SLA C -PU B-8231
hep-ph/9908404
arXiv:hep-ph/9908404v1 18 Aug 1999
Large
I = 23 C ontribution to
Supersym m etry
0
= in
A lexander L.K agana and M atthias N eubertb
Stanford Linear A ccelerator C enter,Stanford U niverity
Stanford,C alifornia 94309,U SA
A bstract
W e show that in supersym m etric extensions of the Standard M odel gluino box
diagram scan yield a large I = 32 contribution to s ! dqq FC N C processes,w hich
m ay induce a sizable C P-violating contribution to the I = 2 isospin am plitude in
K !
decays. T hiscontribution only requires m oderate m ass splitting between
the right-handed squarks u
~R and d~R ,and persistsforsquark m assesoforder1TeV .
L L from K {K m ixing, the resulting
Taking into account current bounds on Im sd
contribution to 0= could bean orderofm agnitudelargerthan them easured value.
(Subm itted to PhysicalR eview Letters)
A ugust 1999
a
O n leave from : D epartm ent ofPhysics,U niversity ofC incinnati,C incinnati,O hio 45221
bOn
leave from :N ew m an Laboratory ofN uclearStudies,C ornellU niversity,Ithaca,N Y 14853
T he recent con rm ation of direct C P violation in K !
decays is an im portant step in testing the C abibbo{K obayashi{M askawa (C K M ) m echanism for C P violation in the Standard M odel. C om bining the recent m easurem ents by the K TeV and
N A 48 experim ents [1] w ith earlier results from N A 31 and E731 [2] gives R e( 0= ) =
(2:12 0:46) 103 . T his value tends to be higher than theoreticalpredictions in the
Standard M odel,w hich centerbelow oraround 1 103 [3]. U nfortunately,itisdi cult
to gauge the accuracy of these predictions, because they depend on hadronic m atrix
elem ents w hich at present cannot be com puted from rst principles. A Standard-M odel
explanation of 0= can therefore not be excluded. N evertheless,it is interesting to ask
how large 0= could be in extensions ofthe Standard M odel.
In the context ofsupersym m etric m odels,it has been know n for som e tim e that it
is possible to obtain a large contribution to 0= via the I = 21 chrom om agnetic dipole
operatorw ithoutviolating constraintsfrom K {K m ixing [4].Ithasrecently been pointed
out that this m echanism can naturally be realized in various supersym m etric scenarios
[5].In thisLetterwe propose a new m echanism involving a supersym m etric contribution
to 0= induced by I = 32 penguin operators.T hese operatorsare potentially im portant
because theire ectisenhanced by the I = 12 selection rule.U nlike previousproposals,
w hich involve left-right dow n-squark m ass insertions, our e ect relies on the left-left
LL
insertion sd
and requires(m oderate)isospin violation in theright-handed squark sector.
T he ratio 0= param etrizing the strength ofdirect C P violation in K !
decays
can be expressed as
0
= iei( 2
0
)
p
!
Im A 2
2j j R eA 2
Im A 0
;
R eA 0
(1)
w here A I aretheisospin am plitudesforthedecaysK 0 ! ( )I, I arethecorresponding
strong-interaction phases, and the ratio ! = R eA 2=R eA 0
0:045 signals the strong
1
3
enhancem ent of I = 2 transitions over those w ith I = 2 . From experim ent,we take
j j= (2:280 0:013) 103 and
= (43:49 0:08) for the m agnitude and phase of
the param eter m easuring C P violation in K {K m ixing,and 2
(42 4) for
0 =
the di erence ofthe S-wave
scattering phases in the two isospin channels. It follow s
that,to an excellent approxim ation, 0= is real.
In the Standard M odel,the isospin am plitudes A I receive sm all,C P-violating contributions via the ratio (VtsVtd)=(VusVud) of C K M m atrix elem ents. T his ratio enters
through the interference ofthe s ! uud tree diagram w ith penguin diagram s involving
the exchange ofa virtualtop quark. A ccording to (1),contributions to 0= due to the
I = 32 am plitude Im A 2 are enhanced relative to those due to the I = 12 am plitude
Im A 0 by a factor of ! 1
22. H owever, in the Standard M odel the dom inant C P0
violating contributionsto = are due to Q C D penguin operators,w hich only contribute
to A 0. Penguin contributions to A 2 arise through electroweak interactions and are suppressed by a power of . T heir e ects on 0= are subleading and ofthe sam e order as
isospin-violating e ects such as 0{ { 0 m ixing.
