ε

Volume 145B, number 5,6

PHYSICS LETTERS

27 September 1984

GLUINO PENGUINS AND ~'/~ J.-M. GI~RARD 1, W. GRIMUS 2 and Amitava RAYCHAUDHURI 3 CERN, Geneva, Switzerland Received 15 June 1984

We calculate the penguin diagrams involving gluinos and discuss their implications on e'/e. The dominant contribution originates in the Kobayashi-Maskawa phase. Fixing this phase from the e parameter, we f'md a smaller value of e'/e than in the non-supersymmetric case alone.

Ever since its experimental discovery in the kaon system, CP-violation has been a fascinating field of theoretical research. Unfortunately, until now no CP-violating effects have been found elsewhere, so the main focus is still on K 0 - K 0. In this paper we will concentrate on the supersymmetry (SUSY) contributions to the CP-violation parameter e'/e for which the latest experimental number is e'/e = - 0 . 0 0 4 6 -+ 0.0053 + 0.0024 [ 1 ]. The very successful standard model offers the nice possibility of explaining CP-violation through the Kobayashi-Maskawa (KM) mechanism and predicts a non-vanishing e'. Recently however it has been claimed that the standard model might not be able to account for e if the top quark turned out to be light [2]. The reason is that the KM angles 02 and 03 are constrained to be rather small [3] because of the relatively long b-lifetime [4] and the small ratio/~ = F(b ~ uev)/ F(b ~ cev) ~< 0.04 [51. Writing e and e' in detail one obtains

e = exp(irr/4)(em + 2~)/2x/~-, em= I m M 1 2 / R e M 1 2 ,

e' = i c o ( I m A 2 / R e A 2 - ~)/x/2~- -ico~/x/~

(1¢ol ~- 1 ) ,

~ = ImAo/ReAo,

(1)

where A0 a n d A 2 are t h e I = 0,2 amplitudes for K 0 ~ rrzr and ~ is given by the so-called penguin diagrams [6,7]. In the standard model Im A2/Re A2 is very small and we will neglect it. There are two major theoretical uncertainties in the calculation of the K 0 - K 0 transition matrix element M12. Firstly, long distance contributions [8] to M12 cannot be calculated reliably and secondly, in the box diagram calculation [9] there appears an inaccurately known parameter B which is the correction factor to the matrix element calculated by vacuum insertion. In general, long distance contributions can modify the CP predictions of the box diagram considerably [10] ; however, arguments have been advanced that there may be important cancellations in the imaginary part of M12 [11]. Both large B and long distance effects inM12 could conspire in such a way as to give enough CP-violation in the standard model. On the other hand, it has been shown that even in the worst case for the standard model, small B [12] (B = 0.33), low top quark mass, rnt, and no long distance contributions, CP-violation can be satisfactorily explained [ 13,14] by going to the supersymmetric extension and including soft breaking terms ,1. In particular, the gluino (~)-squark ( ~ box diagram can account for the missing CP-violation in M12 where the relevant source of CP-viola1 IISN fellow, on leave of absence from Institut de Physique Thdorique, Universit6 de Louvain, Louvain-La-Neuve, Belgium; address after September 1984, Max Planck-Institut f'tir Physik und Astrophysik, Munich, Germany. 2 Address from July 1, Insfitut f'dr theoretische Physik, Boltzmanngasse 5, Vienna, Austria. 3 On leave (1983-1984) from the Department of Pure Physics, Calcutta University, Calcutta, India. ,1 For a review see ref. [15]. 400

0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 145B, number 5,6

PHYSICS LETTERS

27 September 1984

% d(p')

=

.......

% t

=

s(p)

d(p') 4

,-------------7

dj k~

$(P)

d(p')

I k~

a

Ib

~

L-_-_-_-_-_4 ~

s(p)

d(p') .-'-------------~

:

s(p)

dj

"-__."

