CP asymmetries in B ! f0KS decays

Rupak Dutta* and Susan Gardner+

Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA(Received 17 May 2008; published 19 August 2008)

We consider the branching ratio and the CP asymmetries in B! f0ð980ÞKS decay to the end of

determining the deviation of the time-dependent CP asymmetry from sinð2�Þ, �Sf0KS � ��f0KSSf0KS �sinð2�Þ, arising from standard model physics. We obtain �Sf0KS within the context of the QCD

factorization framework for the B! f0ð980ÞKS decay amplitudes assuming the f0ð980Þ is a q �q state

and employing a random scan over the theoretical parameter space to assess the possible range in �Sf0KS .

Imposing the value of the experimental branching ratio within 1� and 3�, respectively, of its central value

as a constraint, we find the range of �Sf0KS to be [0.018, 0.033] for a scan in which the parameters are

allowed to vary within 1� of their central values and the range ½�0:019; 0:064� for a scan in which the

parameters vary within 3� of their central values.

DOI: 10.1103/PhysRevD.78.034021 PACS numbers: 13.25.Hw, 11.30.Er

I. INTRODUCTION

In the standard model (SM), all CP-violating effectsderive from a single, complex phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and predicate a dis-tinctive pattern of CP-violation [1]. For example, in thedecay of a B-meson to a CP-eigenstate f, the time-dependent asymmetry Sf realized from b! sc �c decay

determines sinð2�Þ, where � is given by � ¼argð�VcdV�

cb=VtdV�tbÞ and Vij is a CKM matrix element

[2–4]. This quark-level transition can be studied in a vari-ety of B-meson decays, and departures of the determinedtime-dependent asymmetry from sinð2�Þ could signal thepresence of non-SM physics, which may occur in B� �Bmixing, in the decay amplitude, or in both [5].

In this paper, we consider the decay B! f0ð980ÞKS,which is mediated by the b! sq �q transition at one-looporder in the weak interaction. The decay B! f0ð980ÞKS isone of several penguin-dominated modes which probesinð2�Þ. In contrast, in B! J= KS decay, and relatedcharmonium modes, the b! sc �c transition operates attree level. Were the time-dependent asymmetries in tree-and penguin-dominated modes to differ, then non-SMphysics could be at work in the penguin process [6].Current experimental results suggest that this could bethe case [7], though definite conclusions require both ex-perimental results of improved precision and theoreticalestimates of the subleading SM corrections. The numericalsize of the SM corrections depend on the specific decaymode, mimicking the appearance of non-SM physics [8],so that the needed estimates demand some care. In thiscontext it is worth noting that the ‘‘wrong-phase’’ penguincontribution, proportional to the weak phase of b! su �u, isparticularly small in B! J= KS decay; indeed, the devia-tion of the time-dependent asymmetry SJ= KS from sinð2�Þ

isOð10�3Þ [9,10]—it is suppressed by both CKM and loopeffects. Thus, the comparison of this asymmetry to a ‘‘tree-only’’ determination of sinð2�Þ permits a sensitive assay ofnew physics in B0 � �B0 mixing. Currently this last isconsistent with SJ= KS , as well as with other determinations

of sinð2�Þ which employ information on the sides of theunitarity triangle, at the Oð10%Þ level [11,12]. In the caseof the penguin modes, the wrong-phase penguin is larger asit suffers only CKM, i.e., Oð�2Þ ’ 0:04, suppression. Inthese modes the computed SM deviations from sinð2�Þdetermine a much-needed baseline against which the ex-perimental results can be assessed for new-physics effects,as new-physics-induced deviations from sin2� could cer-tainly be channel dependent as well. Systematic studies ofthe SM corrections exist [13–15] in a variety of modes. TheB! f0ð980ÞKS mode has received less attention, perhapsdue to the ill-known quark structure of the f0ð980Þ [16,17];this is a deficiency we wish to remedy.The f0ð980Þ is a fairly narrow resonance of non-Breit-

