Chakrabarty, S

Ghosh, B

IACS Institutional Repository

Indian J. Phys. 7 5 A  (6), 651 -6 5 6  (2 0 0 1 )U P  Aan international loiirnalRenormalization-group-improved radiatively corrected Higgs masses in the minimalsupersymmetric standard model at LEP2B G h osh  and S Chakrabarty*Department of Pjiysics, Visva-Bharati, Santifiikclan-7^ 1 2^5, Wc l^ Bengal. IndiaR e i e i v e d  I S  D tu e n ih t  r  2 0 t f0  a a e p t e d  I S  J u ly  2 0 0 1Abstract . Recently, using a very simple appioximaiion scheme which includes the most important terms in the ladiative corrections loi the lliggs masses in the Minimal supersymmetric standard model (MSSM), Haber e i a l have estimated the Higgs masses over a very laige traction ol MSSM parametci space The purpcise of this paper is to apply ihc mclhod of solving the renorrnali/alion gioup equations tei lop c)uaik. and hoUoni quaik Yukawa couplings m the two-Higgs doublet model given by Panda and llsmaiii to the above studies of Haber e t u l  Here the effects of the lunmng vacuum expectation values (VFVs) in the two Higgs doubiel model below p  w^/^  ^ taken into account in teims of solutions of RGEs foi theVEVs vj(/i) and It may be mentioned that at mass scales below , the solutions of the non-SUSY (Supersymetry) two Higgs doublet modelare used Interestingly, new le.sults are obtained from systematic studies on the mass of lightest Higgs boson m MSSM which have impoitante m view of the 1.EP2 data and fulufc I.H(' expenmentsKeywords MSSM. renormuli/atmn group equation (RGE) Higgs masses PACS Nos. II U )lo , 12 60JU f^ 10 fq1. In trod u ctionT he standard m o d el o f  particle  p h y sic s  is con sisten t*w iih  all ex p er im en ta l data  e x c e p t  that the p red ic ted  H ig g s  b o so n s , required for g iv in g  m a sse s  to  quarks and lep to n s, have not been  ob serv ed . T h e d isc o v e r y  o f  H ig g s  boson  w ill p rovid e the p ro o f that in the m in im al siandiird m o d el, the H ig g s  m ech an ism  IS the correct d escr ip tion  o f  c lcc tro w ea k  sym m etry  breaking [ I ]. T h e m in im al standard m o d e l H ig g s  sec to r  is  not th eoretica lly  w ell-m o tiv a ted  due to  the n aturalness (or h ierarchy) prob lem . T h e s im p le s t  and  e c o n o m ic a l  m o d e l r e a liz in g  lo w  en erg y  su p ersym m etry  is  the m in im a l su p ersym m etric  standard m o d el (MwSSM) [2] w h ich  can naturally accom m odate elem entary H iggs  b o so n s . In M S S M , a su p ersy m m etric  partner is added to each  quark, lepton  and g a u g e  b oson . M oreover, it con tains ( i ) m in im al g a u g e  g ro u p : SUi3), x  5 1 / ( 2 )  / ; ( i i)  m in im al particlec o n te n t : three g en era tio n s o f  quarks and lep ton s and tw o  H ig g s  d o u b let p lu s their superparlners; ( iii)  an ex a c t d iscrete  R -p arity;( iv )  su p ersy m m etry  b reak in g  param etrised  by e x p lic it  but so ft* Corresponding Authorb reak ing  term s w h ich  in c lu d e s  g a u g in o , sca lar m a sses  and  Irilincar scalar co u p lin g s.T he M S S M  H ig g s sector  is very m uch  constra ined  by the co n cep t o f  supersym m etry . T h e H ig g s  b o so n  