On the interplay of direct and indirect CP violation in the charm sector

0.11 MB·English

by  Marco Gersabeck

On the interplay of direct and indirect CP violation
in the charm sector
2 M Gersabeck1, M Alexander2, S Borghi2,3, VV Gligorov1
1 and C Parkes3,1
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1 EuropeanOrganizationforNuclearResearch(CERN),Geneva, Switzerland
2
2 School ofPhysicsandAstronomy,UniversityofGlasgow,Glasgow,United

n Kingdom
a 3 School ofPhysicsandAstronomy,UniversityofManchester, Manchester,
J UnitedKingdom

7 E-mail: [email protected]
2

] Abstract. Charm mixing and CP violation observables are examined in the
x lightoftherecentlyreportedevidencefromLHCbforCP violationinthecharm
e sector. If the result is confirmed as being due to direct CP violation at the 1%
- level, its effect will need to be taken into account in the interpretation of CP
p violation observables. The contributions of direct and indirect CP violation to
e the decay rate asymmetry difference ∆ACP and the ratios of effective lifetimes
h AΓ and yCP are considered here. Terms relevant to the interpretation of future
[ highprecisionmeasurementswhichhavebeenneglectedinpreviousliteratureare

identified.
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1 PACSnumbers: 13.25.Ft
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Submitted to: J. Phys. G: Nucl. Part. Phys.
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On the interplay of direct and indirect CP violation in the charm sector 2
Charm,afterstrangeandbeauty,isthelastsystemofneutralflavouredmesonswhere
CP violation remains to be discovered. While neutral B mesons are characterised
by their mass splitting leading to fast oscillations and neutral kaons by their width
splitting which results in a short-lived and a long-lived state, neutral charm mesons
have a very small splitting in both mass and width. For charm, compared to the
beauty sector, this leads to rather subtle mixing-related effects in time-dependent as
well as in time-integrated charm measurements, which are examined in detail here.
First evidence for CP violation in the charm sector has recently been reported
by the LHCb collaboration in the study of the difference of the time-integrated
asymmetries of D0 → K+K− and D0 → π+π− decay rates through the parameter
∆A [1]. This measurement is primarily sensitive to the difference in direct CP
CP
violationbetween the two finalstates as discussed further below. Direct CP violation
dependsonthefinalstateandistheasymmetryoftheratesofparticleandantiparticle
decays. It can be caused by a difference in the magnitude of the decay rates or by a
difference in their phase. Indirect CP violation is considereduniversal, i.e. final-state
independent, and is an asymmetry in the mixing rate or in its weak phase.
Indirect CP violationcanbe measuredin time-dependent analyses. To date, two
types of measurements were used to search for indirect CP violation in the charm
sector. One uses the asymmetry of the lifetimes, AΓ, measured in D0 and D0 decays
to the CP eigenstatesK+K− orπ+π− [2,3,4]. The otheris a time-dependentDalitz
plotanalysisofD0andD0decaystoK0π+π− orK0K+K−[5,6]. Anotherobservable
S S
studied related to AΓ is yCP, which is given by the deviation from one of the ratio of
the lifetimes measured in decays to a Cabibbo-allowed,CP averaged,and a Cabibbo-
suppressed, CP eigenstate, final state. Any deviation of a measurement of y from
CP
that of the mixing parameter y would signal CP violation.
In the interpretation of AΓ and yCP, direct CP violation is commonly
neglected [7]. In the light of the new evidence this assumption is no longer justified.
