On the Measurement of the Unitarity Triangle Angle gamma from B^0 -> DK^{*0} Decays
aa r X i v : . [ h e p - ph ] M a r On the Measurement of the Unitarity Triangle Angle γ from B → DK ∗ Decays
Tim Gershon Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom (Dated: October 24, 2018)The decay B → DK ∗ is well-known to provide excellent potential for a precise measurementof the Unitarity Triangle angle γ in future experiments. It is noted that the sensitivity can besignificantly enhanced by studying the amplitudes relative to those of the flavour-specific decay B → D ∗− K + , which can be achieved by analyzing the B → Dπ − K + Dalitz plot.
PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
Among the fundamental parameters of the Stan-dard Model of particle physics, the angle γ =arg ( − V ud V ∗ ub /V cd V ∗ cb ) of the Unitarity Triangle formedfrom elements of the Cabibbo-Kobayashi-Maskawa quarkmixing matrix [1, 2] has a particular importance. It is theonly CP violating parameter that can be measured usingonly tree-level decays, and thus it provides an essentialbenchmark in any effort to understand the baryon asym-metry of the Universe. The precise measurement of γ isone of the main objectives of planned future B physicsexperiments (see, for example, [3, 4, 5]).A method to measure γ with negligible theoretical un-certainty was proposed by Gronau, London and Wyler(GLW) [6, 7]. The original method uses B → DK decays,with the neutral D meson reconstructed in CP eigen-states. It was noted that the method can be extendedto use D meson decays to any final state that is acces-sible to both D and D , and a number of potentiallyuseful modes, including doubly-Cabibbo-suppressed de-cays such as K + π − [8, 9], multibody decays such as K S π + π − [10, 11] and others [12, 13] have been proposed. B bd *0 Ksd DcuW B bd *0 Ksd DucW
FIG. 1: Feynman diagrams for B → DK ∗ , via (left) a¯ b → ¯ cu ¯ s transition and (right) a ¯ b → c ¯ u ¯ s transition. The method can similarly be extended to other B de-cays, such as B → D ∗ K or B → DK ∗ . The use ofneutral B decays was noted as being particularly inter-esting since the amplitudes involving D and D statesmay be of comparable magnitude, as shown in Fig. 1, po-tentially leading to large direct CP violation [14]. Thedecay B → DK ∗ is particularly advantageous since thecharge of the kaon in the K ∗ → K + π − decay unambigu-ously tags the flavour of the decaying B meson, obviatingthe need for time-dependent analysis [15]. This appearsto be one of the most promising channels for LHCb tomake a precise measurement of γ [16, 17]. However, the natural width of the K ∗ meson has, until now, been con-sidered a hindrance to the method, which could be han-dled by the introduction of additional hadronic parame-ters [18, 19, 20, 21].In this paper it is noted that the natural width of the K ∗ meson can be used to enhance the potential sensi-tivity to the CP violating phase γ in the analysis of B → DK ∗ decays. By studying the B → Dπ − K + Dalitz plots with the neutral D meson reconstructed inflavour-specific and CP eigenstate modes, the complexamplitudes of the DK ∗ decays can each be determinedrelative to the flavour-specific D ∗− K + amplitude, illus-trated in Fig. 2, allowing a direct extraction of γ fromthe difference in amplitudes, rather than from the rates.Alternative approaches to measure γ using B → D ∗∗ K decays [22] or using amplitude analyses of B → DKπ decays have been suggested in the literature [23, 24] (thetime-dependent B → D ∓ K S π ± Dalitz plot analysis hasrecently been implemented [25]), however the particularbenefit of the B → Dπ − K + Dalitz plots has not beennoted until now. B bd
Dcd + KsuW