Wigner form which couples to�� andK �K final states. Thequark structure of the f0ð980Þ meson is not well estab-lished. Much discussion has revolved around whether it isbetter regarded as a q2 �q2 state [18] or, perhaps, as a K �Kmolecule [19,20]. The theoretical correction relevant to theinterpretation of Sf0KS as a measurement of sin2� comes

from the presence of the b! su �u transition in the decayamplitude, and interconnected issues complicate its analy-sis. Not only must we consider the possible non-q �q struc-ture of the f0 resonance, but we must also recall that strongfinal-state interactions exist in �� scattering in the I ¼ 0,J ¼ 0 channel [21–23]. Such final-state interactions giverise to the scalar form factor, and particularly the partialwidth �ðf0ð980Þ ! ��;K �KÞ, which can be computedthrough the unitarization of a scattering kernel compatiblewith low-energy constraints [24–26]. We presume thef0ð980Þ resonance to be sufficiently narrow relative tothe energy released in the B decay that we can approximatethe full decay amplitude as the product of the two-body

*rdutta@uky.edu+gardner@pa.uky.edu

PHYSICAL REVIEW D 78, 034021 (2008)

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decay amplitude AðB! f0ð980ÞKSÞ with �ðf0ð980Þ !��;K �KÞ, where we refer to Ref. [27] for a discussion ofthe assumptions implicit to this treatment. In particular, thestrong phases associated with the long-distance physics off0 decay are presumed to be universal and not modified bythe B-decay environment, so that the decay-specific phasesare captured by the application of QCD perturbation theoryin the heavy quark limit to the two-body decay process.Corrections to the picture we employ can be estimated,though not in a systematically improvable way. For ex-ample, so-called final-state rescattering has been estimatedfor various b! s penguin modes [15], and Okubu-Zweig-Iizuka (OZI)-violating effects can also be considered, thelatter contributing significantly to J= ! ��� decay, e.g.[25,26]. We reserve discussion of the impact of the com-puted f0 ! �� and f0 ! K �K partial widths, as well as ofpossible OZI-violating effects, to a subsequent publication[28]. Our particular focus is in the study of the sensitivityof our prediction of the CP asymmetries to the assumedquark structure of the f0ð980Þ; the latter enters in theevaluation of the hadronic matrix elements of the b!su �u transition. A priori one might think such effects tobe most important in the assessment of the deviation ofSf0Ks from sinð2�Þ. We proceed in the same vein as Cheng

and collaborators [16,17], treating the f0ð980Þ as a q �q stateand employing the QCD factorization approach [29,30] forthe hadronic matrix elements. We differ from this earlierwork in the treatment of the B! f0 form factor. We thenperform a random scan over the theoretical parameterspace, after Beneke [14], to assess, in part, the sensitivityof the results to the employed hadronic matrix elementsand hence to the implicitly assumed structure of thef0ð980Þ. We note in passing that the additional diagramsdiscussed in Refs. [16,31] if the f0 is a q

2 �q2 state should besuppressed by the large-energy release of B-meson decay[16]; we neglect them.

Let us give a brief outline of this paper. We begin, inSec. II, with a description of the theoretical framework andbriefly review the role of the b! sq �q transition, with q 2u, d, in the determination of the CP-violating parametersSf and Cf, as well as of the two-body branching ratio. We

present a synopsis of the pertinent QCD factorization for-mulas as well. In Sec. III we present the input parameterswe use in our numerical calculations, reporting theCP-violating parameters which result from the use of ourdefined ‘‘default’’ set of input parameters. It should beemphasized that the factorization theorem from whichthe QCD factorization approach follows holds only atleading power in the heavy quark mass, and the estimateof 1=mb power-suppressed terms, though of apparent phe-nomenological importance, is uncertain. Thus our random

scan over the theoretical parameter space, effected to ex-plore the possible range of �Sf0KS and Cf0KS , takes both

uncertainties in the theoretical inputs and in the assessmentof the Oð�QCD=MBÞ corrections into account. We report

these results in Sec. III as well. We conclude with asummary of our results and an outlook on future work inSec. IV.

