Physics Letters B 548 (2002) 146–152

www.elsevier.com/locate/npe

Does CPT violation affect theBd meson life times and decayasymmetries?

Amitava Dattaa, Emmanuel A. Paschosb, L.P. Singhc

a Department of Physics, Jadavpur University, Kolkata 700 032, Indiab Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany

c Department of Physics, Utkal University, Bhubaneswar 751004, India

Received 23 September 2002; accepted 14 October 2002

Editor: P.V. Landshoff

Abstract

We study indirect CPT violating effects inBd meson decays and mixing, taking into account the recent constraints on theCPT violating parameters from the Belle Collaboration. The life time difference of theBd meson mass eigenstates, expectedto be negligible in the standard model and many of its CPT conserving extensions, could be sizeable (∼ a few percent of thetotal width) due to breakdown of this fundamental symmetry. The time evolution of the direct CP violating asymmetries inone amplitude dominated processes (inclusive semileptonicBd decays, in particular) turn out to be particularly sensitive to thiseffect. 2002 Elsevier Science B.V. All rights reserved.

The suggestion for two distinct lifetimes for theBd or Bs meson mass eigenstates originated in partonmodel calculations [1], which, at that time, werelimited by numerous uncertainties of hadronic (fB ,the bag parameter, top quark mass, . . . ) and weakparameters (CKM matrix elements). Many of these,however, cancel in the ratio

(1)

(�m

�Γ

)d

= 8

9π

(ηt

η

)(mt

mb

)2

f (xt ),

where�md(�Γd) is the mass (width) difference ofthe Bd meson mass eigenstates,ηt , η are calculable

E-mail addresses: adatta@juphys.ernet.in (A. Datta),paschos@physik.uni-dortmund.de (E.A. Paschos),lambodar@iopb.res.in (L.P. Singh).

perturbative QCD corrections,xt =mt/mw and

f (x)= 3

2

x2

(1− x)3lnx

(2)−(

1

4+ 9

4

1

(1− x)− 3

2

1

(1− x)2

).

Following the discovery of mixing in theBd sys-tem [2],�md was measured andmt was the only ma-jor source of uncertainty in the ratio. Using the thenlower bound onmt it was shown [3,4] that�Γd is in-deed very small, while�Γs , the width difference oftheBs meson mass eigenstates could be rather largeas is indicated by the scaling law [4]

(3)

(�Γ

Γ

)s

=(XBs

XBd

)∣∣∣∣VtsVtd

∣∣∣∣2(�Γ

Γ

)d

,

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)02843-5

A. Datta et al. / Physics Letters B 548 (2002) 146–152 147

whereVij ’s are the elements of the CKM matrix and

(4)XBq = 〈Bq |[qγ µ(1− γ5)b

]2|�Bq〉.In the meanwhile, many advances have taken place

with the discovery of the top quark and the determina-tion of its mass [5] and more precise values for CKMmatrix elements. Combining the new values with theabove scaling laws, the width difference among theBd states is(�Γ/Γ )d ≈ 0.0012, which is unobserv-able, but forBs eigenstates(�Γ/Γ )s ≈ 0.045. Morerecent calculations using heavy quark effective theoryand improved QCD corrections [6,7] suggest that cal-culations based on the absorptive parts of the box di-agram improved by QCD corrections give reasonableestimates for bothBd andBs systems.

Nevertheless the possibility that there are loopholesin the above calculations cannot be totally excluded.For example,�Γq (q = d or s) is determined by onlythose channels which are accessible to bothBq and�Bqdecays. Its computation in the parton model may notbe as reliable as the calculation ofΓq , the total widthwhich depends on fully inclusive decays and quark–hadron duality is valid.

In addition to the expected phenomena, one should,therefore, be prepared for unexpected effects and thefinal verdict on this subject should wait for experimen-tal determination of�Γ from theB-factories,B-TeVor LHC-B. Many different suggestions for measuring�Γs have been put forward [3,4,8]. It is believed that(�Γ/Γ )d ∼ 0.1 can be measured atB-factories [9]while (�Γ/Γ )d ∼ 0.001 [10] might be accessible atthe LHC.

