Muge Boz- The Sensitivity of c->u-gamma decay in the Gluino-Axion Model

8/3/2019 Muge Boz- The Sensitivity of c->u-gamma decay in the Gluino-Axion Model

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Turk J Phys26 (2002) , 1 11.c T UBITAK

The Sensitivity of c u decay in the Gluino-AxionModel Muge BOZ

Physics Department, Hacettepe University,06532 Ankara-TURKEY

Received 25.05.2001

Abstract

The sensitivity of c u decay in the recently proposed gluino-axion model is analyzed for a lightsbottom in a restricted parameter space. The decay rate enhances up to 3% with respect to the SMresult in a moderate range of tan values.

1. Introduction

In the standard electroweak theory (SM) the single phase in the CKM matrix CK M is a unique sourcefor both avour and CP violations. In the supersymmetric (SUSY) extensions of the standard model, there

exist novel sources for both avour and CP violations coming from the soft supersymmetry breaking massterms [1]. The new sources of CP violation can be probed via the avour conserving processes such asthe Higgs system [2] and the electric dipole moments (EDM) [3] of the particles. On the other hand, asearching platform for avour violation is the avourchanging processes generally arising at one and higherloop orders [4,5,6].

It is now a wellknown fact that mixing [7,8] and decays [9,10] of D mesons, as well as K and B mesonprocesses [11] are very important tools in searching for the new physics effects via the loops of the newparticles. In this work, our main interest will be the radiative charm decays. The radiative decay D ,which is represented by the radiative decay c u , at the parton level, has been analyzed using differentmethods and levels of precision [12,13]. In Ref. [14], the sensitivity of this decay is analyzed in detail in thecontext of minimal SUSY standard model (MSSM). In Ref. [15], the analysis of c u has been also donein the same context, but in a restricted parameter space with the light scalar bottom quark [16,17]. Forsuch F = 1 decays, the charginos contribute in a manner with no need for avour-changing mixing [18].Therefore, with the aim of saturating the existing EDM bounds, the sfermions of the rst two generations areassumed to be heavy enough( TeV) and approximately degenerate such that their contributions decouplefor any loop of interest. Within such an approximation, it is clear that the contribution of the chargino-squarkloops will be the only essential ones.

It has been observed in Ref. [16] that the light sbottom ( b1 ) could be as light as the bottom quark itself.Moreover, the corresponding parameter space favours the light stop to be degenerate with the top quarkitself:

M t 1 = 175 GeV and M b1 = 5 GeV . (1)

The compatibility of a light bottom squark in the unconstrained MSSM has been recently studied in Ref. [17],where the renormalization group ow up to Grand Unication Scale is considered. Unlike Ref. [17], in

This work was partially supported by the Turkish Scientic and Technical Research Council (T UB ITAK) under the project,No:TBAG-2002(100T108).

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Ref. [15] only chargino, charged Higgs and W-boson loops give contribution to the underlying model, whereasthe gluino mass is not included. In this work, however, we will make our analysis in the recently proposedgluino-axion model [19,20].

The supersymmetric (SUSY) theories, which are designed to solve the hierarchy problems, possess twohierarchy problems: One concerning the strong CP problem, also existing in the S.M, whose source is theneutron EDM exceeding the present bounds by nine orders of magnitude and the other is the puzzle,concerning the Higgsino Dirac mass parameter , which follows from the superpotential of the model. Asimultaneous solution to these two hierarchy problems is achieved in Ref. [19] in the context of supersymmetrywith a new kind of axion [21,22] which couples to the gluino rather than to quarks. In the gluino-axion modelthe invariance of the SUSY Lagrangian and all supersymmetry breaking terms under U (1)R is guaranteedby promoting the ordinary parameter to a composite operator involving the gauge singlet S with unit Rcharge. When the scalar component of the singlet develops vacuum expectation value (VEV) around thePecceiQuinn scale 10

11 GeV, an effective parameter aTeV is induced. Besides, the low energytheory is identical to minimal SUSY model with all sources of soft SUSY phases. Due to all these abilities of the model of Ref. [19], in solving the hierarchy problems, in the analysis below we will adopt its parameter

space.Assuming that all the sfermions of the rst two generations are heavy enough to decouple and all gen-eration changing entries of the sfermion mass matrices vanish, in what follows, we will analyze ( c u )in units of the SM result. In doing this: ( i) the Wilson coefficients are evolved down to the mesonic massscale for a phenomenologically sensible description; ( ii ) all the supersymmetric contributions, as well as thethe contributions of the charged Higgs and W-boson loops, are calculated at the weak scale; ( iii ) due to itsrestricted parameter space, we work in the gluino-axion model [19], where the soft masses needed for thisanalysis are expressed in terms of the parameter through appropriate avour matrices

