CP Violation Beyond the Standard Model in Hadronic B Decays

7 Pages·0.14 MB·English

by  Xiao-Gang He

CP Violation Beyond the Standard Model in Hadronic B Decays
Xiao-Gang Hea
0 Department of Physics, National Taiwan University, Taipei, Taiwan 10764, R.O.C.
0
0
2 Abstract

n In this talk I discuss three different methods, using Bd → J/ψKS,
Ja J/ψKSπ0, Bd → K−π+,π+π− and Bu → K−π0,K¯0π−,π−π0, to ex-
tract hadronic model independentinformation about new physics.
0
3 1. Introduction

TheGold-platedmodeforthestudyofCPviolationintheStandardModel
1
(SM) is B J/ψK . This decay mode can be used to measure sin2β in the
v → S
3 SM without hadronic uncertainties. CDF and ALEPH data obtain1 sin2β =
1 0.91 ± 0.35 which is consistent with sin2β = 0.75+−00..005655 from fitting other
3 experimentaldata2. Thebestfitvalueforγfromthesamefittinggives2γ =
best
1 59.9◦. Rare hadronic B decays also provide important information3. Using
0
factorization calculations for rare hadronic B decays measured by CLEO4, γ
0
0 is determined to be5 114+−2251 degrees. It seems that there is a conflict between
/ γ obtained from rare B decay data and from other fits. At present it is not
h
possibletoconcludewhetherthispossibleinconsistenceisduetoexperimental
p
- andtheoreticaluncertaintiesorduetonewphysics. Itisimportanttofindways
p to test the SMand to study new physics withoutlargehadronicuncertainties.
e
In this talk I discuss three methods, using B J/ψK , J/ψK π06, B
h d S S d
: K−π+,π+π−7 and Bu K−π0,K¯0π−,π−π0→8,9, to extract hadronic mod→el
v →
independent information about the SM and models beyond.
i
X In the SM the Hamiltonian for hadronic B decays is given by, H =
eff
r (4G /√2)[V V∗(c Of(q)(L)+c Of(q)(L)) P V∗V cjO ].
a F fb fq 1 1 2 2 − i=(3−12),j=(u,c,t) jb jq i i
O11,12 are the gluonicand electromagneticdipole operatorsand O1−10 arede-
fined in Ref.10. I use the values of c obtained for the SM in Ref.10 with
i
(c1,c2, ct3,ct4, ct5,ct6, ct8,ct9,c10t,ct11,ct12) =(-0.313,1.150,0.017,-0.037, 0.010, -
0.046, -0.001α , 0.049α , -1.321α , 0.267 α , -0.3, -0.6). cc,u are due
em em em em i
to c and u quarks in the loop and can be found in Ref.10. When going be-
yondtheSM,therearenewcontributions. Itakethreemodelsforillustrations.
Model i): R-parity violation model
In R-parity violating supersymmetric (SUSY) models, there are new CP
violatingphases. HereIconsidertheeffectsdueto,L=(λ′′ /2)UciDcjDck,R-
ijk R R R
ae-mail: [email protected]
1
parity violating interaction. Exchange of d˜ squark can generate the following
i
effective Hamiltonianattree level, H =(4G /√2) V∗V cf(q)[Of(q)(R)
eff F fb fq 1 −
Of(q)(R)]. Here Of(q)(R)= f¯γ Rfq¯γµRb and Of(q)(R) =f¯ γ Rf q¯ γµRb .
2 1 µ 2 α µ β β α
TheoperatorsO (R)havetheoppositechiralityasthoseofthetreeoperators
1,2
O (L) in the SM. The coefficients cf(q) with QCD corrections are given by
1,2
(c c )(√2/(4G V V∗))( λ′′ λ′′∗ /2m2 ). Here m is the squark mass.
1 − 2 F fb fq − fqi f3i d˜i d˜i
B π−K¯0 and B K−K¯0 data constrain cc(q) to be less than O(1)11.
u u
→ → | |
The new contributions can be larger than the SM ones. I will take the values
to be 10% of the corresponding values for the SM with arbitrary phases δf(q)
for later discussions.
