We present the appearance of some new interaction in the MSSM with CP violation.. We give analytic formulae and perform an evaluation of the eíĩecls of these new interactions on some pro
Trang 1VNU lournal o f Science, Mathematics - Physics 25 (2009) 137-142
Charge parity violation in the minimal supersymmetric standard model (M SSM ) and some new interactions
Ha H uy B an g^ N guyen Thi Thu Huong^’% N gu yen Chinh Cuong^
^ Faculty o f Physics, College o f Science, VhỉU
334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
^Faculty of Physics, Hanoi University o f Education, ĩ3 6 X u a n Thuy, Cau Giay Hanoi, Vietnam
Received 6 July 2009
A bstract We present the appearance of some new interaction in the MSSM with CP violation
We give analytic formulae and perform an evaluation of the eíĩecls of these new interactions on
some processes concerning the productions and decays o f squarks We find that these effects
are topically of —3 5 % to + 3 % depending on each process
Key^vorks: MSSM, CP violation.
1 In tr o du ctio n
Test o f th e discrete sym m etries, charge conjugation c , parity p, and time-reversal T, have played
an im portant role in establishing the structure o f Standard M odel (SM) In particular, C P violation has been observed in the eleclrovveak sector o f the SM in the K and B systems It is linked to a single phase in the unitar\' C ab b ib o-K obavashi-M askaw a fC K M ) matrix describing transitions between the three generations o f quarks; see e.g [1] for a detailed review It is im portant to note that this source
o i 'C P violation is strictly flavour non-diagonal
T h e strong sector o f the SM also allows for CP violation through a dim ension-four term OGG,
w hich is o f topological origin Such a term would lead to flavour-diagonal C P violation and hence to clectric dipole m om en ts (E D M s) T h e current experimental limits on the E D M s o f atoms and neutrons [2-4]
d r i\ < 9 1 0 “ “ e c r n ( 9 0 % C L )
| d / / g | < 2 1 0 ' ^ ^ e c 7 T i ( 9 5 % C L )
\dn\ < G \ Q - ^ ^ e c m { 9 ũ % C L )
how ever constrain the strong CP phase to |ớ| < 10“ ^! A com prehensive discussion o f this issue can be
found in [5].W hile 0 appears to be extrem ly tuned, the C K M contribution to the E D M s is several orders
o f m agnitude b elow the expcrimntal bounds, e.g ~ c m Therefore, w hile providing important constraints, the current ED M bounds still leave am ple room for new sources o f C P violation beyond the SM
’ Coưcspondin g author Tcl.: 0 9 8 3 1 9 9 8 7 0
E-mail: n g u ye n h u on gl982@ yah oo.c om
137
Trang 2Such new sources o f CP violation arc indeed vcr>' in teresting in point o f view o f the obserx'cd bar\'on asym m ctr\ o f the Universe
' h with 77/Í, riịị and the num ber densities o f bar>'ons, a n tib a rjo n s an d photons, respcclivelv; s e c [6,7] for rccent reviews T h e necessar>' ingredients for b a rjo g e n e s is [ 8 ] i) baryon num ber violation, ii) c and CP violation and iii) departure from equilibrium are in p rin c ip le present in the SM , h o w e v e r not with sufiicient strength In particular, the am ount o f CP violation is not enough I'his provides a strong motivation to consider CP violation in extensions o f the SM, as review ed e.g in [9]