search w ill play a dom inant role in the LH P2 program m e and LHC\ T here are n u m ero u s w o rk s on  the stu d y  o l H ig g s  m a sse s  w ith in  the  fram ework oi MSwSM 13|.T he paper is o rgan ised  as fo llo w s . T he theory is g iv en  m Section 2. T he R G E s for lop quark, bottom  quark couplings, and v s ( | / )  a lo n g  w ith  their so lu tio n s  for Iw o -H ig g s  d o u b let m od el arc g iv en  in S ectio n  3. T he R G -m o d ified  H ig g s  m a sses  w ith ou t s-quark m ix in g  e ffe c t  in the fram ew ork o f  the m o d e l o f  H a b er  et al is g iv e n  in S e c t io n  4  w h e r e a s  the  corresp on d in g  R G -m o d ified  H ig g s  m a sses  w ith  s-quark m ix in g  is g iv en  in S ec tio n  5. T h e resu lts and d iscu .ssion s arc g iv en  in  S ectio n  6. C o n c lu s io n s  are g iv en  in S ectio n  7.2 , T h eo ryT h e  M S S M  H ig g s  sec to r  co n ta in s tw o  H ig g s  d o u b le t [2 ]. T h e  sc a la r  p a r tic le s  are tw o  C P - e v e n .s c a la r s  and {w ith©2001 lACS6 5 2 B  Ghosh and  S  Chakraharty^  m^o ), a C P -o d d  sca lar and a charged  H ig g s  pair A t the tree le v e l , all H ig g s  m a sse s  d ep en d  on tw o  param eters  and the ratio ^  o f  H ig g s  vacuum  ex p ecta tio n  v a lu es (=  tan p ). T he tree le v e l bound is (m ^o) ^  lc o s2 /8  I , T he H ig g s  m a s s e s  in c r e a s e  1 4 ,5 ], w h en  th e  ra d ia tiv e  c o r r e c t io n s  are  in c lu d ed . T h e  m eth o d  o f  ca lcu la tio n  o f  rad iative correction s are g iv e n  in H aber and H e m p lin g  [6J. F o llo w in g  R ef. [61, the o n e-  lo o p  corrected  H ig g s  m a ss is g iv e n  (sy m b o lic a lly )  by+ A M fu ^+ A M 2m il • (1)M{i + -i-MMl (2)a lo n g  w ith  the w e ll-k n o w n  n orm aliza tion  co n d itio n  viz. vf  +  v |  =  4ml /g ^  = ( 2 4 6  G e V f .(3)(4)(5)(hj) and  b o tto m  (/i^) q u a rk  Y u k a w a  c o u p lin g s  sa t is fy  the  fo l lo w in g  co u p led  ren orm aliza tion  group eq u ation s (R G E s) [7].(6)(7)H ere 0  refers to  the tree le v e l resu lts. T he subscrip t ILL  refers to the o n e -lo o p  lea d in g  log a rith m ic  ap p rox im ation  to the  fu ll o n c -lo o p  ca lcu la tio n  and the sub scrip t 'm ix' refers to  the  c o n tr ib u t io n s  d u e  to  m ix in g  e f f e c t s  o f  th e  th irdg e n e r a t io n  s -q u a r k s . T h e  C P - e v e n  H ig g s  m a s s  sq u a r e d  e ig e n v a lu e s  [6] arcO n the o th er hand, the va cu u m  ex p ec ta tio n  v a lu es  V\{fu) and v2(jU) satisfy  the fo llo w in g  renorm alization  group equations [8 .9 ]16;r dV2clt =  V2 -3 /1 ,16;r^ — =  v, -3hl + dtI c r g i(8)(9)T h e e x p lic it  form u la  for A H aber er al [6] are u sed  fo r  our stu d ies.3. Renormalization g;roup equations (RGEs) for top and bottom quark couplings and their solutions for two-Higgs doublet moddA t the b eg in n in g , w e  like to  m en tio n  that w e  u se  the so lu tio n s  o f  the n o n -S U S Y  tw o  H ig g s  d o u b le t m o d e l b e lo w  the S U S Y  sca le . In this m o d e l, the quark m a ss is the product o f  the H igg.s- quark  Y u k a w a  c o u p lin g  (h ) and  th e a p p ro p ria te  v a cu u m  expectation  value. H aber and H em p lin g  [6] have so lved  the R G E s  for lo p  and b ottom  quark c o u p lin g s  u sin g  so m e  ap p rox im ate  va lu es for m,(jU) and nif,{ij) for tw o  sca le s  v/z. and w h ere the v a lu es  o f  m ,( ^ )  and m^dX)are g iv e n  in the tw o -H ig g s  d o u b le t n o n -su p ersy m m etr ic  m o d el b yIn e q s .  ( 6) -  ( 9 )  w e  h a v e  r =  I n / i .  C / ' ’ =  ( ^ . f  , 8) ,  C /'’' =  i l l  .8) and C,"'' =  ( ^ . | , 0 ) ; i=Y,2L,  3C . F o llo w in g  Parida and U sm an i 17], for in > nif, the so lu tio n s  o f  eq s. (6) and(7 ) a ie( 10)(11)w hereandJ\nm,a,(M) ]a,(m,)2a,(12)(13)T h e fu n c tio n s  A j { f  = t,b)  are o b ta in ed  by  in tegratin g  out the g a u g e  c o u p lin g  co n tr ib u tio n s in eq s . (6, 7 )  and k eep in g  term s u p to  o n e - lo o p  o u t o f  th e tw o - lo o p  ap p roxim ationT h e  ren o rm a liza tio n  g ro u p  e q u a tio n s  fo r  th e to p  quark , bottom  quark and t  lep ton  Y ukaw a c o u p lin g s  in  the tw o -H ig g s  d o u b le t  m o d e l a re  s o lv e d  b y  P a r id a  a n d  U s m a n i [7 ]  fo r  OT, S  )U S  M^ . w h ere  / t  is  th e m a ss  s c a le  and M^ is  the quark  lep ton  u n ifica tio n  s c a le . W e c o n s id e r  th e e ffe c t  o f  m a ss sc a le  ( M ) on  the running V E V s  v , ( p )  and v j f p )  (8 ,9 ] . N o w , th e  topw here1 Ia , i n )  a , ( m , )  2rt m,/  2L'' 5- 3(14)(15)Renorm alization-group-im proved radiatively corrected H iggs m asses etc 6 5 3N e x t, fo l lo w in g  the m eth o d  o f  Parida and co llab orators | 8, 9], the so lu tio n s  o f  eq s . ( 8) and  (9 )  for the V E V s andv j f / t )  arc g iv e n  byv , ( ^ )  =  v’2 ( / n , ) ni ( ^ )  =  ) n |a.iiu) V« ,  (m,)X eX e-V,N o w  eq s. (1 6 )  and (1 7 )  g iv etan /J (^ )  =  Vdnfiimf \(16)(17)( 1H>;T he eq s. ( 10 ), ( 11 ), (1 6 )  and (1 7 )  can be used  in eqs. (3 ) and(4 )  for c a lc u la t in g  and , w h ich  o ccm  in theform ulas for the H ig g s  m a sse s . For a g iv en  tan p{~  - - )  , theand Y ukaw a c o u p lin g s  are co m p u ted  at /./ — ni, u sing  the lo llo w in g  relationsli, {m,) =nifini, )174 sin/?Iih ( ftif)174 s i n /?( 1^ )(20)T lic eqs. (1 0 -2 0 ) arc used in the m odel o f  Haber and H em pling  |(t1 lor the ca lcu la tion  o f  R G -rn od ilied  H ig g s  m asses4 . R G -m o d ifie d  I lig g s  M a sses  w ith o u t .s -q u a rk  m ix in gIn out w ork, w c use  the a n a ly tic  .solutions o f  R G B s g iv en  by Farida ef al |7 -9 ]  w h ich  are su m in a r i/ed  in vSectum 3 Wc apply  th ese  tech n iq u es to the o n e  lo o p  lead in g  logarith m ic C P  ev en  H ig g s squared m a ss m atrix  g iv en  by | 6l= '2Dw here the dom in an t e ffe c ts  of R (T -im provcm ent is g iv en  hyM Ikc -  M |7,/ {m, ip ,), m,, i p, ) ) .  (22)p, =-Jin,Msi/sv ' ■ (23)In our study, w c  vary the param eters o v er  a w id e  range  relevant to L E P 2 and L H C .5. RG-modified Higgs masses with .v-quark mixingH ere, w c  c o n sid er  th e e f fe c ts  a r is in g  from  m ass sp littin g  and  m ix in g  in the third generation  .s-quark sector. W e fo llo w  the fo rm a lism  o f  H ab er er al [6J e x cep t that w e co n sid er  the  so lu tio n s  o f  ren o rm a liza tio n  group  eq u a tio n s g iv e n  by Parida etal 17-9]. W e a lso  co n sid er  the param eters al the I .HC range. At o n c -lo o p , the 5-quark m ix in g  in trod u ces the sh ifts A and {A - M l ) -  , th e  r e le v a n tfonnulas are g iven  in R ef. | 6] For non-zero squark m ixing, M is g en era lized  toA  s  M, \ ^Xm, { p ,  ) ,m, , (p , ))+ AM; , „ , ( m, ( p , ) . mt , { . p , ) ) .  (24)'fh e  d eta ils  o f  ca lcu la tio n s  o f  H ig g s  m ass squared arc g iven  in Ref. 161.6. Results and discussionT he results o f  our system atic  stu d ies on  the ligh test H iggs boson  m a sses (///^ )^ in M S S M  arc g iv en  m Ihgures ( 1- 8) fh e  variation  ol w ith different param eters like , fJ, f.l fj, tan /L m loi m ix in g  as w ell as n o n -m ix in g  ca se s  are stud ied . F'lrst, F igure 1 sh o w s that foi tan / i  =  1.3 in the n o n -m ix in g  ca.se, can vaiy  from about 4()tf» 94  GeV', w hen  M >s varied fiom  20 0  to 2 0 0 0\u\\G eV . In the co n tex t ol LH P2 bou n d s on the mass ol the ligh test H ig g s m M S S M , tan p =~- \ 5 is lu led  out. For large Uu\ p ~ 20  and 30 , varies from 93  to 122  G eV s as varies from about 2 0 0  to 20(H) G eV s for ^  ^  1 1eV , I 'FeV In this ca se . is very large ( I T e V ) and /t -  I I'eVMu -  1 le v  MA -  1 TfjV12011010090>'a>S? 80c70idn|H) - J r, tan(H) - 20 trin(H) - 5030 C—..200 4(X) 600 800 100 1200 1400 1600 1800 2(XK)Msusy (GeV)Figure 1. The ladiativciy corrected liphi CP-cven Higgs mass ( hjJ  is plolled against foi = 1 TcV. - 1 TeV in the luscs of tan /i -  1.5, 20, 50N ex t Figure 2  sh o w s that for M^  ^=  1 (K) GeV, varies from  92to 1(H) G e V  in the n o n -m ix in g  ca se  forM^ ^^ ^^  vary ing  from  about 2(H) to 2(XX) G eV  for tan P =  5 0  and / /  =  2(H) G eV . For M^ =  5 0 0  and 1 (HX) G eV s, can va iy  from 9 3  to 130 G eV s for varyingfrom  2(K) to  2(X)0 G e V s for / i  =  2 0 0  G eV . T h is scenario  h o lds for tan )3 =  50 . It m ay be m en tion ed  that in f'igurc 1 a lso , tan p  =  5 0  g iv e s  s im ila r  resu lts on  m. in c o n fo rm ity  w ith  L E P 2 b ounds.654 B Ghosh and S  ChakrabartyA g a in  F igu re 3 g iv e s  s im ila r  stu d ies  as in F igure 2  ex cep t that tan p  (=  1 .5 ) is  very sm a fl in th is ca se . Interesting ly , the v a lu es  arc m u ch  lo w e r  than the L E P 2  bound on H ereFigure 2. is plotted against for tan )3 == 50, ^  of = 100, 500, 1000 GeV; 200 in the casesva r ies  from  3 2  to  7 0  G e V s  as M is varied  from  2 0 0  to  2 0 0 0\ u \ \G e V s. T h is  study su g g e s ts  that the co m b in a tio n  o f  lo w  tan /J ( = 1 .5 )  and lo w  jti (=  2 0 0 )  arc ru led  out in M S S M . O n the other  hand, in F igure 4 , the param eters are sa m e as in FTgurc 3 e x c e p t  that tan P (=  2 0 )  is  larger in th is ca se . F or =  100  G eV , varies from  9 0  to  UXJ G e V s w h ich  lie  b e lo w  the L E P 2 bounds on  . H en ce , M^ -  is ru led  ou t in M S S M . B u t for Af^ =  5 0 0  and 1(X)0 G eV s, m. can  take v a lu es from  9 2  to 12 9  