eﬀects in the determination of charm distributions, and study the theoretical systematic uncertainties which aﬀect our predictions.
CERN-TH/97-328 hep-ph/9711337

Non-perturbative energy levels of two-fermion bound states

a r X i v :h e p -p h /9809533v 3 17 D e c 1998TTP 98-35Non-perturbative energy levels of two-fermion bound statesViktor Hund and Hartmut Pilkuhn Institut f¨u r Theoretische Teilchenphysik,Universit¨a t,D-76128Karlsruhe,Germany (e-mails:vh &hp@particle.physik.uni-karlsruhe.de)Abstract For the S-states of positronium and muonium,the terms of an expansion of energy levels in powers of the ﬁne structure constant αare also members of a “recoil series”.The ﬁrst two terms of that series are calculated to all orders in α.PACS number:36.10.Dr,03.65.Pm The calculation of energy levels to the order α6for the S-states of positronium and muonium (e −µ+)by Pachucki [1–3]has recently been conﬁrmed for equal masses by en-tirely analytic methods [4].Both calculations use non-relativistic quantum electrodynamics (NRQED)[5].The only two-fermion equation which is solved nonperturbatively is the Schr¨o dinger equation for a particle of reduced mass µ=m 1m 2/m (m =m 1+m 2)in a Coulomb potential V =−α/r (¯h =c =1).Apart from a state-independent function F (µ/m )which is only numerically known for arbitrary m 1/m 2,all terms to order α6and α6log αcan also be arranged in a ﬁnite series in µ/m for the total cms energy E ,E =m +E 1+E 2+E log +E 3,(1)where E i is of the order µ(µ/m )i −1,and E log contains logarithms of mass ratios.In the following,E 1and E 2are derived to all orders in α,and most of the sixth-order terms E (6)3∼α6µ3/m 2of E (6)are conﬁrmed (radiative corrections will be omitted).In combination with the complete expression for E (6),the series (1)increases the precision of QED bound state calculations for systems with m 1=m 2.Our method also reduces the gap between relativistic and non-relativistic expansions.The history of relativistic recoil expansions is long and sad[6–8].Frequently,one starts from a Dirac equation for an electron of mass m1in the potential V(r).The subsequent evaluation of recoil corrections of the order of m1/m2shows that most terms can be taken care of by replacing m1byµin the Dirac equation.Empirically then[6],E1follows from that equation.Similarly,theµ2/m-form has been found to orderα4for the hyperﬁne splitting, which is part of E2.But there has been no indication that to a given order ofα,the series (1)would end after aﬁnite number of terms.For some P-states,hyperﬁne mixing requires in fact inﬁnitely many terms already at the orderα4.The quantitative mixing disagrees with the standard hyperﬁne operator of the Dirac equation.On the contrary,this mixing follows easily from the Breit operators of NRQED.At the end,the Dirac equation approach has been completely abandoned for two-body systems.It is then remarkable that the nonperturbative use of the Schr¨o dinger equation leads to the form(1),in which E1is given precisely by the Dirac equation with reduced mass.For a state of angular momentum j,principal quantum number n and with the abbreviation j+12n−3 /8n3,(2)from which Pachucki’s result[2]follows for j+=1.We have recently derived a relativistic,Dirac-like two-fermion equation from perturbative QED[9],which explains this mystery and renders the calculation of E2trivial,again to all orders inα.The evaluation of E log and of the state-independent F(µ/m)requires the calculation of loop integrals,which remains to be done.The progress arises from the strict use of relativistic two-body kinematics in theﬁrst Born approximation,and from the reduc-tion to8×8components before Fourier transforming.Any system of two free particles of masses m1and m2satisﬁes the cms equation(p2−k2)ψ=0,where the eigenvalue k2can be expressed in terms of E2,m21and m22.It can be brought into the form k2=ε2−µ2E,µE=m1m2/E,ε=(E2−m21−m22)/2E.(3) The free equation is converted into an explicit eigenvalue equation for E2by the substitution ρ=µE r,pρ=p/µE, ε=ε/µE=(E2−m21−m22)/2m1m2.(4) When the interaction is added,the Coulomb potential V(r)is transformed into V(ρ).The resulting dimensionless Dirac equation is[9]β+γ5(σ1+σhf)pρ+V(ρ)− ε ψ=0,(5)σhf=−iσ1×σ2V(ρ)m1m2/E2.(6) Theσ1andσ2are Pauli matrices;the productγ5σ1is normally written asα.For compar-ison,the hyperﬁneσhf of atomic theory sets E=m2and replaces V p by[V,p]/2.To begin with,we evaluate the hyperﬁne energies E hf byﬁrst-order perturbation theory. For orbital angular momentum l=j±12,they areE hf=2m21m22f+1/2(j2+ ε−κD/2)/(4γ3−γ),(7)2κD=2(l−j)j+,γ2=j2+−α2.(8) Special cases of this formula are found in[10].To orderα6and forκD=−j+,the quotient of the last two brackets in(7)isj2+ ε−κD/22jj+ 1+α2 12j2++32n2−j+2)−E(f=j−12)=23in∆E hf.Pachucki’s result for the part E(6)2,hf of E(6)2follows by approximating E2≈m2in the denominator of(7).The remaining two terms of E2appear in the hyperﬁne-averagedshift¯E=34E(f=0).They are conveniently evaluated by a non-relativisticreduction:With the approximation E2≈m2in the hyperﬁne operator,(5)is an explicit eigenvalue equation that permits the standard reduction by elimination of the small components.The resulting Schr¨o dinger equation is(1+p2ρ/2+V(ρ)− εSch)ψSch=0.(10) It has the familiar non-relativistic eigenvalues, εSch−1=−α2/2n2.The equation becomes quite powerful when its centrifugal barrier l(l+1)/ρ2is replaced by an eﬀective barrier l′(l′+1)/ρ2,which includes the lowest-order spin-orbit and hyperﬁne couplings:l′−l≡δl=α2 −1f+1m .(11)The principal quantum number n=n r+l+1gets replaced by n∗=n r+l′+1=n+δl, and the eigenvalues E2Sch follow from(10)asE2Sch−m2=−α2m1m2/n∗2≈−α2m1m2(1−δl/n)2/n2.(12)To orderα6,one obtainsE Sch−m=−α2(1−δl/n)2µ/2n2−α4µ2(1−4δl/n)/8mn4−α6µ3/16m2n6.(13) This expression contains both terms of¯E(6)2and two of the three terms in¯E(6)3[2](note that (δl)2is quadratic in the hyperﬁne interaction).The third term arises from the S-D-mixing in second order perturbation theory and is not calculated here,E(6)SD=−44,and in∆E3,hf with a factor1.There are two more n−5-contributions to∆E3,hf,one from the hyperﬁne part ofδl,δhf=2α2µ2)in(13)(in the bracket followingα4µ2),and one from setting E2=m2−α2m1m2/n2in(7),which eﬀectively enlarges all hyperﬁne eﬀects by a factor1+α2µ/mn2.The total coeﬃcient ofα6µ3/m2n5is then−43+89,in agreement with[1].For higher orders inα,the expansion(1)is conveniently replaced by the simpler expansion for ε−1=(E2−m2)/2m1m2ε=1+ ε1+ ε2+ εlog+ ε3,(14)3in which ε1is pure“Dirac”as in E1,and ε2is pure“hyperﬁne”.In other words,the non-hyperﬁne terms of E2are canceled to all orders inα.There is one more hyperﬁne contribution to∆E3,hf[1]which does not follow from(5):∆E3,hf=2µ3f+1α+16−n−1i=11REFERENCES[1]K.Pachucki,Phys.Rev.A56,297(1997)[2]K.Pachucki,Phys.Rev.Letters79,4021(1997)[3]K.Pachucki and S.G.Karshenboim,Phys.Rev.Letters80,2101(1998)[4]A.Czarnecki,K.Melnikov and A.Yelkhovsky,Phys.Rev.Letters(to be published)[5]W.E.Caswell and G.P.Lepage,Phys.Letters167B,437(1986)[6]G.Erickson,J.Phys.Chem.Ref.Data,6,831(1977)[7]R.Sapirstein and D.Yennie,in:Quantum Electrodynamics,World Scientiﬁc,Singapore1990[8]M.I.Eides and H.Grotch,Phys.Rev.A55,3351(1997)[9]R.H¨a ckl,V.Hund and H.Pilkuhn,Phys.Rev.A57,3268(1998)[10]M.E.Rose,Relativistic Electron Theory,Wiley1961[11]M.Malvetti and H.Pilkuhn,Phys.Rep.C248,1(1994)[12]Review of Particle Physics,European Physical Journal C3,1(1998)[13]Th.Mannel,Acta Physica Polonica B29,1413(1998)5。

Nonperturbative heat kernel and nonlocal effective action

(1.2)
The ﬁrst few orders of this expansion are graphically depicted below as Feynman graphs with the propagator – the Green’s function of the operator (1.1) – and relevant vertices calculated on the background of a generic ϕ Γ1−loop 1 1 = Tr ln F (∇) = 2 2
s
(1.3)
1 1 s + . (1.4) Γ2−loop = 8 12 s The one-loop part (1.3) is peculiar in that it does not explicitly contain the vertices of the classical action (unless it is expanded in powers of the mean ﬁeld ϕ) and given by the functional trace of the logarithm of F (∇). In local ﬁeld theories without loss of generality the operator (1.1) has the form F ( ∇) = − V (x), 3 (1.5)
1. Introduction: heat kernel and eﬀective action 2. Approximation schemes and infrared behavior 2.1. Schwinger-DeWitt technique of local expansion . . . . . . . . . . . . . 2.2. Modiﬁed Schwinger-DeWitt expansion . . . . . . . . . . . . . . . . . . 2.3. Covariant perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 3. Nonperturbative late-time asymptotics 3.1. Leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Subleading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Resummation of covariant perturbation theory . . . . . . . . . . . . . . 4. New nonlocal eﬀective action 5. Inclusion of gravity 5.1. Leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Conformal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Problems with the subleading order . . . . . . . . . . . . . . . . . . . . 6. Discussion and conclusions A. Late time asymptotics in perturbation theory B. Nonperturbative eﬀective action B.1. Small potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Big potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Nonlocal form of the Gibbons-Hawking surface integral D. Metric dependence in the subleading order 3 5 5 7 9 10 11 12 13 15 18 19 23 25 27 30 32 33 35 36 38