G

"

%

k~ G

k~

G

G c

b

d

Fig. 1. The pengluino graphs.

tion in the ~ - d - d interaction is again the KM phase 8. A new phase ¢, the "SUSY phase", can only enter in contributions to M12 through dL--dR mixing but its effect on e is very small. This phase is a combination of possible phases in the gluino mass (~) and in the parameterA which appears in the soft SUSY breaking trilinear scalar couplings [ 15]. In view of this interesting result, it is important to examine also the SUSY contribution to e', i.e., the superpenguins [ 16]. In this paper we make a detailed study of the penguins involving gluinos, the "pengluinos" (fig. I), which constitute the most important SUSY contribution to e' originating in the KM phase 8. Using restrictions for the SUSY phase ¢ from the electric dipole moment of the neutron (EDMN) [17,13], we also look for additional effects of ¢ in e'. We begin by writing down the ~ - d - d interaction:

£'ff'dcl= ix/~gsdti~a ~xa(pLPL + FReR)ildj + h.c.,

eL = ~(1 -- 75),

PR = ½(1 + 75),

(2)

where gs denotes the coupling constant of the strong interactions and the Xa's are the Gell-Mann matrices of SU(3)c. In the basis where the quark mass matrices are diagonal (~/u, d), the down squark mass matrix is

(la2Ll+l~12d+CK~fM2uK

IA I/.~/d '~

(3)

K is the KM matrix,/JL, R and/a come from soft SUSY breaking terms. The parameter c characterises radiative corrections to the tree level squark mass matrix which introduce flavour violation [ 18]. It is in general between 0.1 and 1, and negative. In most of our calculations we will take c = - 1 , which favours SUSY contribution to e. The 6 X 3 matrices FL, r R appearing in eq. (2) are given by [13] (FL, F R ) = U t ( ; i~1

)0e _ i 1~ ,

(4)

where U diagonalises the down squark mass matrix in (3). The mechanism of flavour violation through eq. (2) explained above also leads to interesting results in the B0-B 0 system, i.e. large mixing and enhanced directly measurable CP-violating quantities [ 19]. After having done the momentum integration, we obtain for the graphs of fig. 1 1

(Da)~a= -(i/167r2)g3sC2(G)~xaO]f dx f 0

1-x dy { { ( 2 - -~n) - 1 - 7 - 1 - l n t / ~ . / / ( 4 r r p . 2 ) ] ) 7u

0

+ [x~ + (1 - y ) g+ ~] v"(x~ - y ~ + m)/O/} oj, Dj=M2x + ~2(1 - x ) - x ( 1 - x ) p 2-y(1 - y ) k 2 - 2 k . p x y ,

(5)

401

Volume 145B, number 5,6

PHYSICS LETTERS 1-x

1

(Db)au =-(i/161r2)g 3 [2C2(R ) - C 2 ( G ) ] ~xaO]

f f 0

27 September 1984

dy (((2 - ~n) -1 - 7 - ln[D//(4~rp2)l} 7 u

0

+ [(1 - x) ~b+ y[: + t~] [2xp + (1 - 2y) klU/D]} 0 ] , D] = Fn2x + M2(1 - x ) - x (1 - x ) p2 _ y (1 - y ) k 2 - 2k . p x y ,

(De+ Dd~a = --ig s ~ha [Tu(~ -- md) -1 X(p) + Z(P')(~' - ms) -1 7u] , 1

--iS(p) = (i/161r 2) 2g2C2(R) O] f dx[(1 - x ) ~ + ml ((2 -- ~n) -1 - 7 - ln[E]/(4zrp2)]) O j , 0

E/=(I -x)M2+xFn2-x(1

-x)p

2,

k=p ' -p.

(5 cont'd)

Mj denotes the down squark masses. We have used dimensional regularisation (n = number of dimensions, 7 = Euler constant and/a is a scale parameter with the dimension of mass). Oj, 6j are given by

o;: rVLSPL+ r~spR,

O,: r~d*SR + r~d*PL.