II. THEORETICAL FRAMEWORK

We pattern our analysis after that of Beneke, Ref. [14],and, indeed, adopt a common notation. In particular, weemploy the QCD factorization framework [29,30] for thecomputation of the B! f0KS matrix elements [17] andperform a random scan over the space of possible inputparameters to estimate the uncertainty in their computa-tion. We begin by recalling that the CP asymmetry AfðtÞinto a CP eigenstate f is given by

AfðtÞ ¼ Brð �B0ðtÞ ! fÞ � BrðB0ðtÞ ! fÞBrð �B0ðtÞ ! fÞ þ BrðB0ðtÞ ! fÞ

� Sf sinð�MBtÞ � Cf cosð�MBtÞ: (1)

Here we have neglected �� where �� � �H � �L is thewidth difference of the B eigenstates. We note �MB �MH �ML is the mass difference of the B eigenstates, Sf is

the CP asymmetry generated by the interference of B� �Bmixing and direct decay, and Cf is an asymmetry reflective

of direct CP violation. We recall [1]

Sf ¼2 Im�f

1þ j�fj2; Cf ¼

1� j�fj21þ j�fj2

; (2)

where

� f ¼�q

p

�B

Að �B0 ! fÞAðB0 ! fÞ ; (3)

with the factor ðq=pÞB characterizing B� �B mixing. If wetreat the f0ð980Þ resonance as if it were a stable particle, asin Ref. [17], then f � f0ð980ÞKS, and we write

� f0KS ¼�q

p

�B

�q

p

�K

Að �B0 ! f0 �K0Þ

AðB0 ! f0K0Þ ; (4)

where the factor ðq=pÞK characterizes K � �K mixing. Inthis event, f is a two-body final state, and we write thedecay amplitude as

Að �B! fÞ ¼ �cacf þ �ua

uf / ð1þ e�i�dfÞ; (5)

with df � j�u=�cjðauf=acfÞ and �q � VqbV�qs for q 2 u, c

to determine

�Sf � ��fSf � sinð2�Þ ¼ 2ReðdfÞ cosð2�Þ sin�þ jdfj2ðsinð2�þ 2�Þ � sinð2�ÞÞ1þ 2ReðdfÞ cos�þ jdfj2

; (6)

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where �f is the eigenvalue of the CP-operator associatedwith the eigenstate f. Moreover, we have

Cf ¼ � 2 ImðdfÞ sin�1þ 2ReðdfÞ cos�þ jdfj2

; (7)

so that if jdfj is small, the functions�Sf and Cf show littlecorrelation [14]. The value of �Sf in the SM shows certainsystematic trends with f [14]. For example, if a color-suppressed tree amplitude C contributes to auf, then �Sfis much larger than it would be if it were absent. In thelatter event, the u-quark penguin amplitude Pu drives auf.Generally, df / ð�Cþ PuÞ=Pc. If the parameters describ-ing the B! f decay amplitude make jPcj small relative tojCj, then �Sf can range over a wide array of values. Suchlarge excursions can be controlled, however, by demandingthat the amplitudes be consistent with the empiricalbranching ratios [14]. In the case of current interest, B!f0KS decay, there is no C amplitude, so that we expect�Sf0KS to be small on general grounds, and the impositionof a branching ratio constraint should no longer be crucial.However, we have found exceptional regions in theoreticalparameter space for which Pc is small with respect to Pu

and thus �Sf0KS is large, so that it is, in fact, crucial toapply the branching ratio constraint to eliminate these largeexcursions.