Quantum Correlations in Nuclear Mean Field Theory Through Source Terms

a rXiv:n ucl-t h /9647v14Apr1996nucl-th/9604007Quantum Correlations in Nuclear Mean Field Theory Through Source Terms S.J.Lee ∗Department of Physics,College of Natural Sciences Kyung Hee University,Yongin,Kyungkido 449-701KOREA Abstract Starting from full quantum ﬁeld theory,various mean ﬁeld approaches are derived systematically.With a full consideration of external source depen-dence,the stationary phase approximation of an action gives a nuclear mean ﬁeld theory which includes quantum correlation eﬀects (such as particle-hole or ladder diagram)in a simpler way than the Brueckner-Hartree-Fock ap-proach.Implementing further approximation,the result can be reduced to Hartree-Fock or Hartree approximation.The roll of the source dependence in a mean ﬁeld theory is examined.PACS numbers:21.60.Jz,24.10.Jv,,03.70.+k,11.10.–z,11.15.KcTypeset using REVT E XOne of important problems in nuclear physics is understanding systematically the nuclear systems ranging from the structure of stableﬁnite nuclei to a very hot and dense system which may occur in a high energy heavy ion collision or in a neutron star.A nuclear system is a strongly interacting many body system and thus the description of the system becomes complicated.A modern approach to the study of a nuclear system is based on the relativistic ﬁeld theory in terms of relativistic nucleons interacting each other by exchanging mesons. The simplest approach of such a theory is the so-called relativistic meanﬁeld approximation (RMF)or quantum hadrodynamics(QHD)which describes a nuclear system in terms of nucleonic Diracﬁeld interacting with classical mesonﬁelds[1].In this approach,a nuclear system is composed of relativistic nucleons whose self energy is determined through meson ﬁelds which are generated by the nuclear density.In spite of the success of this simple model in describing various properties of nuclear system with eﬀective interaction[1–9],it is failed to reproduce nuclear saturation prop-erties from the free nucleon-nucleon interaction.The calculations yield correct binding energy with too large density or vice versa forming the Coester band[10,11].Even though the Dirac-Brueckner-Hartree-Fock(DBHF)approximation including ladder diagram[12–17] can achieve the nuclear saturation for a nuclear matter[13,14],it failed in reproducing the ground state properties forﬁnite nuclei using the nucleon-nucleon interaction forming a Coester band[18,19].The DBHF is essentially the RHF with correlation eﬀects(ladder diagrams)determined through the Bethe-Salpeter(BS)equation for a nucleon-nucleon in-teraction.Two coupled self-consistent equations,Dyson’s equation and BS equation,make the DBHF calculations very complicated for aﬁnite nucleus[19].Thus we need to search for a description that is simpler than the DBHF and can consider various quantum correlation eﬀects.On the other hand,various calculations[3,20]show that the exchange term,the vacuumﬂuctuation,and the ladder diagrams can not be treated as perturbation with re-spect to the Hartree term or each other exhibiting non-perturbative characteristics of strong nuclear interaction.For a systematic study of various meanﬁeld approaches,it is necessary to investigate the basic assumptions underlying to each of the meanﬁeld theories starting from the quantum ﬁeld theory.In this paper,we will consider full external source dependence of a stationary phase approximation(SPA)of the path integral for a nuclear system with arbitrary external sources.The added source terms bring the quantum correlations into a meanﬁeld approach in a simple way.Depending on the interaction considered,this approach can give a correction to the DBHF[21]and also reveals the origin of non-perturbative relationships among the various meanﬁeld approximations.Up to few hundreds MeV energy,a nuclear system may be described as a system of nucleons interacting each other via meson exchange.For a nuclear saturation property,we at least need to include a scalar meson for a long range attraction and a vector meson for a short range repulsion.The simplest Lagrangian describing a nuclear system can be written in the form of1L(x)=¯ψ(x)[iγµ∂µ−M]ψ(x)+ﬁeld.More detail diﬀerences between scalar and vector mesonﬁelds are irrelevant for the discussions in the context of this paper.In quantumﬁeld theory,any physical observable can be represented as a function ofﬁeld operators.Introducing arbitrarily small external sources asL(x,J)=L(x)−¯ψ(x)J(x)−¯J(x)ψ(x)−J m(x)ϕ(x),(2) the expectation value of an operatorˆO(ψ,¯ψ,ϕ)can be obtained byˆO(ψ,¯ψ,ϕ) =1∂¯J,i∂∂J m W(J) J=0,(3) where the transition amplitude W(J)can be obtained through a path integral[22–24];W(J)=e iS(J)= Ψf T[e−i H(J)dt] Ψi = D(ψ,¯ψ,ϕ)e i d4x L(x,J) .(4)Here T represents time ordering and J stands for J,¯J,and/or J m.Notice here that¯J(x)is not the conjugate function of J(x);both are independent arbitrary functions.If we use the single particle state representation,then the matrix element L(x,J) = Ψ+|L(x,J)|Ψ− becomes a functional of the single particle wave functionsψ(x)andϕ(x).Now the quantum ﬁeld theory for a nuclear system has been reduced to a problem ofﬁnding the corresponding transition amplitude W(J)or action S(J).For a Lagrangian of the form of Eqs.(1)and(2),we can integrate Eq.(4)over the meson ﬁeldϕ.However the result becomes fourth order in the nucleonﬁeldψand thus further integral is not feasible.On the other hand,we can integrate over the nucleonﬁeldﬁrst, but the resulting nonlinear equation ofϕprevents further integration.Thus we need to use some approximate method.However,since we can not use a perturbative method due to the strong interaction,we are forced to use a classical or semi-classical trajectory as the simplest nonperturbative method[21–25].If we treat all theﬁelds as classical ones,then L(x,J) in Eq.(4)becomes the Lagrangian L of Eq.(2)with c-functionsψ(x),ϕ(x),and J(x).To treat a nuclear ground state as a Slater determinant of occupied single nucleon levels,we should treat the nucleonﬁeld as a quantumﬁeld.Then the external sources J(x) and¯J(x)should also be treated as q-functions having the same quantum characteristics as the nucleonﬁeld with inﬁnitesimal amplitudes to keep the source terms in the transition amplitude[21].Similarly,J m(x)should be a q-function if we consider a quantized meson ﬁeldϕ(x).The stationary phase approximation(SPA)or steepest descent method of the path inte-gral Eq.(4)or equivalently the variation of the Lagrangian with respect to eachﬁeld gives the corresponding Euler-Lagrange equation for a classical trajectory as[iγµ∂µ−M+gΓϕ(x)]ψ(x)=J(x),(5)∂µ∂µ+m2 ϕ(x)=g¯ψ(x)Γψ(x)−J m(x).(6)The equation of motion for¯ψis given as the conjugate of Eq.(5).Using the meson propagator D(x−x′),we can eliminate mesonﬁelds from Eqs.(5)and(6).The result becomesiγµ∂µ−M+g2 d4x′D(x−x′)¯ψ(x′)Γψ(x′) Γ ψ(x)=J(x)+gΓψ(x) d4x′D(x−x′)J m(x′),(7) for a ground state of a nuclear system.This is a nonlinear equation of the nucleonﬁeld. Since the equations ofψ(Eq.(7))and¯ψare coupled nonlinear equations,their solutions would be functions of inﬁnite order in the external source J,i.e.,J,¯J and J ually this source dependence was neglected,and thus the meanﬁeld theory gave the RMF of Walecka [1]or the Hartree-Fock(RHF)approximation only.Since Eq.(7)should be satisﬁed order by order in J,we may expand the solution as a power series of J;ψ(x)=U(x)+ψ1(x)+ψ2(x)+ψ3(x)+ (8)Here U(x)=ψ0(x)is the source independent solution,which is non-zero only for occupied states,andψn(x)is the n-th order term of the external sources J,¯J and J m,i.e.,ψn(x)=ψ(x,J n).The source independent solution U(x)can be obtained from Eq.(7)by neglecting all the external source dependent terms;(iγµ∂µ−M)+g2 d4x′D(x−x′)¯U(x′)ΓU(x′) Γ U(x)=0.(9)Notice that the source independentﬁeld¯U(x)is the conjugateﬁeld of U(x).If we treat mesonﬁeldϕin Eq.(5)as a classicalﬁeld,i.e.,replace the right hand side of Eq.(6)by the corresponding expectation value as it has been done by Walecka,then¯U(x′)ΓU(x′)inside the integral of Eq.(9)becomes a classical quantity ¯U(x′)ΓU(x′) and is independent of U(x)appearing outside the integral.This is the relativistic Hartree(RH)or the RMF of Walecka[1].If we treat the mesonﬁeld as a quantumﬁeld,then both the U(x′)and U(x) are the sameﬁeld and thus we have a nonlinear equation for U.Toﬁnd the solution U(x) using an iterative method,we can linearize this equation asd4x′A(x,x′)U(x′)=0,(10)A(x,x′)= (iγµ∂µ−M)+g2 d4x′′D(x−x′′)¯U(x′′)ΓU(x′′)Γ δ(x−x′)−g2ΓU(x)D(x−x′)¯U(x′)Γ.(11) Here the minus sign of the last term in Eq.(11)originates from the anti-commuting property of the nucleonﬁeld.Notice here that A(x,x′)is hermitian and the self-consistentﬁeld U(x)in a nuclear system is diﬀerent from the Diracﬁeld in a free space.The U(x)and U(x′′)appearing in Eq.(10)through A(x,x′)of Eq.(11)are now treated as independent quantities from U(x′)of Eq.(10)in the iterative calculation for the solution U(x).This is the relativistic Hartree-Fock(RHF)approximation.The second term of A(x,x′)is the direct Hartree contribution to the nuclear self energy and the third term is the exchange Fock contribution to the self energy.In Ref.[2],it was shown that we can obtain RHF using the quantum operator algebra without using the concept of linearlization explicitly.On the other hand,from the n-th order terms of J in Eq.(7),we obtain coupled equations forψn and¯ψn.Their solutions[21]areψn(x)= d4x′B−1(x,x′)J n(x′)−g2 d4x′B−1(x,x′)ΓU(x′) d4x2 d4x3D(x′−x3)¯J n(x2)A−1(x2,x3)ΓU(x3),(12)¯ψn(x)= d4x′¯J n(x′)B−1(x′,x)−g2 d4x2 d4x3¯U(x2)ΓA−1(x2,x3)J n(x3) d4x′D(x2−x′)¯U(x′)ΓB−1(x′,x),(13) whereJ n(x)=−g2n−1n1=0n−1 n2=0n−1 n3=0Γψn1(x) d4x′D(x−x′)¯ψn2(x′)Γψn3(x′)+J(x)δn,1+gΓψn−1(x) d4x′D(x−x′)J m(x′),(14)¯Jn(x)=−g2 d4x′D(x′−x)n−1 n1=0n−1 n2=0n−1 n3=0¯ψn1(x′)Γψn2(x′)¯ψn3(x)Γ+¯J(x)δn,1+g d4x′J m(x′)D(x′−x)¯ψn−1(x)Γ,(15)withn1+n2+n3=n;n1<n,n2<n,n3<n.(16)Notice here that J n and¯J n are given in terms ofψm(x)and¯ψm(x)with m=0,1,...,n−1which are lower order in J.HereB(x,x′)=A(x,x′)−(g2)2ΓU(x) d4x2 d4x3D(x−x3)¯U(x2)ΓA−1(x2,x3)ΓU(x3) D(x′−x2)¯U(x′)Γ.(17)Ifψn(x)and¯ψn(x)were independent[21],then B(x,x′)would be replaced simply withA(x,x′)and only theﬁrst terms survive in Eqs.(12)and(13).Once we solve Eq.(9)for U(x),then we canﬁndψn(x)and¯ψn(x)for all order n.Nowthe mesonﬁeldϕ(x)can be found using Eq.(6).Using these results,we can evaluate thepath integral of Eq.(4)in SPA andﬁnd the corresponding action integral S(J).Since only up to second order terms are needed in calculating propagators of eachﬁeldor the energy and the density of a system,let’s look at up to the second order in the externalsource J.Dropping the integral sign and the irrelevant J-independent terms,the actionintegral becomes up to the relevant second order in J as,S(J)=−¯U(x)J(x)−¯J(x)U(x)−¯J(x′)B−1(x′,x)J(x)−g¯U(x)ΓU(x)D(x−x′)J m(x′)+12g4¯U(x2)ΓA−1(x2,x3)ΓU(x3)D(x3−x4)J m(x4)D(x2−x1)¯U(x1)ΓB−1(x1,x)ΓU(x)D(x−x′)J m(x′) +1Using Eq.(3)with W(J)=e iS(J)of Eq.(18),we canﬁnd the expectation value of any operators up to two body form.The mean nucleonﬁeld and the mean mesonﬁeld becomeψ(x) =1∂¯J(x) W(J)J=0=U(x),(19)ϕ(x) =1∂J m(x) W(J)J=0=d4x′D(x−x′)¯U(x′)ΓU(x′),(20)as expected for a meanﬁeld theory.The mean mesonﬁeld in a nuclear system is generated as a virtual meson due to the(real)nucleonﬁeld.The mesonﬁeld propagator becomes, dropping the obvious integral sign,i∆(x−x′)= Tϕ(x)ϕ(x′) =1∂J m(x)i∂W(J) i∂∂J(x′)W(J)J=0=iB−1(x,x′)+T[U(x)¯U(x′)].(22) The second term is the density dependent propagator which is also appearing in RMF or RHF.Theﬁrst term is the correction of the Feynman propagator iA−1(x,x′)of the RHF. One of the diﬀerences of our method from other meanﬁeld approaches comes through the diﬀerence B(x,x′)−A(x,x′)which corresponds to the ladder diagram included in the DBHF.The second term of Eq.(21)and the second term of Eq.(22)form the RHF while theﬁrst term of Eq.(22)gives a correction to the vacuumﬂuctuation of the RHF.The last three terms of Eq.(21)gives one and two baryon loops with the Feynman propagator iB−1(x,x′) for one of the internal baryon line.This propagator contains higher order correlations(such as ladder diagram)which were missed in the Feynman propagator iA−1(x,x′)of the RHF. The particle-hole correlations can also be considered in this model through the Feynman propagators iA−1(x,x′)and iB−1(x,x′)in Eqs.(18)and(21).If we neglect the coupling term ofψn(x)and¯ψn(x)[21],then there would be no higher order correlation eﬀects in the Feynman propagator than RHF,i.e.,we would have B(x,x′)= A(x,x′).If we neglect the nuclear external source,i.e.,set J(x)=0in Eq.(5),then the third term of Eq.(18)does not exist.This means that the vacuumﬂuctuation comes in through the J dependence in a meanﬁeld theory.On the other hand,if we neglect the mesonic external source,i.e.,set J m(x)=0in Eq.(6),then only theﬁrst four terms of Eq.(18)survive.This approximation gives the RHF with nuclear vacuumﬂuctuation.If we neglect both J(x)and J m(x),then we have the RHF without vacuumﬂuctuation.If we neglectany J dependence ofψin Eq.(6),i.e.,replace¯ψ(x)Γψ(x)in Eq.(6)by¯U(x)ΓU(x),then we get the RHF with theﬁrstﬁve terms in Eq.(18).However if we use the expectation value ¯U(x)ΓU(x) for¯ψ(x)Γψ(x)in Eq.(6),then we have the RH with vacuum eﬀect.Various quantum correlation eﬀects come into a meanﬁeld approach depending on how the sourcedependence ofﬁelds are treated.In conclusion,we have shown that the SPA of the full quantumﬁeld theory with exter-nal sources gives quantum correlation eﬀects which were not included in other meanﬁeld approaches.We also have seen that various assumptions on the external source dependence reduce our SPA to RMF,RH,and RHF.Since these further assumptions on the source dependence are not perturbative,the various meanﬁeld approaches of RMF,RH,RHF, DBHF,and SPA can not have perturbative relationships.The correlation of ladder dia-gram is included in the SPA through the self energy of the propagator iB−1(x,x′)without any further self-consistency condition in contrast to the DBHF where the ladder diagram is included through the self-consistent Bethe-Salpeter equation.Only the HF wave function U(x)is required the self-consistent calculation in the SPA in contrast to the DBHF which uses the Dyson’s equation for the nucleon propagator and the Bethe-Salpeter equation for the eﬀective interaction.We should investigate further for the diﬀerences between the SPA and the DBHF by analyzing more details and by applying the model to a nuclear system [21].To study n-n interaction with this model,we need to consider the action up to4-th order in J.We also need to extend the SPA further to the case with nonlinear mesonﬁeld to consider a nonlinear sigma model or a quark-gluon system.This extension would enable the SPA to consider many body interaction within the meanﬁeld theory.This work was supported in part by the Korea Science and Engineering Foundation under Grant No.931-0200-035-2and in part by the Ministry of Education,Korea through Basic Science Research Institute Grant No.95-2422.I would like to thank the hospitality oﬀered me at the Department of Physics and Astronomy,Rutgers University.REFERENCES[1]J.D.Walecka,Ann.Phys.83,491(1974).[2]S.J.Lee,Ph.D.Thesis,Yale University,1986.[3]ler,Phys.Rev.C9,537(1974);C12,710(1975).[4]B.D.Serot and J.D.Walecka,Adv.Nucl.Phys.16,1(1985).[5]Suk-Joon Lee,J.Fink,A.B.Balantekin,M.R.Strayer,A.S.Umar,P.-G.Reinhard,J.A.,Maruhn,and W.Greiner,Phys.Rev.Letts.57,2916,(1986);59,1171(1987).[6]C.E.Price and G.E.Walker,Phys.Rev.C36,354(1987).[7]W.Pannert,P.Ring,and J.Boguta,Phys.Rev.Lett.59,2420(1987).[8]R.J.Furnstahl and C.E.Price,Phys.Rev.C44,895(1991).[9]C.J.Horowitz and B.D.Serot,Nucl.Phys.A399,529(1983).[10]F.Coester,S.Cohen,B.D.Day,and C.M.Vincent,Phys.Rev.C1,769(1970).[11]H.Kuemmel,K.H.Luhrmann,and J.G.Zabolitzky,Phys.Rep.36,1(1978).[12]M.R.Anastasio,L.S.Celenza,W.S.Pong,and C.M.Shakin,Phys.Rep.100,327(1983).[13]R.Brockmann and R.Machleidt,Phys.Lett.149B,283(1984).[14]B.Ter Haar and R.Malﬂiet,Phys.Rep.149,207(1987).[15]R.Machleidt,Adv.in Nucl.Phys.19,189(1989).[16]H.Shi,B.Chen,and Z.Ma,preprint nucl-th/9412010(1994).[17]H.F.Boersma and R.Malﬂiet,Phys.Rev.C49,233(1994);C49,1495(1994).[18]H.M¨u ther,M.Machleidt,and R.Brockmann,Phys.Rev.C42,1981(1990).[19]R.Fritz and H.M¨u ther,Phys.Rev.C49,633(1994).[20]R.Fritz,H.M¨u ther,and R.Machleidt,Phys.Rev.Letts.71,46(1993).[21]S.J.Lee,to be published.[22]E.S.Aber and B.W.Lee,Phys.Rep.9,1(1973).[23]J.P.Blaizot and H.Orland,Phys.Rev.C24,1740(1981).[24]S.J.Lee,in Selected Topics in Nuclear Physics,Proc.of the Third Symposium onNuclear Physics,Seoul,1990(D.P.Min,ed,Han Lim Won,Seoul,Korea,1991),158.[25]S.J.Lee,Bull.Am.Phys.Soc.31,64(1986).。