(6)

In order to keep track of the origin of the different terms in eq. (5), we have retained the quadratic Casimir operators C2(R) = (N 2 - 1)/2N, C2(G) = N of the fundamental and the adjoint SU(N) representations respectively. In eq. (5), the infinities proportional to C2(G) cancel each other as they must. The reason is the conservation of the gluonic current (SU(3)c is unbroken) and the Ward-Takahashi identity

~r~:gs~x"[z(p)-z(p')],

Fg:i(Da+Db~,

(7)

which is preserved by the renormalised quantities. Because the self-energy Z contains only C2(R), the infinities proportional to C2(G) must cancel each other in F~. But, of course, as the alert reader has already noticed, in this case all infinities are cancelled anyway by the GIM mechanism: F/~*~s : 0 ,

A,B=L,R.

(8)

However, the previous observations are independent of(8) and serve as a check. Finally, we want to remark that on mass shell, using the renormalised vertex r ~ which is obtained by the renormalisation conditions (7) and [20] XR(p) Us(P) = 0 ,

t~d(P') XR(p' ) = 0

(9)

(~, X R denote the transition self-energies s "+ d) is equivalent to summing over all (unrenormalised) diagrams a,b, c, d. Taking the latter option, after having put the wings of the pengluino on shell, we obtain our final result:

u-a(P') i(Da + Db + De + Dd)~ us(P) = (g3/16zr2ffz2) ud(p') -12Xa X {3( 1 k 2'Yu - kt~)[C2(G) A (z]) + C2(R) B(z/)I(F]Ld*I~PL + l~d* F~PR) + io#vk ~'([C2(G) C(z]) -- C2(R) D(z])] [rnd(r{d* ULsPL + r~d* r~PR)

+ ms(~a* r[SPR + lqRd*r{{eL)] + [C2(G)E(zj)- 4C2(R)C(z,)l ~(IVLa*rffPg + URd*IVLSPt)}}Us(P), zj = M2/fft 2. 402

(10)

Volume 145B, number 5,6

PHYSICS LETTERS

27 September 1984

The functions A, B, C, D, E are given by A (z) = ½(1 - z) - 2 [3 - 3z + (2z - 1) In z] ,

B(z) = (1 - z) - 4 [_L~ + 3z - 3z2 + ½z3 - l n z ] , C(z) = ~(1 - z) - 3 [1 - z 2 + 2z In z] , D(z) = ( l - z) - 4 [1 + 1 z _ z 2 + l z 3 + z In z] , E(z) = (1 - z) - 2 [1 - z + z In z] .

(1 I)

Here ms, m d should be considered as constituent quark masses. Since squarks and gluinos are expected to be much heavier than the ordinary hadronic scale, we have fft2x + M2(1 - x ) ~ m 2, m 2, k 2

(12)

for all x E [0, 1]. In eq. (10) we have made an expansion in the latter quantities which are all of the same order and dropped the terms of higher order. In the ordinary penguin there is no ouvkV-term which mixes left- and right-handed quarks and moreover, the ms, dour kV-terms which can in principle be o f the same order as the 7uk2-term are suppressed even if mt is large [20,21]. In the SUSY case we have no reason to expect a suppression of the auvkV-terms and this is borne out by our numerical results. To calculate the phase of the amplitude A 0 ( K 0 -+ rrrr) we have to estimate the matrix elements of the relevant operators. One of them is the well-known operator 05 [22] in the case of which the gluon propagator is cancelled by the k 2 from the vertex whereas the other one containing au~ k v is non-local. To evaluate the matrix elements we use the vacuum insertion m e t h o d [22] with the modification proposed in ref. [23] to also include m o m e n t u m dependence in the pseudoscalar amplitudes in order to be consistent with PCAC. To evaluate the matrix element of the second operator [24] we replace the gluon propagator by MI~ 2, where M H is a typical hadronic mass scale, a procedure which should at least give us the right order of magnitude. In the numerical analysis we will choose M H to be the kaon mass. Now we can write down our results for