The relevance of direct CP violation to a measurement of AΓ has previously been
pointedoutin[8]. However,acloserlookatbothAΓ andyCP isnecessarytoexamine
the contribution of direct and indirect CP violation in these observables as well as
their connection to ∆A .
CP
The mass eigenstates of neutral D mesons, |D1,2i, with masses m1,2 and widths
Γ1,2 can be written as linear combinations ofthe flavoureigenstates |D1,2i=p|D0i±
q|D0i, with complex coefficients p and q which satisfy |p|2 +|q|2 = 1. The average
mass andwidth aredefined asm≡(m1+m2)/2 andΓ≡(Γ1+Γ2)/2. The D mixing
parameters are defined using the mass and width difference as x ≡ (m2 − m1)/Γ
and y ≡ (Γ2 − Γ1)/2Γ. The phase convention of p and q is chosen such that
CP|D0i=−|D0i.
Accordingto[8],thetimedependentdecayratesofD0 andD0 decaystothefinal
state f, which is a CP eigenstate with eigenvalue η , can be expressed as
CP
1
Γ(D0(t)→f)= e−τ |A |2 1+|λ |2 cosh(yτ)+ 1−|λ |2 cos(xτ)
f f f
2
n(cid:0) (cid:1) (cid:0) (cid:1)
+2ℜ(λ )sinh(yτ)−2ℑ(λ )sin(xτ) ,
f f
Γ(D0(t)→f)= 1e−τ A¯ 2 1+|λ−1|2 cosh(yτ)+ 1−|λ−1|2ocos(xτ)
2 f f f
(cid:12)(cid:12) (cid:12)(cid:12) n(cid:16)+2ℜ(λ−f1)s(cid:17)inh(yτ)−2ℑ(cid:16)(λ−f1)sin(x(cid:17)τ) , (1)
o
On the interplay of direct and indirect CP violation in the charm sector 3
(—)
where τ ≡Γt, Af are the decay amplitudes and λf is given by
qA¯ q A¯
λ = f =−η f eiφ, (2)
f CP
pA p A
f (cid:12) (cid:12)(cid:12) f(cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
where ηCP is the CP eigenvalue of t(cid:12)he(cid:12)fi(cid:12)nal(cid:12)state f and φ is the CP violating relative
phasebetweenq/pandA¯ /A . Intro(cid:12)du(cid:12)c(cid:12)ing(cid:12)|q/p|±2 ≈1±A and|A¯ /A |±2 ≈1±A ,
f f m f f d
one can write
|λ±1|2 ≈(1±A )(1±A ), (3)
f m d
where A representsaCP violationcontributionfrommixing andA fromdirectCP
m d
violation and where both A and A are assumed to be small.
m d
Expanding (1) up to second order in τ one can write the effective lifetimes, i.e.
those measured as a single exponential, as
Γˆ(D(—)(t)→f)≈ Γ 1+ 1± 12(Am+Ad)− 81(A2m−2AmAd) ηCP(ycosφ∓xsinφ)
( (cid:20) (cid:21)
∓A (x2+y2)±2A y2cos2φ∓4xycosφsinφ , (4)
m m
)
where terms below order 10−5 have been ignored. The experimental constraints [9]
give x, y, and A for the final states K+K− and π+π− of order 10−2, and A and
d m
sinφ of order 10−1. The sum of measurements of D0 and D0 decays leads to the
definition of the observable y which is given by
CP
Γˆ+Γˆ¯ 1 1
y = −1≈η 1− (A2 −2A A ) ycosφ− (A +A )xsinφ . (5)
CP 2Γ CP 8 m m d 2 m d
((cid:20) (cid:21) )
ThedifferenceofmeasurementsofD0 andD0 decaysleadstotheparameterAΓ which
is defined as
AΓ =(Γˆ−Γˆ¯)(Γˆ+Γˆ¯)−1 ≈ 12(Am+Ad)ycosφ− 1− 18A2m xsinφ−Am(x2+y2)
(cid:20) (cid:18) (cid:19)
η
+2A y2cos2φ−4xycosφsinφ CP . (6)
m
1+y
(cid:21) CP
The weak phase φ has not been assumed to be universal. When averaging
measurements from different channels, a potential decay-dependent weak phase of
the amplitude ratio has to be taken into account [8]. Expanding only up to order
10−4 leads to
1 1
y ≈η 1− A2 ycosφ− (A )xsinφ , (7)
CP CP 8 m 2 m
(cid:20)(cid:18) (cid:19) (cid:21)
and
1 η 1
CP
AΓ ≈ (Am+Ad)ycosφ−xsinφ ≈ηCP (Am+Ad)ycosφ−xsinφ .(8)
2 1+y 2
(cid:20) (cid:21) CP (cid:20) (cid:21)
The difference of y evaluated in (7) from the expression used in literature [7] so
CP
far is the term η 1A2 ycosφ, which can be of similar order as 1A xsinφ and
CP8 m 2 m
should therefore not be ignored. Equation (8) shows that there can be a significant
contribution to AΓ from direct CP violation. Assuming y =1% and cosφ=1, direct
CP violation at the level of Ad/2 = 1% would lead to a contribution to AΓ of 10−4.