Perturbative deformations of conformal field theories revisited

a r X i v :h e p -t h /0609018v 5 1 F eb 208PERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED IGOR KRIZ 1.Introduction Recently,there has been renewed interest in the mathematics of the moduli space of conformal ﬁeld theories,in particular in connection with speculations about el-liptic cohomology.The purpose of this paper is to investigate this space by per-turbative methods from ﬁrst principles and from a purely “worldsheet”point of view.It is conjectured that at least at generic points,the moduli space of CFT’s is a manifold,and in fact,its tangent space consists of marginal ﬁelds,i.e.primary ﬁelds of weight (1,1)of the conformal ﬁeld theory (that is in the bosonic case,in the supersymmetric case there are modiﬁcations which we will discuss later).This then means that there should exist an exponential map from the tangent space at a point to the moduli space,i.e.it should be possible to construct a continuous 1-parameter set of conformal ﬁeld theories by “turning on”a given marginal ﬁeld.There is a more or less canonical mathematical procedure for applying a “P exp ”type construction to the ﬁeld which has been turned on,and obtaining a pertur-bative expansion in the deformation parameter.This process,however,returns certain cohomological obstructions,similar to Gerstenhaber’s obstructions to the existence of deformations of associative algebras [24].Physically,these obstructions can be interpreted as changes of dimension of the deforming ﬁeld,and can occur,in principle,at any order of the perturbative path.The primary obstruction is well known,and was used e.g.by Ginsparg in his work on c =1conformal ﬁeld theories [25].The obstruction also occured in earlier work,see [39,40,41,56,57,58,54],from the point of view of continuous lines in the space of critical models.In the models considered,notably the Baxter model [10],the Ashkin-Teller model [7]and the Gaussian model [42],vanishing of the primary obstruction did correspond to a continuous line of deformations,and it was therefore believed that the primary obstruction tells the whole story.(A similar story also occurs in the case of defor-mations of boundary sectors,see [1,2,11,20,45,46,51,33].)In a certain sense,the main point of the present paper is analyzing,or giving ex-amples of,the role of the hihger obstructions.We shall see that these obstructions can be non-zero in cases where the deformation is believed to exist,most notably in the case of deforming the Gepner model of the Fermat quintic along a cc ﬁeld,cf.[3,49,21,53,38,59,60,13,14,16].Some discussion of marginality of primary ﬁeld in N =2-supersymmetric theories to higher order exists in the literature.Notably,Dixon,[18]veriﬁed the vanishing for any N =(2,2)-theory,and any linear combi-nation of cc,ac,ca and ac ﬁeld,of an amplitude integral which physically expresses the change of central charge (a similar calculation is also given in Distler-Greene12IGOR KRIZ[17]).Earlier work of Zamolodchikov[63,64]showed that the renormalizationβ-function vanishes for theories where c does not change during the renormalization process.However,weﬁnd that the calculation[18]does not guarantee that the primaryﬁeld would remain marginal along the perturbative deformation path,due to subtleties involving singularities of the integral.The obstruction we discuss in this paper is an amplitude integral which physically expresses directly the change of dimension of the deformingﬁeld,and it turns out this may not vanish.We will return to this discussion in Section3below.This puzzle of having obstructions where none should appear will not be fully explained in this paper,although a likely interpretation of the result will be dis-cussed.Very likely,our eﬀect does not impact the general question of the existence of the non-linearσ-model,which is widely believed to exist(e.g.[3,49,21,53,38, 59,60,13,14,16]),but simply concerns questions of its perturbative construction. One caveat is that the case we investigate here is still not truly physical,since we specialize to the case of ccﬁelds,which are not real.The actual physical defor-mations of CFT’s should occur along realﬁelds,e.g.a combination of a ccﬁeld and its complex-conjugate aaﬁeld(we give a discussion of this in case of the free ﬁeld theory at the end of Section4).The case of the corresponding realﬁeld in the Gepner model is much more diﬃcult to analyze,in particular it requires regulariza-tion of the deforming parameter,and is not discussed here.Nevertheless,it is still surprising that an obstruction occurs for a single ccﬁeld;for example,this does not happen in the case of the(compactiﬁed or uncompactiﬁed)freeﬁeld theory.Since an n’th order obstruction indeed means that the marginalﬁeld gets de-formed into aﬁeld of non-zero weight,which changes to the order of the n’th power of the deformation parameter,usually[25,39,40,41,56,57,58,54],when obstruc-tions occur,one therefore concludes that the CFT does not possess continuous deformations in the given direction.Other interpretations are possible.One thing to observe is that our conclusion is only valid for purely perturbative theories where we assume that allﬁelds have power series expansion in the deformation parame-ters with coeﬃcients which areﬁelds in the original theory.This is not the only possible scenario.Therefore,as remarked above,our results merely indicate that in the case when our algebraic obstruction is non-zero,non-perturbative corrections must be made to the theory to maintain the presence of marginalﬁelds along the deformation path.In fact,evidence in favor of this interpretation exists in the form of the analysis of Nemeschansky and Sen[49,30]of higher order corrections to theβ-function of the non-linearσ-model.Grisaru,Van de Ven and Zanon[30]found that the four-loop contribution to theβ-function of the non-linearσmodel for Calabi-Yau manifolds is non-zero,and[49]found a recipe how to cancel this singularity by deforming the manifold to metric which is non-Ricciﬂat at higher orders of the deformation parameter.The expansion[4]used in this analysis is around the0 curvature tensor,but assuming for the moment that a similar phenomenon occurs if we expanded around the Fermat quintic vacuum,then there are noﬁelds present in the Gepner model which would correspond perturbatively to these higher order corrections in the direction of non-Ricciﬂat metric:bosonically,suchﬁelds would have to have critical conformal dimension classically,since theσ-model LagrangianPERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED3 is classically conformally invariant for non-Ricciﬂat target K¨a hler manifolds.How-ever,quantum mechanically,there is a one-loop correction proportional to the Ricci tensor,thus indicating thatﬁelds expressing such perturbative deformations would have to be of generalized weight(cf.[34,35,36,37]).Fields of generalized weight, however,are not present in the Gepner model,which is a rational CFT,and more generally are excluded by unitarity(see discussions in Remarks after Theorems2, 3in Section3below).Thus,although this argument is not completely mathemati-cal,renormalization analysis seems to conﬁrm ourﬁnding that deformations of the Fermat quintic model must in general be non-perturbative.It is also noteworthy that theβ-function is known to vanish to all orders for K3-surfaces because of N=(4,4)supersymmetry.Accordingly,we alsoﬁnd that the phenomenon we see for the Fermat quintic is not present in the case of the Fermat quartic(see Section7 below).It is also worth noting that other non-perturbative phenomena such as in-stanton corrections also arise when passing from K3-surfaces to Calabi-Yau3-folds ([13,14,16]).Finally,one must also remark that the proof of[49]of theβ-function cancellation is not mathematically complete because of convergence questions,and thus one still cannot exclude even the scenario that not all non-linearσmodels would exist as exact CFT’s,thus creating some type of“string landscape”picture also in this context(cf.[19]).In this paper,we shall be mostly interested in the strictly perturbative picture. The main point of this paper is an analysis of the algebraic obstructions in cer-tain canonical cases.We discuss two main kinds of examples,namely the freeﬁeld theory(both bosonic and N=1-supersymmetric),and the Gepner models of the Fermat quintic and quartic,which are exactly solvable N=2-supersymmetric con-formalﬁeld theories conjectured to be the non-linearσ-models of the Fermat quintic Calabi-Yau3-fold and the Fermat quartic K3-surface.In the case of the freeﬁeld theory,what happens is essentially that all non-trivial gravitational deformations of the freeﬁeld theory are algebraically obstructed.In the case of a free theory compactiﬁed on a torus,the only gravitational deformations which are algebraically unobstructed come from linear change of metric on the torus.(We will focus on gravitational deformations;there are other examples,for example the sine-Gordon interaction[62,12],which are not discussed in detail here.)The Gepner case deserves special attention.From the moduli space of Calabi-Yau3-folds,there is supposed to be aσ-model map into the moduli space of CFT’s. In fact,when we have an exactly solvable Calabi-Yauσ-model,one gets operators in CFT corresponding to the cohomology groups H11and H21,which measure deformations of complex structure and K¨a hler metric,respectively,and these in turn give rise to inﬁnitesimal deformations.Now the Fermat quintic(1)x5+y5+z5+t5+u5=0in C P4has a model conjectured by Gepner[22,23]which is embedded in the tensor product of5copies of the N=2-supersymmetric minimal model of central charge 9/5.The weight(1/2,1/2)cc and acﬁelds correspond to the100inﬁnitesimal deformation of complex structure and1inﬁnitesimal deformation of K¨a hler metric of the quintic(1).Despite the numerical matches in dimension,however,it is not quite correct to say that the gravitational deformations,corresponding to the moduli space of Calabi-Yau manifolds,occurs by turning on cc and acﬁelds.This4IGOR KRIZis because,to preserve unitarity,a physical deformation can only occur when we turn on a realﬁeld,and theﬁelds in question are not real.In fact,the complex conjugate of a ccﬁeld is an aaﬁeld,and the complex conjugate of an acﬁeld is a caﬁeld.The complex conjugate must be added to get a realﬁeld,and a physical deformation(we discuss this calculationally in the case of the freeﬁeld theory in Section4).In this paper,we do not discuss deformations of the Gepner model by turning on realﬁelds.As shown in the case of the freeﬁeld theory in Section4,such deformations require for example regularization of the deformation parameter,and are much more diﬃcult to calculate.Because of this,we work with only with the case of one cc and one acﬁeld.We will show that the ac deformation(corresponding to constant K¨a hler metric change)and its“mirror-symmetric”cc-deformations are algebraically unobstructed,but at least one cc deformation,whose real version corresponds to the quintics(2)x5+y5+z5+t5+u5+λx3y2=0for small(but not inﬁnitesimal)λis algebraically obstructed.(One suspects that similar algebraic obstructions also occur for the remainingﬁelds,but the computa-tion is too diﬃcult at the moment.)It is an interesting question if non-linearσ-models of Calabi-Yau3-folds must also contain non-perturbative terms.If so,likely,this phenomenon is generic,which could be a reason why mathematicians so far discovered so few of these conformal ﬁeld theories,despite ample physical evidence of their existence[3,49,21,53,38, 59,60].Originally prompted by a question of Igor Frenkel,we also consider the case of the Fermat quartic K3surfacex4+y4+z4+t4=0in C P3.This is done in Section7.It is interesting that the problems of the Fermat quintic do not arise in this case,and all the inﬁnitesimally criticalﬁelds exponentiate in the purely perturbative sense.This dovetails with the result of Alvarez-Gaume and Ginsparg[5]that theβ-function vanishes to all orders for critical perturbative models with N=(4,4)supersymmetry,and hence from the renormalization point of view,the non-linearσmodel is conformal for the Ricciﬂat metric on K3-surfaces. There are also certain diﬀerences between the ways mathematical considerations of moduli space and mirror symmetry vary in the K3and Calabi-Yau3-fold cases, which could be related to the behavior of the non-perturbative eﬀects.This will be discussed in Section8.To relate more precisely in what setup these results occur,we need to describe what kind of deformations we are considering.It is well known that one can obtain inﬁnitesimal deformations from primaryﬁelds.In the bosonic case,the weight of theseﬁelds must be(1,1),in the N=1-supersymmetric case in the NS-NS sector the critical weight is(1/2,1/2)and in the N=2-supersymmetric case the inﬁnitesimal deformations we consider are along so called ac or ccﬁelds of weight (1/2,1/2).For more speciﬁc discussion,see section2below.There may exist inﬁnitesimal deformations which are not related to primaryﬁelds(see the remarksPERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED5 at the end of Section3).However,they are excluded under a certain continuity assumption which we also state in section2.Therefore,the approach we follow is exponentiating inﬁnitesimal deformations along primaryﬁelds of appropriate weights.In the“algebraic”approach,we as-sume that both the primaryﬁeld and amplitudes can be updated at all points of the deformation parameter.Additionally,we assume one can obtain a perturba-tive power