Nonuniversality of indirect CP asymmetries in D → ππ, KK decays

Amol Dighe,1,* Diptimoy Ghosh,2,† and Bhavik Kodrani1,‡1Tata Institute of Fundamental Research, Mumbai 400005, India

2INFN, Sezione di Roma, Piazzale A. Moro 2, I-00185 Roma, Italy(Received 21 August 2013; published 22 May 2014)

We point out that, if the direct CP asymmetries in the D → πþπ− and D → KþK− decays are unequal,the indirect CP asymmetries as measured in these modes are necessarily unequal. This nonuniversality ofindirect CP asymmetries can be significant with the right amount of new physics contributions, a scenariothat may be fine-tuned but is still viable. A model-independent fit to the current data allows differentindirect CP asymmetries in the above two decays. This could even be contributing to the apparent tensionbetween the difference CP asymmetries ΔACP measured through the pion-tagged and muon-tagged datasamples at the LHCb. This also implies that the measurements of AΓ and yCP in the πþπ− and KþK− decaymodes can be different, and averaging over these two modes should be avoided. In any case, the completeanalysis of CP violation measurements in the D meson sector should take into account the possibility ofdifferent indirect CP asymmetries in the πþπ− and KþK− channels.

DOI: 10.1103/PhysRevD.89.096008 PACS numbers: 11.30.Er, 13.25.Ft

I. INTRODUCTION

The study of CP violation in decays of K and B mesonshave yielded path-breaking results over the past half acentury. Through these measurements, the Kobayashi–Maskawa paradigm of CP violation has been tested frommany directions and has emerged vindicated so far. Thesetests act as indirect probes of new physics beyond theStandard Model and so far have not yielded any conclusiveevidence for a deviation from the Standard Model (SM)predictions. A positive identification of deviations fromthe SM is often limited by the uncertainties in the SMpredictions themselves, especially in the processes thatinvolve decays of mesons, due to hadronic uncertainties.The measurements of D − D mixing and the CP

violation in D decays have started yielding interestingresults only in the past decade. One of the reasons for theDdecays to have come to the forefront so late is that themixing as well as CP violation in the D sector is expectedto be small. In the SM, the contribution to the D − Dmixing box diagram is suppressed—for intermediate u andc quarks, due to their small masses, and for an intermediateb quark, due to the small Cabibbo–Kobayashi–Maskawa(CKM) matrix elements. This leads to very small values forboth the dispersive as well as absorptive parts of the D − Dmixing amplitude. Indeed, the measurements give x≡Δm=Γ and y≡ ΔΓ=ð2ΓÞ to be less than Oð1%Þ [1–3].Moreover, since the phases of the relevant CKM matrixelements are very small, the CP violation is also expectedto be not more than Oð0.1%Þ [4].

The SM calculations of the mixing and CP asymmetriesin the neutral D meson system are difficult due to the massof the charm quark—it is not light enough to enable the useof the chiral perturbation theory and not heavy enough toguarantee convergence of the ð1=mcÞ expansion in theheavy quark effective theory. Moreover, the nonperturba-tive long-distance contributions to the mixing as well asdecay amplitudes may be dominant—since the strongcoupling is not very small at the scale of the D mesonmass, and the short-distance contributions do not have thebenefit of an intermediate t quark as in the case of Bmesons. Therefore, D decays are not a good place forprecision measurements of the SM parameters. However,they can still be used as probes of new physics (NP), if theNP effects can be large compared to the SM ones [4,5].Modes like D=D → πþπ− and D=D → KþK− can besensitive to the presence of such NP [6,7].While the measurements of the mixing parameters x and

y are consistent with the SM estimates of x, y ∼Oð1%Þ[8,9], recent measurements of CP-violating quantities havegiven us a reason to consider the presence of NP con-tributions. TheCP violation inD → πþπ− andD → KþK−

decays was constrained by E791 [10], FOCUS [11], CLEO[12], and the B factories [13,14]. Recently CDF [15,16] andLHCb [17–19] presented the measurements for the “differ-ence CP asymmetry” ΔACP, the difference between the CPasymmetries in the above two decay modes, that wasexpected to cancel out some of the systematic uncertainties.These results, obtained using the pion-tagged samples,indicated a value of Oð0.5%Þ for ΔACP. These measure-ments disfavored a vanishing CP asymmetry and were alsoaway from the SM prediction (leading order in 1=mc [21])ofOð0.05%–0.1%Þ by more than ∼2σ. Moreover, the latestLHCb results with the pion-tagged sample [18] and themuon-tagged sample [19] have central values with opposite

*amol@theory.tifr.res.in†diptimoy.ghosh@roma1.infn.it‡bhavik@theory.tifr.res.in

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signs and differ by ∼2.2σ from each other. The average ofthese two LHCb measurements has also been recentlyreported [20], which is consistent with vanishing ΔACP.(However, as we shall point out in this paper, such anaverage need not be the right observable to look for.)Several attempts have been made to check if the

observed large CP asymmetries can be accommodatedwithin the SM. It has been claimed that QCD penguinoperators, with large strong phases, may give rise to asignificant enhancement [21–24]. The breaking of SUð3Þ[25], or U-spin [26,27] symmetries, or of the naive 1=Nccounting [28], may also be a reason for the observed largeΔACP. However, the jury is still out on whether thesecontributions can account for the data without the need togo beyond the SM.Specific NP models that can enhance the CP asymme-

tries have also been extensively studied. These includethe fourth quark generation [27,29], supersymmetricgluino-squark loops [6], littlest Higgs model with T parity[30,31], flavor violation in the up sector [32,33], modelswith a color-sextet diquark [34], models giving rise to thet-channel exchange of a weak doublet with a special flavorstructure [35], the nonmanifest left-right symmetric model[36], or models with warped extra dimensions [37]. Asurvey of the effect of NP models that may contribute to thedifference CP asymmetry was performed in Ref. [38]. Itpoints out that the CP violation may be generated at the treelevel with models that involve flavor-changing couplings ofZ, Z0 bosons, new charged gauge bosons, a flavor-changingheavy gluon, scalar octets, a scalar diquark, or a two-Higgsdoublet with minimal flavor violation. Models with GIM-unsuppressed fermion and scalar loops, or those withchirally enhanced magnetic penguin operators, can alsocontribute to the CP asymmetry at the loop level. It hasbeen observed [39] that NP models in which the primarysource of flavor violation is linked to the breaking of chiralsymmetry are natural candidates to explain the CP asym-metries, via enhanced chromomagnetic operators. Many ofthese models also affect the measurements of otherD decaychannels, as well as the D − D mixing and ϵ0=ϵ in the Ksector, and hence the masses of new particles and couplingsin most of these models are severely constrained [40].Identifying whether the enhancement in the CP violation

in D → πþπ−, KþK− is from the SM or NP is notstraightforward; however, some information may beobtained from related decay modes. It was pointed outin Ref. [41] that, since the enhancement due to non-perturbative physics should only affect exclusive modes,an enhancement in the inclusive modes will point defini-tively to NP. One could also look at modes related to πþπ−,KþK− by isospin symmetry, since this symmetry is notexpected to be broken significantly. Such a comparisoncould distinguish between a large penguin amplitude andan enhanced chromomagnetic dipole operator, for exam-ple, Ref. [42].