Model ii): SUSY with large gluonic dipole interaction
In SUSY models with R-parity conservation,potentiallargecontributions
to B decays may come from gluonic dipole interaction by exchanging gluino
at loop level with left- and right-handed squark mixing. In the mass insertion
approximation12, cnew = (√2πα /(G V V m ))(m /m )δLRG(m2/m2),
11 − s F tb ts g˜ q˜ b qb g˜ q˜
where m is the gluino mass, and δLR is the squark mixing parameter. G(x)
g˜ qb
is from loop integral. cnew is constrained by experimental data from b sγ
11 →
which, however, still allows cnew to be as large as three times of the SM con-
11
tribution in magnitude with an arbitrary CP violating phase δ 12. I will
dipole
take cnew to be 3 times of the SM value with an arbitraryδ .
11 dipole
Model iii): Anomalous gauge boson couplings
Anomalous gauge boson couplings can modify c (mainly on ct ) with
i 7−10
the sameCP violating sourceasthat for the SM9,13. The largestcontribution
may come from the WWZ anomalous coupling ∆gZ. To the leading order
1
in QCD corrections, the new contributions to cAC due to ∆gZ are given
7−10 1
by9,13, (ct ,ct ,ct ,ct ) = α ∆gZ(1.397, 0.391,-5.651,1.143). LEP
7AC 8AC 9AC 10AC em 1
data constrain14 ∆gZ to be within 0.113 < ∆gZ < 0.126 at the 95% c.l..
1 − 1
c can be very different from those in the SM.
iAC
2. Test new physics using sin2β from B J/ψK ,J/ψK π0
S S
→
The usual CP violationmeasure for B decays to CP eigenstates is Imξ =
Im (q/p)(A∗A¯/A2) , where q/p = e−2iφB is from B0–B¯0 mixing, while A,
A¯ a{re B, B¯ dec|ay| am} plitudes. For B J/ψK , the final state is P-wave
S
→
hence CP odd. Setting the weak phase in the decay amplitude to be 2φ =
0
Arg(A/A¯), one has , Imξ(B J/ψK )= sin(2φ +2φ ) sin2β .
→ S − B 0 ≡− J/ψKS
For B J/ψK∗ J/ψK π0, the final state has both P-wave (CP odd)
S
→ →
andS-andD-wave(CP even)components. IfS-andD-wavehaveacommon
2
weak phase φ˜ and P-wave has a weak phase φ 6,
1 1
Imξ(B J/ψK π0) = Im e−2iφB[e−2iφ1 P 2 e−2iφ˜1(1 P 2)]
S
→ { | | − −| | }
≡ (1−2|P|2)sin2βJ/ψKSπ0, (1)
where P 2isthefractionofP-wavecomponent. IntheSMonehasφ =β and
B
2φ =|2φ| (2φ˜ ) =Arg[ V V∗ c +a(a′)c / V∗V c +a(a′)c ]=0, in
the0Wolfe1nstei1nphaseco{nvcbenctsio{n1. Hereaa2n}d}a′{arceb pcasr{am1 eterswh2ic}h}indicate
therelativecontributionfromOc(s)(L)comparedwithOc(s)(L)fortheP-wave
2 1
′
and (S-, D-) wave. In the factorization approximation 1/a = 1/a = N (the
c
number of colors). Therefore sin2βJ/ψKS = sin2βJ/ψKSπ0 = sin2β. |P|2 has
beenmeasuredwithasmallvalue15 0.16 0.08 0.04byCLEOwhichimplies
± ±
that the measurement of sin2β using B J/ψK π0 is practical although
S
→
there is a dilution factor of 30%.
When one goes beyond the SM, sin2βJ/ψKS = sin2βJ/ψKSπ0 is not nec-
essarily true. The difference ∆sin2β ≡ sin2βJ/ψKS −sin2βJ/ψKSπ0 is a true
measure of CP violation beyond the SM without hadronic uncertainties.
Let us now analyze the possible values for ∆sin2β for the three models
describedintheprevioussection. BecauseB J/ψK ,J/ψK∗ aretreedomi-
S
→
natedprocesses,Modelsii)andiii)wouldnotchangetheSMpredictionssignif-
icantly. ∆sin2β isnotsensitivetonewphysicsinModelsii)andiii). However,
for Model i), the contributions can be large. The weak phases are given by
2φ =2φ (2φ˜ )=Arg[ V V∗ c +ac +( )cc(s) 1 a(a′) / V∗V c +
ac0+( )1cc(s)1∗ 1 a(a{′) cb ]c.s{Ta1king t2he n−ew cont{rib−utions}t}o}b{e 1c0b%cso{f1the
2
SM one−s, one o{bt−ains φ}}=}φ φ˜ 0.1sinδc(s). From this, ∆sin2β
0 1 1
≈ − ≈ ≈
4((1 P 2)/(1 2P 2))cos2φ (0.1sinδc(s)) 0.5cos2φ sinδc(s). φ may
B B B
−| | − | | ≈
bedifferentfromtheSMoneduetonewcontributions. Usingthecentralvalue
sin2β = 0.91 measured from CDF and ALEPH, ∆sin2β 0.2sinδc(s)
J/ψKS ≈
which can be as large as 0.2. Such a large difference can be measured at B
factories. Information about new CP violating phase δc(s) can be obtained.