series expansion in the deformation parameter,and we do not allow counterterms of generalized weight or non-perturbative corrections.We describe a cohomological obstruction theory similar to Gerstenhaber’s theory[24]for associa-tive algebras,which in principle controls the coeﬃcients at individual powers of the deformation parameter.Obstructions can be written down explicitly under certain conditions.This is done in section3.The primary obstruction in fact is the one which occurs for the deformations of the freeﬁeld theory at gravitationalﬁelds of non-zero momentum(“gravitational waves”).In the case of the Gepner model of the Fermat quintic,the primary obstruction vanishes but in the case(2),one can show there is an algebraic obstruction of order5(i.e given by a7point function in the Gepner model).It should be pointed out that even in the“algebraic”case,there are substan-tial complications we must deal with.The moduli space of CFT’s is not yet well deﬁned.There are diﬀerent deﬁnitions of conformalﬁeld theory,for example the Segal approach[52,31,32]is quite substantially diﬀerent from the vertex operator approach(see[36]and references therein).Since these deﬁnitions are not known to be equivalent,and their realizations are supposed to be points of the moduli space,the space itself therefore cannot be deﬁned until a particular deﬁnition is selected.Next,it remains to be speciﬁed what structure there should be on the moduli space.Presumably,there should at least be a topology,so than we need to ask what is a nearby conformalﬁeld theory.That,too,has not been answered.These foundational questions are enormously diﬃcult,mostly from the philo-sophical point of view:it is very easy to deﬁne ad hoc notions which immediately turn out insuﬃciently general to be desirable.Because of that,we only make min-imal deﬁnitions needed to examine the existing paradigm in the context outlined. Let us,then,conﬁne ourselves to observing that even in the perturbative case,the situation is not purely algebraic,and rather involves inﬁnite sums which need to be discussed in terms of analysis.For example,the obstructions may in fact be undeﬁned,because they may involve inﬁnite sums which do not converge.Such phenomenon must be treated carefully,since it doesn’t mean automatically that perturbative exponentiation fails.In fact,because the deformed primaryﬁelds are only determined up to a scalar factor,there is a possibility of regularization along the deformation parameter.We brieﬂy discuss this theoretically in section3,and then give an example in the case of the freeﬁeld theory in section4.We also brieﬂy discuss suﬃcient conditions for exponentiation.The main method we use is the case when Theorem1gives a truly local formula for the inﬁnitesimal amplitude changes,which could be interpreted as an“inﬁnitesimal isomorphism”in a special case.We then give in section3conditions under which such inﬁnitesimal isomorphisms can be exponentiated.This includes the case of a coset theory,which6IGOR KRIZdoesn’t require regularization,and a more general case when regularization may occur.In the ﬁnal sections 5,6,namely the case of the Gepner model,the main prob-lem is ﬁnding a setup for the vertex operators which would be explicit enough to allow evaluating the obstructions in question;the positive result is obtained using a generalisation of the coset construction.The formulas required are obtained from the Coulomb gas approach (=Feigin-Fuchs realization),which is taken from [29].The present paper is organized as follows:In section 2,we give the general setup in which we work,show under which condition we can restrict ourselves to deformations along a primary ﬁeld,and derive the formula for inﬁnitesimally deformed amplitudes,given in Theorem 1.In section 3,we discuss exponentiation theoretically,in terms of obstruction theory,explicit formulas for the primary and higher obstructions,and regularization.We also discuss supersymmetry,and in the end show a mechanism by which non-perturbative deformations may still be possible when algebraic obstructions occur.In section 4,we give the example of the free ﬁeld theories,the trivial deformations which come from 0momentum gravitational deforming ﬁelds,and the primary obstruction to deforming along primary ﬁelds of nonzero momentum.In section 5,we will discuss the Gepner model of the Fermat quintic,and in section 6,we will discuss examples of non-zero algebraic obstructions to perturbative deformations in this case,as well as examples of unobstructed deformations.In Section 7,we will discuss the (unobstructed)deformations for the Fermat quartic K 3surface,and in Section 8,we attempt to summarize and discuss our possible conclusions.Acknowledgements:The author thanks D.Burns,I.Dolgachev,I.Frenkel,Doron Gepner,Y.Z.Huang,I.Melnikov,K.Wendland and E.Witten for explanations and discussions.Special thanks to H.Xing,who contributed many useful ideas to this project before changing his ﬁeld of interest.2.Infinitesimal deformations of conformal field theoriesIn a bosonic (=non-supersymmetric)CFT H ,if we have a primary ﬁeld u of weight (1,1),then,as observed in [52],we can make an inﬁnitesimal deformation of H as follows:For a worldsheet Σwith vacuum U Σ(the worldsheet vacuum is the same thing as the “spacetime”,or string,amplitude),the inﬁnitesimal deformation of the vacuum is (3)V Σ= x ∈ΣU Σx u .Here U Σx u is obtained by choosing a holomorphic embedding f :D →Σ,f (0)=x ,where D is the standard disk.Let Σ′be the worldsheet obtained by cutting f (D )out of Σ,and let U Σx u be obtained by gluing the vacuum U Σ′with the ﬁeld u inserted at f (∂D ).The element U Σx u is proportional to ||f ′(0)||2,since u is (1,1)-primary,so it transforms the same way as a measure and we can deﬁne the integral (3)withoutcoupling with a measure.The integral (3)is an inﬁnitesimal deformation of thePERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED7 original CFT structure in the sense thatUΣ+VΣǫsatisﬁes CFT gluing identities in the ring C[ǫ]/ǫ2.The main topic of this paper is studying(in this and analogous supersymmetric cases)the question as to when the inﬁnitesimal deformation(3)can be exponenti-ated at least to perturbative level,i.e.when there exist for each n∈N elementsu0,...,u n−1∈H,u0=uand for every worldsheetΣU0Σ,...,U nΣ∈ H∗⊗Hsuch that(4)UΣ(m)=mi=0U iΣǫi,U0Σ=UΣsatisfy gluing axioms in C[ǫ]/ǫm+1,0≤m≤n,(5)u(m)=m i=0u iǫiis primary of weight(1,1)with respect to(4),0<m≤n,and (6)dUΣ(m)8IGOR KRIZ Then we calculatedUΣ(m)dǫassuming(6)forΣ=D,so the assumption we need is(7)dUΣdǫ.The composition notation on the right hand side means gluing.Granted(7),wecan recover dUΣ(m)dǫfor the unit disk D.Now in the case of the unitdisk,we get a candidate for u(m−1)in the following way:Assume that H is topologically spanned by subspaces H(m1,m2)ofǫ-weight(m1,m2)where m1,m2≥0,H(0,0)= U D .Then U D(m)is invariant under rigid notation,so(8)U D(m)∈ˆ k≥0H(k,k)[ǫ]/ǫm+1.We see that if A q is the standard annulus with boundary components S1,qS1with standard parametrizations,then(9)u(m−1)=limq→01dǫexists and is equal to the weight(1,1)summand of(8).In fact,by(7)and the deﬁnition of integral,we already see that(6)holds.We don’t know however yetthat u(m−1)is primary.To see that,however,we note that for any annulusA=D−D′where f:D→D′is a holomorphic embedding with derivative r,(7) also implies(for the same reason-the exhaustion principle)that(6)is valid withu(m−1)replaced by(10)U A u(m−1)||r||2=u(m−1)which means that u(m−1)is primary.PERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED9 We shall see however that there are problems with this formulation even in the simplest possible case:Consider the free(bosonic)CFT of dimension1,and the primaryﬁeld x−1˜x−1.(We disregard here the issue that H itself lacks a satisfactoryHilbert space structure,see[32],we could eliminate this problem by compactifying the theory on a torus or by considering the state spaces of given momentum.)Let us calculateU1D= D exp(zL−1)exp((12)2 k≥1x−k˜x−kz)= (v L,v R)u−v L−w L,−v R−w R z v L10IGOR KRIZof operators which act on w are discrete.Even more strongly,we assume that (17)Y(u,z,z)where(18)Y i(u,z)= u i;−v L−w L z v L,˜Yi(u,z v Rwhere all the operators Yi(u,z)commute with all˜Y j(v,z,t,z)Y(v,t,t)Y(u,z,z−t)v,t,z)=∂z),Y(˜L−1u,z,∂z)and(21)V wL,w Ris the weight(w L,w R)subspace of(L0,˜L0).Remark:Even the axioms outlined here are meant for theories which are initial points of the proposed perturbative deformations,they are two restrictive for the theories obtained as a result of the deformations themselves.To capture those deformations,it is best to revert to Segal’s approach,restricting attention to genus 0worldsheets with a unique outbound boundary component(tree level amplitudes). Operators will then be expanded both in the weight grading and in the perturbative parameter(i.e.the coeﬃcient at each power of the deformation parameter will be an element of the product-completed state space of the original theory).To avoid discussion of topology,we simply require that perturbative coeﬃcients of all compositions of such operators converge in the product topology with respect to the weight grading,and the analytic topology in each graded summand.In this section,we discuss inﬁnitesimal perturbations,i.e.the deformed theory is deﬁned over C[ǫ]/(ǫ2)whereǫis the deformation parameter.One case where such inﬁnitesimal deformations can be described explicitly is the followingPERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED11 Theorem1.Considerﬁelds u,v,w∈V where u is primary of weight(1,1).Next, assume thatZ(u,v,z,t)= α,βZα,β(u,v,z,t)whereZα,β(u,v,z,t)= i Zα,β,i(u,v,z,t)˜Zα,β,i(u,v,t)and for w′∈W∨ofﬁnite weight,w′Zα,β,i(u,v,z,t)(z−t)αzβ(resp.w′˜Zα,β,i(u,v,t)zβ)is a meromorphic(resp.antimeromorphic)func-tion of z on C P1,with poles(if any)only at0,t,∞.Now write(22)Y u,α,β(v,t,z,t,z,soY u(v,t,t)+ǫ α,βY u,α,β(v,t,k+1k=−1(thusﬁxing the integration constant),and analogously with the˜’s.Let then(27)Cα,β,i=ωα,β,i,∞−ωα,β,i,t, Dα,β,i=ωα,β,i,∞−ωα,β,i,0,˜Cα,β,i=˜ωα,β,i,∞−˜ωα,β,i,t,˜Dα,β,i=˜ωα,β,i,∞−˜ωα,β,i,012IGOR KRIZ(see the comment in the proof on branching).Letφα,β,i=π n uα,β,i,−n˜uα,β,i,−nt)w= iφ′α,β,i Y(v,t,t)w−Y(v,t,t)w= iφ′α,β,i Y(v,t,t)w−Y(v,t,PERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED13Proof:Let us work on the scaled real worldsheetΣ′.Letηα,β,i=Zα,β,i(u,v,z,t)dz,˜ηα,β,i=˜Zα,β,i(u,v,t)dt)(31) ∂0ωα,β,i,0˜η=−Y(φα,β,i v,t,14IGOR KRIZrespectively.On the other hand,the sum of the last two terms,looking at points x+,x−for each x∈c,can be rewritten as(35) c+Cα,β,i(−e−2πiα+1)˜ηα,β,i= c−Dα,β,i(−e2πiβ+1)˜ηα,β,i.Now recall(27).Choosing˜ωα,β,i,∞as the primitive function of˜ηα,β,i,we see that for the end point x of c−,(36)˜ωα,β,i,∞(x+)−˜ωα,β,i,∞(x−)=˜ωα,β,i,t(x+)−˜ωα,β,i,t(x−)=(e−2πiα−1)˜ωα,β,i,t(x−)=(e−2πiα−1)˜ωα,β,i,∞(x−)+(e−2πiα−1)˜Cα,β,i.Similarly,for the beginning point y of c−,(37)−˜ωα,β,i,∞(y+)+˜ωα,β,i,∞(y−)=−˜ωα,β,i,0(y+)+˜ωα,β,i,0(y−)=−(e2πiβ−1)˜ωα,β,i,0(y−)=−(e2πiβ−1)˜ωα,β,i,∞(y−)−(e2πiβ−1)˜Dα,β,i.Then(36),(37)multiplied by Cα,β,i are the integrals(34),while the integral(35) is(38)−Dα,β,i(1−e2πiβ)˜ωα,β,i,0(y−)+Cα,β,i(1−e−2πiα)˜ωα,β,i,0(x−).Adding this,we getCα,β,i˜Cα,β,i(−1+e−2πiα)+Dα,β,i˜Dα,β,i(1−e2πiβ),as claimed.3.Exponentiation of infinitesimal deformationsLet us now look at primary weight(1,1)ﬁelds u.We would like to investigate whether the inﬁnitesimal deformation of vertex operators(more precisely world-sheet vacua or string amplitudes)along u indeed continues to aﬁnite deformation, or at least to perturbative level,as discussed in the previous section.Looking again at the equation(6),we see that we have in principle a series of obstructions similar to those of Gerstenhaber[24],namely if we denote by(39)L n(m)=mi=0L i nǫi,L0n=L na deformation of the operator L n in Hom(V,V)[ǫ]/ǫm,we must have(40)L n(m)u(m)=0∈V[ǫ]/ǫm+1for n>0(41)L0(m)u(m)=u(m)∈V[ǫ]/ǫm+1.PERTURBATIVE DEFORMATIONS OF CONFORMAL FIELD THEORIES REVISITED 15This can be rewritten as(42)L n u m =−i ≥1L i n u m −i (L 0−1)u m =−i ≥1L i 0um −i .(Analogously for the ˜’s.In the following,we will work on the obstruction for the chiral part,the antichiral part is analogous.)At ﬁrst,these equations seem very overdetermined.Similarly as in the case of Gerstenhaber’s obstruction theory,however,of course the obstructions are of cohomological nature.If we denote by A the Lie algebra L 0−1,L 1,L 2,... ,then the system(43)L n (m )u (m −1)(L 0(m )−1)u (m −1)is divisible by ǫm in V [ǫ]/ǫm +1,and is obviously a coboundary,hence a cocycle with respect to L 0(m )−1,L 1(m ),... .Hence,dividing by ǫm ,we get a 1-cocycle of A .Solving (42)means expressing this A -cocycle as a coboundary.In the absence of ghosts (=elements of negative weights),there is another sim-pliﬁcation we may take advantage of.Suppose we have a 1-cocycle c =(x 0,x 1,...)of A .(In our applications,we will be interested in the case when the x i ’s are given by (42).)Then we have the equationsL ′k x j −L ′j x k =(k −j )x j +k ,where L ′k =L k for k >0,L ′0=L 0−1.In particular,L ′k x 0−L ′0x k =kx k ,or(44)L k x 0=(L 0+k −1)x k for k >0.In the absence of ghosts,(44)means that for k ≥1,x k is determined by x 0with the exception of the weight 0summand (x 1)0of x 1.Additionally,if we denote the weight k summand of y in general by y k ,then(45)c =dymeans(46)(x 0)k =(k −1)y,(47)(x 0)1=0.The rest of the equation (45)then follows from (44),with the exception of the weight 0summand of x 1.We must,then,have(48)(x 1)0∈ImL 1.Conditions (47),(48),for x k =− i ≥1L i k um −i ,are the conditions for solving (42),i.e.the actual obstruction.。