In this Letter we wish to emphasize that apart fromdynamical surprises in the decay mechanism, a pos-sible breakdown of the CPT symmetry contributes to�Γ/Γ . The currently available constraints on CPT vi-olating parameters [11,12] certainly allow this possi-bility. If this happens its effect will be more visible anddetectable in theBd system which, in the electroweaktheory, is expected to have negligible(�Γ/Γ )d . Inother words the scenario with(�Γ/Γ )d large not onlydue to hitherto unknown dynamics but also due toa breakdown of CPT is quite an open possibility. Inthe case of(�Γ/Γ )s CPT violation may act in tan-dem with the already known electroweak dynamics toproduce an even larger effect.

There are several motivations for drawing outa strategy to test CPT symmetry. From the experi-

mental point of view all symmetries of nature mustbe scrutinized as accurately as possible, irrespectiveof the prevailing theoretical prejudices. It may be re-called that before the discovery of CP violation, therewas very little theoretical argument in its favour.

There are purely theoretical motivations as well.First of all the CPT theorem is valid for local, renor-malizable field theories with well defined asymptoticstates. It is quite possible that the theory we are deal-ing with is an effective theory and involving smallnonlocal/nonrenormalizable interactions. Further theconcept of asymptotic states is not unambiguous inthe presence of confined quarks and gluons. It hasbeen suggested that physics at the string scale may in-deed induce nonlocal interactions in the effective lowenergy theory leading to CPT violation [13]. More-over, modification of quantum mechanics due to grav-ity may also lead to a breakdown of CPT [14].

One of the major goals of theB-factories runningat KEK or SLAC is to reveal CP violation in theB system. The discrete symmetry CPT has not yetbeen adequately tested for theB meson system,although there are many interesting suggestions to testit [15,16]. In all such works, however, the correlationbetween�Γ and CPT violation was either ignored ornot adequately emphasized. It will be shown belowthat�Γ can in general be numerically significant evenif CPT violation is not too large.

We consider the time development of neutral mes-onsM0 (which can beK0 orD0 orB0

d orB0s ) and their

antiparticles�M0. The time development is determinedby the effective HamiltonianHij = Mij − i

2Γij withMij andΓij being the dispersive and absorptive partsof the Hamiltonian, respectively [17]. CPT invariancerelates the diagonal elements

(5)M11 =M22 and Γ11 = Γ22.

A measure of CPT violation is, therefore, given by theparameter

(6)δ = H22 −H11√H12H21

which is phase convention independent. In orderto keep the discussion simple we shall study theconsequences of indirect CPT violation only. Sinceindirect CPT violation is a cumulative effect involvingsummations over many amplitudes, it is likely that itsmagnitude would be much larger than that of direct

148 A. Datta et al. / Physics Letters B 548 (2002) 146–152

violation in a single decay amplitude. It is furtherassumed that CPT violation does not affect the off-diagonal elements ofHij . These assumptions can bejustified in specific string models [13], where termsinvolving both flavour and CPT violations receivenegligible corrections due to string scale physics.A further consequence of this assumption is that theusual SM inequalityM12 � Γ12 holds even in thepresence of CPT violation.

The eigenfunctions of the Hamiltonian are definedas

|M1〉 = p1∣∣M0⟩ + q1

∣∣ �M0⟩,(7)|M2〉 = p2

∣∣M0⟩ − q2∣∣ �M0⟩

with the normalization|p1|2+|q1|2 = |p2|2+|q2|2 = 1.We summarize the consequences of the symmetries.We define

(8)η1 = q1

p1=

[(1+ δ2

4

)1/2

+ δ

2

][H21

H12

]1/2

,

(9)η2 = q2

p2=

[(1+ δ2

4

)1/2

− δ

2

][H21

H12

]1/2

and note that CPT violation is contained in the firstfactor, while indirect CP violation is in the secondfactor with the square root. In many expressionswe need the ratioω = η1/η2 = q1p2

p1q2which is only

a CPT violating quantity. CPT conservation requiresImω = 0, Reω= 1 andη1 = η2.