FIG. 2: Feynman diagram for the flavour-specific B → D ∗− K + decay. Experimentally, the decay B → D K ∗ has beenstudied by the B factories, with the world average of itsbranching fraction being B ( B → D K ∗ ) = (4 . ± . × − [26, 27, 28]. Initial studies of the B → D π − K + Dalitz plot also indicate the sizeable presence of the B → D ∗− K + decay [29]. Limits on the branching frac-tion of the B → D K ∗ decay have been set [26, 27, 28],the most restrictive limit being B ( B → D K ∗ ) < . × − at 90% confidence level. First attempts to ob-tain constraints on γ from B → DK ∗ decays have beenmade using neutral D meson decays to K S π + π − [30] andto suppressed final states such as K − π + .To illustrate the method, consider first of all the Dalitzplot of the B → D π − K + decay, in which the D is re-constructed in the K + π − final state. Initially, this istreated as a flavour-specific decay (hence the flavour ofthe D meson is indicated – the notation D is used to indi-cate a neutral charm meson that is some admixture of D and D ). The effect of the doubly-Cabibbo-suppressed D → K + π − amplitude will be considered later. Recallthat the charge of the prompt kaon in the D π − K + finalstate unambiguously identifies the flavour of the decaying B meson, so that it is not necessary to consider effectsdue to B – B mixing [31].The B → D π − K + Dalitz plot will, of course, con-tain π − K + resonances such as K ∗ (892), K ∗ (1430) and K ∗ (1430). One advantage of the Dalitz plot approachis that the hadronic parameters of each resonance can bedetermined, avoiding the complications that arise due tothe use of effective hadronic parameters in the quasi-two-body DK ∗ analysis [18, 19, 20, 21]. Furthermore, CP vi-olation effects can be studied simultaneously in all of thecontributing π − K + resonances, enhancing the sensitivityto γ . However, more importantly, the Dalitz plot will alsocontain significant contributions from D π − resonancessuch as D ∗− (2400) and D ∗− (2460) (contributions from D ∗− (2010) K + are not considered, since the D ∗− (2010)is too narrow to interfere with other resonances). Thecrucial point is that for such resonances the flavour ofthe D meson is unambiguously identified by the chargeof the accompanying pion, independent of the D decaymode. Resonances of D K + are not possible (at least,not as simple quark-antiquark mesons), and the presenceof any D ∗∗ + s -type contributions to the Dalitz plot wouldindicate the presence of amplitudes involving the D me-son.It is sufficient to consider a toy model of the Dalitz plotcontaining only K ∗ (892) and D ∗− (2460) resonances.The amplitude of the D K ∗ decay relative to that ofthe D ∗− K + decay can be determined, as illustrated inFig. 3 (left), where the relative phase between the twoamplitudes is denoted by ∆. The complex amplitudesof any other contributions to the Dalitz plot can be andshould be determined simultaneously, so as not to biasthe extraction of the amplitudes of interest, but this doesnot affect the principle of the measurement. Since theneutral D meson is flavour-specific, all contributions tothe Dalitz plot are dominated by the b → cus tree-leveltransition, ie. all have the same weak phase. Therefore,no direct CP violation is expected and the same relationbetween amplitudes should be obtained for B decaysand for the conjugate B decays.Consider now the amplitudes that will be determinedwhen a similar analysis is applied to the Dπ − K + Dalitzplot when the neutral D meson is reconstructed in CP -even eigenstates such as D → K + K − . (As mentionedlater, CP -odd decays such as D → K S π can also beincluded in the analysis if they are experimentally acces- sible.) Since the D ∗− K + amplitude is flavour specific,the reference amplitude remains the same. In Fig. 3(right) this amplitude is denoted as √ A ( D ∗− CP K + )where D ∗− CP denotes that the neutral D meson producedin the decay of the D ∗− is reconstructed in a CP -eveneigenstate. Neglecting trivial phase factors, | D CP i = √ (cid:0)(cid:12)(cid:12) D (cid:11) + (cid:12)(cid:12) D (cid:11)(cid:1) so that the relation √ A ( D ∗− CP K + ) = A ( D ∗− K + ) holds.In the absence of contributions from D K ∗ onewould expect to find exactly the same amplitude for DK ∗ relative to that for D ∗− K + as found for flavour-specific D decays. The extracted relative ampli-tude therefore contains information about the ratio ofthe B → D K ∗ and B → D K ∗ amplitudes, r B = (cid:12)(cid:12) A ( B → D K ∗ ) /A ( B → D K ∗ ) (cid:12)(cid:12) , their relativestrong phase difference δ B , and their relative weak phasedifference γ . This is illustrated in Fig. 3 (right), for both B and B decays, where the sign of the weak phasedifference between the amplitudes is flipped.It is clear that the triangle constructions shown inFig. 3 (right) are exactly those regularly drawn to illus-trate the GLW method [6, 7], except rotated by a con-stant angle ∆. To reiterate the advantage of the approachoutlined here, in the typical quasi-two-body DK ∗ anal-ysis, one must reconstruct these triangles from measure-ments only of the lengths of the long sides and the base;in this approach, one determines directly the positions ofthe apexes of the triangles. Thus this approach providessignificant additional information to constrain γ , as wellas resolving ambiguities in the strong phase difference.As a further elaboration of this point, note that the rateand asymmetry measurements in the usual GLW analysiscan be translated into measurements of the parameters x ± = r B cos( δ B ± γ ) conventionally used in studies of B → D ( ∗ ) K ( ∗ ) decays with subsequent multibody D de-cays such as D → K S π + π − [32, 33]. However, this isonly possible if asymmetries and rates for both CP -evenand CP -odd D decays have been measured, and further-more no constraints on y ± = r B sin( δ B ± γ ) are obtained(except indirectly from a constraint on r B = x ± + y ± ).With Dalitz plot analysis of the Dπ − K + Dalitz plots,both problems are solved: from the relation x + + iy + = r B e i ( δ B + γ ) = ( √ A ( D CP K ∗ )) / ( √ A ( D ∗− CP K + ))( A ( D K ∗ )) / ( A ( D ∗− K + )) − √ A ( D CP K ∗ ) A ( D K ∗ ) − , (1)both x + and y + can be obtained using only CP -evenand flavour specific D decays reconstructed in B decays,with x − + iy − similarly obtained from the conjugate B decays. In Eq. 1, the fact that both D CP K ∗ and D K ∗ amplitudes must be determined relative to D ∗− K + ismade explicit. (If CP -odd D decays are also used, toadd statistics and to provide a useful experimental cross-check, the right-hand side of the last two relations ofEq. 1 will be multiplied by a minus sign.) Note thatall relevant normalization factors and subdecay branch-ing fractions are automatically taken into account sincethe complex amplitudes A ( D CP K ∗ ) and A ( D K ∗ ) ofEq. 1 are both obtained relative to the flavour-specific D ∗− K + amplitude. Therefore this approach, which doesnot require reconstruction of CP -odd D decay modes,appears highly promising for LHCb where reconstructionof states such as K S π will be extremely challenging inthe hadronic environment. Indeed, previous studies ofthe potential of LHCb to measure γ from B → DK ∗ decays [16, 17] have shown a strong dependence of thesensitivity on the unknown value of the hadronic param-eter δ B . Since the origin of this dependence is related tothe absence of information from CP -odd D decays, it isto be expected that it will be appreciably reduced usingthe Dalitz plot analysis suggested here. ReIm −1 +1 +2−1+1+2 D ) *0 K D fi A(B ) + K D fi A(B
ReIm −1 +1 +2−1+1+2 g g B d ) *0 K CP D fi A(B2 ) *0 K CP D fi B A(2 ) + K D fi A(B2
FIG. 3: Argand diagrams illustrating the measurementsof relative amplitudes and phases from analysis of theDalitz plots of (left) D π − K + and (right) D CP π − K + .In these illustrative examples the following values areused: ˛˛ A ( B → D K ∗ ) /A ( B → D ∗− K + ) ˛˛ = 1 .
5, ∆ =arg ` A ( B → D K ∗ ) /A ( B → D ∗− K + ) ´ = 20 ◦ , γ = 75 ◦ , δ B = 45 . ◦ and r B = 0 .