Effective Interactions for the Three-Body Problem

arXiv:nucl-th/0404028v1 9 Apr 2004
Techniques in popular use in the nuclear three-body problem include the Faddeev[1], Green’s Function Monte Carlo (GFMC), and correlated hyper-spherical harmonics expansion methods (see, for instance, refs.[2, 3, 4]). These methods have been used with realistic phenomenological potentials (e.g. Av18[5],, CD-Bonn[6], etc. . .) with strong short-range interactions, yielding binding energies accurate to within 0.1%[7]. All of these methods treat the bare N N interactions in Hilbert spaces that are eﬀectively inﬁnite. Because the complexity of the Hilbert spaces grows very rapidly with nucleon number A, the extension of such methods to heavier systems becomes increasingly diﬃcult. This motivates another approach: Solving A-body problems in more tractable ﬁnite “included spaces,” while accounting for the missing physics of the “excluded space” through an eﬀective interaction. For shell-model-inspired eﬀective theories that deﬁne included spaces in terms of harmonic oscillator (HO) bases, the missing physics includes both high-momentum and long-wavelength interactions. The former are missing because the included space contains only low-energy oscillator shells, while the latter are connected with the ﬁnite extent of the basis states, which are overconﬁned in an HO potential. Shell-model calculations often employ two-body eﬀective interactions, but these are generally determined phe-