The time development of the states is determinedby the eigenvalues

λ1 =H11 + √H12H21

[(1+ δ2

4

)1/2

+ δ/2

],

(10)λ2 =H22 − √H12H21

[(1+ δ2

4

)1/2

+ δ/2

]

which can be parametrized asλ1,2 = m1,2 − i2Γ1,2.

The quantities which occur in the asymmetries areλ1 − λ2 = �m − i

2�Γ and Γ = 12(Γ1 + Γ2). To

leading order in(Γ12/M12) they are expressed in termsof the CPT parameter

y =(

1+ δ2

4

)1/2

as follows:

�m=m1 −m2 = 2|M12|(

Rey + 1

2Re

Γ12

M12Imy

),

(11)

�Γ = Γ1 − Γ2 = 2|M12|(

ReΓ12

M12Rey − 2 Imy

).

In the CPT conserving limity = 1 and the contributionto �m is large, overwhelming CPT violating correc-tions. The CPT conserving contribution to�Γ , on theother hand, is suppressed by ReΓ12

M12. The purely CPT

violating term dominating�Γ remains, therefore, anopen possibility. In order to get a feeling for the mag-nitude of�Γ/Γ , we use the smallδ approximationand obtain|�Γ/Γ | = 0.5× (�m/Γ )× (Reδ× Im δ).

Most of the measurements of�m/Γ have been car-ried out by assuming CPT conservation. If CPT is vio-lated its magnitude could be somewhat different (see,e.g., Kobayashi and Sanda in [15]). Recently the BelleCollaboration has determined�m with and withoutassuming CPT symmetry [12]. The two results,�m=0.463± 0.016 and 0.461± 0.008± 0.016 ps−1, re-spectively, do not differ appreciably from each other orfrom the average�m given by the particle data group(PDG). We shall, therefore, use throughout the Let-ter�m/Γ = 0.73, which is perfectly consistent withthe PDG value. The relevant limits on CPT violat-ing parameters from Belle are|mB0 − m�B 0|/mB0 <

1.6×10−14 and|ΓB0 −Γ�B 0|/ΓB0 < 0.161, which im-plies |Reδ| < 0.54 and| Imδ| < 0.23. A choice likeReδ × Imδ ∼ 0.1, consistent with the above bounds,would then yield�Γ/Γ of the order of a few percents,larger than the SM estimate by an order of magnitude.Moreover�Γ/Γ will be well within the measurablelimits of LHC B, shouldδ happen to be much smaller.

The Belle limits are derived under the assumptionthat �Γ/Γ is negligible. We emphasize that fora refined analysis of CPT violation�m/Γ and�Γ/Γalong with δ should be fitted directly from the data.Such a combined fit may open up the possibility thatδ could be somewhat larger than the above bounds.In our numerical analysis values consistent with thebounds as well as somewhat larger values will beconsidered.

The time development of the states involves, now,the time factors

(12)f−(t)= e−iλ1t − e−iλ2t ,

(13)f+(t)= e−iλ1t +ωe−iλ2t ,

(14)f+(t)= ωe−iλ1t + e−iλ2t .

A. Datta et al. / Physics Letters B 548 (2002) 146–152 149

The new feature is the presence of the factorω in thesecond and third of these equations.