The organization of this work is as follows. In Sect. 2, starting from the low energy Lagrangian of themodel, we dene the soft masses which we need for this analysis. We work on the sbottom and charginocontributions in Sects 2.1 and 2.2. In Sec. 2.3, armed with full diagonalization of the squark and charginosectors, we discuss the one-loop contributions to the Wilson coefficients. A numerical study of the theoreticalpredictions in comparison with that of the SM is performed in Sec. 3. We summarize our conclusions inSec. 4.

2. The Model

In the gluino-axion model of Ref. [19], the soft terms of the low energy Lagrangian are identical to thosein the general MSSM:

LsoftMSSM = Q

M 2Q Q + ucM 2u c uc + dc

M 2dc dc + L

M 2L L

+ ec M 2e c ec + Au Q H u uc + AdQ H d dc+ Ae L H d ec + h.c. + M 2H u |H u |2+ M 2H d |H d|2 + ( BH u H d + h.c. )+ M 3 a3

a3 + M 2

i2

i2 + M 1 1 1 + h.c. , (2)

except for the fact that the soft masses are proportional to the the parameter via appropriate avourmatrices. (Note that here, and in what follows, we will neglect the effects of axion, axino and saxinointeractions as their couplings are severely suppressed [19].) The avour matrices form the sources of CPviolation and intergenerational mixings in the squark sector. The phases of the trilinear couplings ( Au,d,e ),the gaugino masses ( M 3 ,2 ,1), and the effective parameter

v2s /M Pl e i QCD / 3a TeV e i QCD / 3 , (3)are the only phases which contribute to CP violation observables. In this formula for parameter, vs

1011 GeV is the PecceiQuinn scale, and QCD is the effective QCD vacuum angle. In the gluino-axion model

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all the soft masses are expressed in terms of the parameter. In our analysis only the chargino and bottomsquark loops contribute to the process, so that we will not need to express all the soft masses in terms of the parameter. Instead we will only need to specify the bottom squark soft masses, and the gaugino masses,namely:

(i) the bottom squark soft masses:

M 2Q = k2Q ||2 , (4)

where kQ is a real parameter;(ii ) the gaugino masses:

M 2 = k2 , (5)

where k2 is a complex parameter.

2.1. Sbottom Sector

We start by analyzing the squark sector where the elements of the sbottom mass matrix M2b,ij can beexpressed in the weak basis ( bL , bR ) as follows:

M2b =M 2b,11 mb(Ab tan )

mb(Ab tan ) M 2b,22, (6)

where

M 2b,11 = k2Q ||2 + m2b + cos2 M 2Z (T bz Qb sin2 w ) ,

M 2b,22 = k2Q ||2 + m2b + cos2 M 2Z Qb sin2 w , (7)

with Qb = 1/ 3, T bz = 1/ 2 and sin w = (1 M 2W /M 2Z )1/ 2 . Moreover, the sbottom soft masses M 2Q areexpressed in terms of the parameter(4). Additionally, the mass-squared eigenvalues are given by

M 2b1 ( b2 ) =12

2k2Q ||2 + 2 m2b + T bz cos2M 2Z

(+) cos2M 2Z (T bz 2Qb sin2 w ) 2 + 4 m2b (Ab R b)2 . (8)In the present scenario the light stop and sbottom masses are known, which allows one to solve one of the unknowns in (8) in terms of these known masses. We choose to solve the sbottom leftright entries(Ab tan ), that is:

(Ab tan ) =1

m2bM 2b1 k2Q ||2 m2b (1/ 2) T bz cos2M 2Z

2

14 cos2M 2Z (T bz 2Qb sin2 w ) 2 . (9)

One notes that the LR mixing parameter depends only on the sbottom soft mass M 2Q which is proportionalto the parameter (4). As usual, the sbottom matrix M2b can be diagonalized via the unitary rotation:

bLbR

= cos b sin b

sin b cos b b1b2

, (10)

which relates the weak basis ( bL , bR ) to the mass eigenstates ( b1, b2), whereas the mixing angle b is denedas

cos b =mb(Ab tan )

m2b(Ab tan )2 + M 2b,11 M 2b1

, (11)

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where M 2b,11 is the 11matrix element of M2b dened in (7), M 2b1 is given by (8), and the numerator and thedenominator of (11) are expressed in terms of (9).