【doc】高能所2002年发表的专著及在学术刊物上公开发表的论文

高能所2002年发表的专著及在学术刊物上公开发表的论文附录3.发表论文目录141附录3.高能所2002年发表的专着及在学术刊物上公开发表的论文作者(以原序发表干U物名称序号专着或论文题目单位排列)卷号(年)页lFirstMeasurementofthe实验物理J.Z.Baieta1.Phys.Rev. BranchingFractionoftheDecay中心BESCollab.D65(2002)052004v/(2s,Z-+一2MeasurementoftheCrOSSSection实验物理J.Z.Baieta1.Phys.Rev.Lett.f0ree一hadronsat中心BESCollab.88(2002)101802Center-of-massEnergiesfrom2to5GeV3AMeasurementof(2)实验物理J.Z.Baicta1.Phys.Lett.中心BESCollab.B550(2002)24?32ResonanceParamcters4/.3(z7/"一)分支比实验物理BES合作组高能物理与核物理中心26(2002)8?16的测定5Decaysofthe/to∑..实验物理BES合作组高能物理与核物理FinalState中心26(2002)93?996中性和带电D介子单举半轻子实验物理BES合作组高能物理与核物理(电子)衰变分支比的测量中心26(2002)547?5567D和D.介子的遍举实验物理BES合作组高能物理与核物理分支比上限的测定中心26(2002)1093?11028北京谱仪II中性径迹测量误差的实验物理王君等高能物理与核物理确定中心26(2002)116-1219用联合D.和D单双标记测定实验物理荣刚等高能物理与核物理分支比的方法中心26(2002)207-215l0TEX0N0低能中微子实验中的实验物理李金等高能物理与核物理CsI(T1)晶体探测器中心26(2002)393-401l1TELESIS在3y衰变末态实验物理许国发等高能物理与核物理分析中的应用中心26(2002)462?470l2TEX0N0中微子实验屏蔽效果实验物理陈栋梁等高能物理与核物理的MonteCarlo研究中心26(2002)626?631l3在BEPCII/BESIII上寻找r/和实验物理李刚等高能物理与核物理中心26(2002)645-651态的MonteCar1o研究l4EnergyCalibrationofCsI(TI)实验物理岳骞等高能物理与核物理CrystalforQuenchingFactor中心26(2002)728?734 MeasurementinDarkMatterSearchl5最优化的北京谱仪取数时间实验物理苑长征等高能物理与核物理中心26(2002)759?765l42中国科学院高能物理研究所2002年《年报》豁幸●}案}謦謦沓尊l_专蓝l霉1'奄譬,鼍善童警{孳鼍上{点《l,作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页l6MeasurementofQuenchingFactor实验物理岳骞等高能物理与核物理forNuclearRecoilsinCsI(TI)中心26(2002)855—860Crystall7北京谱仪(BESII)的飞行时间计实验物理彭海平等高能物理与核物理数器(1_0F)蒙特卡罗模拟的改进中心26(2002)86l一869l8TEXONO反应堆中微子能谱的实验物理陈栋梁,李金等高能物理与核物理计算中心26(2002)889—894l9R值测量中束流相关本底的扣除实验物理鄢文标等高能物理与核物理方法中心26(2002)998—100320北京谱仪(BESII)的飞行时间计实验物理彭海平等高能物理与核物理数器(TOF)时间和分辨律的修中心26(2002)1078-1086正2lt粲能区物理及对加速器和探测实验物理苑长征等高能物理与核物理器设计的要求中心26(2002)1201—120822BES--III主漂移室输出信号的模实验物理王铮等高能物理与核物理拟中心26(2002)1297—130123高速串行数据通信VME插件的实验物理章平等核电子学与探测技术研制中心22(2002)44—4624BESII快速数据重建实验物理荣刚等核电子学与探测技术中心22(2002)105—11025数字式随机脉冲产生器实验物理富洪玉等核电子学与探测技术中心22(2002)162—16526微阴极条感应室实验物理李金等核电子学与探测技术中心22(2002)193—19927BESIII系统环境的网络监测模型实验物理宋立温等核电子学与探测技术中心22(2002)272—27528CMS阴极条室的张力和丝距测实验物理姜春华等核电子学与探测技术量中心22(2002)335—33729北京谱仪PC机物理分析平台介实验物理刘天容等核电子学与探测技术绍中心22(2002)367—37030新型智能CAMAC机箱控制器实验物理张永吉,赵京伟核电子学与探测技术中心等22(2002)382—3843l基于DSP的多通道波形取样电实验物理王铮核电子学与探测技术路设计中心22(2002)409—41132~(3770)扫描实验中利用快速实验物理郭义庆等核电子学与探测技术重建数据测量数据样本的积分亮中心22(2002)420—423度33基于WEB的BES数据存储系统实验物理叶梅等核电子学与探测技术模型中心22(2002)449—45234StudiesofPrototypeCsI(TI)实验物理Y.Liu(刘延)etNIMA482(2002)125 CrystalScin—tillatorsforLow中心a1.EnergyNeutrinoExperiments鬟一附录3.发表论文目录143作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页35NuclearRecoilMeasurementsin实验物理M.Z.Wang,Phys.Lett.CsI(TbCrystalforColdDark中,J.Li(李金)eta1.B536(2002)203Mat'terDetection36ProbingNeutrinoOscillation实验物理Y.F.Wang(王Phys.ReD65(2002)0730 jointlyinLongandveryLong中心贻芳)etal2lBaselineExperiments37OntheOptimumLongBaseline实验物理Y.F.Wang(王Phys.Rev.D65(2002)0730 f0rtheNextGenerationNeutrino中心贻芳)etaI.06OscillationExperiments38StudyonthePropertyoft天体物理G.M.ChenProceedingsofthethird HadronicDecays中心jointmeetingofChinesephyscistsworldwide,P.185,worldscientific200239Bs.dintechnicolormodel天体物理ZhaohuaPhys.Lett.B546(200)withscalars中心Xiong,JinMin22l-227Y ang40Loopeffectsandnondecoupling天体物理ZhaohuaXiongPhys.Rev.D66, propertyofsupersymmetricQCD中心et.a1.015007(2002)ingb--->tH.4lGreatScintillatingPropertiesofa天体物理GouQuanBuChin.Phys.Lett.V o1.19,Y ALO3:Cecrystal中心et.a1.No.7(2002)92942GreatScintillationgPropertiesofa天体物理苟全朴,李祖物理7期929.930Y A103:CeCrystal中心毫,吕雨生等43非重子暗物质粒子的研究进展天体物理盛祥东,何会物理9期31卷中心林.戴长江577-58044CirX.1的时变性质天体物理屈进禄等河北师大26,1,10中心月45致密星的X射线辐射时延现象天体物理屈进禄,宋黎天文学进展刊物20卷, 中心明,吴枚等第2期143.15546两类长Y暴的里叶功率谱天体物理申荣锋,宋黎明天文43卷第4期中心342-34547L3宇宙线实验触发系统和触发天体物理李忠朝,郁忠高能物理和核物理26. 效率的测量中心强,过雅南等172.17948～掏小型数据获取系统及天体物理何会林,戴长核电子学与探测技术CaF2(Eu)性能的测量中心江,盛祥东22卷第l期27.3049新型"CaF2(Eu)+液闪"复合天体物理盛祥东,戴长高能物理和核物理26 WIMP探测器的实验研究中心江,何会林卷3期273.27850Chandraobservationofsupernova天体物理E.J.Lu.TheAstrophysics568; remnantG54.1+0-3:Aclosecousin中心Q.D.Wang,L49.L52 oftheCRABnebulaL.M.Song,●,●,,,,f,},,};●,,,,,.}●I,I:}::I1,!1,tt●}t{,t,{.^I1.,'l_4l44中国科学院高能物理研究所2002年《年报》作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页5lBlobejectionfrom天体物理Jian?Min,wangTheApjsadvection.dominatedaccretion中心dengl38:249.263flow.H.themultiwavelengthpropertiesoflightcurves52Comptelobservationsofthe天体物理S.ZhangdengAstronomy386,843?853 gamma-rayblazarpks1622?297中心53KilohertzQuasi—periodic天体物理Wenfei.YuTheAstrophysical osciliationfrequency中心deng567;L67-L70anticorrelatedwithmilihertzquasi—periodicoscillationfluxin4u1608.5254TheAccretionratesandspectral天体物理Jian?MinWangTheApjs energydistributionsofbllacetae中心dengobjects55Bs,d+?inTechnicolor天体物理ZhaohuaPhysicslettersb546 modelwithSCalarS中心Xiong,JinMin221.227Jang56Discoveryofa136millisecond天体物理F.J.LudengAstrophysics574:7卜74 radioandX?raypulsarin中心supernovaremnantG54.1+0.357Chandraobsevationofsupemova天体物理FJ.Lu.AstrophysicalremnantG54.1+0.3:aclosecousin中心L..M..song568;L-49一L52 ofthecrabnebula58低轨道空间闪烁探测器异常区开天体物理张承模,梁晓核电子学及探测技术关机设计及控制中心华,徐玉朋等第22卷第6期59Furtheranalysisofdi?gluonfusion理论室DongshengDu,HighEner.Phys.&Nuc1. mechanismforthedecaysofDeshanY angPhys.V.26,1(2002)B—T1'KandGuohuaiZhu60Phenomenologicalanalysisof理论室DongshengDu,Phys.Rev.D65,074001B--,PPdecayswithQCDHaijunGong,(2002)factorizationJunfengSunetal_6lPhenomenologicalanalysisof理论室DongshengDu,Phys.Rev.D65,094025 charmlessdecaysB—}PVwithHaUunGong,(2002),Erratum,ibid. QCDfactorizationJun~ngSunetD66,079904(2002)a1.62D—}7mdecayswithfinalstate理论室MedinaPhys.Lett.B536,34 interactionsAblikim,(2002)DongshengDu,MaozhiYang63CPasymmetryinx-sleptondecay理论室WeiminYang,Phys,Rev.D65,1I5005 intheminimalSUSYstandardDongshengDu(2002)model哮}謦j-1謦1.唾}..专簪■奄枣t鼍t.t蹙垂《l《'譬jj鼍t毒'1.●il-耄羹囊萋:附录3.发表论文目录145EIIEllI重IIIE|IlIEE宦E量III￡EIIlIIIlIIIIIlEEIIIIEE譬EElEIElIIE氍EE群匿臀鐾鞋越匪匪暖壁E_曩EI-iiiI-重匿匪髭|曲B量E匡__譬I|l苍I重IIIIIIIII量lIEE,』作者(以原序发表干U物名称序号专着或论文题目单位排列)卷号(年)页64IsthetruncatedSU)理论室QishuY ah,Phys.Rev.D65,105009non-abeliangaugetheoryinextraDongshengDu(2002)dimensionsrenormalizable?65Branefluctuationandelectroweak理论室QishuY ah,Phys.Rev.D65,094034 chirallagrangianDongshengDu(2002)66Thesystematicstudyof理论室Zheng-TaoNuc1.Phys.BB7/"formfactorsinpQCDWei.Mao-Zhi642(2002)263 approachanditsreliabilityY ang67Calculationofpureannihilation理论室C.D.LuEuro.Phys.J.C24, typedecayB-yDsl2I-126(2002)68B一ⅡP.Ⅱ(I)Decaysin理论室C.D.LU.M.-Z.Euro.Phys.J.C23, PerturbativeQCDApproachY ang275-287(2002)69Constraintontheanglealphafrom理论室C.D.Ltt.z.-j.Phys.Rev.D66, theexperimentalmeasurementsofXiao074011(2002) directCPviolationofB0pipi—decay70Determiningthefactorization理论室Zhi-zhongXingHEP&NP26 parameterandstrongphase(2002)100differencesinB—Dzdecays7lFakeCPTviolationin理论室Zhi-zhongXingJ.Phys.G28 disappearanceneutrino(2002)7oscillations72TexturezerosandMajorana理论室Zhi-zhongXingPhys.Lett.B phasesoftheneutrinomassmatrix530(2002)15973Possibleimplicationsofsmallor理论室Zhi-zhongXingHEP&NP26 largeCPviolationinB—jKs(2002)197decays74RephasinginvariantsofCPandT理论室W.L.GUO.Phys.Rev.D violationinthefour-neutrinoZhi-zhongXing65(2002)073020 mixingmodels75Model-independentconstraintson理论室Zhi-zhongXingPhys.Rev.D65 theneutrinomassspectrumfrom(2002)077302theneutrinolessdoublebetadecay76Nearlytri-bimaximalneutrino理论室Zhi-zhongXingPhys.Lett.B mixingandCPviolation533(2002)8577Cantheleptonflavormixing理论室Zhi—zhongXingPhys.Rev.D matrixbesymmetric?65(2002)113010l46中国科学院高能物理研究所2002年《年报》作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页78AfuIldeterminationofthe理论室Zhi-zhongXingPhys.LeR.B neutrinomassspectrumfrom539(2002)85two.zerotexturesoftheneutrinomassmatrix79Possibleeffectsof理论室Z.Chang,Phys.Rev.D66 noncommutativegeometryonZhi-zhongXing(2002)056009 weakCPviolationandunitaritytriangles80Apredictiveansatzforneutrino理论室Zhi-zhongXingPhys.Lett.B mixingandleptogenesis545(2002)3528lNoncommutativityofleptonmass理论室Zhi-zhongXingFortsch.Phys. matrices:flavormixingandCP50(2002)569violation82Hierarchicalneutrinomassesand理论室Zhi-zhongXingPhys.LeR.B largemixinganglesfromthe550(2002)l78 Fritzschtextureofleptonmassmatrices83Unitarityquadranglesoffour理论室W.L.Guo.Phys.Rev.D66neu~inomixingZhi-zhongXing(2002)09730284Poynt!ngVector,EnergyDensity理论室C.-G.Huang,'Phys.Rev.A65 andEnergyV elocityinAnomalousY.Z.Zhang(2002),015802DispersionMedium85ThermodynamicsofdeSitter理论室C.GHuang.Phys.Rev.DuniverseL.Liu,Bobo65(2002),.08350lWang86Negativegroupvelocityand理论室Chao-GuangJ.Optics.A: distortionofpulseinanomalousHuang,PureApp1.Optics.dispersivemediumYuan-ZhongZhang4(2002),263.87Thermodynamicsof理论室Bo-boWang,Class.Quant.Reissner-NordstromblackholesinChao-GuangGray.19(2002),2491.Y ork'SFormalismHuang88Propagationofrectangularpulse理论室Chao-GuangCommun.Theor.Phys.38 inananomalousmediumHuang,(2002),224.Yuan-ZhongZhang89V acuumBlackHoleMass理论室mun.Theor.Phys.38 FormulaisaV anishingNoetherGUO,C.G.(2002),309ChargeHuang,K.Wu90topquarkspincorrelationatlinear理论室Z.H.Lin,Phys.Rev.D65. colliderswithanomalousHan,Huang,0l4008(2002)couplingsJ.X.Wang.X.Zhang,}◆F辱}tF孽.,..'薹髻茎董;耋一附录3.发表论文目录147作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页9lStabilityoftheClassicalSolution理论室YinYu-dongHEP&NP,V26 ofGaugeTheoryHuangTao(2002)28-3492S~ongCouplingsofHeavy理论室Z.H.Li.Phys.Rev.D65 MesonstOaLightV ectorMesonsT.Huang,(2002)076005inQCDJ.Z.Sun.Z.H.Dai93MuonAnomalousMagneticinthe理论室T.Efeng,Commun.111eor. SupersymmetricModelswithandT.Huang,Phys.37(2002)179withoutRi【ght-handedNeutrinosX.Q.LietaI.94SIGMAMESON【NJ/PSI理论室WujunHuo,Phys.Rev.D65 DECAYSXinminZhang,(2002)097505TaoHuang95ImpactoftheSoft-Gluon理论室X.YWu.CPL19(2002)1596 ExchangesonB一>pipiDecaysZ.H.Li,J.Y.Cui,T.Huang96Re-examinedRadiativeDecay理论室Zuo-HongLi,JPhysicsG inLightConeQCDXiang-Y aoWu,28(2002)2583SumRuIeTaoHuang97NewapproachtOconstructingthe理论室Da-QingLiu,HighEnergyPhys.and opermoronthela~icefortheJi-MinWu.Nuc1.Phys.26(2002)222 calculationofglueballmassesYingChen98StudyonthebaryonstateX理论室ShenQixing,Commun.Theor.Phys., producedintheprocessYuHong,Li37(2002)pp189-192J/gP+X.P+PDemin99Onthemassrelationofameson理论室De-MinLi,ModemPhys.Lett., nonetHongYu.17(2002)163-169Qi-XingShenl00Themassrelationbetweenthe理论室De-MinLi,ModemPhys.Lett., pureandphysicalgroundHongYu,17(2002)619-626 pseudoscalarglueballsQi-XingShenl0lNumericalcalculationofenergies理论室BinDUan,Eur,Phys.J.D ofsomeexcitedstatesinaXiao-Y ahGu,19(2002)9-12heliumatomZhong-QiMalO2Non.relativisticLevinson's理论室Shi—HaiDong,Phys.Rev.A65(2002) theoreminSD$dimensionsZhong-QiMa042717(1-6)l03Thehiddensymmetryfora理论室Shi-HaiDong,Am.J.Phys.70 quantumsystemwithaninfinitelyZhong-QiMa(2002)520-52I deepsquare?wellpotentiall04The(2+1)Diracequationwiththe理论室Shi-HaiDong,Found.Phys.Lett. deltapotentialZhong-QiMa15(2002)171-178●●●{'j●●f},,,!l{;.'+.}_.,},●.,●t,t}々jl48中国科学院高能物理研究所2002年《年报》蝣1簟毒Ili手睾每叁鸯夸誊;;:'一作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页l05Quantumthree-bodysysteminD理论室Xiao-Y anGu,J.Math.Phys.43(2002) dimensionsBinDuan,2895-2906Zhong-QiMal06Algebraicapproachtothe理论室Shi-HaiDong,Inter.J.Mod,Phys.E11 pseudoharmonicoscillatorin2DZhong-QiMa(2002)155-160l07ExactsolutionstotheDirac理论室Xiao-Y anGu,Inter,J.Mod.Phys.E11, equationforaCoulombpotentialZhong-QiMa,(2002)335-346inD+ldimensionsShi-HaiDongl08ExactsolutionstoD-dimensional理论室Lu-YaWang,Found.Phys.Lett. SchrodingerequationwithaXiao-Y anGu,15(2002)569-576 pseudoharmonicoscillatorZhong-QiMaeta1.l09ExactsolutionsoftheDirac理论室Shi-HaiDong,Inter,J.Mod.Phys. equationwithXiao-Y anGu,E11(2002)483-489 aCoulombplusscalarpotentialinZhong-QiMa2+ldimensionseta1.l10群论习题精解理论室马中骐着科学出版社(北京)2002年出版l1lStudyofthestructureof理论室R.G.Ping,Phys.Rev.D66,baryonsfromJ/—÷BBbardecaysH.C,Chiang,054020(2002) intheciuarkmodelB.S.ZOUl12TheoreticalformalismandMonte理论室W.H.Liang,J.Phys.G28,333(2002) CarlostudyofpartialwavePN.Shen.analysisforfPP∞J.X.Wang,etal,l13Partondistributionofproton理论室Y.J.Zhang,Phys.Lett.B528, inasimplestatisticalmodelB.S.Zou,228(2002)L,M.Y angl14Commentonprotonium理论室C.Amslereta1.,Phys.Rev.D66:058101, annihilationintopi0pi0atrestinB.S.ZOU2002aliquidHydrogentargetl15i=0.C=-1mesonsfrom理论室A.AnisovichetPhys,Lett.B542:19-28,l940-mevto24l0-meva1..B.S.Zou2002l16Combinedanalysisofmeson理论室A.mnisovichPhys.Lett.B542:8-18, channelswithi=1.C=-1frometa1..B.S.Zou2002l940-mevto2410一mevl17High-EnergyProton-Induced理论室C.GDuan.Chin,Phys.Lett. DimuonProductionfromNucleiH.M.Wang.19(2002)485GL.Lil18~(400-1200)物理本质的探讨理论室李龙,邹冰松,高能物理与核物理厉光烈26(2002)346l19p-ADrell-Y an过程中的核效应理论室乔秀梅,段春高能物理与核物理贵,厉光烈26(2002)1036.附录3.发表论文目录149作者(以原序发表刊物名称序号专着或论文题目单位排列)卷号(年)页l20AGB星演化过程中的核合成理论室厉光烈,张妙静原子核物理评论和张波19(2002)301l2lATwoStepMechanismfor理论室YWYu.PHighEnergyPhys.and FormingDibaryonWang.Z.YNuc1.Phys.26(2002)46(f2f2):O+Zhangeta1.l22Multi-s~angeHypernucleiina理论室P.Wang.HNuc1.Phys.ChiralSU(3)QuarkMeanFieldGuO.Z.YA705(2o02)455 ModelZhangeta1.l23ChiralSymmetryRestorationof理论室EWang,H.J.Phys.G28(2002)2265 NuclearMatterGuO.YB.Dong.Z.YZhangYu124AnAnalysisofQuarkEffectin理论室Z.YZhang.HighEnergyPlays.and TwoBaryonSystemYuNuc1.Phys.26(2002)712125Di..mega(f2f2):0+理论室YWYu.EPhys.Rev.Wang.Z.YC66(2002)015205PriductionCrossSectionZhangeta1.Calculationl26D.NBoundStateinSU(3)Quark理论室YuZ.HighEnergyPhys.and ModelZhang,Q.B.LiNuc1.Phys.26(2002)1041l27Baryon+Baryon理论室YuCommen.Theo~Phys.cross(QQ):0+37(2002)443SectionCalculation128f)f20+DibaryonProducti.ns理论室mun.Theor.Phys.Wang.YWYu38(2002)483inCentralAu+AuCollisionsatRHICEnergy=130GeV129Dynamicalsymmetrybreaking,理论室Z.GWang.S.L.Phys.Lett. confinementwithflat-bottomWanB536(2002)24lPotentiall30Calculationofsomepropertiesof理论室Z.GWangJ.Phys.GNuc1.Part. thevaccuumPhys.28(2002)1l3lMassiveQuarkPropagatorinthe理论室M.Huang,ChinesePhys.Lett.COlO卜supercOnductingPhaseEEZhuang,19(2002)644W.Q.Chaol32AnalyticSolutionofGroundState理论室ZhaoWeiqin,HEP&NP26(2002)299 forCoulombPlusLinearPotentialJuChangsheng,.,口::i-,■■,;,■r■!:,l::;.tt;I:l50中国科学院高能物理研究所2002年《年报》作者(以原序发表千IJ物名称序号专着或论文题目单位排列)卷号(年)页l33MesonEff~ctsonChiral理论室HuangMei,Commun.Theor.Phys. CondensateatFiniteDensityZhuang38(2002)181Peng~i,ChaoWeiqinl34V ariationa1.IterativeSolutionof理论室ZhaoWeiqinCommun.Theor.Phys. GroundStateforCentralPotential38(2002)15l35DifferentV ersionsofPerturbative理论室ZhaoWeiqin,Commun.TheoEPhys. ExpansionBasedontheJuChangsheng38(2002)271Sin?。