The aim of this paper is not to check whether the SM orany specific NP model explains the data. Rather, we chooseto take the data at face value and learn in a model-independent way what they tell us about the CP violationin the D − D mixing and decay. To this end, we perform afit to the data with four complex parameters, M12, Γ12, Rπ ,and RK. Here, the mixing parameters M12 and Γ12 are thecomplex-valued dispersive and absorptive components,respectively, of the effective D − D mixing Hamiltonian.The other two parameters,

Rπ ≡ AðD → πþπ−ÞAðD → πþπ−Þ and RK ≡ AðD → KþK−Þ

AðD → KþK−Þ ;

are the ratios of decay amplitudes of a pure D and D to theCP eigenstates.The data on D − D mixing and CP asymmetries in the

πþπ− and KþK− channels form the main input for the fit.The ingredients for the fit also include the asymmetriesAΓðπÞ, AΓðKÞ, yCPðπÞ, and yCPðKÞ, constructed from theratios of effective lifetimes measured in the CP-eigenstatemodes D=D → πþπ−, KþK−, and the (almost-)flavor-specific modes D → Kþπ− and D → K−πþ [43], whichhave been reported by FOCUS [44], CLEO [12], Belle[45], BABAR [46,47], and LHCb [48]. While the measure-ments of AΓðπÞ, AΓðKÞ and yCPðπÞ are consistent with zeroto within 2σ, the asymmetry yCPðKÞ has been found to benonzero to more than 4σ [45–47]. More recently, Belle [49]and BABAR [50] have reported values of AΓ and yCPaveraged over the πþπ− and KþK− samples, where theyCPðavgÞ has been found to be nonzero to more than ∼4σ ineach experiment. These asymmetries may be represented interms of the combinations of the same parameters consid-ered above; hence, the information content in thesemeasurements is also relevant in determining the favoredparameter values and, in fact, is commonly used [3].The classification of CP violation in neutral meson

systems is normally described in two languages. Onemay talk in terms of CP violation in only mixing (deviationof jq=pj from unity), in only decay (deviation of jRfj fromunity, where Rf ≡ Af=Af), and in the interference ofmixing and decay (imaginary part of λf ≡ ðq=pÞRf).This is the standard notation used in the discussion of Bdecays. On the other hand, one may use the language ofdirect vs indirect CP violation, which has its origins inthe analyses of K decays. While we personally prefer theformer formulation due to its clarity in distinguishing thesource of the CP violation, the latter one has been used inmost of the literature on the CP asymmetries in D decaysthat is the focus of this paper. Indeed, the recent exper-imental data [15–19] have been interpreted in terms of thedirect and indirect CP asymmetries (Adir

CP and AindirCP ,

respectively) in the ππ and KK decays. We therefore shallrefer to both the notations, at the risk of some repetition inpresenting our results and interpretations.

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The interpretation of ΔACP in terms of its direct andindirect components often [16–19] takes Aindir

CP ðπÞ ¼AindirCP ðKÞ. We point out that if this condition were strictly

valid it would also imply AdirCPðπÞ ¼ Adir

CPðKÞ, independentof the origin of the CP asymmetry. This is clearly not thecase, even in the limit of flavor SUð3Þ, where these twoquantities have the same magnitudes but opposite signs.Therefore, the assumption of exactly equal Aindir

CP in theπþπ− and KþK− channels is, strictly speaking, not accu-rate. In practice, with certain “natural” expectations aboutthe amplitudes and phases of NP contributions, the non-universality of Aindir

CP may turn out to be so small that it maybe neglected [6,7], since the difference Adir

CPðπÞ − AdirCPðKÞ is

less than Oð0.01Þ and is expected to contribute to thenonuniversality Aindir

CP ðπÞ ¼ AindirCP ðKÞ only to the second

order, making it Oð10−4Þ. However, while searching forphysics beyond the SM, the analysis of data should beperformed without prejudice to theoretical expectations,and alternative scenarios, however unlikely they may seemat first, should be taken into account. We thereforereanalyze the current data without the approximationAindirCP ðπÞ ¼ Aindir

CP ðKÞ. Our fit, in fact, shows that thepreferred parameter space allows significantly differentvalues for Aindir

CP in πþπ− and KþK− decays. Such adifference could also contribute to the seemingly differentΔACP values measured through the pion-tagged and muon-tagged data samples at the LHCb [18,19]. That thisnonuniversality also leads to the nonuniversality of AΓand yCP has been indirectly alluded to in Ref. [51].Our paper is organized as follows. In Sec. II, we present

the analytical expressions for the time-dependent CPasymmetries and their direct and indirect components,AdirCP and Aindir

CP , inferred from the data. We also relate AΓand yCP, the quantities obtained from the measurements ofeffectiveD decay rates in different channels, to the relevantCP-violating quantities. In Sec. III, we perform a χ2 fit tothe data and obtain the favored values for the parametersof interest. Section IV is devoted to the feasibility andimplications of a significant nonuniversality of Aindir

CP .Section V summarizes our results and recommends takingthe possible nonuniversality in Aindir

CP into account for futureanalyses of neutral D decay data.

II. D − D MIXING AND DECAY: FORMALISM

We follow the analysis of D − D mixing and decay as inRefs. [4,51]. Since the notations vary from analysis toanalysis, we repeat the relevant steps to clarify our notation.In the ðD; DÞ flavor basis, the effective HamiltonianH ¼ M − iΓ=2 is not diagonal. The off-diagonal elementsof the dispersive and absorptive components, i.e., M12 andΓ12, are responsible for the D − D mixing. The masseigenstates are given by

jDLi ¼ pjDi þ qjDi; jDHi ¼ pjDi − qjDi; (1)

where

jqj2 þ jpj2 ¼ 1;

�qp

�2

¼ M�12 − i

2Γ�12

M12 − i2Γ12

: (2)

The deviation of jq=pj from unity corresponds to CPviolation in mixing. Note that, as opposed to the mixingin the neutral B systems (B − B, Bs − Bs), wherejΓ12j ≪ jM12j, here we have the possibility of Γ12 beingof the same order asM12 or even a few times larger [43]. If,in addition,M12 and Γ12 have significantly different phases,then jq=pj can differ substantially from unity, and theeffects of this CP violation in mixing need to be taken careof in the analysis.With the mass difference and the decay width difference

of the interaction eigenstates DH;L defined as

Δm ¼ mH −mL; ΔΓ ¼ ΓH − ΓL; (3)

the time evolutions of the mass eigenstates are

jDH;LðtÞi ¼ e−iðmH;L−i2ΓH;LÞjDH;Li: (4)