2.2. Chargino Sector

We start with the 2 2 mass matrix [23] of the charginos which are the mass eigenstates of chargedgauginos and Higgsinos:

M C =k2 2M W cos

2M W sin , (12)

where the gaugino masses M 2 are expressed in terms of the parameter (5). The mass matrix of thecharginos (12) can be diagonalized via the biunitary transformation:

C R M C C L = diag( m 1 , m 2 ) , (13)

where C L and C R are 2 2 unitary matrices and m 1 , m 2 are the masses of the charginos 1 , 2 such thatm 1 > m 2 . For convinience, the following explicit parametrization can be chosen for the chargino mixingmatrices:

C L (R ) =cos L (R ) sin L (R )

sin L (R ) cos L (R ), (14)

where the angle parameters L (R ) can be determined from 13. After a straightforward calculation, one gets:

tan2 L (R ) =8M W k22 2 cos2 + 2 sin2 + 2k2 sin2 k22 2 2 (+)2 M 2W cos 2 , (15)

and nally, the masses of the charginos are given by

m2 1(2) =12

k22 2 + 2 + 2 M 2W + ( )[(k22 2 2)2 + 4 M 2W cos2 2

+ 4 M 2W (k22 2 + 2 + 2 k2 2 sin2 )]1/ 2 . (16)

Following the full diagonalization of the sbottom and the chargino sectors, in what follows, we will startworking on the contributions to the Wilson coefficients from the chargino, charged Higgs and W-boson loops.Between these three types of contributions, the rst one is of SUSY origin in which chargino and sbottomloops contribute while the other two are of S.M.

2.3. One Loop Contributions to The Wilson CoefficientsAnalysis of the c u decay follows closely that of the b s decay [11,24]. The effective Hamiltonian,describing a F = 1 transition, is dened at the scale = M W with the standard quark operators

H eff = GF 2

8

i =1C iOi , (17)

where Oi=1 , ,8 are the standard quark operators [25]. For a phenomenologically sensible description, theWilson coefficients C i are evolved down to the mesonic mass scale and, in solving the the Wilson coefficients,two successive steps are followed: rst the Wilson coefficients are evolved from = M W down to = mblevel using ve active quark avors, then these coefficients are run down to = mc level for four active quarkavours. The details of this analysis, at m = mc level, are studied in Ref. [15]. However, we summarize ourcalculational procedure in Appendix A for completeness.

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As all other radiative avourchanging decays, c u decay is described in terms of the coefficient C 7evaluated at = mc :C 7(mc) = C

(0)7 (mc ) + C 2(M W )C

(2)7 (mc ) + C 7(M W )C

(7)7 (mc)

+ C 8(M W )C (8)7 (mc ). (18)

The explicit expressions of each term in (18) are dened in Appendix A. One notes that C 7(mc ) dependson three Wilson coefficients: C 2(M W ), C 7(M W ) and C 8(M W ) and the determination of these short-distancecoefficients reects the properties of the model under consideration. Here, C 2(M W ) = 1, and C 7(M W ) andC 8 (M W ) can be expressed as follows:

C 7(8) (M W ) = C 7(8) (M W ) + C

H 7(8) (M W ) + C

W 7(8) (M W ). (19)

Clearly, these coefficients receive contributions from chargino, charged Higgs and W-boson loops. C 7(8) (M W )is the pure supersymmetric contribution, where chargino ( ) and bottom squark ( b) loops contribute to theprocess. C H

7(8)(M W ) and C W

7(8)(M W ) are the charged Higgs, and W boson loop contributions respectively.

The chargino contribution to C 7(8) (M W ) can be expressed as

C 7(8) (M W ) =2

i =1

2

k =1

M 2W m2 i

m2 iM 2bk

C L 1i S b1k +hbg2

C L 2i S b2k

C L 1iS b1k +hbg2

C L 2iS b2k K 7(8)1 (r ki )

h tg2

m im t

C R 2i S b1k K 7(8)2 (r ki ) , (20)

where

h t = m t g22M W sin ,hb =

mbg22M W cos , (21)

and

r ki =m2 iM 2bk

. (22)

The contributions of charged Higgs bosons are given by:

C H

7(8) (M W ) = 12

r h K 7(8)1 (r H ) + tan

2 K 7(8)2 (r H ) , (23)

where

r H =m2b

M 2H . (24)

Finally, the standard W-boson contributions lead to

C W 7(8) (M W ) = 3 r W K 7(8)1 (r W ) , (25)

where

r W =m2b

M 2W . (26)

The loop functions K 7(8)1 (20), (23) and K 7(8)1 (25) are dened in Appendix B.