山东大学博士开题报告.pdf

Hard, Infrared safe, calculable Universal, extremely difficult (if not impossible) to calculate
Short-distance and long-distance are separately gauge invariant.
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Factorization
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Motivation
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Factorization
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H(n)→H(n-1)
gauge link (A+)
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Hadronic Tensor = Hard × Gauge Invariant Matrix (4×4)
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a r X i v :c o n d -m a t /9812094v 1 7 D e c 1998Perturbative and non-perturbative parts of eigenstates and local spectral density ofstates:the Wigner band random matrix modelWen-ge WangDepartment of Physics,South-east University,Nanjing 210096,ChinaInternational Center for the Study of Dynamical Systems,University of Milan at Como,via Lucini 3,22100Como,ItalyDepartment of Physics and Centre for Nonlinear Studies,Hong Kong Baptist University,Hong Kong,China(February 1,2008)A generalization of Brillouin-Wigner perturbation theory is applied numerically to the Wigner Band Random Matrix model.The perturbation theory tells that a perturbed energy eigenstate can be divided into a perturbative part and a non-perturbative part with the perturbative part expressed as a perturbation expansion.Numerically it is found that such a division is important in understanding many properties of both eigenstates and the so-called local spectral density of states (LDOS).For the average shape of eigenstates,its central part is found to be composed of its non-perturbative part and a region of its perturbative part,which is close to the non-perturbative part.A relationship between the average shape of eigenstates and that of LDOS can be explained.Numerical results also show that the transition for the average shape of LDOS from the Breit-Wigner form to the semicircle form is related to a qualitative change in some properties of the perturbation expansion of the perturbative parts of eigenstates.The transition for the half-width of the LDOS from quadratic dependence to linear dependence on the perturbation strength is accompanied by a transition of a similar form for the average size of the non-perturbative parts of eigenstates.For both transitions,the same critical perturbation strength λb has been found to play important roles.When perturbation strength is larger than λb ,the average shape of LDOS obeys an approximate scaling law.PACS number 05.45.+bI.INTRODUCTIONThe average shape of energy eigenfunctions (EFs),which characterizes the spreading of the eigenstates of a perturbed system over the eigenstates of an unperturbed system,is very important in a wide range of physical ﬁelds,from nuclear physics and atomic physics to con-densed matter physics.For example,in a recent approach for the description of isolated Fermi systems with ﬁnite number of particles,a type of “microcanonical”partition function has been introduced and expressed in terms of the average shape of eigenfunctions [1].However,up to now only some of the general features of the shape have been known clearly.For example,when perturbation is not strong,generally the shape has a Breit-Wigner form (Lorentzian distribution)[2–5],but in a larger range on the interaction strength,without any simple analytical expression known for the shape,only some phenomeno-logical expressions have been suggested [6–11].Another quantity,the so-called local spectral density of states (LDOS),has also attracted lots of attention recently [6,7,12–16].This quantity,known in nuclear physics as the “strength function”,gives information about the “decay”of a speciﬁc unperturbed state into other states due to interaction.In particular,the width of the LDOS deﬁnes the eﬀective “lifetime”of the unper-turbed state.For suﬃciently weak coupling,the average shape of LDOS is usually close to the Breit-Wigner form with the half-width having a quadratic dependence onthe interaction strength.On the other hand,when inter-action is strong,the shape is model dependent and the half-width is linear in the interaction strength.Numerically it is already known that for Hamiltonian matrices with band structure generally both the aver-age shape of EFs and that of LDOS can be divided into two parts:central parts and tails with exponen-tial (or faster)decay.However,an analytical deﬁnition for such a division has not been achieved yet.A possi-ble clue for this problem comes from a generalization of the so-called Brillouin-Wigner perturbation theory [17]introduced in Ref.[18]for studying long tails of EFs,which tells that analytically EFs can be divided into per-turbative and non-perturbative parts with perturbative parts being able to be expanded in a perturbation expan-sion by making use of the corresponding non-perturbative parts.The relationship between central parts and non-perturbative parts of EFs was not studied in Ref.[18],while it is more important.The purpose of this paper is to study if the above mentioned generalization of Brillouin-Wigner perturba-tion theory (GBWPT)can provide deeper understanding for properties of EFs and LDOS,especially the relation-ship between the two divisions mentioned above for EFs.As a ﬁrst step,here we study the so-called Wigner band random matrix (WBRM)model,which is one of the sim-plest models for the application of the GBWPT.The WBRM was ﬁrst introduced by Wigner 40years ago [19]for the description of complex quantum sys-tems as nuclei.It is currently under close investigation [6,12,13,20–25]since it is believed to provide an adequate description also for some other complex systems(atoms, clusters,etc.)and as well as for dynamical conservative systems with few degrees of freedom,which are chaotic in the classical limit.Unlike the standard Band Ran-dom Matrices(see,for example,[26,27]),the theory of WBRM is not well developed.Numerically it is known that diﬀerent averaging procedures for EFs give diﬀerent results for their average shape.Speciﬁcally,when aver-aging is done with respect to centers of energy shell,for strong interaction central parts of averaged EFs are close to a form predicted for the LDOS by the semicircle law. On the other hand,for the case of averaging with respect to centroids of EFs,central parts of averaged EFs are obviously narrower than the corresponding LDOS when the so-called Wigner parameter is large and an ergodicity parameter is small[13].Comparatively,more properties are known for the av-erage shape of the LDOS of the WBRM.For the tails,a corrected analytical expression has been derived in Ref.[6]for weak perturbation and numerically found correct for even strong perturbation[6,11,12].A more general analytical approach for the LDOS of the WBRM is given in Ref.[12],where an expression for the LDOS is given in terms of a function satisfying an integral equation.In suf-ﬁciently weak and extremely strong perturbation cases, the expression gives the well known Breit-Wigner form and semicircle law,respectively.For the transition from the Breit-Wigner form to the semicircle form,although the expression also predicts correct results for the LDOS, the physical explanation for it is not so clear yet.There-fore,it is of interest to study if the GBWPT can throw new light on the above mentioned properties of the EFs and the LDOS of the WBRM.This paper has the following structure.For the sake of completeness,in section II we give a brief presentation of the GBWPT.Some predictions for properties of EFs of the WBRM are also given in the section.Numerical results for the size of non-perturbative parts of EFs are discussed in section III.It is shown that for the average shape of EFs the averaged non-perturbative part is in-deed related to the central part.A region predicted in section II,which is between the non-perturbative part and the(exponentially or faster decaying)tails of an EF, is also studied numerically.Section IV is devoted to a dis-cussion for the relation between the average shape of EFs and that of the LDOS.In section V,some properties of the average shape of LDOS,such as the transition from the Breit-Wigner form to the semicircle form and the dependence of its half-width on perturbation strength, are studied numerically and found to be closely related to properties of the average size of the non-perturbative parts of the corresponding EFs.Finally,conclusions are given in section VI.II.GENERALIZATION OF BRILLOUIN-WIGNER PERTURBATION THEORYFor the sake of completeness,in this section weﬁrst give a brief presentation of the above mentioned gener-alization of Brillouin-Wigner perturbation theory(GB-WPT)[18].Then,we give a brief discussion for some properties of the EFs of the WBRM.Generally,consider a Hamiltonian of the formH=H0+λV(1) where H0is an unperturbed Hamiltonian,V is a pertur-bation and the parameterλis for adjusting the strength of the perturbation.For the WBRM with dimension N and band width b,the Hamiltonian matrix considered here is chosen of the formH ij=(H0+λV)ij=E0iδij+λv ij(2) whereE0i=i(i=1,···,N)(3) are eigenenergies of the eigenstates of H0labeled by|i . Oﬀ-diagonal matrix elements v ij=v ji are random num-bers with Gaussian distribution for1≤|i−j|≤b ( v ij =0and v2ij =1)and are zero otherwise.Here basis states have been chosen as the eigenstates|i of H0in energy order.Eigenstates of H,labeled by|α ,are also ordered in energy,H|α =Eα|α ,Eα+1≥Eα.(4) In order to obtain the GBWPT,let us divide the set of basis states into two parts,{|i ,i=p1,p1+1, (2)and{|j ,j=1,···p1−1,p2+1,···N}.This also di-vides the Hilbert space into two sub-spaces,for which the corresponding projection operators areP≡p2i=p1|i i|and Q≡1−P.(5)Subspaces related to the projection operators P and Q will be called in the following the P and Q subspaces,re-spectively.For an arbitrary eigenstate|α ,which is split into two orthogonal parts|t ≡P|α and|f ≡Q|α by the projection operators P and Q,by making use of the stationary Schr¨o dinger equation,it can be shown that|α =|t +1Eα−H0QλV,(7)the iterative expansion of Eq.(6)is of the form |α =|t +T |t +T 2|t +···+T n −1|t +T n |α .(8)Then,one can see that if the projection operators P and Q are such chosen thatlim n →∞α|(T †)n T n |α =0,(9)Eq.(8)gives|α =|t +T |t +T 2|t +···+T n |t +···.(10)Here the eigenvalue E αhas been treated as a constant.Eq.(10)is a generalization of the so-called Brillouin-Wigner perturbation expansion (GBWPE)(for BWPE,see,for example,[17,28]).Equation (9)is the condition for the GBWPE (10)to hold.Generally,if P and Q subspaces are such chosenthat |E α−E 0j |is large enough compared with bλV forany basis state |j in the Q subspace,|T nα>will van-ish when n →∞.Therefore,generally there are many choices for the P and Q operators ensuring Eq.(10)to hold.Among them,the projection operator(s)P with the minimum number of basis states and the correspond-ing operator(s)Q with the maximum number of basis states will be denoted by P αand Q α,respectively,and the corresponding |t and |f parts of the state |α will be denoted by |t α ≡P α|α and |f α ≡Q α|α ,respec-tively.The |t α part of the state |α will be called the non-perturbative (NPT)part of |α ,and the |f α part called the perturbative (PT)part of |α .Correspond-ingly,the eigenfunction of the state |α ,i.e.,its compo-nents in the basis states,can also be divided into NPT and PT parts.Eq.(10)tells that |f α can be expressed in terms of |t α ,E α,λV and H 0.In what follows,generally the GBWPE (10)will be used to discuss the perturbative parts of eigenstates only.To achieve a more explicit condition for Eq.(9)to hold,let us write T n in the following form,T n=1E α−H 0Q (12)is an operator in the Q subspace.Eq.(11)shows that it is the properties of λU that determines if T n |α vanishes when n approaches to ∞.Indeed,writing eigenstates of the operator U in the Q subspace as |ν ,U |ν =u ν|ν ,(13)it is easy to see that if all the values of |λu ν|are less than 1,then T n |α vanishes when n goes to inﬁnity.Therefore,generally the condition for Eq.(10)(also Eq.(9))to holdis that the corresponding λU operator in the Q subspace does not have any eigenvector |ν with |λu ν|≥1.As an application of the above results to the WBRM,let us study the component of an eigenstate |α on a basis state |j in the Q αsubspace,C αj ≡ j |α ,i.e.,a component of the perturbative part |f α .In what follows,for convenience,we use |j to indicate a basis state in the Q αsubspace,|i to indicate a basis state in the P αsubspace and |k for the general case.Sup-pose j is larger than p 2for the P αsubspace,speciﬁcally,p 2+mb <j ≤p 2+(m +1)b with m being an inte-ger greater than or equal to zero.Noticing that a result of the band structure of the Hamiltonian matrix of the WBRM is that j |(QV )n |t α is zero when n is less than (m +1),from Eqs.(10)and (11)we have C αj = j |(T m +1+T m +2+···)|t α= j |11−λu ν|ν .(16)Then,substituting Eq.(16)into Eq.(14),one hasC αj =11−λu νj |ν ](λu ν)m .(17)Since |λu ν|<1for all ν,the behavior of the long tails of the EFs of the WBRM is more or less like exponential decay as discussed in Ref.[6].In fact,from the viewpoint of the GBWPT,the proof and arguments given in Ref.[6]for behaviors of long tails of the LDOS of the WBRM are still valid when perturbation is strong.What is of special interest here is the behavior of C αj for j in the regions of (p 2,p 2+b ]and [p 1−b,p 1),i.e.,for j in the regions of size b just beside the non-perturbative part of the eigenstate.For these j ,m =0in Eq.(17)and it is not necessary for |C αj |to decay exponentially.That is to say,the decaying speed of |C αj |for these j should be slower than that for j larger than p 2+b or less than p 1−b .The two regions of j ∈(p 2,p 2+b ]and j ∈[p 1−b,p 1)will be called the slope regions of the eigenstate |α (the reason for such a name will be clear from numerical results in the next section).According to Eq.