4. These are in line with expecta-tion and current measurements, though ∆ and δ B are uncon-strained at present. The precise gain in sensitivity to γ compared to thequasi-two-body analysis is difficult to estimate, since itdepends on how precisely the relative phase ∆ can bemeasured. Dalitz plot analyses of B → Dπ − K + havenot yet been carried out, so there is no experimental in-formation with which to assess this issue. However, astudy of B → D π + π − shows that the relative phase be-tween D ∗− π + and D ρ can be well-measured [34]. Thisinterference can also be exploited to obtain weak phaseinformation, as recently noted [35]. Furthermore, studiesof Kπ resonances produced in B decays have revealed arich structure (see, for example, [36, 37]). These resultsprovide confidence that the phase ∆ can be accuratelydetermined, and that the Dalitz plot B → Dπ − K + analysis advocated in this paper promises a substantialimprovement over the quasi-two-body B → DK ∗ ap- proach. Moreover, the analysis advocated herein obtains γ with only a single unresolved ambiguity ( γ → γ + π , δ B → δ B + π ), whereas the quasi-two-body approach suf-fers an eight-fold ambiguity (note that other methods toreduce the ambiguities exist).The discussion above has neglected the fact that neu-tral D decays to π − K + are not completely flavour-specific, due to the existence of doubly-Cabibbo-suppressed D → π − K + amplitudes. The ratioof this suppressed amplitude to its Cabibbo-favouredcounterpart has been precisely measured to be r D = (cid:12)(cid:12) A ( D → π − K + ) /A ( D → π + K − ) (cid:12)(cid:12) = (5 . ± . δ D = (22 +11 −
12 +9 − ) ◦ [42, 43] (a more pre-cise constraint is found from a global fit including mea-surements of charm mixing parameters [28]). When the D → π − K + decay mode is used there will therefore bea contribution from the B → D K ∗ amplitude withmagnitude suppressed by r B × r D compared to that of B → D K ∗ ; the strong phase and weak phase differ-ences will be δ B + δ D and γ , respectively ( CP conser-vation in D decay is assumed). This could, if neglected,potentially bias the extracted value of γ . Any bias wouldbe small, due to the factor of r B × r D , but could nonethe-less be significant in an era of precision measurements.In the analysis where the suppressed D decay ampli-tudes are neglected, one has four observables (which can,for convenience, be taken to be ( x + , y + , x − , y − ) and threeunknowns ( r B , δ B , γ )). Introducing suppressed ampli-tudes adds two more parameters ( r D , δ D ) but also addstwo new observables, since one can now measure CP vi-olating differences between the B → D K ∗ decay am-plitude and its conjugate (both measured relative to theflavour-specific D ∗ K amplitudes). Furthermore, externalconstraints on these new parameters can be used in theanalysis. Therefore, it is still possible to extract γ with aprecision that should not be significantly worse than thatwhen the suppressed amplitudes are neglected. (A moreprecise measurement of δ D would, however, be useful.)One may consider whether studying the B → Dπ − K + Dalitz plot with the D meson reconstructedin the suppressed modes will add additional useful in-formation. Although this appears promising, there willbe a complication since the flavour-specific D ∗ K ampli-tude that has, until now, been used as a reference willno longer be one of the larger contributions to the Dalitzplot. There could, potentially, be D ∗∗ + s -type resonancesof DK + that could provide an alternative flavour-specificreference, though these would be