Using Eqs. (1) and (4), the time evolution of an initial D orD state becomes

jDðtÞi ¼ gþðtÞjDi − qpg−ðtÞjDi; (5)

jDðtÞi ¼ gþðtÞjDi − pqg−ðtÞjDi; (6)

where the coefficients g�ðtÞ are

g�ðtÞ ¼1

2ðe−imHt−1

2ΓHt � e−imLt−1

2ΓLtÞ: (7)

We use the standard convention

Af ¼ hfjHjDi; Af ¼ hfjHjDi (8)

to denote the amplitudes for the decay of D and D mesonsto a final state f. The time-dependent decay rate of an initialD meson to the final state f can then be written as

dΓðDðtÞ → fÞdt

¼ NfjhfjHjDðtÞij2; (9)

where Nf is the time-independent normalization factor.Using Eqs. (5), (6), and (7), we get

dΓðD0ðtÞ→ fÞdt

¼Nf

2e−ΓtjAfj2½ð1þjλfj2ÞcoshðyΓtÞ

þð1− jλfj2ÞcosðxΓtÞþ2ReðλfÞsinhðyΓtÞ−2 ImðλfÞsinðxΓtÞ�; (10)

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where Γ ¼ ðΓH þ ΓLÞ=2, and λf ¼ ðq=pÞðAf=AfÞ.Similarly,

dΓðDðtÞ→ fÞdt

¼Nf

2e−Γt

����pqAf

����2

½ð1þ jλfj2ÞcoshðyΓtÞ

− ð1− jλfj2ÞcosðxΓtÞþ 2ReðλfÞ sinhðyΓtÞþ 2 ImðλfÞ sinðxΓtÞ�: (11)

The expressions above are applicable for both the finalstates, f ¼ πþπ− and f ¼ KþK−, that are the focus ofthis paper.

A. Direct and indirect CP asymmetries

The time-dependent CP asymmetry for the decay proc-ess D → f is

ACPðtÞ ¼dΓðDðtÞ→fÞ

dt − dΓðDðtÞ→fÞdt

dΓðDðtÞ→fÞdt þ dΓðDðtÞ→fÞ

dt

: (12)

From Eqs. (10) and (11), we get

ACPðtÞ ¼ðj qp j2 − 1ÞΩþ þ ðj qp j2 þ 1ÞΩ−

ðj qp j2 þ 1ÞΩþ þ ðj qp j2 − 1ÞΩ−; (13)

where

Ωþ ≡ ð1þ jλfj2Þ coshðyΓtÞ þ 2ReðλfÞ sinhðyΓtÞ;Ω− ≡ ð1 − jλfj2Þ cosðxΓtÞ − 2 ImðλfÞ sinðxΓtÞ: (14)

Since x, y≲Oð1%Þ [3] and Γt ∼Oð1Þ or less, the aboveexact expression may be simplified by expanding in thesmall parameters x and y and keeping the leading terms.For convenience, we also use the notation

jq=pj2 ¼ 1þ ζ: (15)

Given the current 95% bounds 0.44 < jq=pj < 1.07 [3], wecannot take ζ to be a small quantity. The expansion in smallparameters x and y to linear order allows us to writeEq. (13) in the form1

ACPðtÞ ¼ AdirCP þ t

τDAindirCP ; (16)

where τD is the lifetime of the D meson. Here, AdirCP and

AindirCP are given as

AdirCP ¼ 1 − jλfj2 þ ζ

1þ jλfj2 þ ζ¼ 1 − jAf=Afj2

1þ jAf=Afj2¼ 1 − jRfj2

1þ jRfj2; (17)

AindirCP ¼−2

����qp����2 ½ð1þjλfj2Þx ImðλfÞþð1− jλfj2ÞyReðλfÞ�

ðj qp j2þjλfj2Þ2:

(18)

The above expression for AindirCP reduces to the one com-

monly used [3] in the limit Rf ¼ 1 in both the decay modes,since for CP-even final states like πþπ− and KþK− wehave λf ¼ −jðq=pÞRfjeiϕ [6,7]. Our expression is moregeneral and needs to be used if the possibility of direct CPviolation is to be taken into account. Even if the measureddirect CP violation is very small, i.e., jRfj ≈ 1.00, it ispossible that the phase of Rf is different for the two finalstates. This would make the value of λf different for thetwo final states. Indeed, as will be seen in Sec. III, themeasurements indicate jRπj ≈ jRKj ≈ 1.00 to within 1%,while ArgðRπÞ − ArgðRKÞ can be large. Such a scenariowould, of course, need the NP contribution to be of a veryspecific magnitude and phase. This important issue will bediscussed later in Sec. IV in detail.Note that the effective lifetimes of the decay modes

D → f and D → f differ from τD by terms of Oðx; yÞ.However, this does not change Adir

CP, and the change in AindirCP

due to this is quadratic in x, y. Hence, this difference can beneglected in our linear expansion. Integrating Eq. (16) overthe observed normalized distribution of the proper decaytime as measured in the D → f decay, we get

hACPi ¼ AdirCP þ hti

τDAindirCP : (19)

Here, hti is the average decay time that can be measuredseparately for each D → f decay mode. For the πþπ− andKþK− decay modes,

hACPðπÞi ¼ AdirCPðπÞ þ

htðπÞiτD

AindirCP ðπÞ; (20)

hACPðKÞi ¼ AdirCPðKÞ þ

htðKÞiτD

AindirCP ðKÞ; (21)

where htðπÞi, htðKÞi are average decay times of D mesonsfor decays into the πþπ− and KþK− states, respectively.These average times are characteristics of specific experi-ments, the values for which have been shown in Table I.Note that for the LHCb data hti ¼ ðhtðKÞi þ htðπÞiÞ=2and Δhti ¼ htðKÞi − htðπÞi.Subsequently, the difference CP asymmetry can be

obtained as [51]

ΔACP ¼ hACPðKÞi − hACPðπÞi

¼ AdirCPðKÞ − Adir

CPðπÞ þhtðKÞiτD

AindirCP ðKÞ

−htðπÞiτD

AindirCP ðπÞ: (22)

1Note that this indeed is the definition of AindirCP used in the

experimental analysis of data [16,19].

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As can be seen from Eqs. (17) and (18), AindirCP depends on

the final state f through its λf. Hence, in general, indirectCP asymmetries in D → πþπ− and D → KþK− can bedifferent. Hence, the equality Aindir

CP ðπÞ ¼ AindirCP ðKÞ, as is

generally used in the analyses of these channels, is onlyapproximate. Note that this statement is independent of themechanism of CP violation, since our analysis has beencompletely model independent.We would like to make a subtle point here.