(10),the explicit expression for Cαj in terms of elements of the Hamiltonian matrix can be written asCαj=p2i=p1(λV jiEα−E0jλV kiEα−E0jλV klEα−E0l···)Cαi(18)where Cαi= i|tα .This expression can be written in a simpler form by making use of the concept of path, in analogy to that in the Feynman’s path integral theory [29],which can be done as follows.For(q+1)basis states {|j ,|k1 ,···|k q−1 ,|i }with only|i in the Pαsubspace, we term the sequence j→k1→···→k q−1→i a path of q paces from j to i,if V kk′corresponding to each pace is non-zero.Clearly,paths from j to i are determined by the structure of the Hamiltonian matrix in H0represen-tation.Attributing a factor V kk′/(Eα−E0k)to each pace k→k′,the contribution of a path s to Cαj,denoted by fαs(j→i),is deﬁned as the product of the factors of all its paces.Then,Cαj in Eq.(18)can be rewritten asCαj=p2i=p1sfαs(j→i)Cαi.(19)An advantage of expressing Cαj in terms of contribu-tions of paths is that it makes it easier to understand the important role played by the size of the non-perturbativepart of the EF in determining its shape.Such a size for the state|α will be denoted by N p,N p≡p2−p1+1. (N p is,in fact,a function ofα,but,for brevity,the sub-scriptαwill be omitted.)Two values ofλare of special interest whenλincreases from zero.One is the smallest λfor N p=2,indicating the beginning of the invalidity of the ordinary Brillouin-Wigner perturbation theory.The otherλis for the case of N p=b.The value of thisλis of interest since topologically the structure of paths for the case of N p≥b is diﬀerent from that for N p<b. For example,when N p≥b,paths starting from j<p1 can never reach a j′which is greater than p2.That is to say,no path can cross the non-perturbative region. On the other hand,when N p<b,paths can cross the non-perturbative region.One can expect that such a dif-ference should have some eﬀects on the shapes of both EFs and LDOS.III.NUMERICAL STUDY FOR THE PERTURBATIVE AND NON-PERTURBATIVE PARTS OF EIGENFUNCTIONSIn this section we study numerically the division of EFs into perturbative(PT)and non-perturbative(NPT) parts.Both the individual shape and the average shape of EFs will be studied.The boundary of the NPT part of a state|α ,i.e.,thevalues of p1and p2,are calculated in the following way. For a Hamiltonian matrix as in Eq.(2),weﬁrst diago-nalize it numerically and get all its EFs.Then,for an eigenstate|α with energy Eα,weﬁnd out all the pairsof p1and p2for which the value of α|(T†)n T n|α could become smaller than a small quantity(say,10−6)when n goes to inﬁnity.Finally,for the pair of(p1,p2)thus ob-tained with the smallest value of N p=p2−p1+1and the pairs of(p1+1,p2)and(p1,p2−1),we calculate eigenval-ues of the corresponding U operators in Eq.(12)in order to check out if all the|λuν|for the pair of(p1,p2)are less than1while some of the|λuν|for each of the other two pairs(p1+1,p2)and(p1,p2−1)are larger than1.The shape of an eigenstate|α in the unperturbedstates can be deﬁned asWα(E0)= k|Cαk|2δ(E0−E0k).(20)In order to obtain the average shape of eigenstates,dif-ferent individual distributions Wα(E0)should beﬁrst shifted into a common region.For this,we express Wα(E0)with respect to Eαbefore averaging.Forαfrom α1toα2,with Wα(E0)expressed as Wα(E0−Eα),the average shape of the eigenstates,denoted by W(E0s),can be calculated.Numerically,W(E0s)are thus calculated: For a state|α ,some value of E0s and all k satisfying |E0s−(E0k−Eα)|<δE w/2,whereδE w is some quantity chosen for dividing E0s into small regions,we calculate the sum of k|Cαk|2,then take the average of the sum overαfromα1toα2.The result thus obtained is the av-erage value of W in the region(E0s−δE w/2,E0s+δE w/2) and is also denoted by W(E0s).Similarly,the values of W(E0s±mδE w)(m=1,2,···)can also be calcu-lated.The average shape of eigenstates can also be di-vided into a NPT part and a PT part by the averaged boundary of the NPT parts of the states|α ,denoted by p a1≡ p1−Eα and p a2≡ p2−Eα .The size of the NPT part of the average shape of eigenstates is N p ≡ p2−p1+1 .The values ofλfor N p beginning to be larger than1and for N p =b will be denoted by λf andλb,respectively.Now let us present the numerical results.Theﬁrst one is for the case of N=300,b=10andλ=0.6.This is a case for which N p can be equal to both1and2.The values of|Cαk|2=| k|α |2for four EFs in the middle energy region(α=148−151)are given in Fig.1.The two vertical dashed-dot straight lines for each case indi-cate the positions of p1and p2of the NPT part of the eigenstate,i.e.,the boundary of the NPT part.Since E0k=k,p1and p2are also the unperturbed eigenen-ergies of the corresponding basis states|k=p1 and |k=p2 .Forα=148and149,since p1=p2=αthere is only one vertical dashed-dot line in each case. Forα=150and151,there are,in fact,two NPT partsfor each case.In Fig.1we give one of them for each case only,i.e.,(p 1,p 2)=(149,150)for α=150(the other one is (150,151))and (p 1,p 2)=(151,152)for α=151(the other one is (150,151)).Numerically we have found that only for small λis it possible for an eigenstate to have more than one NPT parts.The average shape of EFs for αfrom 130to 170is given in Fig.2with the correspond-ing values of p a 1and p a2indicated by vertical dashed-dotlines.δE w =1for calculating this W (E 0s )function.As mentioned in section II,another perturbation strength of interest is for the case of N p being able to equal the band width b of the Hamiltonian matrix.An example for this case is λ=1.4.For this value of λ,there is only one NPT part for each state |α ,e.g.,N p =9,10,7,9for α=148,149,150,151,respectively.Individual EFs for α=148−151are presented in Fig.3with their boundaries of the NPT parts.Although for an individual EF the value of |C αk |2could be still large outside the NPT region (Fig.3),the main body of the average shape of EFs lies obviously in the averaged NPT region as shown in Fig.4(a).For this averaged EF we still have δE w equal to 1and αfrom 130to 170.An interesting feature of the distribution lnW in the sloperegions [p a 1−b,p a1)and (p a 2,p a 2+b ]in Fig.4(b)is that,as predicted in section II,the decaying speed in the two slope regions is obviously slower than that in the long tail regions.From Fig.4(a)it is quite clear why we call theregions “slope”.In Ref.[13]the average shape of EFs for diﬀerent val-ues of the Wigner parameter q ,q =(ρv )24√2.(23)Here by the NPT region of an EF we mean the region of E 0k between the boundaries (dashed-dot lines)of the NPT part of the EF as shown in Fig.5.Then,since as shown in Fig.5centroids of the EFs are generally not far from the centers of the NPT regions,they are generally not far from the corresponding eigenenergies.Therefore,in this case the averaging with respect to the centroids of the EFs and the averaging with respect to the eigenenergies do not have much diﬀerence,and the average shape of EFs obtained by the two methods should be more or less similar.However,since there are still some diﬀerences between the centroids of the EFs and the eigenenergies,results of the two averaging methods can not be quite close to each other.In fact,as shown in Ref.[13],results of the method of averaging with respect to centroids of EFs should be a little narrower than those of the other method.The average shape of EFs for αfrom 130to 170and λ=4.0is given in Fig.6with δE w =3.0.(In this pa-per numerically we study the averaging with respect to eigenenergies only.)For this averaged EF the slope re-gions are even clearer.The sizes of the two slope regions in Fig.6are larger than b,the predicted one for a sin-gle EF,since the values of N p for the NPT parts of the eigenstates |α taken for averaging are not the same.In fact,the variance of the N p for the states is about 1.56.For studying the case of localization in energy shell as in Ref.[13],the dimension N =300is too small.There-fore,we increase the value of N to 900and study the case of b =10and λ=30.0,for which the parameters q and λe are equal to 90and 0.24,respectively,as those in Ref.[13].The values of |C αk |2(dots)of four EFs in the middle energy region are given in Fig.7with the cor-responding boundaries (p 1and p 2)of the NPT parts of the EFs (indicated by vertical dashed-dot lines).Clearly the main body of each of the EFs occupies only part of the NPT region.Then,resorting to Eq.(23),it is easy to see that for the average shape of EFs the result of averaging with respect to centroids of EFs should be ob-viously narrower than that of averaging with respect to eigenenergies.This phenomenon is called localization in energy shell in Ref.[13].In our opinion,it is,in fact,localization of EFs in their NPT regions.Although,as shown in Fig.7,when λ=30.0many components of the EFs are quite small in the NPT parts,they can still be distinguished from those in the PT parts.In Fig.8we present the values of |C αk |2in Fig.7in loga-rithm scale,from which the diﬀerence between the |C αk |2in the NPT parts and those in the PT parts is quite clear.In fact,it can be seen from Fig.8that |C αk |2for small components in the NPT parts also decay ex-ponentially on average,but the decaying speed is slower than that for components in the PT parts.Similar re-sults have also been found for other eigenfunctions and the centroids of the EFs have been found scattered in the NPT regions.Therefore,averaging with respect to eigenenergies(around the centers of the NPT regions)is reasonable when studying the average shape of EFs.The average shape of EFs in the middle energy region forαfrom420to480is shown in Fig.9(δE w=10).The two slope regions are also obvious.The widths of the two slope regions,which can be seen from theﬁgure is about2δE w=2b,are also larger than that predicted for a single EF due to the fact that the variance of N p for these states|α is about5.14.In order to have a clear picture for the variation of the average size N p of NPT parts of EFs with the perturba-tion strengthλ,we plot it in Fig.10(a)(for N=300and αfrom130to170).The two solid straight lines in the ﬁgure show that N p has a good linear dependence onλin the two regions ofλ∈(2.0,4.5)andλ∈(4.5,7.0).The variation of N p forλless than2is given in the inset of theﬁgure in more detail.The solid curve in the inset is aﬁtting curve of a quadratic form:(7.5(λ−0.4)2+1.0). Since in this case the value ofλf is a little larger than0.4, i.e.,forλ≤0.4the size N p of the NPT parts of the EFs is equal to one,the region ofλ<0.4has been neglected whenﬁtting the N p by the quadratic curve.From the inset it can be seen that theﬁtting curveﬁts the values of N p quite well in the region ofλ∈(0.4,1.5).Inter-estingly,the value ofλb for N p =b is between1.45and 1.5,that is to say,in the region ofλ∈(λf,λb),the values of N p have a quadratic dependence onλ.The variance of N p,denoted by∆N p,is given in Fig.10(b)(circles),which shows that∆N p is small com-pared with N p.From theﬁgure it can be seen that for λ<5the value of∆N p increases withλon average. Then,since,as mentioned above,the value of∆N p is about5.14whenλ=30,it does not change much in the region ofλ∈(5,30),that is to say,the relative value of the variance,namely∆N p/N p,becomes even smaller as λbecomes larger.IV.RELATIONSHIP BETWEEN THE SHAPE OF EIGENFUNCTIONS AND OF LDOSThe existence of a relationship between the average shape of EFs and that of LDOS in some particular cases has already been noticed numerically by several authors (see,for example,[13,16,18]).Here for the WBRM we show that some analytical arguments can be given for it by making use of the GBWPT.The so-called local spectral density of states(LDOS) for an unperturbed state|k is deﬁned asρk L(E)= α|Cαk|2δ(E−Eα)(24) where Cαk= k|α .Making use of the division of EFs into NPT and PT parts,the non-perturbative(NPT)and perturbative(PT)part of the LDOS can be deﬁned in the following way.Let us consider all the perturbed states |α for which Cαk is in the NPT parts of their eigenfunc-tions.Supposeαp1is the smallest one among theseαandαp2is the largest one among them,i.e.,|αp1 has the smallest eigenenergy and|αp2 has the largest.A property of the WBRM is that for any state|α withαp1≤α≤αp2,generally to say,Cαk is in the NPT part of its eigenfunction.Then,the non-perturbative(NPT)part of the LDOSρk L(E)can be deﬁned as the sum in Eq.(24)overα∈[αp1,αp2],and the perturbative(PT) part can be deﬁned as the sum over the restα.The size of the NPT part of the LDOS is deﬁned as(αp2−αp1+1) and will be denoted by Nαp.The average shape of LDOS,denoted byρL(E s),can be obtained in a way similar to that for the EFs discussed in section III,except that for the LDOS the distributions ρk L(E)are expressed as functions of(E−E0k)before av-eraging.Since E0k is also the centroid ofρk L(E),such an averaging method is the most natural one.The average shape of LDOS can also be divided into a NPT part and a PT part by the averaged boundary of the NPT parts of the individual LDOS.Before discussing the relation between the average shape of EFs and that of LDOS,let usﬁrst draw some conclusions for properties of EFs from the numerical re-sults discussed in section III.Firstly,based on the nu-merical results given in section III for the average shape of EFs,we can make such an approximate treatment for the average values of|Cαi|2in the NPT parts of EFs: for a givenλnot extremely large,they can be treated as a constant denoted by t2λ,i.e.,|Cαi|2on the edges of the NPT parts become smaller than those in the middle regions whenλincreases and the above approx-imate treatment may fail in case of extremely largeλ. Secondly,another result of the last section is that the variance∆N p is small compared with N p.This means that when discussing approximate relations it is reason-able to treat the size N p of NPT parts of eigenstates as a constant for aﬁxedλ.As results of the above two approximations and the approximate relation(23),it can be shown that,under the corresponding conditions,(a)the average value of |Cαk|2in the NPT parts of LDOS is also about t2λ,(b) the size of the NPT part of a LDOS is close to that of an EF,i.e.,Nαp≈N p,(c)usually E0k is in the mid-dle region of the NPT part of the LDOSρk L(E),i.e.,E0k−Eαp1≈Eαp2−E0k.Then,one can see that the following approximate relation holds for the NPT part。

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