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Figure 1 . Variation of R (%) with < 0 for tan = 4 (solid curve), tan = 10 (dashed curve), tan = 15(dottedcurve), tan = 20 (dot-dashed curve) with k 2 = 2 .5 and kQ = 1 .25.

but for convinience, we summarize the data starting from k2 = 2 .5 for different values of kQ in Table 1. Itis also noticable that for values of k2 larger than 3, the data for R(%) are not favorable.In Fig. 1, the variation of R(%) is shown for < 0 , with four different values of tan = 4 (solid curve),tan = 10 (dashed curve), tan = 15 (dottedcurve) and tan = 20 (dotdashed curve). In between thedifferent values of k2 and kQ , we concentrate on one particular k2 , kQ s pair and make our analysis for thecase where k2 = 2 .5 and kQ = 1 .25. Moreover, since

R(%) remains constant at about

0.1% in the rangebetween 500 GeV to 250 GeV, we discard this region of the parameter space which does not make sensefor our analysis. Instead, we concentrate on the interval as ranges from 250 GeV to 80 GeV. Onenotes that as increases, the supersymmetric contribution decreases in accordance with the decoupling of the supersymmetric contribution.

As the gure suggests, for tan = 10 (dashed curve) Rincreases from 0.2% up to 3% level, and fortan = 4 (solid curve) it increases up to 2.5% level as ranges from 250 GeV to 80 GeV. Thoughthe behaviour is similar, for larger tan , Rdecreases faster. Indeed it can be at most 2% for tan = 15and 1.25% for tan = 20. It is observed in this analysis that for larger tan values, tan > 20, theenhancement in the rate remains always below 1%, thereby leaving a small window of observation.As is noted immediately from the gure, although the maximum value of R(%) decreases with increasingtan , the maximum value of R(%) for tan = 10 is greater than the one for tan = 4. This effect canbe easily understood by observing the dependency of tan on the lowest value of as noted before. (The

maximum value of Ris 2.7 for 88 GeV, it decreases to 1.8 at 95 GeV for tan = 10, whereas for tan = 4,its value at 95 GeV is 2.4 but for 88 GeV there is no solution).Fig. 2 is similar to Fig. 1, except for the choice of positive parameter. Being an approximate mirror

of Fig. 1, Ragain decreases with increasing .From the analysis of these two gures showing the variation of R(%) in a moderate range of tan values,in general, one can arrive at a conclusion that theoretically, this region is favorite since the enhancement ismore than 1% for most of the parameter space, that is, b s decay is as well in the allowed band of CLEOdata.

4. Conclusions

In this work, we have analyzed c

u decay in the gluino-axion model for a light scalar bottom

quark. The scalar bottom quark enters the chargino contribution, and thus, inuences the decay rate. For

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Figure 2 . The same as Fig. 1 but > 0.

a moderate range of tan values (where there is no danger of violating the CLEO result on b s decayrate), we nd that the rate is enhanced up to 3% with respect to the SM.Before closing, we would like to point out that: Although the numerical analysis of the branching ratio based on real parameters, namely real gauginomass and the parameter, in general the phases of these parameters are crucial for determining the

CP asymetry of the decay. The phases of these parameters contribute to direct CP violation as wellas D 0 D 0 mixing as in the Kaon and B-meson systems. However, the CP asymetry of this decay hasnot been measured yet, so the complex parameters are not analyzed in this work.

Besides such direct measurements, another important clue for CP violation in charm-quark systems(e.g D-meson) is the formation of P-wave charmoniums; especially the phases of the trilinear couplings,the A-parameters dened in (2), can be directly probed by the observation of such CP odd charmoniumlevels at lepton colliders [26]. Moreover, it is known that the A-parameters are never constrained byany other observable such as the Higgs system and EDMs.

Acknowledgements

The author thanks D. A. Demir for useful conversations.