Currentmeasurementsyieldasensitivityofafew10−3 [2,3,4]. Futuremeasurements
On the interplay of direct and indirect CP violation in the charm sector 4
at LHCb and future B factory experiments are expected to reach uncertainties of
the level of 10−4, i.e. that of the direct CP violation contribution. More precise
measurements maywell changethe approximationsmade in (7) and(8), in particular
a measurement of A .A .
m d
Intime-integratedmeasurementstherateasymmetryismeasuredwhichisdefined
as
Γ(D0 →f)−Γ(D0 →f)
A ≡ . (9)
CP Γ(D0 →f)+Γ(D0 →f)
Introducing
adir ≡ |Af|2−|A¯f|2 = 1− AA¯ff 2 = −Ad ≈−1A , (10)
CP |Af|2+|A¯f|2 1+(cid:12)(cid:12)A¯f(cid:12)(cid:12)2 2+Ad 2 d
(cid:12)Af(cid:12)
and using (1), (9) becomes (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
hti hti
ACP ≈adCiPr −AΓ(1−(adCiPr)2) τ ≈adCiPr −AΓ τ , (11)
where hti denotes the average decay time of the observed candidates. Terms in hti2
are below order 10−4, givencurrent experimentalconstraints, and have been ignored.
A common way to reduce experimental systematic uncertainties is to measure
thedifferenceintime-integratedasymmetriesinrelatedfinalstates. Forthetwo-body
final states K+K− and π+π−, this difference is given by
∆A ≡A (K+K−)−A (π+π−)
CP CP CP
=adir(K+K−)−adir(π+π−)
CP CP
ht(K+K−)i ht(π+π−)i
−AΓ(K+K−) +AΓ(π+π−) . (12)
τ τ
Assuming the CP violating phase φ to be universal [10] this can be rewritten as
hti ∆hti
∆A ≈∆adir 1+ycosφ + aind+adirycosφ (13)
CP CP τ CP CP τ
!
(cid:16) (cid:17)
where ∆X ≡ X(K+K−)−X(π+π−), ∆X ≡ X(K+K−)−X(π+π−), and aind =
CP
−(A /2)ycosφ+xsinφ. The ratio hti/τ is equal to one for the lifetime-unbiased B
m
factory measurements [11, 12] and is 2.083±0.001 for LHCb [1] and 2.53±0.02 for
CDF[13],thusleadingtoacorrectionof∆adir oftheorderof10−2. Thefactor∆hti/τ
CP
multiplyingtheindirectCP violationiszerofortheBfactorymeasurementsandranges
from 0.098±0.003 to 0.26±0.01 for LHCb and CDF, respectively. Therefore, ∆A
CP
islargelyameasureofdirectCP violationwhileanobviouscontributionfromindirect
CP violation exists. The contribution from direct CP violation to AΓ pointed out in
(8) leads to a term proportional to y. This term may be of similar size as the term
proportional to ∆hti and should therefore be taken into account.
In summary, the mixing and CP violation parameters yCP, AΓ and ∆ACP have
been discussed in the light of the recent evidence for CP violation in the D0 sector.
Theparametery isleastaffectedbydirectCP violation,however,itcontainsaterm
CP
whichhasbeenneglectedintheliteraturesofarandwhichcanbeofthesameorderas
the constribution proportionalto x. A measurementof AΓ can exhibit a contribution
of direct CP violation at the level of 10−4, comparable to the expected future
experimental sensitivity. The direct CP violation term in the ∆A measurement
CP
contains a contribution proportionalto y. The interpretation of future high precision
measurements of these observables will need to take account of these contributions.
On the interplay of direct and indirect CP violation in the charm sector 5
Acknowledgments
The authors would like to thank Tim Gershon, Bostjan Golob, Alex Kagan and
Vincenzo Vagnoni for helpful discussions and comments. Furthermore, the authors
thank the LHCb collaboration whose work inspired this paper. MG and VVG are
supportedbya MarieCurieAction: “Cofundingofthe CERNFellowshipProgramme
(COFUND-CERN)” of the European Community’s Seventh Framework Programme
undercontractnumber(PCOFUND-GA-2008-229600). MA,SBandCPacknowledge
the support of the STFC (United Kingdom).
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