= - f Im ~°/Re 7 ,

(13)

where = GF/(zr~¢/2) G A - (as/3Tn2)[C2(G)A (zj) + C2(R ) B(z])]

-- (as/~2)([C2(G) C(z]) - C2(R ) O(zj)l(r

d*

+ [C2(G)E(z/) - 4C2(R) C(zi)] (I~d* r/Rs --

r'Ra*r Lb ~ }

(r'La*r[ + r d*

- r~a*r~)(ms

- mQ)

× (3mQ/M2H)[f+ + (fK/f~r)(l + 2m2/m2)]/[--f+ + (fK/f~r)(1 + 2m2/m2)].

(14)

GA comes from the contribution of the ordinary penguin and in our calculation we use the exact expression of ref. [20]. fK --~f~r= 93 MeV and ma is the mass of the 0 + scalar meson which we take to be 700 MeV. f i s defined as the fraction o f the total amplitude A 0 which can be attributed to penguins [7]. a s is the strong coupling constant which appears in the loop and will be fixed at 0.1. mQ denotes the u, d constituent quark mass; we will choose mQ = 300 MeV and ms = 500 MeV. f+ is defined in ref. [24] and its numerical value is f+ ~. 1. In evaluating the necessary matrix elements we neglect m JKm. 2 2 As far as the zk/= 1/2 rule is concerned [22], penguin contributions are negligible for reasonable values o f the squark and gluino masses [25] (in the numerical examples below, the real part o f the SUSY penguins does not ex403

Volume 145B, number 5,6

PHYSICS LETTERS

27 September 1984

ceed 5% of that of the conventional penguin). It is therefore legitimate to take the value of the parameter f from non-SUSY considerations. We choose f = ~ to be in agreement with ref. [26]. Before turning to the discussion of the numerical results, we briefly describe our procedure [19,2]. Given the b o t t o m lifetime r B and the ratio/~, the KM angle 03 is tixed and 02 is obtained as a function of the KM phase G. Now, for a fLxed rot, the experimental value of e is used to fix 6, where the theoretical expression for e is given by the sum of the ordinary W-box and the gluino box diagrams. The contribution of ~ is neglected because of the smallness of e'/e (see numerical results below). As in the standard case [26], for every rB there exist two solutions for G for both of which sin G > 0. We will vary rB but will always keep/~ = 0.03. Let us now mention the SUSY input values and the range of the top quark mass. To get a significant SUSY contribution to e we choose a large c (c = - 1 ) and rather light SUSY mass parameters, namely ffz = 40 GeV,/a L = 50 GeV, 2 _ tZ2R= 100 GeV 2 and/~ = 40 GeV. As it can be easily seen from Md~, one obtains an upper bound: m 2 in order to keep SU(3)c X U(1)em unbroken (i.e. a l l M2 > 0). This is a consequence o f c < 0. Therefore, in this case, mt ~ 45 GeV to have a reasonably heavy bL, which is the lightest down squark in this scenario. Finally, we should stress again that we always take B = 0.33. As far as our numerical results are concerned, one should keep in mind that there is some uncertainty involved in the ratio of the matrix elements in eq. (14) which might affect the relative size of the 7u- and cr~vkV-contribu tions; nonetheless the general features should remain unchanged. It turns out that e'/e is somewhat diminished by the SUSY contribution. This point deserves some elaboration.

-la2/c

¢'/¢ 0,026.

""'.. 0.022

0.020

c'/¢

......... Ord nary :O.S ....

'...

,.....

•...

"-,4.0

0.024 rdinary ....

0.022

'...

0.020 c=-1/2, A=3

'-... 0.018

0.018

~,~"'...... ,~ "...

0.016

0.016

~x

0.016.

"\.

c:-1, A=3

13 ,

0.01/, .......