H ere we point out that in supersym m etric extensions ofthe Standard M odelpotentially large,C P-violating contributionscan arise from avor-changing strong-interaction
1
processesinduced by gluino box diagram s.W hereasin the lim itofexactisospin sym m etry in the squark sector these graphs only induce I = 21 operators at low energies,in
the presence ofeven m oderate up-dow n squark m asssplitting they can lead to operators
w ith large I = 32 com ponents.In theterm inology ofthestandard e ective weak H am iltonian,this im plies that the supersym m etric contributions to the W ilson coe cients of
Q C D and electroweak penguin operatorscan beofthesam e order.Speci cally,both sets
f 2 w ith m
f a generic supersym m etric m ass,com pared w ith
ofcoe cients scale like s2=m
2
2
s W =m W and
W =m W ,respectively,in the Standard M odel.T hese contributionscan
be m uch larger than the electroweak penguins ofthe Standard M odeleven for supersym m etric m asses oforder 1TeV .O n the other hand,supersym m etric contributions to
the W ilson coe cients proportionalto electroweak gauge couplings are param etrically
suppressed and w illnot be considered here.
W e nd thatthe relevant S = 1 gluino box diagram slead to the e ective H am iltonian
i
G F X4 h q
eqi( )Qe qi( ) + h.c.;
ci( )Q qi( )+ c
He = p
2 i= 1
w here
Q q1 = (d s )V
A
(q q )V + A ;
Q q2 = (d s )V
A
(q q )V + A ;
Q q3 = (d s )V
A
(q q )V
A
Q q4 = (d s )V
A
(q q )V
A
;
f ,Qe qi are operatorsofoppoare localfour-quark operatorsrenorm alized ata scale
m
sitechirally obtained by interchanging V A $ V + A ,and a sum m ation overq = u;d;:::
and over color indices ; is im plied. In the calculation ofthe coe cient functions we
use the m ass insertion approxim ation,in w hich case the gluino{quark{squark couplings
are avor diagonal. Flavor m ixing is due to sm alldeviations from squark-m ass degeneracy and is param etrized by dim ensionless quantities iAjB ,w here i;j are squark avor
indices and A ;B refer to the chiralities ofthe corresponding quarks (see,e.g.,[4]). In
general, these m ass insertions can carry new C P-violating phases. C ontributions involving left-right squark m ixing are neglected, since they are quadratic in sm allm ass
LR LR
insertion param eters,i.e.,proportionalto sd
qq . W e de ne the dim ensionless ratios
xLu ;R
=
m u~L ;R
m g~
!2
xLd ;R
;
=
m d~L ;R
m g~
!2
;
w here m u~L ;R and m d~L ;R denote the average left-orright-handed squark m asses in the up
and dow n sector,respectively. SU (2)L gauge sym m etry im plies that the m ass splitting
between theleft-handed up-and dow n-squarksm ustbetiny;however,wew illnotassum e
such a degeneracy between the right-handed squarks. For the W ilson coe cients c qi at
2
f we then obtain
the supersym m etric m atching scale m
cq1 =
2 LL
1
p s sd 2
f(xLd ;xRq )
2 2G F m g~ 18
cq2 =
2 LL
1
7
p s sd 2
f(xLd ;xRq )+ g(xLd ;xRq ) ;
6
2 2G F m g~ 6
2 LL
cq3 =
cq4 =
p s sd
2 2G F m 2g~
5
g(xLd ;xRq ) ;
18
1
5
f(xLd ;xLq )+
g(xLd ;xLq ) ;
9
36
2 LL
7
1
p s sd 2
f(xLd ;xLq )+
g(xLd ;xLq ) ;
3
12
2 2G F m g~
w here
xyln y
x[2x2
+
1)2(x y)2
(x
f(x;y) =
x(x + 1 2y)
(x 1)2(y 1)(x y) (y
g(x;y) =
xy2 ln y
x[ 2x + (x + 1)y]
+
(x 1)2(y 1)(x y) (y 1)2(x y)2
(x + 1)y]lnx
;
1)3(x y)2
x2[x(x + 1) 2y]lnx
:
(x 1)3(x y)2
eqi areobtained by interchanging L $ R in theexpressions
T heresultsforthecoe cients c
for cqi.