3. Test new physics using rate differences between B π+K−,π+π−
d
→
Theoretical calculations for rare hadronic B decays have large uncertain-
ties. Information extractedusing model calculations is unreliable. I now show
that hadronic model independent information about CP violation can be ob-
tained using SU(3) analysis for rare hadronic B decays.
The operatorsO1,2, O3−6,11,12,andO7−10 transformunderSU(3)symme-
try as ¯3 +¯3 +6+15, ¯3, and ¯3 +¯3 +6+15, respectively. These properties
a b a b
enable one to write the decay amplitudes for B PP in only a few SU(3) in-
→
variantamplitudes. There are relations between different decays. When small
annihilation contributions are neglected, one has
3
A(B π+π−)=V V∗T +V V∗P,
d → ub ud tb td
A(B π+K−)=V V∗T +V V∗P.
d → ub us tb ts
Due to different KM matrix elements involved in these decays, although
the amplitudes have similarities, the branching ratios are not simply related.
However, when considering the rate difference, ∆(PP) = Γ(B PP)
d
Γ(B¯ P¯P¯),thesituationisdramaticallydifferentbecausethesimp→lerelatio−n
d
→ ∗ ∗ ∗ ∗
Im(V V V V )= Im(V V V V ). In the SU(3) limit,
ub ud tb td − ub us tb ts
∆(π+π−)= ∆(π+K−). (2)
−
This non-trivial equality dose not depend on detailed models for the hadronic
physics and provides test for the SM7. Including SU(3) breaking effect from
factorizationcalculation,onehas,∆(π+π−) fπ2∆(π+K−). Althoughthere
≈−fK2
is correction, the relative sign is not changed.
When going beyond the SM, there are new CP violating phases leading
to violation of the equality above. Among the three models described in the
first section, Models i) and ii) can alter the equality significantly,while Model
iii) can not because the CP violating source is the same as that in the SM.
To illustrate how the situation is changed in Models i) and ii), I calculate
the normalized asymmetry A (PP) =∆(PP)/Γ(π+K−) using factoriza-
norm
tion approximationfollowing Ref.16. The new effects may come in sucha way
that only B π+K− is changed but not B π+π−. This scenario leads
d d
→ →
to maximal violation of the equality discussed here. I will take this iscenario
for illustration.
TheresultsareshowninFigure1. ThesolidcurveistheSMpredictionfor
A (π+K−) as a function of γ. For γ A (π+K0) 10%. It is clear
norm best norm
≈
from Figure 1 that within the allowed range of the parameters, new physics
effects can dramatically violate the equality discussed above. Experimental
study of the rate differences ∆(π+π−,π+K−) can provide model independent
information about CP violation beyond the SM in the near future.
4. Test new physics using SU(3) relation for B π−K¯0,π0K−,π0π−
u
→
Inowdiscussanothermethodwhichprovidesimportantinformationabout
new physics using B π−K¯0,π0K−,π0π−. Using SU(3) relation and fac-
u
→
torization estimate for the breaking effects, one obtains8,9
A(B π−K¯0)+√2A(B π0K−) = ǫA(B π−K¯0)ei∆φ(e−iγ δ ),
u u u EW
→ → → −
4
0.2
0.1
0
-0.1
-0.2
50 100 150 200 250 300 350
Figure 1: Anorm(π+K−) vs. phases γ, δu(s) and δdipole for the SM (solid), Mod-
els i) (dashed) and ii) (dotted). For Models i) and ii), γ = 59.9◦ is used.
Anorm(π+K−) = −(fπ/fk)2Anorm(π+π−) is satisfied in the SM. For Models i) and ii)
−(fπ/fk)2Anorm(π+π−)isapproximatelythesameasthatintheSM(dot-dashed curve).