If we treat �B0 ! f0ð980Þ �K0 decay as a two-body pro-cess, then the decay rate is

� ¼ p

8�M2B

jMj2; (8)

whereM � Að �B0 ! f0 �K0Þ, as we shall describe in detail,

and

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðM2

B � ðMf0 þMK0Þ2ÞðM2B � ðMf0 �MK0Þ2Þ

q2MB

: (9)

We recall that the branching ratio is given by �=�B, where�B is the total decay width of the B meson. In reality, thef0ð980Þ is not a stable particle; rather, it is a resonancewhich decays to both �� and K �K final states. We wish toinvestigate the role of finite-width effects explicitly in asubsequent publication [28]: we neglect them here. Tobegin, we rewrite the two-body decay amplitude as

Að �B! f0 �K0Þ ¼ Anu�u þ Anc�c þ Asu�u þ Asc�c; (10)

where the superscript ðn; sÞ refers to the nonstrange andstrange quark components of the f0ð980Þ, respectively, sothat

df0KS ¼���������u�c

���������Anu þ AsuAnc þ Asc

�: (11)

We now proceed to calculate the An;su;c amplitudes.

A. �B ! f0 �K decay in QCD factorization

The decay amplitudes of exclusive hadronic B-mesondecays can be systematically analyzed in a combinedexpansion of inverse powers of the heavy quark mass mb

and the strong coupling constant �s [32–34]. The QCDfactorization approach, in specific, permits the rigorouscomputation of these amplitudes, for certain two-bodyfinal states, in leading power in Oð�QCD=mbÞ and in a

power series in �sðÞ, where �OðmbÞ [29]. Its startingpoint is the effective weak Hamiltonian for charmlesshadronic B decay, consisting of a sum of the products ofCKM matrix elements, local operators Qi, and Wilsoncoefficients CiðÞ, evaluated in next-to-leading-order(NLO) precision in �sðÞ [35]. Some dissension in theliterature exists concerning the ingredients of the leadingpower analysis [36], particularly in regards to the charm-quark penguin contributions [37,38]: in the QCD factori-zation approach any nonfactorizable charm-quark penguincontributions appear in Oð�QCD=mbÞ corrections.

Although recent work suggests that such charm-quarkpenguin effects may be needed to explain the B-decaydata to light charmless mesons [39], note also Ref. [40],in our context we can safely neglect such developments, asthey cannot make�Sf0KS larger. B-meson decays to scalar-

and pseudoscalar-meson final states have recently beenstudied by Cheng and collaborators in a series of papers[15–17]; they employ the QCD factorization approach andtreat the f0ð980Þ resonance as a q �q state. Their resultsconnect to those of Beneke and Neubert for B-mesondecays to pseudoscalar- and vector-meson final states[30], as the amplitudes for the scalar- and pseudoscalar-meson channels follow from these earlier results upon aseries of replacements [17]. This means that the twist-3light-front distribution amplitudes are assumed to be de-termined by two-particle configurations only. The ampli-tude for �B0 ! f0 �K

0 decay is given in Ref. [17], so that weidentify, after Eq. (10),

Anu ¼ �GFffiffiffi2

p��au4 � rKa

u6 �

1

2ðau10 � rKa

u8Þ�f0K

�

� fKFBf00 ðM2

K0ÞðM2B �M2

f0Þ

þGFffiffiffi2

p��b3 � 1

2b3EW

�f0K

�fB; (12)

Anc ¼ �GFffiffiffi2

p��ac4 � rKa

c6 �

1

2ðac10 � rKa

c8Þ�f0K

�

� fKFBf00 ðM2

K0ÞðM2B �M2

f0Þ

þGFffiffiffi2

p��b3 � 1

2b3EW

�f0K

�fB; (13)

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Asu ¼ �GFffiffiffi2

p��au6 �

1

2au8

�Kf0

��rf0 �fsf0F

BK0 ðM2

f0ÞðM2

B �M2K0Þ

þGFffiffiffi2

p��b3 � 1

2b3EW

�Kf0

�fB; (14)

Asc ¼ �GFffiffiffi2

p��ac6 �

1

2ac8

�Kf0

��rf0 �fsf0F

BK0 ðM2

f0ÞðM2

B �M2K0Þ

þGFffiffiffi2

p��b3 � 1

2b3EW

�Kf0

�fB: (15)