In general, CP violation in extensions o f the SM can be e ith e r explicit or spontaneous Ex plicit CP violation occurs through phases in the Lagrangian, w h ic h cannot be rotated aw ay b v field redefinitions This is the standard case in the M S SM , on w h ich I w ill concentrate in the following Spontaneous CP violation, on the other hand, occurs if an extra H iggs field develops a c o m p le x vac uum expectation value T his can lead to a vanishing ỊÈ term as w ell as to a com plex C K M matrix Spontaneous C P violation is a very interesting and elegant idea, but difficult to realize in S U S Y and obviously not possible in the M SSM (where the Higgs potential co nserves CP) T here has, how ever, been ver\' interesting new w ork on left-right symmetric models and S U S Y G UTs For instance, m odels based on supersym m etric SO(IO) may provide a link with the n e u trin o seesaw and leptogenesis 1 do not follow this further in this talk but refer to [9] for a review
It is noted that C P violation in the M SSM alone is a large field willi a vast am ount o f literature;
In this pape.; w e present the appearance o f new vertices co n ce rn in g w ith interaction o f squarks in the
M SSM once CP violation taken into account T hen w e give anal>tic form ulae and numerical results to evaluate the efTects o f these new interactions to some processes co n ce rn in g w ith squarks productions and decays
138 //.//, Ban^ ct al ' \'NU Jounuii o f Scicncc Síaihetnaíics - /Vỉv.v/cv 25 (2009) Ỉ 3 7 - N 2
2 C P violation in the M S S M and the appcarancc o f som e n ew sq u a r k interaction vertices
The M inim al S upersym m etric Standard Model (M S S M ) is coiisidcrcd the most attractive ex te n sion o f the Standard Model M any phenom enological studies on S U S Y particle searches have been performed in the M S S M with real SUSY parameters In general, how ever, som e o f the S U S Y pa rameters may be com plex, in particular the higgsino mass p ara m e te r /i, the gaugino mass param eters
A/i 2 3 and the trilinear scalar coupling paramc* "s A j o f the s fc rm io n s / ,
thus inducing explicit C P violation in the model Not all o f th e phases in eq (1) are, however,
physical T he physical com binations indeed are A r g {M ^ ỊÌ) and A r g { A j ^ ) T h e y can
-afTect sparticle masses and couplings through their m ixing,
-induce CP m ixing in the Higgs sector through radiative corrections,
-influence CP-even observables like cross sections and b r a n c h in g ratios,
-lead to interesting CP-odd asym m etries at colliders
Non-trivial phases, although constrained by E D M s, can h e n c e significantly influence the coliider phenom enology o f Higgs and SUSY particles, and also the prop erties o f neutralino dark matter
Trang 3/ / / / Banịỉ, cl aỉ / VNU Journal o f Science, Mathematics - Physics 25 (2009) 137-142 139
In the M S S M cver> q uark has tw o scalar partners, the squarks and (] r , corresponding to the
left a n d the right hclicily states o f a quark In general and (Ịĩỉ m ix to form mass eigenstates q\
and i/2 (with ), the siz e o f the mixing being proportional to the m ass o f the corresponding
quark q [10] and so ncliiiible cxccpt for the third generation T h e mass matrix in the basis { q i ỉ ị u ) is