The decays of an original|B0〉 or |�B 0〉 state toa flavor eigenstate|f 〉 vary with time and are givenby

(15)Pf (t)= ∣∣⟨f ∣∣B0(t)⟩∣∣2 = ∣∣f+(t)

∣∣2N∣∣⟨f ∣∣B0⟩∣∣2,(16)�Pf (t)= ∣∣⟨f ∣∣�B 0(t)

⟩∣∣2 = ∣∣f+(t)∣∣2N∣∣⟨f ∣∣�B 0⟩∣∣2,

Pf (t)= ∣∣⟨f ∣∣B0(t)⟩∣∣2

(17)= |η1|2∣∣f−(t)

∣∣2N∣∣⟨f ∣∣�B 0⟩∣∣2,�Pf (t)= ∣∣⟨f ∣∣�B 0(t)

⟩∣∣2(18)= ∣∣f−(t)

∣∣2N |ω|2∣∣⟨f ∣∣B0⟩∣∣2/|η1|2,

whereN−1 = |1 + ω|2 and the matrix elements onthe right-hand sideg = 〈f |B0〉, g = 〈f |�B 0〉, . . . , arecomputed att = 0 and have no time dependence. Fromthese expressions it is evident that the five unknowns,Γ , �m, �Γ and Reδ and Imδ (or equivalently Reωand Imω), must be determined from the data. We em-phasize that�Γ must be treated as a free parameter,since in addition to the CPT violating contributions itmay also receive contributions from new dynamics. Inaddition taking linear combinations of these decays,we can produce exponential decays accompanied byoscillatory terms which help in separating the variouscontributions. It may be recalled that the time depen-dent techniques for extracting these probabilities andthe associated electroweak parameters from data arenow being used extensively.

Different schemes for testing CPT violation sug-gested in the literature [15,16] often involve observ-ables specifically constructed for this purpose. Herewe wish to point out that some of the observables in-volving the above probabilities, which are now beingroutinely measured at BaBar and Belle are also suffi-ciently sensitive to CPT violation and have the poten-tial of either revealing the breakdown of this funda-mental symmetry or improving the limit on the CPTviolating parameter. One such observable is the directCP violating asymmetry inBd and�Bd decays to flavorspecific channelsf andf , respectively [18], but withf different fromf . The following ratio is at the center

of current interest

adirCP(t)=

|〈f |B0(t)〉|2 − |〈f |�B 0(t)〉|2|〈f |B0(t)〉|2 + |〈f |�B 0(t)〉|2

(19)= |f+(t)|2|g|2 − |f+(t)|2|g|2|f+(t)|2|g|2 + |f+(t)|2|g|2

.

In the SM or in any of its CPT conserving exten-sions, f+(t) = f+(t) and the asymmetry is time in-dependent in general. The time independence holdseven if�Γ happens to be large due to new dynamicsor direct CPT violation and/or new physics influencethe hadronic matrix elements. Time evolution of thisasymmetry is, therefore, a sure signal of indirect CPTviolation. Flavour specificB decays involving a singlelepton or a kaon in the final state are possible candi-dates for this measurement.

This consequence is even more dramatic for decaysdominated by a single amplitude in the SM, in whichcase|g| = |g| and adir

CP(t) vanishes at all times. Purelytree level decays arising from the subprocessb →uiuj dk (i �= j ), penguin induced processesb →dididk, dominated by a single Penguin operator orinclusive semileptonic decaysb → Xl+ν (l = e or µandX is any hadronic final state) are examples of suchdecays. The last process is particularly promising.A single amplitude strongly dominates the decaynot only in the SM but also in many extensionsof it. The large branching ratio (∼ 20% for l = e

andµ) and reasonably large efficiency of detectingleptons is sufficient to ensure the measurement of thisasymmetry atB-factories, provided it is of the orderof a few percent.

For this class of decays the matrix elements alongwith their theoretical uncertainties cancel out in theratio. Consequently in presence of indirect CPT vio-lation the time dependent asymmetry is the same forall one-amplitude dominated processes and the statis-tics may be improved by including several channels. Ifa difference in the time dependence of various modesis observed, the assumption of one amplitude domi-nance will be questionable and new physics beyondthe standard model leading to|g| �= |g|, in addition toindirect CPT violation, may be revealed.