The M1M2 subscripts mean that the quantities in bracketsare to be interpreted as api ðM1M2Þ, in which theM1 mesoncontains the spectator quark from the B-meson, andbiðM1M2Þ, in which the M1 meson carries an antiquarkfrom the weak vertex and the M2 meson contains a quarkfrom the weak vertex. The QCD factorization framework isrigorous in leading order in Oð�QCD=mbÞ, but it also

includes estimates of 1=mb-suppressed corrections. We

note that �rf0 ¼ 2Mf0=mbðÞ and rK ¼ 2M2

K0=ðmbðÞ�ðmsðÞ þmdðÞÞÞ but that these chirally enhanced termsare counted as terms of leading power in the 1=mb expan-sion in QCD factorization. The quark masses are running

masses defined in the MS scheme. Certain 1=mb correc-tions can suffer end point divergences, and their estimate isuncertain. The biðM1M2Þ terms, for example, reflect anni-hilation contributions, which are a class of1=mb-suppressed corrections to the decay rate. Moreover,the api ðM1M2Þ terms contain Oð�sÞ corrections to theoperator matrix elements, which can, in turn, contain1=mb-suppressed corrections. The Oð�sÞ corrections en-code nonperturbative input through integrals over the light-cone distribution amplitudes. The coefficients api ðM1M2Þ,where p ¼ u, c, can be expressed as [29,30]

api ðM1M2Þ ¼ CiðÞ þ Ci�1ðÞNc

þ Ci�1ðÞNc

CF�sðÞ4�

��ViðM2Þ þ 4�2

NcHiðM1M2Þ

�þ Ppi ðM2Þ;

(16)

where the upper signs apply when i is odd, the lower signsapply when i is even, and CF ¼ ðN2

c � 1Þ=2Nc with Nc ¼3. The quantities ViðM2Þ, HiðM1M2Þ, and PiðM2Þ reflectvertex corrections, hard spectator interactions, and penguincorrections, respectively. The function HiðM1M2Þ containsan end point divergence in its power-suppressed terms—we refer to Refs. [29,30] for all omitted details. For worktowards a theory of the power corrections, we refer thereader to the developments in Refs. [32–34]. The coeffi-cients bi relevant to our calculation can be expressed as[29,30]

b3ðM1M2Þ ¼ CFN2c

½C3Ai1 þ C5ðAi3 þ Af3Þ þ NcC6A

f3�;

b3;EWðM1M2Þ ¼ CFN2c

½C9Ai1 þ C7ðAi3 þ Af3Þ þ NcC8A

i3�;(17)

where in the annihilation amplitudes Ai;fn � Ai;fn ðM1M2Þthe superscripts i and f refer to gluon emission from initial-and final-state quarks, respectively. For the calculation of

Ai;fn ðM1M2Þ, we include the corrections coming from �K2 ,the second Gegenbauer moment of the kaon, for consis-tency with our analysis of the f0ð980Þ, and we provideexplicit expressions in the Appendix.The expressions for �B! f0 �K decay contain nonpertur-

bative hadronic input through the meson decay constantsand B! M form factors; these quantities are sensitive tothe assumed quark structure of the hadrons. Assuming thef0ð980Þ resonance can be written as a q �q state, we write

jf0ð980Þi ¼ cos�js�si þ sin�jn �ni; (18)

where jn �ni � ðju �ui þ jd �diÞ= ffiffiffi2

p. The empirical observa-

tion of both �ðJ= ! f0!Þ and �ðJ= ! f0�Þ suggeststhat the f0ð980Þ has both strange and nonstrange compo-nents, so that � is nonzero [16]. We define the B! f0 formfactor as

hf0ðp0Þj �d��5bjBðpÞi ¼ �i��P �M2

B �M2f0

q2q

�

� FBf01 ðq2Þ

þM2B �M2

f0

q2qF

Bf00 ðq2Þ

�; (19)

where P ¼ ðpþ p0Þ and q ¼ ðp� p0Þ, and the f0decay constant as

hf0jq �qj0i ¼ Mf0�fqf0 (20)

for q 2 ðn; sÞ. Defining jfq0 i � jq �qi, we have

hfs0js�sj0i ¼ Mf0~fsf0 and hfn0 ju �uj0i ¼

1ffiffiffi2

p Mf0~fnf0 ;