i^iven by 10
/ , ^ ^ 2
<ÌL Clqĩnq m
<1R /
<72 / with
r n j = M '^ + m \ CO S2/3(/|^ - e , sin^ 9 \v ) +
Q r;i:„ = i \ / ? + T7|| cos 2/?e, sii/^ Ow) + rtii,
'ÌL
2
(3)
(4)
Here Ị'ị is the third com ponent o f the weak isospin and the electric charge o f the quark q M ộ Q ịj and A t I, arc soft S U S Y -b re a k in g param eters and ta ii/3 = V 2 / V \ with V Ị (V2) being the vacuum
expectation value o f the Higgs field / / ỹ (H ^ ) A ccording to eq (2), M 'ị is diagonalized by a unitary
matrix R'' T he w eak eigenstates <7i and ~12 are thus related to their mass eigenstates q i and ĨỊiì by
K, QR
With com plex param eters, w e have
n'< = e ầ ^ i c o s O q e 2‘^'ỉsiiiớự ''
Let us turn to the im pact o f the CP phase ộq to the squark interaction O ur terminology and
n o t a t i o n ar c aa in R c f [ l 1] T h e r e l e v a n t p a r t s o f t he L a g r a i i g i a n f or s q u a r k i n t e r a c t i o n s
(<■' Havor indices) are given by
£ „ n = n : e „ A , ụ r ị , U ] , R ị , R ] ^ ) q ] l ) - i e e , A ~ 0 , , q ^ ^ >%
i ' , , , - ( R ĩ , n ỵ ) c ‘'„ ĩ r „ 'd ‘% =
where ĩ„ = ;ỉ; ' ;ỉ] , + ;ỉ? ,í ỉ’,,^
In case o f CP conserv ing M S S M (say Ộ,Ị = 0), R ’’ is real and vve have
(7) (8)
=
Hercfore, only interaction m odes w ith i = j exist, (for example: Ỉ2 Ỉ2 9, bỵ bị 7 ) If CP
tion is taken into account {ộfỊ Ỷ 0 )> using (6) w e have
\
Thus, in this case, new interaction vertices w ith i Ỷ j appear, e.g ?2 ^ 7 , b^bxQ- T hese new couplings are nonzero and depend on ộq, h e n c e giving contributions to the decay w idth and production cross
section o f squark as vve will see below
Trang 4140 //.// Bang et al / VNU Journal o f Sciencc, Mathematics ' Physics 25 (2009) Ĩ37-Ỉ42
3 Numerical results and d is c u s sio n s
Our terminologies and nota tions arc as in Ref [12, 13] Firstly, let us co n sid er the proc;esscs
[M ~ — q^qj {( ~ ) in w h ich " is one o f the exchanged bosons
(a)
-h
(b)
Qx
<n
(Z,
Qi
(Ii
Fig 1 F e y m a n d ia g r a m s for t h e p r o c e s s — > Qicjj ( £ '^ £ = e ~ ^ e f-i ) ), ( a ) C P c o n s e r v i n g c a s e ,
( b ) C P v i o l a t i n g case.
The cross sections for th e s e processes read [12,13]
and
(10)
(11 (
^ ■ T v v + T „ n + '^^ 7 V //j ,
w h e r e
r7i^ ,a^,/;^ ,sm /?|(G ^),j|(c+ f4 c?,.i + d \ c , j d z ) s s - M ị
In case o f CP conserving, there is only z exchange in the interaction m ode w ith i ^ j W hereas, in
case o f CP violating, there are both z and 7 contributions (see F ig.l) For the range 0Ĩ Ộ — [0,0.1
we find that the term propotional to ỗij (w ith I Ỷ j ) w hich arises from new vertices QiQj'y (i Ỷ j ) can contribute from (- 1% ) to (+ 1% ) to the cross sections o f these processes, nam ely e “^ e“ -+ u t j ,
e '^ e ^ —> bibj, ịx~ —+ u t j , b^bj (i Ỷ j ) ' For a w ider range o f Ộ — [0 , 1] w e find that these
term s can contribute up to from (- 3.5% ) to (+ 3% ) to the cross sections o f these above processes
Trang 5In adtdilion, n ew vertices t ị ĩ ị l - , also allow for the d eca y s o f squarks to photon and gluon
correction in case o f CP conserving MSSM [14], At tree level, the amplitude o f the above decay has
the gemeral form
A/"(^7T - + Ả : 2 ) % ( ^ ' 3 ) , ( 1 2 )
with; A’l, k '2 and Ả’3 are the four - momenta o f gj', and V ( V = 7 ,ổ )»