expected to be broaderthan one would wish for such a reference amplitude. If itwere possible to use such a reference, its phase relative to D ∗ K could be determined in the Dalitz plot where the D meson is reconstructed in CP eigenstates. Thus it mightbe possible to use information about the suppressed D decay modes, additional to that on the rates, to furtherimprove the sensitivity to γ .In passing, it is worthwhile to note that the method de-scribed above can easily be extended to B → D ∗ π − K + decays, where the neutral D ∗ meson can be reconstructedin decays to either Dπ or Dγ [44]. However, in this casethere will be an additional complication due to the dif-ferent helicity amplitudes that are possible in the B → D ∗ K ∗ decay [45]. The method can also be extendedto use other D decays, including multibody decays suchas D → K S π + π − [19] or others [13] or single-Cabibbo-suppressed decays such as D → K ∗± K ∓ [12, 46]. An-other possible extension would be to use DK + ( nπ ) − finalstates [23].Finally, it should be noted that the method discussedabove does not, unfortunately, work well when applied tocharged B decays. The K ∗ + produced in B + → DK ∗ + can decay to K + π or K π + . In the former case, Dπ resonances do not identify the flavour of the D meson.While D ∗∗ + s -type resonances of DK + are possible, theamplitudes for B + → D ∗∗ + s π decays are expected tobe rather small (by extrapolation from published resultson B + → D + s π , for example [47]). The DK + π Dalitzplot can, however, benefit from a possible alleviation ofthe suppression of the D K + π amplitude [24]. In thecase that the K ∗ + decays to K π + , neutral DK reso-nances are not possible (at least, not as simple quark-antiquark mesons), and amplitudes for B + → D ∗∗ + K are expected to be negligible (by extrapolation from lim-its on B + → D ( ∗ )+ K decays, for example [48]). Thus,there is no significant flavour-specific amplitude to pro-vide the necessary reference point by which to obtaininformation about γ in the Dalitz plot analysis.In summary, it has been shown that a potentiallysignificant improvement in the measurement of γ canbe achieved by measuring the complex amplitudes of B → DK ∗ decays relative to that of the flavour-specificdecay B → D ∗− K + , which can be achieved by analyz-ing B → Dπ − K + Dalitz plots. Compared to previouslysuggested techniques to measure γ from B → DK ∗ de-cays, this approach helps to resolve ambiguities, solvesproblems related to interferences between various reso-nances while avoiding the need for the introduction ofeffective hadronic parameters and provides a potentiallysignificant overall improvement in the sensitivity whilereducing its dependency on currently unknown parame-ters. This method can be used at LHCb and other future B physics experiments to make a precise measurement ofthis fundamental parameter of the Standard Model.I am grateful to Alex Bondar, Amarjit Soni and JureZupan for discussions, and to Tom Latham, Paul Harri-son, Jim Libby, Owen Long, Giovanni Marchiori, AchilleStocchi and Vincent Tisserand for discussions and com-ments on the manuscript. This work is supported by theScience and Technology Facilities Council (United King-dom). [1] N. Cabibbo, Phys. Rev. Lett. (1963) 531.[2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. (1973) 652.[3] T.E. Browder, JHEP 02 (2008) 110 [arXiv:0710.3799[hep-ph]].[4] M. Artuso et al. , Eur. Phys. J. C (2008) 309[arXiv:0801.1833 [hep-ph]].[5] T.E. Browder, T. Gershon, D. Pirjol, A. Soni and J. Zu-pan, arXiv:0802.3201 [hep-ph].