It is straightforward to relate the quantities cqi to the W ilson coe cients appearing
in the e ective weak H am iltonian ofthe Standard M odelasde ned,e.g.,in [6].W e nd
( t = VtsVtd)
(
t)C 3 =
cu3 + 2cd3
;
3
(
t)C 4 =
cu4 + 2cd4
;
3
(
t)C 5 =
cu1 + 2cd1
;
3
(
t)C 6 =
cu2 + 2cd2
3
for the Q C D penguin coe cients,and
3
(
2
t)C i+ 6
= cui
cdi
ci ; i= 1:::4
(2)
for the coe cients of the electroweak penguin operators. T he supersym m etric contributions to the electroweak penguin coe cients vanish in the lim it ofuniversalsquark
m asses. H owever, for m oderate up{dow n squark m ass splitting the di erences c i =
cui
cdi are of the sam e order as the coe cients c qi them selves. In this case gluino
box contributions to Q C D and electroweak penguin operatorsare ofsim ilar m agnitude.
T his conclusion is unaltered w hen additionalcontributions to C 3:::6 from Q C D penguin
diagram s w ith gluino loops are taken into account. Because the electroweak penguin
operators contain I = 32 com ponents their contributions to 0= are strongly enhanced
3
and thusareexpected to bean orderofm agnitudelargerthan thecontributionsfrom the
Q C D penguin operators. In this Letter,we focus only on these enhanced contributions.
ei)from thesuperT herenorm alization-group evolution ofthecoe cients c i (and c
f
sym m etric m atching scale m dow n to low energies iswellknow n. In leading logarithm ic
approxim ation,one obtains [6]
c 1( ) =
1=
0
f );
c1(m
"
c 1( )
=
c 2( )+
3
c 3( )+ c 4( ) =
c 3( )
c4( ) =
8=
2=
4=
0
0
#
f )
c 1(m
f )+
c 2(m
;
3
0
f )+ c 4(m
f )];
[ c 3(m
f )
[ c 3(m
f )];
c4(m
2
f ),and 0 = 11
w here = s( )= s(m
n . It is understood that the value of 0 is
3 f
changed at each quark threshold. W e use the two-loop running coupling norm alized to
s(m Z ) = 0:119 and take the quark thresholds at m t = 165G eV ,m b = 4:25G eV and
e3;4 at
m c = 1:3G eV .In Table1 wegivetheim aginary partsofthecoe cients c 1;2 and c
f = m g~ = m d~ = m d~ = 500G eV
the scale = m c,obtained for the illustrative choice m
L
R
and three (larger) values ofm u~R . Since the m ass splitting between the left-handed u
~L
~
e
and dL squarks is tiny, we can safely neglect the coe cients c 3;4 and c1;2 in our
analysis. N ote that for xed ratios of the supersym m etric m asses the values of the
f 2 ,i.e.,signi cantly larger values could be obtained for sm aller
coe cients scale like m
m asses. For com parison,the last colum n contains the im aginary parts of c 1:::4 in the
Standard M odelcom puted from (2) using Im t = 1:2 104 and the next-to-leading
orderW ilson coe cientsC i com piled in [6]. W e observe thatforsupersym m etric m asses
LL
oforder 500G eV ,and for a m ass insertion param eter Im sd
oforder a few tim es 10 3
(see below ),the W ilson coe cient c 2 can be signi cantly larger than the value ofthe
corresponding coe cient in the Standard M odel,w hich is proportionalto C 8. T his is
interesting,since even in the Standard M odelthe contribution ofC 8 to 0= issigni cant.