(21)

so that �fsf0 ¼ ~fsf0 cos� and �fnf0 ¼ ~fnf0 sin�. Similarly we

define FBf00 ¼ sin�FBfd

0

0 =ffiffiffi2

p, where F

Bfd0

0 describes the

form factor to the jd �di piece of the f0ð980Þ final state.In order to calculate the B! f0 form factor we assume

the f0 to be a q �q state and use the constituent quark model(CQM) of Refs. [41–44], which combines heavy quarkeffective theory with chiral symmetry in the light-quarksector. In Refs. [41,42], Gatto et al. study the D! ��!3� andDs ! f0� amplitude using the CQMmodel for theD! � andDs ! f0 form factors and find good agreementwith E791 data. This model describes interactions in termsof effective vertices between a light quark, a heavy quark,

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and a heavy meson. The model depends on both its UVandIR cutoffs. The UV cutoff � is set by the spontaneouschiral symmetry breaking scale�, which is ofOð1 GeVÞ.This model does not include confinement, so that one hasto introduce an IR cutoff �. The constituent quark mass mis determined by solving the Nambu Jona-Lasinio gapequation, which, in turn, depends on the UVand IR cutoffs.We choose � ¼ 1:25 GeV as described in Ref. [44]. Thusfor fixed�, as the IR cutoff varies,m varies accordingly, asexplicitly illustrated in Ref. [45]. For the default value ofthe B! f0 form factor, we use m ¼ 0:3 GeV, � ¼0:3 GeV, and �H ¼ 0:4 GeV. The parameter �H �MH �MQ, where MH is the mass of the heavy meson

and MQ is the mass of the constituent heavy quark.

Explicit expressions for the polar and direct contributionsto the form factor are given in Refs. [41,42]. To assess therange of the B! f0 form factor, we vary � in the range[0.25, 0.35], as per the associated variation in m given inRef. [45], and vary �H in the range [0.3, 0.5] as given in

Refs. [41–44]. We find that the variation in FBfd0 is mainly

driven by the variation in �. However, the error rangereported in Refs. [41,42] was apparently determined byvarying �H alone [46]. The default value of the B! f0form factor, as well as its uncertainty, are given inSec. III A.

The uncertainty in the calculation of the decay ampli-tude comes from both the statistical and systematic errorspresent in the QCD factorization approach. The uncertain-ties associated with different input parameters such as thescalar meson decay constants, the form factors, the quarkmasses, and the Gegenbauer moments of light-cone distri-bution amplitudes are of the first kind, whereas the uncer-tainties associated with the 1=mb-suppressed correctionsare of the second kind. The theoretical uncertainties in thelatter case come from the hard spectator and the weakannihilation contributions which contain end point diver-gences. The end point divergences XH in the hard spectatorand XA in the annihilation terms are parameterized as [29]

XH;A ¼ ln

�MB

�h

�ð1þ �H;Ae

i�H;AÞ; (22)

where we assume �H;A � 1 and �h ¼ 0:5 GeV. We note

�H;A are unknown, strong-interaction phases.

We wish to determine the impact of the various theoreti-cal uncertainties in the B! f0K decay amplitude on thevalue of �Sf0KS in a quantitative way. To realize this, we

begin by defining a ‘‘default model.’’ This consists of usingthe central values of the inputs given in Refs. [17,29], aswell as setting �A ¼ �H ¼ 0 in the parametrization of theend point divergences. With this in place we thus determinethe default values of the An;su;c amplitudes. To gauge the sizeof the uncertainties, we perform a random scan of theallowed theoretical parameter space. That is, we include

the uncertainties coming from all the input parameters. Forthe parameter scan, we choose the range of the inputparameters by taking either 1�, 2�, or 3� deviationsfrom the central values. Once we have the maximal andminimal values of the inputs in the chosen ranges, we drawrandom values of the input parameters within that range.Similarly, to include the uncertainties coming from XA andXH, we vary �H;A from 0 to 1 and �H;A from 0 to 2�. Thatis, we draw random values of �H;A in the range 0 to 1 and

�H;A in the range 0 to 2�.