In Fig. 1 we present the asymmetry for a oneamplitude dominated process as a function of timefor Imδ = 0.1 and Reδ = 0.1 (solid curve, here�Γ/Γ = 0.004), 0.5 (dotted curve,�Γ/Γ = 0.02)

150 A. Datta et al. / Physics Letters B 548 (2002) 146–152

Fig. 1. The time evolution of the time dependent asymmetry(adir

CP(t)) for Im δ = 0.1 for any one amplitude dominated process

(in all figurest is in units ofτB/10, whereτB is the averageB0 lifetime). See text for the details.

and 1.0 (dashed curve,�Γ/Γ = 0.04). Both the timeevolution and the nonvanishing of the asymmetry areclearly demonstrated.

The correlation between adirCP and �Γ calls for

a more detailed analysis. As has been noted�Γ issignificantly different from the SM prediction only ifIm δ × Reδ �= 0. The numerator and the denominatorof the asymmetry are determined to be

D(t)= Pf (t)− �Pf (t)= [(|ω|2 − 1

)(e−Γ2t − e−Γ1t

)(20)− 4 Imωe−Γ t sin�mt

]N

and

S(t)= Pf (t)+ �Pf (t)= [(|ω|2 + 1

)(e−Γ2t + e−Γ1t

)(21)+ 4 Reωe−Γ t cos�mt

]N.

A nonvanishing asymmetry can arise in various ways:

(i) Imω �= 0, which requires Imδ �= 0.0;(ii) Reω �= 1.0 and�Γ/Γ as small as in the SM.

Or from a combination of the two possibilities. It istrivial to expressω in terms ofδ and confirm that bothD(t) andS(t) are modified from the SM predictionthroughδ. When both numerator and denominator ofthe asymmetry are measured accurately, one can deter-mine separately real and imaginary parts of delta. This

Fig. 2. The variation ofD(t) = Prob(B → f ) − Prob(�B → f )as function of time Imδ = 0.1 for any one amplitude dominatedprocess. See text for the details.

Fig. 3. The variation ofS(t) = Prob(B → f ) + Prob(�B → f ) asfunction of time. This quantity is sensitive to Reδ only. For com-parison we have plotted for Reδ = 0.0 (SM) (the dotted curve) andReδ = 0.5 (the solid curve).

may indicate, albeit indirectly, that�Γ is unexpect-edly large.

In order to have an idea of how large the effectscan be, in Fig. 2 we plotD(t) as a function of timefor the values Reδ = Im δ = 0.1 (the solid curve).D(t) vanishes for Imδ = 0 and has a relatively weakdependence on Reδ, as illustrated by also plotting onthe same figure the cases with Reδ = 0.5 (the dottedcurve) and 1.0 (the dashed curve). A similar studyof S(t) is presented in Fig. 3. This quantity is fairlyinsensitive to Imδ.

In order to estimate roughly the number of taggedB-mesons needed to establish a nonzeroD(t), weassume that att = 0 there is a sample ofN0 taggedB0d and �Bd0. Let of number of semileptonicB0

d (�B 0d )

A. Datta et al. / Physics Letters B 548 (2002) 146–152 151

decays in the time intervalt = (1.0 ± 0.1) × τB ben(t) (n(t)) (we assume the lepton detection efficiencyto be∼ 1). By requiring

(22)|n(t)− n(t)|√n(t)+ √

n(t)� 3.0,

we obtain for Reδ = 0.1 and Imδ = 0.1,N0 ≈ 2.0 ×106, a number which is realizable atB-factories afterseveral years of run and certainly at the LHC. Includ-ing other flavour specific channels likeB0

d →K+ +X,which has a larger branching ratio (≈ 70%), a measur-able asymmetry may be obtained with a smallerN0.

In the presence of indirect CPT violation, the timeintegrated asymmetry is obtained by integrating thenumeratorD(t) and the denominatorS(t). This leadsto

adirCP=

((|ω|2 − 1)�ΓΓ

− 4 Imωx

(1+ x2)

)

(23)×(

2(|ω|2 + 1

) + 4 Reω

(1+ x2)

)−1

with x = �m/Γ . In the standard model and forprocesses dominated by one amplitude the integratedasymmetry vanishes. In extensions of the SM in whichthe decays are no longer dominated by one amplitudethe integrated asymmetry may be nonzero [19]. Thusa nonzero integrated asymmetry points either to newphysics (coming from additional amplitudes) or toindirect CPT violation. In Fig. 4 we present thevariation of this observable with Imδ for Reδ = 0.1(solid line) and 0.75 (dashed line).

Experimental studies of CPT violating phenomenacan be combined with experiments that search for

Fig. 4. The variation of the time integrated asymmetry adirCP for a one

amplitude dominated process vs Imδ. See text for further details.

a �Γ/Γ . For example, one can consider untaggedB mesons decaying to a specific flavour [3,4]. Theobservable

S1(t)= Pf (t)+ �Pf (t)which in the absence of CPT violation have a time de-pendence governed by two exponentials. If now CPTviolation is also included, then an oscillation is super-imposed on the exponentials. The original articles [3,4] consideredBs decays but the same properties holdfor Bd meson decaying semileptonically or to specificflavour final states.

Looking at flavour non-specific channels there areresults for adir

CP (also denoted byCππ ) from BaBar[20] and Belle [21] for the channelB → π+π−.In the SM using naive factorization this asymmetryturns out to be small [22]. It is interesting to notethat although the Babar result is fairly consistent withthe SM prediction, the Belle result indicates a muchlarger asymmetry. It should, however, be noted thatthere are many theoretical uncertainties. Neither themagnitude of the penguin pollution nor the magnitudeof the strong phase difference between the interferingamplitudes can be computed in a full proof way. DirectCP violation in flavour specific, charmless decays havealso been measured [23]. Here the data is not yet veryprecise and the theoretical uncertainties are also large.In view of these uncertainties it is difficult to drawany conclusion regarding new physics effects. Thisunderlines the importance of inclusive semileptonicdecays which are theoretically clean and the branchingratios are much larger than any of the above exclusivemodes.Bd decays to CP eigenstates have been observed

and established CP-violation via time dependent mea-surements [24,25]. The golden example isB0 → ψKswhere the time dependent asymmetry is proportionalto sin2β , whereβ (also denoted byφ1) is an angleof the unitarity triangle. The current averaged value ofthis parameter is sin 2β = 0.78± 0.08. It is straight-forward to obtain the asymmetry in the presence ofindirect CPT violation. An attempt to fit the data as inthe SM would lead to an effective sin 2β which is timedependent. We have checked that with Reδ = 0.1 andIm δ = 0.1, this effective sin 2β varies between 0.74and 0.84. We therefore conclude that if sin 2β is deter-mined with an accuracy of 5% or better, some hint ofindirect CPT violation may be obtained. However, this

152 A. Datta et al. / Physics Letters B 548 (2002) 146–152

observation cannot establish CPT violation unambigu-ously. Since CPT conserving new physics may changethe phase of theBd– �Bd mixing amplitude and/or thedecay amplitudes and lead to similar effects.

Many other observables specifically constructedfor the measurement of CPT violation [15,16] havebeen suggested in the literature. It will be interestingto compare the sensitivities of these observables tothe CPT parameterδ with that of the observablesconsidered in this Letter, which are already beingmeasured in the context of CP violation.

In summary, we wish to emphasize again that anunexpectedly large life time difference ofBd mesons,which is predicted to be negligible in the SM andmany of its CPT conserving extensions, may revealindirect CPT violation. Time dependence of the directCP violating asymmetry for flavour specific decays,which is time independent and vanishes for decaysdominated by only one amplitude may establish CPTviolation as well as a large life time difference.The theoretically clean inclusive semileptonic decayshaving relatively large branching ratios might beparticularly suitable in this context.

Acknowledgements

We wish to thank the Bundesministerium fur Bil-dung and Forschung for financial support under con-tract No. 05HT1PEA9. One of us (E.A.P.) thanksMr. W. Horn for useful discussions. A.D. thanksthe Department of Science and Technology, Gov-ernment of India for financial support under projectNo. SP/S2/k01/97 and Abhijit Samanta for help incomputation.

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