In estim ating the supersym m etric contribution to 0= we focusonly on the (V A )
(V + A ) operators associated w ith the coe cients c 1 and c 2,because their m atrix
elem entsare chirally enhanced w ith respectto those ofthe (V A ) (V A )operators.
T he penguin operators contribute to the im aginary part of the isospin am plitude A 2.
T he real part is, to an excellent approxim ation, given by the m atrix elem ents of the
standard current{current operators in the e ective weak H am iltonian. Evaluating the
m atrix elem ents of the four-quark operators in the factorization approxim ation, and
(2)
param etrizing nonfactorizable corrections by hadronic param eters B i as de ned in [6],
we obtain
Im A susy
2
R eA 2
(2)
Im [ c 2(m c)+ 31 c 1(m c)]B 8 (m c)
3
m 2K
:
(2)
2 m 2s(m c) m2d(m c)
jVusVudjz+ (m c)B 1 (m c)
Follow ing com m on practice we have neglected a tiny contribution proportionalto the
(2)
(2)
di erence B 7
B8 . In the above form ula z+ is a com bination ofW ilson coe cients.
4
LL
ei(m c)in unitsof10 4 Im sd
Table1: Valuesofthecoe cients c i(m c)and c
4
RR
and 10 Im sd ,respectively,for com m on gluino and dow n-squark m asses of
500G eV and di erentvaluesofm u~R .T helastcolum n show sthecorresponding
values in the Standard M odelin units of10 7 .
m u~R [G eV ]
c 1(m c)
c 2(m c)
e3(m c)
c
e4(m c)
c
750
0:05
2.12
0:50
0.56
1000 1500
0:08
0:12
3.19
4.16
0:76
1:01
0.87
1.17
SM
0.42
1:90
20.64
7:63
(2)
T he product z+ B 1 = 0:363 is schem e independent and can be extracted from experim ent. N ote that at leading logarithm ic order the scale dependence ofthe com bination
c 2 + 13 c 1 cancels the scale dependence ofthe running quark m asses,and hence the
(2)
hadronic param eter B 8 is scale independent.
Putting everything together,we nd forthe supersym m etric I = 32 contribution to
0
=
0
"
500G eV
19:2
m g~
#2 "
# 34 "
f
s(m
)
0:096
21
130M eV
m s(m c)
w here
#2
(2)
B 8 (m c)X (xLd ;xRu ;xRd )Im
LL
sd
;
(3)
2
32
[f(x;y) f(x;z)]+ [g(x;y) g(x;z)]:
27
27
T he existence of this contribution requires a new C P-violating phase L de ned by
LL
LL
Im sd
jsd
jsin L . T he m easured values of m K and in K {K m ixing give bounds
LL 2
LL 2
on R e( sd ) and Im ( sd
) ,respectively,w hich can be com bined to obtain a bound on
LL
Im sd as a function of L . U sing the m ost recent analysis ofsupersym m etric contributionsto K {K m ixing in [7],we show in Figure 1 the resultsobtained form d~L = 500G eV
and three choices ofm g~.
LL
It is evident that the bound on Im sd
depends strongly on the precise value of
0
L . To address the issue of how large a supersym m etric contribution to = one can
reasonably expect via the m echanism proposed in this Letter,it appears unnaturalto
take the absolute m axim um of the bound given the peaked nature of the curves. To
LL
be conservative we evaluate our result (3) taking forIm sd
one quarter ofthe m axim al
allowed values show n in Figure 1,noting however that a larger e ect could be obtained
forthe specialcase ofj L jvery close to 90 . O ur results forj 0= jare show n in Figure 2
as a function of m u~R for the case m d~L = m d~R = 500G eV and the sam e three values
ofm g~ considered in the previous gure. T he choice m d~L = m d~R is m ade for sim plicity
only and does not a ect our conclusions in a qualitative way. Except for the special
X (x;y;z)=
5
0.05
0.04
0.03
0.02
0.01
0
0
25 50 75 100 125 150 175
phiL
LL
Figure1: U pperbound on jIm sd
jversustheweak phase j L j(in degrees)for
2
m d~L = 500G eV and (m g~=m d~L ) = 1 (solid),0.3 (dashed)and 4 (short-dashed).
case ofhighly degenerate right-handed up-and dow n-squark m asses,the I = 32 gluino
box-diagram contribution to 0= can by far exceed the experim entalresult,even taking
into account the bounds from m K and . Indeed,even form oderate splitting Figure 2
LL
im pliessubstantially strongerboundson jIm sd
jthan thoseobtained from K {K m ixing.