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Appendix AThe solutions of the QCD renormalization group equations (RGE) can be expressed as:

C k () = U 5(4)kl (, M W )C l (M W ), (28)

where U 5kl and U 4kl are the evaluation matrices dened in ve and four avour contexts, respectively, andcan be determined by the following expressions [27]:

U 5 (mb, M W )kn = Okl a lb O 1kn ,

b = s (M W ) s (mb)

,

U 4(mc , m b)kn = Okl a lbc O 1kn ,

bc = s (mb) s (mc )

. (29)

In the above expression, a l Dll2 0 with 0 =

11 2n f3 , in which f can be taken as 4(5) as needed, and

D = O 1 (eff )T O is the diagonalized 8 8 anamolous dimension matrix.The explicit expressions for each term in (18) can be obtained after the solution of (29):

C (0)7 (mc) =8

i =1

a ib f i (bc ) , (30)

where the numbers a i are given by

a i = 0.609 , 0.696 , 0.423 , 0.261 , 0.146 , 0.522 , 0.899 , 0.409 , (31)and

f 1(bc) = 2 .2996 0 .640bc , f 2(bc) = 1.09 0 .640bc ,f 3(bc) = 0.001 0 .845bc + 0 .027 0 .420bc 0.0004 0 .132bc

0.032 0 .347bc + 0 .483 0 .560bc 0.516 0 .640bc ,f 4(bc) = 0 .0002 0 .845bc 0.0006 0 .420bc 0.0010 0 .132bc

+0 .4 0 .240bc + 0 .107 0 .347bc 0.546 0 .560bc 0.388 0 .640bc ,

f 5(bc) = +0 .00004 0 .845bc 0.00009 0 .42bc 0.0057 0 .132bc+0 .006 0 .347bc + 0 .084

0 .560bc 0.091 0 .640bc ,

f 6(bc) = +0 .017 0 .845bc + 0 .016 0 .420bc + 0 .00042

0 .132bc

+0 .032 0 .347bc 0.084 0 .560bc 0.038 0 .640bc ,f 7(bc) = 0 .0065 0 .845bc + 0 .00044 0 .420c + 0 .00006 0 .132bc

+0 .004 0 .347bc 0.267 0 .560bc + 0 .238 0 .640bc ,f 8(bc) = 0.0004 0.845bc + 0 .00096 0 .420bc + 0 .0008 0 .132bc

+0 .728 0 .347bc 3.744 0 .560bc + 2 .365 0 .640bc . (32)The second term C (2)7 (mc ) is given by

C (2)7 (mc) =2

i =1

r ibcgi (b) , (33)

and the numbers r i read as

r i = 0.560 , 0.640 , (34)

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and

g1(b) = 0 .304 0.899b

0.466 0 .423b

0.111 0 .146b

+4 .872 0 .409b 4.599 0 .609b ,g2(b) = 0.304 0 .899b + 0 .466 0 .423b + 0 .111 0 .146b ,

4.872 0 .409b + 4 .599 0 .609b . (35)For the third term, C (7)7 (mc), in (18) we nd

C (7)7 (mc) = C 7(M W )0 .696b

0 .640bc . (36)

Finally, C (8)7 (mc ) in (18) is given by:

C (8)7 (mc ) = C 8 (M W )2

i =1

s ibc h i (b) , (37)

where

s i = 0.560 , 0.640 , (38)

and

h1(b) = 5.333 0 .609b ,h2(b) = 2 .667 0 .609b + 5 .333

0 .609b 2.667 0 .696b . (39)

Appendix BThe loop functions K 71 (r ) and K 72 (r ) are dened as:

K 71(2) (r ) = I 1(2) (r ) 13

J 1(2) (r ) ,

K 81(2) (r ) = J 1(2) (r ) . (40)

where the functions I 1 (r ), I 1(r ) and J 1(r ), J 2 (r ), can be expressed as:

I 1(r ) =1

12(1 r )4(2 + 3 r 6r 2 + r 3 + 6 r lnr ) ,

I 2(r ) =1

12(1 r )3(3 + 4 r r 2 2lnr ),

J 1(r ) =1

12(1

r )4

(1 6r + 3 r 2 + 2 r 3 6r 2lnr ),J 2(r ) =

112(1 r )3

(1 r 2 + 2 r lnr ) . (41)

K 71(2) (r ) = J 1(2) (r ) 13

I 1(2) (r ) ,

K 81(2) (r ) = I 1(2) (r ) . (42)

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Muge Boz- The Sensitivity of c->u-gamma decay in the Gluino-Axion Model