G eV s for  vary in g  from  2 0 0  to  2 0 0 0  G e V s. T h u s, large tan P and lo w  se e m  to  b e  favored  by M S S M .tan(B) = 1 5Msusy (QeV)Figure X  is plotted against for tan /9 s; 1.5,  ^ ~ 200 in the oases of Af, = 100, 500, 1000 GcV.A g a in , F igu re  5 g iv e s  v aria tion  o f  w ith  for  =  1 0 0 0  G e V s , tan p  =  2 0  in the n o n » m ix in g  and m ix in g  c a se s . In the n o n -m ix in g  ca se , v ar ies from  9 2  to  130  G e V s  for A/ ,Msusy (GeV)Figure 4. is plotted against lor tan P  = 20, of Af, = 100. 500, 1000 GcV: 200 in the casesvarying  from  2 0 0  to 2 0 0 0  G e V s. In the m ix in g  ca se  for the sam e  va lu es o f  A/^ ( =  KXK) G e V ) and tan P (=  2 0 ), there is no change  in the variation  o f  w h en  I^h ^suss • T h ese  resu lts are a llo w e d  by L E P 2 b ou n d s on  the lig h test H ig g s  b oson  m ass.MA -  1 TeV, tan(B)  ^20rton mixing mixing...130125120115S  1101051009590200 400 600 800 1000 1200 1400 1600 1800 2000 Msyiy (GeV)Figure 5. is plotted against for = 1 TeV, tan ^ = 20, = 200and /^// = ^ s u s y  in the mixing and non-mixing cases.F igure 6 a lso  g iv e s  s im ila r  stu d ie s  to  that g iv e n  in  F igure 5 e x c e p t that here tan p  is  very  sm a ll (=  1 .5 ). T h e  v a lu es  o f  in the n o n -m ix in g  ca se , vary from  4 2  to  102 G e V s for varying  from  2 0 0  to 2 0 0 0  G eV s in ca se  o f  A/^ =  1 TeV. = 2 0 0 G e V s a n d  th ese  resu lts are n o t p o ss ib le  a cco rd in g  to  L E P 2  data. In the  m ix in g  c a se  for the sa m e  v a lu e s  o f  Af^ (=  1 T eV ) and tan p  (=K enorm alizotion-group-im proved radiatively corrected Hi}^gs m asses etc 655I 5), varies from  about 5 2  to 106 G eV s for /y =  iVf jpcase of v a ry in g  from  2 0 0  to 2(KX) G e V s. H en ce , probably  the com b in ation  o f  large A/^ and sm all tan p  are not p o ss ib le  in M SSM . T h u s, F ig u re  6 sh o w s  that for large M^ ( =  1 T eV ) and very sm all tan P (=  1.5 ), the resu lts in both the n o n -m ix in g  and  m ixing c a se s  are u n a llo w ed  in M S S M  by L E P 2 data. H en ce , w e  should restrict tan P to  large v a lu es .11010090MA = ITeV, tan(B) = 1 5O£1280 ■70605040non-mixingmixingMA= lOOTaV, tan(B) = 20100 r n*mlxing mixing _O’'200 400 600 800 1000 1200 1400 1600 1000 2000 Msusy (GeV)Figure 7. is plotted against for = 100 GeV, tan P  g  = 200 and = ^ ^ u s \  in the mixing and non-mixing cases20.P ( l -5 0 )fo r M ^  =  200GcV, p  :r. 2CX)G eVsand P h ^  ^ ^  I TeV  in the non -m ix in g  and m ix in g  ctises. In the non -m ix in g  case, varies from  75 to  120 G e V s  as tan p  is varied  from  1 to 50. H ere, is very sen s it iv e  to  tan /} for tan )3 varying  from  I 10. T he resu lts on  are not so  se n s it iv e  to  tan p  for tan p varying from  10 to 50 . For ex a m p le , in the m ix in g  case , varies from  118 to 119  G eV  as tan P varies from 11 to 50. For I <  tan p  < 3, varies from 75 to 102 G eV s w hich  are ruled out in M S S M . T lie se  resu lts on  the variation  o f  ni, w ith  M for tan P = 20, M. -  1 T eV  are co m p arab le  w ith  the resu lts o f  other authors {6. 