3 V V c +c V f A(π+π0)
δ = | cb|| cs| 9 10, ǫ=√2| us| K | |,
EW −2 V V c +c V f A(π+K0)
ub us 1 2 ud π
| || | | | | |
where∆φisthedifferenceofthefinalstaterescatteringphasesforI =3/2,1/2
amplitudes. For f /f = 1.22 and Br(B± π±π0) = (0.54+0.21 0.15)
10−54, one obtainsKǫ=π0.21 0.06. → −0.20± ×
Neglecting small tree con±tribution to B π−K¯0, one obtains
u
→
(r2 +r2)/2 1 ǫ2(1 δ2 )
cosγ =δ + − − − − EW , r2 r2 =4ǫsin∆φsinγ,
EW − 2ǫ(cos∆φ+ǫδ ) +− −
EW
where r2 =4Br(π0K±)/[Br(π+K0)+Br(π−K¯0)]=1.33 0.454.
±
±
The relation between γ and δ is complicated. Howeverit is interesting
EW
to note that even in the most general case, bound on cosγ can be obtained9.
For ∆=(r2 +r2)/2 1 ǫ2(1 δ2 ) ( )>0, we have
+ − − − − EW ≥ ≤
∆ ∆
cosγ ( )δ , or cosγ ( )δ . (3)
EW EW
≤ ≥ − 2ǫ(1+ǫδ ) ≥ ≤ − 2ǫ( 1+ǫδ )
EW EW
−
The bounds on cosγ as a function of δ are shown in Fig. 2 by the
EW
solid curves for three representative cases: a) Central values for ǫ and r2; b)
±
Centralvalues for ǫ and 1σ upper bound r2 =1.78;and c) Centralvalue for ǫ
±
and 1σ lower bound r2 =0.88. The bounds with cosγ 1 for a), b) and c)
±
| |≤
are indicated by the curves (a1, a2), (b) and (c1, c2), respectively. For cases
a) and c) there are two allowed regions, the regions below (a1, c1) and the
regions above (a2, c2). For case b) the allowed range is below (b).
5
1
0.75
a2 c2 c1
0.5
0.25 a1 b
0
-0.25
-0.5
-0.75
-1
0 0.5 1 1.5 2
Figure 2: cosγ vs. δEW. The solutions for the cases a), b) and c) are indicated by the
dashed,dot-dashedanddottedcurvesrespectively.
When the decay amplitudes for B± Kπ, B± π±π0 and the rate
→ →
asymmetries for these decays are determined to a good accuracy, γ can be
determined, one obtains,
(1 cos2γ)[1 ( ∆ ǫδ )2] (r+2 −r−2)2 =0. (4)
− − 2ǫ(δ cosγ) − EW − 16ǫ2
EW
−
To have some idea about the details, I analyze the solutions of cosγ as a
function ofδ for the three cases discussedearlierwith a givenvalue for the
EW
asymmetry A =(r2 r2)/(r2 +r2)=15%.
asy +− − + −
IntheSMforr = V /V =0.08and V =0.2196,δ =0.81. The
v ub cb us EW
| | | | | |
central values for r± and ǫ prefers cosγ <0 which is different from the result
obtainedinRef.2 byfittingotherdata. Theparameterδ issensitivetonew
EW
physics in the tree and electroweak sectors. Model i) has large corrections to
the tree level contributions. However, the contribution is proportional to the
sum of the coefficients of operators Ou(s)(R) which is zero in Model i). The
1,2
above method does not provide information about new physics due to Model
i). This method would not provide information about new physics due to
Model ii) neither because the gluonic dipole interactiontransforms as ¯3 which
does not affect δ . Model iii) can have large effect on δ . In this model
EW EW
δ =0.81(1+4.33∆gZ)whichcanvaryintherange0.40 1.25. Constraint
EW 1 ∼
on cosγ can be very different from the SM prediction.
For case a), in the SM cosγ < 0.18 which is inconsistent with cosγ
best
≈
0.5. In Model iii) cosγ can be consistent with cosγ . For case b), cosγ is
best
less than zero in both the SM and Model iii). If this is indeed the case, other
types of new physics is needed. For case c) cosγ can be close to cosγ for
best
both the SM and Model iii). Improved measurements of ǫ and r± can provide
6
important information about γ and new physics.
5. Conclusion
From discussions in previous sections, it is clear that using B J/ψK ,
d S
J/ψK π0, B π−K+,π+π− and B− π0K−,π−K¯0,π0π−→important
S u
→ →
information free from uncertainties in hadronic physics about the Standard
Model and models beyond can be obtained. These analyses should be carried
out at B factories.
Acknowledgements
This work was partially supported by National Science Council of R.O.C.
undergrantnumberNSC89-2112-M-002-016. IthankDeshpande,Hou,Hsueh
and Shi for collaborations on materials presented in this talk.
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