0.012

0.012

0.010

0.010

O. 008

"~.

0.008

0.006

0.006

0.006.

0.004

0.002

0.002

c=-1, A=I

0 I

i 29

~

i 1

33

L 35 37 mt (GeV)

1 39

/.1

6.3

c=-1, A:O

0

6-5 -0.002

Fig. 2. e?/e as a function of the top quark mass in the standard model for/3 = 0.5 (dotted lines) and in the SUSY model (characterized in the text) for fl = 0.5 (dot-dashed lines), 1.0 (full lines) and 1.5 (dashed lines). The bottom lifetime e B = /3 × 10-12 s. In this plotwe choose the flavour violation parameter c = - 1 and the d L - d R mixing parameter IAI = 3.

404

0.6

I

0.8

t 1.0

i

1.2

1.14

~ 1.6

1.18

2.10

2.2

x B (10"12sec)

Fig. 3. e?/e as a function of the bottom lifetime TB for a top quark mass equal to 40 GeV for the SUSY model (for different choices o f the parameters c andA) and the standard modeL

Volume 145B, number 5,6

PHYSICS LETTERS

27 September 1984

The SUSY contribution to e'/e consists of three terms which correspond to 7,, ms, dotty k v and ~na, v k v in eq. (10) and are of the same order as the ordinary contribution. It can be checked that for negative c, the last one, which is proportional to d E - d R mixing, has the same sign as the ordinary contribution, while the other two are of opposite sign. Therefore f o r A = 0, the SUSY contribution to e'/e is negative, whereas for [A [ = 3, it turns out that the last term dominates and the net SUSY contribution is positive. However, the total (ordinary + SUSY) e'/e is always less than the value obtained from the non-SUSY theory alone. This is due to the fact that when SUSY contributions are included, the e parameter can be fitted with a somewhat smaller KM phase ~ and even the ordinary contribution to e'/e is thereby reduced. The s u s Y phase ¢ can only have an effect through the fho, vlf-term and f o r A :/: 0. But the numerical calculations show that with the bound on ¢ from the EDMN ( I¢1 5 10-2), it has at most an effect of 10% on e'/e. Since this would shift the curves only marginally we do not show it in the plots. In fig. 2 we exhibit the behaviour o f e'/e as a function o f m t . We have taken [A [ = 3 and have p l o t t e d curves for several values of the b o t t o m lifetime rB. Since e'/e is a double-valued function of rB, it must also be so as a function of rot. This is indeed so in fig. 2 where for comparison we have also presented the ordinary curve for rB = 0.5 X 10 - 1 2 s; for the range of mr in this figure, the other two values o f r B are excluded in the ordinary case by the e parameter. In fig. 3 the top quark mass is fixed at 40 GeV and the behaviour as a function of rB for different choices of IA[ and c is shown. F o r A = 0 there is a remarkable, although accidental, cancellation between the SUSY and the ordinary contributions and e'/e is even slightly negative for the input values we have taken. In some situations therefore super-Kobayashi-Maskawa CP-violation mimics the superweak model predictions. Finally, we should point out that we have used/~ = 0.03. F o r smaller/~, the curves lie within the corresponding ones that we have presented. In conclusion, we can say that in addition to the nice features of CP-violation induced by gluinos in K 0 - g . 0 mixing, this mechanism is not endangered by EDMN [19] and e'/e. The prediction for e'/e is smaller than that in the standard case for negative values of the f l a v o u r v i o ~ t i o n parameter c, which is preferred by radiative corrections. Furthermore, our calculations indicate that dE--dR mixing is o f significance as far as e'/e is concerned. After we had finished this work, we received a preprint by Langacker and Sathiapalan [27] which also discusses the gluino contribution to e'/e. Our study is however more complete because we include d L - d R mixing and the ms, dO~v k v-term. It is a great pleasure to thank A. Masiero and D.V. Nanopoulos for their interest in this work. W.G. enjoyed many informative discussions with G. Ecker on penguins and CP-violation.