101.200 400 600 800 1000 1200 1400 1600 1800 2000 Msusy (GeV)Figure 6. is plotted agam.si toi I TeV, lan P  I 5, -  200and P h “ m the mixing and non-nuxing ca.sesFigure 7 g iv e s  in the n o n -m ix in g  and m ix in g  ca.ses for -  I (X) GeV, tan P =  2 0  for varying from 2(K) to 2(XX) G c V  In both the ca ses , the m a x im u m  v a lu es  of lie s  b etw een  9 0  and  99 G eV s. H en ce , th ese  resu lts  are ruled out by the L E P 2  data, su g g estin g  that the co m b in a tio n  o f  lo w  and large tan P arc not p o ss ib le . L astly , F igu re 8 g iv e s  the variation  o f  w ith  tan120 115 110 105 f-510001  9590 85 h 80 7570 I—  0MA =- 200GeV. Msusy -  1 TeVn-mixmgmixing10 15 20 25 30 35Msusy (GeV40 45 50Figure 8. is plotted against tan P  lor -  200 GeV, P n  -  IVV, p  ~  200 111 the mixing and non-mixinp ca.ses1-17. ConclusionsT hus, on the basis o f  our stu d ies w ith  the R G -im p ro v em en t in the H ig g s  sector  o f  the M S S M , w e  o b serv e  the fo llo w in g  ‘(i) th e  l ig h te s t  H ig g s  m a s s  in the M S S M  d e p e n d s  sh a rp ly  on  the p a ra m eters  lik e  tan P and  M .\u\\(ii) for A f v a r y i n g  from  2 0 0  to  20(X) G eV , very sm all v a lu es o f  tan P (=  1 .5 ) and ~  UK) G e V ) arc ruled  out.(iii) for M ^, v a ry in g  upto 2  T eV, large v a lu es  o f  tan P (2(L-5b) and (5(X>- I (XX) G e V ) are a llow ed  by I .BP2 data.(iv) the e ffe c ts  o f  running V E V s on  the H ig g s  m a sses  arc sm all { • - few  G ev s).W e find  that the upper lim it o f  the v a lu e  o f  is equal to  130  G e V  (F ig u r e  2 ) . T h e  H ig g s  m a s s e s  u n d er  ce r ta in  se t s  o f  param eters lie  w e ll w ith in  th e  reach  o f  the L E P 2  H ig g s  search . W e h o p e  that so m e  o f  the result.s g iv e n  in our stu d ies w ill be  ver ified  by the future ex p er im en ts.656 B Ghosh and S Chakrahart\Referencestl) J F Gunion, H E Haber, G L Kane and S Dawson T h e  H u n te r 's  G u id e  (New York : Addison-Wcsicy) (1990)[2J H P Nilles P h y s . R e p , 110 1 (1984) , H E Haber and G L Kane P h y s  R e p  117 75 (1985)[1] Z Kunszi and F Zwimer N u c l  P h y s  B385 3 (1992)[4] HE Haber and R Hempling P hy,\ R e v  U t i  66 1815 (1991)[51 Y Okada, M Yamaguchi and T Yanagida P r o ^  T h e o r  P h y s 85 1(1991); J Ellis, G Ridolti and F Zwirner P h y s  L e t t . B257 (1991)16| H E Haber and R Hempling P h y s  R e v . D48 4280 (1993), H f: Haber, R Hempling and A H Hoang Z P h y s . C I S  539 (1997)[71 M K Panda and A Usmam P h y s  R e v  1)54 3663 (1996)[8] M K Panda and B Purkayastha E ur, P h y s  J . C14 159 (2()()())[9] C R Das and M K Panda Eur. P h y s . J  C  (2001) (hep-ph/00l0(K)4i[10] J A Casas, J R Espinosa and H E Haber N u c l  P h y s B526 3 (199m

Renormalization-Group-Improved Radiatively Corrected Higgs Masses in the Minimal Supersymmetric Standard Model at LEP2

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/5981/1/Renormalization-Group-Improved%20Radiatively_By%20B%20Ghosh.pdf

Renormalization-Group-Improved Radiatively Corrected Higgs Masses in the Minimal Supersymmetric Standard Model at LEP2

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository