Recent progress in calculations of the total cross section for top quark
pair production near threshold is reviewed. Different top quark mass definitions
adequate for threshold studies are discussed. A relation between
the potential subtracted mass and the IS mass is studied. The potential
subtracted IS mass is defined which incorporates attractive features of both
schemes

Jeżabek, Marek

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Top Quark Pair Production in e+e- Annihilation Near Threshold  Author: Marek Jeżabek   Citation style: Jeżabek Marek. (1999). Top Quark Pair Production in e+e- Annihilation Near Threshold. "Acta Physica Polonica. B" (1999, no. 11, s. 3317-3324). T O P  QUARK PA IR PR O D U C T IO N  IN e+e“ ANNIHILATION N EA R  THRESHO LD *M . J e z a b e kInstitute of Nuclear Physics Kawiory 26a, 30-055 Cracow, Poland and Institute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland(Received November 4, 1999)Recent progress in calculations of the to tal cross section for top quark pair production near threshold is reviewed. Different top quark mass def­initions adequate for threshold studies are discussed. A relation between the potential subtracted mass and the IS mass is studied. The potential subtracted IS mass is defined which incorporates attractive features of both schemes.PACS numbers: 12.15. y. 12.38.Bx, 14.65.Ha1. In tro d u c tio nRecently an impressive progress has been achieved in calculations of cross sections for top quark pair production in e+ e-  annihilation near threshold. A future linear collider (LC) operating at energies close to  t i  threshold will be an ideal machine to study properties of the top quark. Prospects th a t LC will be built during the next decade stim ulate growing interest in precise theoretical description of this reaction. In this article I concentrate only on new developments in the years 1998-99. Older calculations are described in reviews, see e.g. [1-7] and references cited therein. In Sec. 2 a considerable increase of precision is described due to  new mass definitions [8, 9] which are more adequate than  the pole mass [10,11] for threshold studies of the to ta l cross section. The potential subtracted IS mass is proposed as a com­bination of the potential subtracted [8] and the IS [9] mass definitions for the top quark. In Sec. 3 a brief review of recent calculations of higher order corrections is presented.* Presented a t th e  Summer Institu te  ’99, Yamanashi, Japan, August 15-21, 1999, and a t the  XXIII International School of Theoretical Physics “Recent Developments in Theory of Fundam ental Interactions”, U stron, Poland, September 15-22, 1999.2. M ass definitions2.1. Pole mass and potential subtracted massO ne of th e  m ain  goals of top  quark  physics a t LC is a precise de term ina­tion  of top  quark  m ass. E xpected  lum inosities and  beam  energy resolutions are so good th a t  a m easurem ent of th is  m ass w ith  precision b e tte r  th an  100 MeV is conceivable from  experim ental po in t of view. Even b e tte r  preci­sion of theore tical stud ies is therefore m andatory . A t present th ere  are a few sources of theore tical uncertain ties. O ne of them , re la tiv istic  corrections are considered in th e  following section. In th is  section we use non-relativ istic  ap­proxim ation  and consider th e  top  quark  as a s tab le  partic le  characterized  by th e  pole m ass m poie- In th is approxim ation  in teractions betw een t  and  t  are described by an in stan taneous chrom ostatic  p o ten tia l which in m om entum  space is conventionally w ritten  asV h )  =  - C F ^ M ,  (1)w here q2 =  \q\2 denotes th e  square of (th ree)m om entum  tran sfer q and CF =  4 /3 . T his form ula looks qu ite  sim ilar to  th e  well known Coulom b po ten tia l. However, for our purposes we need a b e tte r  precision and  cannot neglect q dependence of th e  function  a y .  