Equation (18) indeed agrees with the statement madein the literature [6] that in the absence of direct CPviolation the indirect CP violation is universal. However,this statement needs to be interpreted with caution. It istrue only if the absence of direct CP violation is taken tomean Rf ¼ 1, both in magnitude as well as phase, for allmodes (in any consistent phase convention). The absenceof observable direct CP violation, however, only requiresjRfj ¼ 1, which is not enough to guarantee this univer-sality. On the other hand, the universality of indirect CPviolation only needs Rf to be equal (in magnitude as wellas phase) for all relevant decay modes, and its magnitudeis immaterial.The measured values of hACPðπÞi, hACPðKÞi, and ΔACP

are shown in Table II. The assumption of AindirCP ðπÞ ¼

AindirCP ðKÞ may also be responsible for the apparent discrep-

ancy between the values of ΔACP measured at the LHCbthrough the pion-tagged and the muon-tagged samples.Note that the values of htðπÞi ¼ hti − Δhti=2 for thetwo samples are different, and so are the values ofhtðKÞi ¼ hti þ Δhti=2. Indeed, we can write the differenceδðΔACPÞ≡ ðΔACPÞπ − ðΔACPÞμ as

δðΔACPÞ ¼�htðKÞiπ − htðKÞiμ

τD

�AindirCP ðKÞ

−�htðπÞiπ − htðπÞiμ

τD

�AindirCP ðπÞ

¼�ðΔhtiÞπ − ðΔhtiÞμ

2τD

�½Aindir

CP ðKÞ þ AindirCP ðπÞ�

þ�htiπ − htiμ

τD

�½Aindir

CP ðKÞ − AindirCP ðπÞ�: (23)

The term on the last line would be missed if one assumesAindirCP ðπÞ ¼ Aindir

CP ðKÞ. We shall revisit this quantitativelyduring our numerical analysis in the next section.

B. CP-violating observables through effective lifetimes

The expansion of Eq. (10) to first order in x, y yields

dΓdt

ðDðtÞ → fÞ

≈Nf

2e−ΓtjAfj2½1þ yΓt × ReðλfÞ − xΓt ImðλfÞ�: (24)

With zf ≡ x ImðλfÞ − yReðλfÞ, this could be written in theform [43]

dΓdt

ðDðtÞ → fÞ ∝ e−Γtð1 − zfΓtÞ: (25)

The effective lifetime in the D → f mode is then

τf ≈ ð1 − zfÞ=Γ: (26)

Since zf depends on the decay mode D → f in general, theeffective lifetimes measured in different modes can be

TABLE I. Experimental values of various quantities that appearin the determination of Adir

CP and AindirCP . We take τD ¼ ð0.41�

0.0015Þ ps [5].Quantity Value Reference

htðπÞi=τD 2.4� 0.03 CDF 2011a, CDF2012 [15,16]

htðKÞi=τD 2.65� 0.03 CDF 2011a, CDF2012 [15,16]

Δhti=τD 0.0983� 0.0022� 0.0019 LHCb 2011a

(π tagged) [17]0.1119� 0.0013� 0.0017 LHCb 2013

(π tagged) [18]0.018� 0.002� 0.007 LHCb 2013

(μ tagged) [19]hti=τD 2.0826� 0.0077 LHCb 2011a

(π tagged) [17]2.1048� 0.0077 LHCb 2013

(π tagged) [18]1.062� 0.001� 0.003 LHCb 2013

(μ tagged) [19]aThese data are not used for the fit.

TABLE II. Experimental values of CP asymmetries measuredat the experiments.

Quantity Value (%) Reference

hACPðπÞi −4.9� 7.8� 3.0 E791 1997a [10]4.8� 3.9� 2.5 FOCUS 2000a [11]1.9� 3.2� 0.8 CLEO 2001a [12]0.04� 0.69 CDF 2011a [15]

−0.24� 0.52� 0.22 BABAR 2008 [13]0.55� 0.36� 0.09 Belle 2012 [14]

hACPðKÞi −1.0� 4.9� 1.2 E791 1997a [10]−0.1� 2.2� 1.5 FOCUS 2000a [11]0.0� 2.2� 0.8 CLEO 2001a [12]

0.00� 0.34� 0.13 BABAR 2008 [13]−0.24� 0.41 CDF 2011a [15]

−0.32� 0.21� 0.09 Belle 2012 [14]ΔACP −0.82� 0.21� 0.11 LHCb 2011a (π -tagged) [17]

−0.62� 0.21� 0.10 CDF 2012 (π -tagged) [16]−0.34� 0.15� 0.10 LHCb 2013 (π -tagged) [18]0.49� 0.30� 0.14 LHCb 2013 (μ -tagged) [19]

aThese data are not used for the fit.

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different. These differences may be used to constructobservables that are sensitive to CP violation in D decays.For the decay D → f, Eq. (11) may also be written in

another convenient form:

dΓdt

ðDðtÞ → fÞ ¼ Nf

2e−ΓtjAfj2½ð1þ jλ−1f j2Þ coshðyΓtÞ

þ ð1 − jλ−1f j2Þ cosðxΓtÞþ 2Reðλ−1f Þ sinhðyΓtÞ− 2 Imðλ−1f Þ sinðxΓtÞ�: (27)

After neglecting terms that are quadratic or higher powersin x, y, one gets

dΓdt

ðDðtÞ → fÞ ∝ e−Γtð1 − zfΓtÞ; (28)

with zf ≡ x Imðλ−1f Þ − y × Reðλ−1f Þ, so that the effectivelifetime for this mode becomes

τf ¼ ð1 − zfÞ=Γ: (29)

When the final state f is a CP eigenstate fCP like πþπ−or KþK−, a CP-violating quantity can be constructed fromthe difference of the effective lifetimes of D → fCP andD → fCP [43]:

AΓðfCPÞ≡ τfCP − τfCPτfCP þ τfCP

¼ 1

2ðzfCP − zfCPÞ

¼ 1

2ðx½ImðλfCPÞ − Imðλ−1fCPÞ�

− y½ReðλfCPÞ − Reðλ−1fCPÞ�Þ (30)

clearly vanishes in the limit of CP conservation sinceλfCP ¼ �1 in that case. This expression reduces to the oneused in Ref. [3] in the limit Rf ¼ 1, as expected. Therelation AΓ ¼ −Aindir

CP used in Ref. [3] is also valid only inthis approximation, the actual relation being

AΓ ¼ −1

4AindirCP

�jRfj þ

1

jRfj�

2

: (31)

Although the form of Eq. (30) seems different from the onegiven in Ref. [51], it is a result of expansions up to differentorders in small quantities. In particular, we do not assumeζð≡jq=pj2 − 1Þ to be small and keep terms to a higherpower in it.The quantities AΓðπÞ and AΓðKÞ have been measured

separately [45,46,48] and as an average over the two modes[49,50]; however, the errors are not small enough for anonzero measurement. See Table III.For flavor-specific decays, where D → f is allowed but