(14)
T h e s e interesting results will aíTect the decay pattern o f squarks and can not be neglected in he detailed
s tu d y o f s q u a rk s
/ / H Ban^ et al / VNU Journal o f Science, Mathematics - Physics 25 (2009) Ỉ37-Ì42 141
<7? k ,
^’3
7 (g)
Fig 2 Feyman diagrams for the decay q f 7 (5
)-Qi _Q i - - -
■g
(ik
g
Fig 3 Example diagrams for real gluon emission in squark decays into vector bosons
(a) CP conserving case, (b) CP v io latin g case
Moreover, in o th e r decay modes o f squarks ^ Qj V ( V is gauge bosons z , 7 , g, W"^),
at lepton colliders £~^£~ —V q^Qj, the infrared divergences occur w h e n the one loop vertex corrections
are included In Oder to cancel these infrared divergences, w e need to add real gluon em issions (see
Trang 6Fic-3), and this will lead to the ap p carance o f the new terms propotional to (with ? -Ạ j ) in the
fonnulea o f dccay w idths and cross sections We take some evaluation for the raimc o f 0 ~ [0,0.1 and find that these term s can contribute from (- 1%) to (+ 0.5% ) to the decay w idths o f the processes
—* (f^ 4- V', from (- 1.2%) to (- 0 5 % ) to the decay widlli o f the processes ỹp —^ f H and from (- 0.4%) to (+ 0.1% ) to the cross section o f the processes e ^ e ~ Ĩ J j , e~^e~ hJ)j, ĩĩĩj^
» b^hJ Particularly, there are som e cases with large contribution, from (- 2.2% ) to (- 1.5%),
for example the decay For a w ider range o f — [0 , 1] we find that these terms can
contribute from (-1% ) to (+0.5% ) to the cross sections o f the processes e'^e~ u t j , e ^ ( r hibj^
In conclusion, we have represented the appearance o f new vertices concerning with squark interaction We also deduced the form ulea and evaluated analytically the contributions o f these new interactions to s o m e o f the d e c a y a n d p ro d u c tio n o f s q u a rk s and found that th e y are tvpicallv o f {- 3.5% ) to (+3 % ) dep en d in g on particular process T his could have an im portant im p'ication in the determination o f the M S SM at future linear colliders
A c k n o w le d g m e n ts T his work w as supported in part by Project on Natural Sciences o f Vietnam National University (I'h c Strong S cientific Group on Theoretical Physics)
142 //.// Banịr eí a i / VNU Jo u n w ! o f Science, Maíhemaíics - Physics 25 (2009) ỉ 37-142
References
[1] A.J Buras, arXiv:hcp-pli/0505175.
[2] B.C Regan el ail., ỉh iy s Rev Lett 88 ( 2 0 0 2 ) 0 7 1 8 0 5 .
[3] M v Romalis, v c Grifiith H.N i'orlson Phys R e v L ett 86 (2001) 2505.
[4] P.O lỉarris, ct a l l , P hys Rev Lett 82 ( 1 9 9 9 ) 904.
Ỉ51 M Pospeỉov A Ritz A n n a ỉs P hvs 318 (2 0 0 5 ) 119 larXiv; hcp-ph/050423 1Ị.
[6] M Dine, A Kuscnko Rev M od r h y s 16 (2 0 0 4 ) 1 |arXiv: hcp-ph/0303()65|.
|7] J M C'linc, arXiv: hcp-ph/0609145.
| 8 | A.D Sakharov P tsm a yji E ksp Tear, h'iz 5 (1967) 32.
19) T Ibrahim, p Nath arXiv:hcp-ph/0705.2008.
|1 0 ] J Ellis S Rudaz / V ụ x L ett h 128 ( 1 9 9 3 ) 248.
[ I l l A Barll cl al., r h y s Lett, ỈÌ 4 1 9 ( 1 9 9 8 ) 243.
[12] N TT Huong IỈ.II Bang, N , c Cuong, D.T.L Thuy, ỉnl J o f T h e o r r h y s 4 6 (1) (2007) 4
[13] N c Cuong IỈ.H Bang N T T Iliiong, D.T.L Thuy IC T P p re p rin t IC (2 0 0 5 ) 035.
[14] A Bartl et ciL arXiv: h cp -p h /9710286.