In QCD  th e  coupling a y  is running  and  a t p resent its  evolution is known up  to  two-loop accuracy [12,13] in p e rtu rb a tiv e  calculations. T h e  coupling a y  can be  expressed in term s of the  conventional strong  coupling constan t a w  and  th e  re la tion  including term s a 3 has been derived in [12]. T h e  coupling a— (q) is also runn ing  and the  first four coefficients (/3q, • • • , / 3 ) are known for its  renorm alization  group ¡3 function. Two-loop accuracy m eans th a t  all these coefficients are included in th e  renorm alization  group equation  for and  a y  is calcu lated  including also term s a 3 . We shall also use a y  ca lculated  in one-loop accuracv, i.e.MS "including one order less in th e  re la tion  betw een a y  and a —  as well as in th e  renorm alization  group equation  for a  (q). C onventionally  a  (Mz) ,  i.e. th e  value of th e  strong  coupling constan t a t Z°  peak  is used as a s ta rtin g  po in t for th e  evolution.T he coupling ay (q )  grows w ith  decreasing q and around  1 GeV becomes com parable to  or larger th a n  1 , and  eventually  a t som e po in t even infinite. In th is  range of q we canno t tru s t  p e rtu rb a tiv e  expansions and are forced to  use som e n o n -p ertu rb a tiv e  m ethods or ex tra  phenom enological in p u t to  calcu late  th e  po ten tia l V(q).  U nfo rtunate ly  th e  Lippm ann-Schw inger equa­tion  for th e  energy levels of t t  system  contains an in tegral over m om entum  transfers including th e  dangerous region of low q. Therefore, we have to  e s tim ate  how m uch are th e  energies of topon ium  s ta te s  affected by contri­b u tions from  th is  region. In o ther words we have to  es tim ate  theoreticaluncerta in ties due to  present poor knowledge of th e  n o n -p e rtu rb a tiv e  QCD  poten tia l. Let qm be  a m om entum  tran sfer such th a t  for q > qm a per­tu rb a tiv e  form ula Vpert(q) is sufficiently accu ra te  w hereas for q < qm some n o n -p e rtu rb a tiv e  expression should be used. In th e  following discussion we assum e th a t  m poie =  175 GeV and qm =  3 GeV. As th e  n o n-pertu rba tive  po ten tia l we choose th e  one proposed by R ichardson [14]. R ichardson po­ten tia l depends on a n o n -p e rtu rb a tiv e  p aram ete r A r  which after Fourier transfo rm ation  to  th e  position  space can be  determ ined  from  th e  slope of th e  linear confining po ten tia l. A successful descrip tion  of bb and  cc nS s ta tes  is ob ta ined  for A r  =  0.4 GeV. A form ula for th e  R ichardson po ten tia l as well as a descrip tion  of its  num erical im plem entation  are given in [15], see A ppendix  A therein . For th e  p e rtu rb a tiv e  p a r t of th e  p o ten tia l we use av(q)  calcu lated  w ith  one-loop accuracy and a _ ( M z )  =  0.118 . In th e  pole m ass scheme th e  binding energies of topon ium  s ta te s  are defined as¿ T le =  M r -  2m pole , (2)where M r denotes th e  rest m ass of th e  s ta te  r .  T hus -Efgle is th e  binding energy of th e  15  s ta te .In Table I th e  values of -Efgle are given for a few values of Ar . As already explained th e  realistic values are ob ta ined  for Ar around  0.4 GeV. In theTABLE IBinding energies and energy shifts for toponium resonances in the pole mass and in the potential subtracted mass scheme for different values of Ar and p f  =  5 GeV.A r E l T 2 öm(ßf ) E f l k f ) E U k f ) 4 f (M /) E l l k f )0.01 -  2.273 0.827 -  1.446 0.152 0.559 0.6680.1 -  2.616 1.171 -  1.445 0.163 0.602 0.7860.2 -  2.785 1.340 -  1.445 0.171 0.629 0.8480.4 -  2.956 1.511 -  1.445 0.184 0.679 0.9490.6 -  3.014 1.567 _  1.447 0.197 0.725 1.0421.0 -  2.928 1.474 -  1.454 0.228 0.828 1.234range 0 .2 -0 .6  GeV th e  varia tion  of -Efgle is reasonably  m odera te  and  one can conclude th a t  determ ina tion  of m poie from  a m easurem ent of IS  s ta te  m ass M i s  is possible w ith  theore tical u n certa in ty  of order 100 MeV due to  con tribu tions from  th e  n o n -p ertu rb a tiv e  region. Even for a very d rastic  change of th e  phenom enological p o ten tia l and  A r  =  0.01 GeV th e  change in m poie for fixed M i s  is ab o u t 350 MeV. All th is  m eans th a t  IS toponium  s ta te  is too  sm all to  be affected significantly by m om entum  transfers below qm . In fact th e  s itu a tio n  is m uch b e tte r  th an  it follows from  a m odera tedependence of E f f e on T r .  T h e  la tte r  is due to  a dependence of m poie on sm all m om entum  transfers. Beneke proposed [8 ] to  replace m poie by the  p o ten tia l su b trac ted  (PS) m assmps(pf )  = m poie - d m ( i i f ) ,  (3)where8m (n f ) = ~  J ^ F ( g )  • (4)<KUfIn Eq. (4) f i f  is an  a rb itra ry  p aram ete r larger th a n  qm , i.e. p,f should be chosen in th e  region of m om entum  transfers w here th e  p e rtu rb a tiv e  expan­sion is sufficiently accurate . F urtherm ore  it can be  d em onstra ted  th a t  such a definition corresponds to  a m ass p aram ete r which is no t sensitive to  sm all m om entum  transfers [8 ]. Let us define th e  energy sh ift1 for a s ta te  r  in the  PS scheme asE Y s ( p f ) = M r ^ 2 m PS(pf ).  (5)It follows th a tE f s ( p f ) =  E ? ole +  28m (n f ) . (6)In Table I th e  values of 28m(p,f) are given for p,f =  5 GeV and  a few values of T r .  As expected  th is  q u an tity  also changes w ith  T r .  It is rem arkable, however, th a t  th e  variations of 2 8m(p,f) and  -Efgle cancel each o ther and  the  energy shift for IS bound  s ta te  becom es surprisingly  stable, see Table I. T he energy shifts for nS bound  s ta te s  up to  n = 4  are also given in Table I. I t is seen th a t  th e  dependence on T r  is reasonably  sm all for 2S s ta te . However for 3S and  IS a significant dependence on T r  persists which m eans th a t  these rad ia l excita tions are spatia lly  large enough to  be affected by low m om entum  transfers.Com ing back to  IS  s ta te  we observe th a t  th e  precision which can be achieved in determ ina tion  of m ps is dom inated  by th e  m easurem ent of M i s , cf. Eq. (5). C o n trary  to  a w idespread belief in th is  case th e  large w id th  of th e  top  quark  does no t help a t all by cu ttin g  off n o n -p ertu rb a tiv e  dynam ics a t low m om entum  transfers and large sp a tia l d istances. In th e  real world the  topon ium  IS  resonance has th e  w id th  of ab o u t 3 GeV which is a large num ber w hen com pared to  100 MeV precision to  be achieved in determ ina tion  of the  top  quark  m ass. O f course, for larger energies, say 2 GeV or m ore above IS  level th e  top  w id th  helps. However, for a m easurem ent located  in energy close to  IS s ta te  a real problem  is how to  unravel its  effects.1 ‘Binding energy’ would be a misleading terminology because, depending on the values of f i f  the energy shifts for some or even all bound states can be positive.2.2. Potential subtracted I S  massR ecently  H oang and T eubner have observed [9] th a t  it is very convenient to  perform  calculations in a scheme in which th e  m ass of th e  top  quark  is defined sim ply as one half of M i s  in th e  lim it of zero top  w idth . T hey  proposed th e  nam e IS m ass for such m ass p aram eter and dem onstra ted  th a t  it is a sho rt d istance one, i.e. unlike th e  pole m ass is no t sensitive to  dynam ics a t large d istances. I th in k  th a t  it is useful to  consider their p roposal as a condition on th e  m ass p aram ete r /¿y in PS scheme. In fact it is s traigh tforw ard  to  find /¿is such th a tE f l Y  i s )  =  0 .  ( 7 )O ne can also show th a t  /¿is is in th e  p e rtu rb a tiv e  regim e and  corresponds to  a typical m om entum  transfer for IS  bound  s ta te . T h e  corresponding p o ten tia l su b trac ted  m assmpis = mps(pis) = \Mis  (8)is by definition a IS m ass. I t is also clear why IS  m ass is a b e tte r  scheme th an  2S or 3S m ass schemes. T he energies of higher rad ia l excita tions are sim ply m ore affected by n o n -p ertu rb a tiv e  sm all m om entum  transfers.A very good s tab ility  of =  5 GeV) guaran tees th a t  th e  m ass pa­ram eter / ¿ i s  does not depend on n o n -p ertu rb a tiv e  param eters  like ¿ 1 r .  On th e  o ther hand  it depends on th e  dynam ics in th e  p e rtu rb a tiv e  regime. In p articu la r /¿is depends on th e  value of a — (M ^) and  on th e  order of p e r tu r­b ative calculations. For exam ple: a t one-loop accuracy and  for ex— (M z ) =0.15, 0.18 and  0.21 th e  corresponding values of /¿is are equal to  13.26 GeV, 13.63 GeV and  14.00 GeV, respectively.2.3. Rem,arksIt is evident th a t  th e  sho rt d istance m asses discussed in th is  section are superior and  m ore convenient th a n  th e  pole m ass in stud ies of th e  to ta l cross section near th reshold . Does it m ean th a t  th e  pole m ass is a to ta lly  useless concept which should be abandoned for perm anen tly  confined quarks? I believe th a t  th e  answer to  th is  question  will be  no. I do so because th ere  are o ther cross sections and d istrib u tio n s which can be m easured in experim ental stud ies near th reshold . T hey  are less inclusive th a n  th e  to ta l annih ilation  cross section and in consequence m ore difficult from  theore tical po in t of view. I t is n o t precluded th a t  th e  pole m ass can be  a good p aram ete r to  describe som e of them . For exam ple: it is p lausible th a t  th e  invariant m ass d is trib u tio n  of top  quark  decay p ro d u c ts  has a m axim um  close to  th e  pole m ass ra th e r th a n  to  IS m ass. O f course th e  corresponding m ass param etercan be ex trac ted  w ith  a lim ited  accuracy. A t som e level of precision it will be  necessary to  decide if a pion, which is slow in t t  center of m ass, belongs to  decay p ro d u c ts  of t  or to  decay p ro d u c ts  of t  and th is  m ay be im possible even in principle.3. Top w id th  and  higher o rder correctionsFor center-of-m ass energies close to  th e  t t  th resho ld  th e  top  quarks are p roduced  w ith  nonrela tiv istic  velocities v -C 1. T herefore nonrela tiv istic  approxim ation  is a good s ta rtin g  point. However a high precision determ i­n ation  of th e  top  quark  m ass requires a system atic  s tu d y  of higher order corrections including re la tiv istic  and  rad ia tive  corrections. In com parison to  bound  s ta te  problem s like spectroscopy of positron ium  or hydrogen-like ions, which have been stud ied  in Q ED , a novel fea tu re  of t t  p roduction  near th resho ld  is a very large w id th  of th is  system . In th e ir pioneering work Fadin and  Khoze [16] showed how to  inco rporate  th e  top  w id th  into th eo re t­ical descriptions. T hey  proposed to  use G reen function  ra th e r th an  binding energies and  wave functions for ind ividual resonances. T heir Leading O rder approach (LO) was fu rth e