References [1] University of Chieago-Saclay Collab., B. Winstein, results presented at CERN (June 1984). [2] P.H. Ginspaxg, S.L. Glashow and M.B. Wise, Phy~ Rev. Lett 50 (1983) 1415; A.J. Buras, W. Srominski and H. Steger, Nuel. Phys. B238 (1984) 529. [3] K. Kleinknecht and B. Renk, Phy~ Lett. 130B (1983) 459. [4] MAC CoUab., E. Fernandez et al., Phys. Rev. Lett. 51 (1983) 1022; Mark II Collab., N.S. Lockyer et aL, Phy~ Rev. Lett. 51 (1983) 1316. [5] CLEO Collab., A. Chen et al., Phy~ Rev. Lett. 52 (1984) 1084. [6] J. Ellis, M.K. GaiUard and D.V. Nanopoulos, Nucl. Phy~ B109 (1976) 213. [7] F.J. Gilman and M.B. Wise, Phys. Lett. 83B (1979) 83. [8] L. Wolfenstein, Nucl. Phys. B160 (1979) 501. [9] M.K. Gaillard and B.W. Lee, Phys. Rev, D10 (1974) 897. [10] J.F. Donoghue and B.R. Holstein, Phy~ Rev. D29 (1984) 2088; G. Ecker, University of Vienna preprint UWThPh-1984-21 (1984). [11] C.T. Hill, Phy~ Lett. 97B (1980) 275. [12] J. Donoghue, E. Golowich and B. Holstein, Phy~ Lett. l19B (1982) 412. [13] J.-M. G6rard, W. Grimus, A. Raychaudhuri and G. Zoupanos, Phy~ Lett. 140B (1984) 349. 405

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[14] J.-M. G6rard, W. Grirnus, A. Masiero, D.V. Nanopoulos and A. Raychaudhuri, Phys. Lett. 141B (1984) 79. [15] E.g, H.P. Nilles, University of Geneva preprint UGVA-DPT 1983/12-412. [16] S.-P. Chia and S. Nandi, Phys. Rev. D27 (1983) 1654; E. Franco and M. Mangano, Phys. Lett. 135B (1984) 445. [17] J. Ellis, S. Ferrara and D.V. Nanopoulos, Phys. Lett. l14B (1982) 231; W. Buchmiiller and D. Wyler, Phys. Lett. 121B (1983) 321; J. Polchinski and M.B. Wise, Phys. Lett. 125B (1983) 393. [18] M.J. Dunearg NucL Phys. B221 (1983) 285; J.F. Donoghue, H.P. Nilles and D. Wyler, Phys. Lett. 128B (1983) 55. [19] J.-M. G6rard, W. Grimus, A. Masiero, D.V. Nanopoulos and A. Raychaudhuri, CERN preprint TH.3920. [20] S.-P. Chia, Phys. Lett. 130B (1983) 315. [21] Y. Meurice, Ann. Soe. Scient. Bruxelles 96 (1982) 28. [22] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Pis'ma Zh. Eksp. Teor. Fiz. 22 (1975) 123 [JETP Lett. 22 (1975) 55] ; Zh. Eksp. Teor. Fiz. 72 (1977) 1275 [Soy. Phys. JETP 45 (1977) 670] ; Nuel. Phys. B120 (1977) 315. [23] J.F. Donoghue, University of Massachusetts preprint UMHEP-196 (1984). [24] A.I. Sanda, Phyg Rev. D23 (1981) 2647; N.G. Deshpande, Phys. Rev. D23 (1981) 2654. [25] P. Langaeker and B. Sathiapalan, Phys. Lett. 144B (1984) 395. [26] F.J. Gilman and J.S. Hagelin, Phys. Lett. 126B (1983) 111. [27] P. Langaeker and B. Sathiapalan, Phys. Lett. 144B (1984) 401.

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