[6] M. Gronau and D. London, Phys. Lett. B (1991)483.[7] M. Gronau and D. Wyler, Phys. Lett. B (1991) 172.[8] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. (1997) 3257 [arXiv:hep-ph/9612433].[9] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. D (2001) 036005 [arXiv:hep-ph/0008090].[10] A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev.D (2003) 054018 [arXiv:hep-ph/0303187].[11] A. Poluektov et al. [Belle Collaboration], Phys. Rev. D (2004) 072003 [arXiv:hep-ex/0406067].[12] Y. Grossman, Z. Ligeti and A. Soffer, Phys. Rev. D (2003) 071301(R) [arXiv:hep-ph/0210433].[13] J. Rademacker and G. Wilkinson, Phys. Lett. B (2007) 400 [arXiv:hep-ph/0611272].[14] I. I. Y. Bigi and A. I. Sanda, Phys. Lett. B (1988)213.[15] I. Dunietz, Phys. Lett. B (1991) 75.[16] K. Akiba and M. Gandelman, CERN-LHCB-2007-050.[17] K. Akiba et al. , CERN-LHCB-2008-031.[18] M. Gronau, Phys. Lett. B (2003) 198[arXiv:hep-ph/0211282].[19] S. Pruvot, M. H. Schune, V. Sordini and A. Stocchi,arXiv:hep-ph/0703292.[20] S. Pruvot, Th`ese de Doctorat, Universit´e Paris-Sud XIOrsay, LAL 07-76.[21] V. Sordini, Th`ese de Doctorat, Universit´e Paris-Sud XIOrsay, LAL 08-59.[22] N. Sinha, Phys. Rev. D (2004) 097501[arXiv:hep-ph/0405061].[23] D. Atwood, G. Eilam, M. Gronau and A. Soni, Phys.Lett. B (1995) 372 [arXiv:hep-ph/9409229].[24] R. Aleksan, T. C. Petersen and A. Soffer, Phys. Rev. D (2003) 096002 [arXiv:hep-ph/0209194].[25] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D (2008) 071102 [arXiv:0712.3469 [hep-ex]].[26] P. Krokovny et al. [Belle Collaboration], Phys. Rev. Lett. (2003) 141802 [arXiv:hep-ex/0212066].[27] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D (2006) 031101 [arXiv:hep-ex/0604016].[28] E. Barberio et al. [Heavy Flavor Averaging Group],arXiv:0808.1297 [hep-ex].[29] B. Aubert et al. [BABAR Collaboration], Phys. Rev.Lett. (2006) 011803 [arXiv:hep-ex/0509036].[30] B. Aubert et al. [BABAR Collaboration],arXiv:0805.2001 [hep-ex].[31] M. Gronau, Y. Grossman, Z. Surujon and J. Zupan,Phys. Lett. B (2007) 61 [arXiv:hep-ph/0702011].[32] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D (2008) 034023 [arXiv:0804.2089 [hep-ex]].[33] K. Abe et al. [Belle Collaboration], arXiv:0803.3375 [hep-ex]. [34] A. Kuzmin et al. [Belle Collaboration], Phys. Rev. D (2007) 012006 [arXiv:hep-ex/0611054].[35] T. Latham and T. Gershon, J. Phys. G (2009) 025006[arXiv:0809.0872 [hep-ph]].[36] A. Garmash et al. [Belle Collaboration], Phys. Rev. Lett. (2006) 251803. [arXiv:hep-ex/0512066].[37] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D (2008) 012004 [arXiv:0803.4451 [hep-ex]].[38] L. M. Zhang et al. [Belle Collaboration], Phys. Rev. Lett. (2006) 151801 [arXiv:hep-ex/0601029].[39] B. Aubert et al. [BABAR Collaboration], Phys. Rev.Lett. (2007) 211802 [arXiv:hep-ex/0703020].[40] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. (2008) 121802 [arXiv:0712.1567 [hep-ex]].[41] C. Amsler et al. [Particle Data Group], Phys. Lett. B (2008) 1. [42] J. L. Rosner et al. [CLEO Collaboration], Phys. Rev.Lett. (2008) 221801 [arXiv:0802.2264 [hep-ex]].[43] D. M. Asner et al. [CLEO Collaboration], Phys. Rev. D (2008) 012001 [arXiv:0802.2268 [hep-ex]].[44] A. Bondar and T. Gershon, Phys. Rev. D (2004)091503(R) [arXiv:hep-ph/0409281].[45] N. Sinha and R. Sinha, Phys. Rev. Lett. (1998) 3706[arXiv:hep-ph/9712502].[46] A. K. Giri, B. Mawlong and R. Mohanta, Phys. Rev. D (2007) 093008 [arXiv:0706.2533 [hep-ph]].[47] B. Aubert et al. [BABAR Collaboration], Phys. Rev.Lett. (2007) 171801 [arXiv:hep-ex/0611030].[48] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D72