r developed in [17-19]. In p articu la r Q C D  s ta tic  p o ten tia l was included a t one-loop accuracy level and N ext-to-L eading O rder (NLO) con tribu tions in nonre la tiv istic  expansion were calcu lated  to  forw ard­backw ard asym m etry  [20] and top  quark  po larization  [21,22]. T hese early stud ies were done in th e  pole m ass scheme, so a considerably  b e tte r  accu­racy  can be ob ta ined  by using one of th e  sho rt d istance m asses discussed in Sec. 2. F u rth er progress canno t be achieved w ithou t perform ing calcu­lations a t N ext-to-N ext-to-L eading  O rder (NNLO) including corrections of o rder v 2, a sv and  a 2. T his problem  is very com plicated  because in calcula­tions of G reen function  re la tiv istic  and  rad ia tive  corrections do n o t factorize and  have to  be consider sim ultaneously.D uring last two years a num ber of papers appeared  presenting calcula­tions a t NNLO level and  using com pletely different techniques [9,23-31]. In p articu la r in [9, 28-31] a com plete NNLO resu lts are presented. Q u a lita ­tively all these calculations agree qu ite  well. NNLO corrections produce an im p o rtan t shift in th e  binding energies, i.e. in th e  position  of th e  th reshold , and  a significant increase of th e  norm alization  for th e  to ta l cross section. However, a ra th e r large u n certa in ty  rem ains in th e  norm alization  due to  scale dependence in NNLO corrections. At a m ore q u an tita tiv e  level a de­tailed  com parison is difficult because different m ass definitions are used by different groups.An im p o rtan t new theore tical developm ent is th e  so-called P o ten tia l N on-R elativ istic Q CD  [32]. In th is fram ew ork a system atic  s tu d y  of QCD p o ten tia l in even higher orders can be accom plished. In p articu la r calcula­tions of quarkonium  spec trum  a t order a 3 l n a s have been presented [32].I would like to  th an k  M asahiro Y am aguchi and Y ukinari Sum ino for th e ir hospitality , excellent organization and very stim ulating  atm osphere a t Sum m er In s titu te  ’99 in Y am anashi, Jap an . I con g ra tu la te  H enryk Czyż for organizing X X III Silesian U niversity  sum m er school in U stroń . T his work is p a r tly  su p p o rted  th e  by Polish S ta te  C om m ittee for Scientific R esearch (K BN) under gran ts 2P03B08414 and  2P03B14715.R E FE R E N C E S[1] J.H. Kühn, in: F.A. Harris, S.L. Olsen, S. Pakvasa and X. Tata (eds.), Physics and Experiments with Linear e+e-  Colliders, World Scientific, 1993.[2] M. Jeżabek, N ucl Phys. B  (Proc. Suppl.) B37, 197 (1994).[3] M. Jeżabek, Acta Phys. Pol. B26, 789 (1995).[4] J.H. Kühn, Theory of Top Quark Production and Decay, lectures delivered at 23rd SLAC Summer Institute, Stanford, CA, 10-21 Jul 1995, hep-ph/9707321.[5] Y. Sumino, Acta Phys. Pol. B28, 2461 (1997).[6] E. Accomando et al., Phys. Rep. 299, 1 (1998).[7] T. Teubner, Acta Phys. Pol. B30, 1941 (1999).[8] M. Beneke, Phys. Lett. B434, 115 (1998).[9] A.H. Hoang, T. Teubner, hep-ph/9904468.[10] R. Tarrach, Nucl. Phys. B183, 384 (1981).[11] A.S. Kronfeld, Phys. Rev. D58, 051501 (1998).[12] Y. Schröder, Phys. Lett. B447, 321 (1999).[13] M. Peter, Phys. Rev. Lett. 78, 602 (1997).[14] J.L. Richardson, Phys. Lett. B82, 272 (1979).[15] M. Jeżabek, J.H. Kühn, M. Peter, Y. Sumino, T. Teubner, Phys. Rev. D58,014006 (1998).[16] V.S. Fadin, V.A. Khoze, JE TP  Lett. 46, 525 (1987); Yad. 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Top Quark Pair Production in e+e- Annihilation Near Threshold

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Top Quark Pair Production in e+e- Annihilation Near Threshold

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