D → f is not, λf vanishes, and Eq. (10) gives

dΓdt

ðDðtÞ → fÞ ≈ Nfe−ΓtjAfj2; (32)

when terms with quadratic and higher powers of x, y areneglected. The average lifetime for such processes is clearlyτFS ≈ 1=Γ. (Note that while taking D → π−Kþ to be aflavor-specific mode the doubly Cabibbo-suppressed decayD → πþK− has been neglected.) The quantity

yCP ≡ τFSðτfCP þ τfCPÞ=2

− 1 ≈1

2ðzfCP þ zfCPÞ

¼ 1

2ðx½ImðλfCPÞ þ Imðλ−1fCPÞ�

− y½ReðλfCPÞ þ Reðλ−1fCPÞ�Þ (33)

is not necessarily CP violating; however, it forms animportant input for disentangling zfCP and zfCP from themeasurement of AΓ. Note that the expression clearlyreduces to the one used in Ref. [3] in the limit of Af ¼Af (or Rf ¼ 1), and in the absence of CP violation, i.e.,ϕ ¼ 0, one has yCP ¼ y.Taking the CP eigenstates to be πþπ− and the flavor-

specific final state to be πK, the measurements of yCPðπÞhave so far been consistent with zero to within 2σ[12,45,46], while taking the CP eigenstates to be KþK−,nonzero measurements of yCPðKÞ to more than the 4σ levelhave been obtained [45–47]. Averaging over πþπ− andKþK− modes [49,50] also gives a nonzero value at morethan the 3σ level.

III. NUMERICAL ANALYSIS

We now perform a χ2 fit to the data on theD − Dmixingand decay with an aim toward disentangling the

TABLE III. Measured values of AΓ and yCP.

Quantity Value (%) Reference

AΓðπÞ −0.28� 0.52� 0.15 Belle 2007 [45]−0.049� 0.73 BABAR 2008 [46]

AΓðKÞ 0.15� 0.35� 0.15 Belle 2007 [45]0.39� 0.46 BABAR 2008 [46]

−0.59� 0.59� 0.21 LHCb 2011 [48]AΓðavgÞ −0.03� 0.20� 0.08 Belle 2012 [49]

0.09� 0.27 BABAR 2013 [50]yCPðπÞ 0.5� 4.3� 1.8 CLEO 2001 [12]

1.44� 0.57� 0.25 Belle 2007 [45]0.46� 0.65� 0.25 BABAR 2008 [46]

yCPðKÞ 3.42� 1.39� 0.74 Focus 2000 [44]−1.9� 2.9� 1.6 CLEO 2001 [12]1.25� 0.39� 0.25 Belle 2007 [45]1.60� 0.46� 0.17 BABAR 2008 [46]1.12� 0.26� 0.22 BABAR 2009 [47]0.55� 0.63� 0.41 LHCb 2011 [48]

yCPðavgÞ 1.11� 0.22� 0.11 Belle 2012 [49]0.72� 0.18� 0.12 BABAR 2013 [50]

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contributions from CP violation in mixing and in decay.The fit is performed to the four model-independent com-plex parameters M12, Γ12, Rπ , and RK. Here, one has to becareful about the data to be included. We use the followingprescription:

(i) For the data on hACPðπÞi; hACPðKÞi, and ΔACP, weuse the experimental data as shown in Table IIdirectly. The average decay times as given in Table Iare used. Note that the data marked with a * areshown for the sake of completeness, but they are notused in the fit, either because they give too weakconstraints or because they have been used in laterresults by the same collaboration. This helps avoiddouble counting the same data.

(ii) For the data on AΓ and yCP, we do not use theCOMBOS fit [3] directly since it assumes equal valuesof these quantities in the πþπ− and KþK− channels.Whereas, as can be seen from Sec. II B, the differ-ence between AΓðπÞ and AΓðKÞ, as well as betweenyCPðπÞ and yCPðKÞ, is of linear order in x, y whenλπ ≠ λK , and hence is relevant for our analysis here.We therefore use the data on AΓðπÞ, AΓðKÞ, yCPðπÞ,and yCPðKÞ separately, as shown in Table III. Wehave to contend with the problem that the mostrecent Belle and BABAR results [49,50] only reportvalues of AΓ and yCP that are averaged over theπþπ− and KþK− modes. While using these data, wetake the averaged values to be weighted averages,with weights proportional to the number of events inthe two modes. However, it would have beendesirable to have the values of AΓðπÞ, AΓðKÞ,yCPðπÞ, and yCPðKÞ separately, directly from theexperiments, for a more accurate analysis.

(iii) For the input fromD − Dmixing also, we do not usethe HFAG [3] fit for x, y, jq=pj, φ directly here since,in addition to theD − Dmixing data, it uses the dataon ACP, AΓ and yCP that have already been usedabove, which would have given rise to doublecounting of the data. We therefore only use the datafrom [1,2] and the COMBOS average [3] for RM ≡ðx2 þ y2Þ=2 from the semileptonic D decays. Tokeep the number of fit parameters limited, we do notuse the data on mixing parameters from the Kπ orKππ channel.

The χ2 function is taken to be

χ2 ¼X32i¼1

ðXi − Xexpi Þ2

ðσXiÞ2 : (34)

Here, Xi, Xexpi , and σXi

with (i ¼ 1; 2; 3;…; 32) representthe theoretical values, experimental values, and corre-sponding experimental uncertainties, respectively, of theobservables given in Tables II, III and IV. We add thestatistical and systematic errors in quadrature and take allthe measurements to be independent and uncorrelated. The

MINUIT [52] subroutine is used for the minimization of χ2

in the multidimensional parameter space. The best-fitvalues of the fit parameters are

jM12j ¼ 0.0059 ps−1; ArgðM12Þ ¼ 3.37;

jRπj ¼ 1.002; ArgðRπÞ ¼ 3.82;

jΓ12j ¼ 0.0207 ps−1; ArgðΓ12Þ ¼ 3.39;

jRKj ¼ 1.000; ArgðRKÞ ¼ 3.20:

Note that the phases of M12 and Γ12 are conventiondependent, however, the difference between them is inde-pendent of phase conventions. Since this difference issmall, the magnitude of q=p at the best-fit point is stillclose to unity. It is also observed that the values of best fitfor jRKj as well as jRπj do not deviate much from unity, sothat the direct CP violation in both these decay modes isexpected to be rather small. However, the phases of Rπ andRK at the best-fit point are significantly different. This willbe relevant in our discussion later.The fit is rather good: at the best-fit point,

χ2=dof ¼ 27.6=24. It is interesting that, even if we imposea further restriction of jλπj ¼ jλKj, which would correspondto jRπj ¼ jRKj, the fit still stays almost as good, withχ2=dof ¼ 28.1=25. The values of the best-fit parameters arealso very similar. This indicates that around the best-fitpoint, the CP violation through mixing alone as well as CPviolation through decay alone is very small, so the CPviolation observed is mainly through the interferencebetween mixing and decay. Further insisting on identicalvalues for the magnitudes as well as phases of λπ and λK ,however, worsens the fit to χ2=dof ¼ 37.3=26. This indi-cates the tension of the data with the scenario of equal CPviolation in the two decays. As indicated in the previoussection, this implies significantly different values of Aindir

CP inthe two decays.To make quantitative statements about the significance

of our observations above, we show the parameter spacesfavored by the data in Fig. 1. The contours shown inthe figure correspond to Δχ2 ¼ 2.3 (68% C.L.) and

TABLE IV. Experimental input for D − D mixing parameters.