T his nding isin contrastto thecom m only held view thatsupersym m etric contributions
to the electroweak penguin operatorshave a negligible im pacton 0= . In thiscontext,it
is worth noting that a large m ass splitting between u
~R and d~R can be obtained,e.g.,in
G U T theoriesw ithoutSU (2)R sym m etry and w ith hypercharge em bedded in the uni ed
gauge group,w ithout encountering di culties w ith naturalness [8].
T he allowed contribution to 0= in Figure 2 increases w ith the gluino m ass(for xed
squark m asses) because the K {K bounds becom e weaker in this case. Ifallsupersym m etric m assesare rescaled by a com m on factor ,and the boundsfrom K {K m ixing are
rescaled accordingly,the valuesfor 0= scale like 1= m odulo logarithm ice ectsfrom the
f ). T herefore,even for larger squark m asses oforder 1TeV the
running coupling s( m
0
new contribution to = can exceed the experim entalvalue by a large am ount,im plying
LL
nontrivialconstraints on Im sd
.
Before concluding,we note thatin the above discussion we have m ade no assum ption
RR
RR
regarding the m ass insertion param eter Im sd
jsd
jsin R for right-handed squarks.
RR
LL
In m odelsw here j sd jisnothighly suppressed relative to j sd
j,m uch tighterconstraints
LL
LL R R
on Im sd can bederived by applying the severe boundson theproduct sd
sd obtained
from the chirally-enhanced contributions to K {K m ixing. In analogy w ith Figure 1,we
LL
jas a function of L and R ,w hich is sharply peaked
obtain an upper bound on jIm sd
LL
R R 1=2
along the line L + R = 0 m od and scales like j sd
= sd
j . A s above,we take one
6
25
20
15
10
5
0
400
600
800 1000 1200 1400
muR in GeV
Figure 2: Supersym m etric contribution to j 0= j (in units of 10 3 ) versus
(2)
m u~R , for m s(m c) = 130M eV , B 8 (m c) = 1, m d~L = m d~R = 500G eV , and
(m g~=m d~L )2 = 1 (solid), 0.3 (dashed) and 4 (short-dashed). T he values of
LL
jIm sd
jcorresponding to the three curves are 0.011,0.005 and 0.027,respectively (see text). T he band show s the average experim entalvalue.
quarterofthepeak valueobtained using the resultscom piled in [7].C onsidering thecase
LL
m d~L = m g~ = 500G eV forexam ple,we nd thatthe upperboundson jIm sd
ja reduced
RR
LL
RR
LL
by a factorranging from 3% in the lim it w here j sd
j= j sd
jto 8% forj sd
j= 0:1j sd
j.
0
3
In the lattercase,the supersym m etric contribution to = can stillbe oforder10 ,i.e.,
com parable to the m easured value. M oreover,signi cantly largervaluescan be obtained
for specialpoints in m odulispace,w here the weak phases obey L + R
0 m od .
In sum m ary,wehaveshow n thatin supersym m etricextensionsoftheStandard M odel
gluino box diagram s can yield a large I = 32 contribution to 0= ,w hich only requires
m oderate m asssplitting between the right-handed squarks,i.e.,(m u~R md~R )=m d~R > 0:1.
In a large region ofparam eter space,the m easured value of 0= im plies a signi cantly
LL
stronger bound on Im sd
than is obtained from K {K m ixing.
Acknow ledgm ents: W e are gratefulto YuvalG rossm an and JoA nne H ewettforuseful
discussions. A .K .issupported by the U nited StatesD epartm entofEnergy underG rant
N o.D E-FG 02-84ER 40153,and M .N .under contract D E{A C 03{76SF00515.
7
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8