Quantity Value Reference

x ð0.80� 0.29þ0.09þ0.10−0.07−0.14 Þ% Belle 2007 [1]

y ð0.33� 0.24þ0.08þ0.06−0.12−0.08 Þ% Belle 2007 [1]

jq=pj 0.86þ0.30þ0.06−0.29−0.03 Belle 2007 [1]

φ ð−14þ16þ5þ2−18−3−4 Þ degrees Belle 2007 [1]

x ð0.16� 0.23�0.12� 0.08Þ%

BABAR 2010 [2]

y ð0.57� 0.20�0.13� 0.07Þ%

BABAR 2010 [2]

RM ¼ ðx2 þ y2Þ=2 0.0130� 0.0269 [3]

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Δχ2 ¼ 4.61 (90% C.L.). The following observations maybe made from the figure:

(i) jΓ12j > jM12j in the whole of the region allowed to90% C.L.. Indeed, the data seem to favor the regionwith jΓ12j equal to a few times jM12j.

(ii) The phase between M12 and Γ12 is compatible withzero, although deviations of up to ≈0.9 radians arepossible. The SM prediction for this phase would beOð0.1%Þ, so much larger values for this phase arestill allowed by the data.

(iii) The 90% allowed values for both jRπj and jRKj, forboth πþπ− and KþK− modes, are consistent withunity. While a deviation of ≈1% from unity isallowed for jRπj, the value of jRKj is restricted tobe within ≈0.5% of unity. This indicates that thedirect CP violation Adir

CP is restricted to a fraction of apercent.

(iv) The relative phase between Rπ and RK is a con-vention-independent, physical quantity. Data seem

to prefer different phases for Rπ and RK . Thisindicates that, although jRπj ≈ jRKj is allowed,Rπ ¼ RK is disfavored. This would further implyλπ ≠ λK , and consequently Aindir

CP ðπÞ ≠ AindirCP ðKÞ.

The derived quantities x, y, jq=pj and φ≡ Argðq=pÞthat are commonly used to describe the D − D mixing areshown in Fig. 2. The figure indicates the following:

(i) The values of both x and y are positive to 90% C.L..(ii) Although jq=pj is consistent with unity, a variation

in the range (0.7, 1.3) is still allowed to 90% C.L..This implies that significant CP violation throughmixing is allowed. Note that our fit gives highervalues of jq=pj as compared to the one in Ref. [3];however, there are differences in the two fit proce-dures. We have used only a subset of the data usedtherein but have taken care of possibly differentvalues of AΓ and yCP in the πþπ− and KþK− modes.

(iii) The phase φ is restricted to be in the rangeð−0.9; 0.3Þ to 90% C.L.. This quantity is of course

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.005 0.01 0.015 0.02 0.025 0.03

|Γ12

| [ps

-1]

|M12| [ps-1]

2.34.61

Best fit pointSM

|M12| = |Γ12|

0

π/2

π

3π/2

2π

0 π/2 π 3π/2 2π

Arg

(Γ12

)

Arg(M12)

2.34.61

Best fit pointArg(M12) = Arg(Γ12)

0.99

0.995

1.00

1.005

1.01

1.015

0.99 0.995 1.00 1.005 1.01 1.015

|RK|

|Rπ|

2.34.61

Best fit point|Rπ| = |RK|

0

π/2

π

3π/2

2π

0 π/2 π 3π/2 2π

Arg

(RK)

Arg(Rπ)

2.34.61

Best fit pointArg(Rπ) = Arg(RK)

FIG. 1 (color online). Parameter spaces favored by the current data on D − D mixing and decay. While displaying constraints on twoparameters, the values of the other parameters are varied over all their allowed ranges to minimize χ2. The SM value of ðjM12j; jΓ12jÞ istaken from Ref. [30].

AMOL DIGHE, DIPTIMOY GHOSH, AND BHAVIK KODRANI PHYSICAL REVIEW D 89, 096008 (2014)

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phase-convention dependent and is physically mean-ingful only when compared with the phases of Rπ orRK . So we shall not discuss it further.

Let us now explore the extent of CP violation throughthe interference of mixing and decay. This may be para-metrized through the imaginary part of λπ ≡ ðq=pÞRπ andλK ≡ ðq=pÞRK . While the phases of q=p, Rπ and RK shownabove are convention dependent, the phases of λπ and λKare physical quantities, independent of the phase conven-tion used. We show the allowed ranges of the magnitudesand phases of λπ and λK in Fig. 3:

(i) The magnitudes of λπ and λK are highly correlated:jλπj ≈ jλKj. This is expected, since jλfj ¼ jðq=pÞRfj,wherein the quantity jq=pj is common to both thedecay modes, and jRπj and jRKj are both very closeto unity.

(ii) On the other hand, different phases for λπ and λKseem to be preferred. This leads to a difference in the

values of λπ and λK and hence different values ofAindirCP in the two modes.

With the allowed ranges of parameters as determinedabove, we present the allowed values of Adir

CP and AindirCP in

the form of a scatter plot in Fig. 4. We can observe thefollowing:

(i) At the best-fit point, we have

AdirCPðπÞ ¼ −0.0024; Aindir

CP ðπÞ ¼ 0.0021;

AdirCPðKÞ ¼ −0.0001; Aindir

CP ðKÞ ¼ −0.0008;

so that the data favors different indirect CP asym-metries in these two modes.

(ii) While the direct CP violation AdirCP in the πþπ− mode

is restricted to be less than a percent, that in theKþK− mode is restricted even more severely, to beless than half a percent. These allowed values are

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

y [%

]

x [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best fit point

-π/2

0

π/2

0.6 0.8 1 1.2 1.4

Arg

(q /

p)

|q / p|

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best-fit point(1,0)

FIG. 2 (color online). Mixing parameters in D − D mixing, applicable to both πþπ− and KþK− decays.

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

|λK|

|λπ|

2.34.61

Best fit point

0

π/2

π

3π/2

2π

0 π/2 π 3π/2 2π

Arg

(λK)

Arg(λπ)

2.34.61

Best fit pointArg(λπ) = Arg(λK)

FIG. 3 (color online). Constraints on the magnitudes and phases of λπ and λK .

NONUNIVERSALITY OF INDIRECT CP ASYMMETRIES … PHYSICAL REVIEW D 89, 096008 (2014)

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still much larger as compared to the SMexpectations.

(iii) IndirectCP violation to the extent of half a percent isstill allowed for πþπ−, while in the case of KþK− itcan be maximum up to a quarter of a percent. Moreimportantly, different values of Aindir

CP in these modesare highly preferred by the data.

(iv) The shaded band shows that region in which theapparent discrepancy between the ΔACP measure-ments from the pion-tagged and muon-tagged sam-ple is resolved to within 1σ. As expected, theresolution favors significantly different values ofAindirCP , which is consistent with the results of our fit,

almost to ∼1σ. Referring back to Eq. (23), the largecoefficient of the ½Aindir

CP ðKÞ − AindirCP ðπÞ� (see Table I)

allows such an explanation of the apparent discrep-

ancy through a moderate difference in the indirectCP asymmetries in the two decay modes.

Before continuing, we also present the information onthe quantities

zπ≡xImðλπÞ−y×ReðλπÞ; zπ≡xImðλ−1π Þ−y×Reðλ−1π Þ;zK≡xImðλKÞ−y×ReðλKÞ; zK≡xImðλ−1K Þ−y×Reðλ−1K Þ;

(35)

obtained from the measurements of the quantities AΓ andyCP. It may be observed from Fig. 5 that the favored regionsin the ðzπ − zπÞ and ðzK − zKÞ parameter space are quitedifferent; they have only a small overlap. This is inconsonance with our overall observation that the dataindicate unequal amounts of CP violation in πþπ− and

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

AC

Pdi

r (K)

[%]

ACPdir(π) [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best fit point(0,0)

ACPdir(π) = ACP

dir(K)

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

AC

Pin

dir (K

) [%

]

ACPindir(π) [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best fit point(0,0)

ACPindir(π) = ACP

indir(K)

FIG. 4 (color online). Constraints on direct and indirect asymmetries in D → ππ and D → KK from the data in Tables II, III, and IV.The yellow (gray) band in the plot on the right corresponds to the values that will reconcile the ΔACP measurements through the pion-tagged and muon-tagged samples at the LHCb to within 1σ.

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5

z_ π [%

]

zπ [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best-fit point

zπ = z_

π

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5

z_ K [%

]

zK [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best-fit point

zK = z_

K

FIG. 5 (color online). Bounds on the parameters zπ , zπ , zK , and zK from the measurements of AΓðπÞ, AΓðKÞ, yCPðπÞ, and yCPðKÞ.

AMOL DIGHE, DIPTIMOY GHOSH, AND BHAVIK KODRANI PHYSICAL REVIEW D 89, 096008 (2014)

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KþK− modes. It is therefore important that the measure-ments of AΓðπÞ, AΓðKÞ, yCPðπÞ, and yCPðKÞ be availableseparately, without averaging over the πþπ− and KþK−

modes, and analyzed without the assumption of theirequality.

IV. FEASIBILITY AND IMPLICATIONS OFNONUNIVERSAL AINDIR

CP

Our analysis as such does not depend on whether wehave only SM, or whether NP is present in addition.However within the SM, even given the uncertaintiesdue to long-distance contributions, it is very difficult toget CP violation of the order of 1% or larger. Indeed, ifjq=pj actually differs substantially from unity, it will need alarge phase difference between M12 and Γ12, which doesnot seem possible in the SM. Also, getting significantlydifferent phases for the quantities Rπ and RK , as indicatedby the data, is not something the SM can do. We thereforeinterpret our results in terms of the demands they make onNP models and the observations required for identifyingsuch NP. To compare our results with previous analyses,here we present our arguments in terms of the language andnotation used in Refs. [5–7].In the notation of the Particle Data Group [5], the decay

amplitudes Af and Af [see Eq. (8)] are written as

Af ¼ ATfe

þiϕTf ½1þ rfeiðδfþϕfÞ�;

Af ¼ ATfe

−iϕTf ½1þ rfeiðδf−ϕfÞ�; (36)

where ATf is the leading tree amplitude with its correspond-

ing phase ϕTf , while rf is the ratio of the subleading to the

leading amplitude, with the corresponding strong and weakphase differences δf and ϕf, respectively. This notationmay be matched to ours by using

Rf ¼Af

Af¼ e−2iϕ

Tf

�1þ rfeiðδf−ϕfÞ

1þ rfeiðδfþϕfÞ

�: (37)

The approximation rf ≪ 1 used in Refs. [5,6] leads touniversal indirect CP violation (¼ am þ ai) that is inde-pendent of rf, δf, and ϕf. As a result, the quantities AΓ andyCP are also universal, i.e., identical for the πþπ− andKþK− modes.Although the approximation rf ≪ 1 is valid in the SM, it

is not guaranteed to be true in the presence of NP. Higher-order terms in rf give nonuniversal contributions to Aindir

CPthrough different values for λf [7]. The value of rf isbounded by the CP-violation measurements themselves:since Adir

CPðfÞ ∼Oð0.01Þ, we have jRfj ∼ 1þOð0.01Þ, andhence rf sin δf sinϕf ∼Oð0.01Þ, for both the decaymodes. On the other hand, in order to have a significantnonuniversality in Aindir

CP through λπ ≠ λK , we needArgðλπÞ ≠ ArgðλKÞ, since our fit suggests equal magni-tudes for these two quantities (see Fig. 3). Using the results

in Ref. [7], the relevant nonuniversality may be expressedin terms of

ArgðλKÞ − ArgðλπÞ ≈ 2rπ cos δπ sinϕπ − 2rK cos δK sinϕK;

(38)

so that at least one of the two quantities rf cos δf sinϕfshould be significantly large, i.e., Oð0.1–1Þ. For both theabove constraints to be satisfied simultaneously, weneed sinϕf ∼Oð1Þ, rf sin δf ∼Oð0.01Þ, and rf cos δf ∼Oð0.1–1Þ for at least one of the final states πþπ−and KþK−.The above argument suggests that for the condition

implied by our best-fit point to be met one needs anenhanced subleading amplitude with a large relative weakphase and a small relative strong phase compared to theleading term, for at least one of the two final states. Whilethe first two conditions may be generically satisfied by aNP model that is not too constrained from other measure-ments, the smallness of the relative strong phase may seemto be rather fine-tuned, since the NP operators typicallydiffer from the leading ones in their color and chiralitystructure [7]. Explicit calculations or direct measurementsof these strong phase differences are not available; recentanalysis of the ππ scattering data in the context of SM [53]indicates large strong phases, but note that what is neededhere is a small difference in the strong phases of the leadingand subleading contributions. On the other hand, themagnitudes and phases of the SM and NP contributionsshould conspire such that jRfj ≈ 1.00 to within 1%. The NPscenario required here is thus rather fine-tuned, but is notruled out, and hence should not be ignored. Further, notethat if the difference between the LHCb measurements withthe pion-tagged and muon-tagged samples is real this isperhaps the only mechanism that can account for it. In fact,the measurement of nonuniversality of Aindir

CP itself wouldgive direct information on the smallness or largeness of thisphase that appears in all the CP-violating observables.The NP scenario described above, which gives large

nonuniversality in AindirCP , is mandated if the best-fit point

obtained from our fit indeed turns out to be the right onewith future data. However, even if the future data were toindicate smaller nonuniversality in Aindir

CP than the best-fitpoint obtained our fit, our analysis in Sec. II still staysrelevant. It gives generalized expressions for the CPasymmetry ACP and the related quantities AΓ, yCP [seeEqs. (17), (18), (30), and (33)] that are valid even withnonuniversal Aindir

CP . The expressions prevalent in the stan-dard literature [3,5] do not take this possibility into account.The nonuniversality of AΓ and yCP was explicitly

calculated in Ref. [51], albeit with different expansionparameters than the ones considered here. We believe thatour expansion parameters (x and y) are more well motivatedsince they have been measured to be small. Moreover,while obtaining the final expression for ΔACP, Ref. [51]

NONUNIVERSALITY OF INDIRECT CP ASYMMETRIES … PHYSICAL REVIEW D 89, 096008 (2014)

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used the assumption of universality of the phase of λf, thusrestricting its domain of validity.

V. SUMMARY AND CONCLUSION

The recent measurements of the CP asymmetries inD=D → πþπ−, KþK− modes and the difference in the CPasymmetries in these modes (the so-called difference CPasymmetry) have yielded values differing from the SMexpectations. Moreover, the difference CP asymmetriesmeasured at the LHCb through the pion-tagged and themuon-tagged samples differ substantially. We examinethese data in a model-independent manner to discern thenature of CP violation involved therein and to find aresolution for the above discrepancy.By performing a fit to the data on D − D mixing, CP

asymmetries inD=D → πþπ−,KþK− modes, as well as therelated quantities AΓ and yCP in these channels, we find thefollowing: (i) the CP violation through decay only in boththe modes is restricted to be less than Oð0.5%Þ, althoughthis limit still allows values much larger than thosepermitted by the SM; (ii) the CP violation through mixingonly, on the other hand, can be quite large—the value ofjq=pj can differ from unity by ∼Oð10%Þ; and (iii) the CPviolation through the mixing-decay interference mayplay an important role in the D=D → πþπ−, KþK− modes.The phases of the quantities λπ and λK tend to differsubstantially.In the language of direct and indirectCP asymmetries, as

used while presenting the recent experimental data, thedirect CP asymmetries in both πþπ− and KþK− modes isrestricted to be less than Oð0.5%Þ, and these two asym-metries can have different values. The indirect CP asym-metries in these two modes also can differ substantially incertain scenarios. Indeed, we demonstrate that, mathemati-cally speaking, different direct CP asymmetries imply

different indirect CP asymmetries, unless there are acci-dental cancellations. We therefore emphasize that takingthe indirect CP asymmetries in these two channels to beequal is an approximation, one that may be valid but thatshould be checked with data.It turns out that the possibility of nonuniversal indirect

CP asymmetry also allows a partial reconciliation betweenthe seemingly different difference CP asymmetries mea-sured through the pion-tagged and muon-tagged samples atthe LHCb. The analyses for these decay modes in terms ofthe direct and indirect CP asymmetries should therefore beperformed without the usual assumption of equal indirectCP asymmetries in the two modes.Our formalism also allows us to express the quantities

AΓ and yCP in a symmetric form, AΓ ¼ ðz − zÞ=2 andyCP ¼ ðzþ zÞ=2, without having to assume equal magni-tudes as well as phases for AðD → fÞ=AðD → fÞ for theπþπ− and KþK− channels. This also indicates that the dataon these quantities in the πþπ− and KþK− channels shouldbe presented separately, since these quantities can bedifferent in these two modes and an averaging might loseinformation critical for ascertaining the presence and natureof any NP present.A significant nonuniversality in Aindir

CP would require thesubdominant amplitude in D → f decay to be comparablein magnitude to the dominant one, as well as a small strongrelative phase and a large weak relative phase betweenthese two amplitudes. This scenario appears fine-tuned,given the theoretical expectation of large relative strongphases; however, there is no direct calculation or meas-urement of the strong phase, so it is not ruled out.Therefore, it is important to take into account the possibilityof such a NP that could give rise to significant nonun-iversality in Aindir

CP , AΓ, and yCP. The generalized analysisand expressions presented in this paper then need to be used

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

AC

Pdi

r (K)

[%]

ACPdir(π) [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best fit point(0,0)

ACPdir(π) = ACP

dir(K)

-1.0

-0.5

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

AC

Pin

dir (K

) [%

]

ACPindir(π) [%]

2.3 < ∆ χ2 < 4.61∆ χ2 < 2.3

best fit point(0,0)

ACPindir(π) = ACP

indir(K)

FIG. 6 (color online). Constraints on direct and indirect asymmetries in D → ππ and D → KK from the current data, with the newLHCb 2013 [54] results added. The yellow (gray) band in the plot on the right corresponds to the values that will reconcile the ΔACPmeasurements through the pion-tagged and muon-tagged samples at the LHCb to within 1σ.

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instead of the ones in standard literature that assumeuniversality of Aindir

CP . Indeed, such a generalized analysiswill be useful in measuring the extent of the nonuniversalityitself and testing of the theoretical expectation of a largestrong relative phase.Since using the generalized analysis (by removing just

one assumption) offers the possibility of explaining thecurrent data as well as probing NP signals with future data,this opportunity should not be missed. This implies that theaveraging of AΓ, yCP values in πþπ−, KþK− modes at the Bfactories, as well as the averaging of ΔACP in pion-taggedand muon-tagged modes in LHCb, should be avoided. Withlarge amounts of data around the corner from the LHCupgrade and the super-B factory, statistics will cease to bethe limiting factor, and one can probe possible NP in theCP violation in D decays in a clean way.

ACKNOWLEDGMENTS

We would like to thank Marco Gersabeck, Alex Kagan,Yossi Nir, Luca Silvestrini, and Rahul Sinha for usefulcomments and scintillating discussions on the first version

of the manuscript. D. G. would like to thank SatoshiMishima, Ayan Paul, and Luca Silvestrini for usefuldiscussions. The research by D. G. leading to these resultshas received funding from the European Research Councilunder the European Union’s Seventh FrameworkProgramme (FP/2007-2013)/ERC Grant No. 279972.D. G. also acknowledges the hospitality of the SLACTheoretical Physics Group where this work was completed.

Note Added.—While this article was under review, theLHCb Collaboration announced undated results on AΓ [54]:AΓðKÞ ¼ ð−0.035� 0.062� 0.012Þ%, AΓðπÞ ¼ ð0.033�0.106� 0.014Þ%. Adding these measurements to our fit,we find that (see Fig. 6) the ability of nonuniversal Aindir

CP tohelp reconcile the ΔACP measurements through the pion-tagged and muon-tagged samples at the LHCb is ratherrestricted by the new data. However, the last word on theCP violation in D meson system has yet to be written [55],and a complete analysis should take into account thepossibility of different indirect CP asymmetries in theπþπ− and KþK− channels.

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