III. RESULTS AND DISCUSSION

A. Inputs

For definiteness, we summarize the parameter choices ofour default model. We employ a renormalization scale ofOðmb=2Þ; namely, we choose ¼ 2:1 GeV, so that thevalue of the strong coupling constant in NLO at this scale is�sð2:1 GeVÞ ¼ 0:286. To realize this, we work with Nf ¼5 throughout, after Ref. [29], and choose �ð5Þ ¼0:225 GeV, which corresponds to �sðMZÞ ¼ 0:118. As acheck of the accuracy of NLO precision in this context, wecompare our results with those computed at ¼ 4:2 GeV,for which we note �sðmbÞ ¼ 0:224. For the electromag-netic coupling constant, we use ��1 ¼ 129 and neglect theQ2 evolution of � as in Ref. [29]. To realize the Wilsoncoefficients, we follow the procedures of Ref. [29], usingmtðmtÞ ¼ 167 GeV,MW ¼ 80:4 GeV, and sin2�W ¼ 0:23[29]1 and verify the leading-order (LO) and NLO Wilsoncoefficients they report through explicit computation. Wealso use the values �A ¼ �H ¼ 0 to fix the default valuesof the hard spectator and annihilation terms. We fix thevalue of the b-quark mass at the scale ¼ 2:1 GeV to bembð2:1 GeVÞ ¼ 4:88� 0:08 GeV, which corresponds tombðmbÞ ¼ 4:2� 0:07 GeV [47] using the two-loop ex-pression for the running quark mass [35]. The error inmbð2:1 GeVÞ is calculated using the maximum and theminimum value of mb at the ¼ mb scale. Similarly thevalue of msðmbÞ ¼ 0:077� 0:017 GeV corresponds tomsð2:1 GeVÞ ¼ 0:090� 0:020 GeV, after Ref. [30]. Forthe meson masses we use MB ¼ 5:2795ð5Þ GeV, MK0 ¼0:497 648ð22Þ GeV, and Mf0 ¼ 0:980ð10Þ GeV, as given

in Ref. [47], where the uncertainty in the last digits isindicated by the number in parentheses. Similarly, themean life of B meson is B ¼ 1:530ð9Þ � 10�12 s [47],and GF ¼ 1:166 37ð1Þ � 10�5 GeV�2 [47]. The errors inthese empirical quantities are unimportant for our pur-poses, so that we neglect them in our random scan. TheCKM matrix elements, taken from [48], are

1Repeating our calculation with mt ¼ 172:7 GeV, as perRef. [47], and following the procedures described in text yield�Sf0KS ¼ 0:0269, Cf0KS ¼ �0:005 57, and Brðf0KSÞ ¼ 13:6�10�6 in place of the values reported in Eq. (31). We can safely

neglect this update in our parameter scans.

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jVubj ¼ 0:003 57þ0:000 17�0:000 17; jVusj ¼ 0:226 53þ0:000 75

�0:000 77

jVcbj ¼ 0:0405þ0:0032�0:0029; jVcsj ¼ 0:973 16þ0:000 18

�0:000 18

j�u=�cj ¼ 0:0205� 0:0019; j�cj ¼ 0:0394� 0:0031

� ¼ 76:8þ30:4�31:5; � ¼ 21:5þ1:0

�1:0; (23)

where � and � are reported in degrees. We use the un-certainties associated with the CKM matrix elements tocalculate the uncertainties in j�cj and j�u=�cj, assumingthe errors are uncorrelated for simplicity. We note the angle� is the phase associated with �u=�c, namely �u=�c �j�u=�cje�i�, and that � is the unitarity-triangle angle wedefined in Sec. I. In our scan over parameter space, we varyour inputs within 2� and 3� of their default values as well.In the case of the CKM parameters we use the ranges asreported in Ref. [48] for jVikj, �, and � for�2� and�3�as appropriate; it is worth noting, in particular, that � whenranged over a 3� variation is never negative.

The other inputs are taken from Refs. [17,29,49] and aregiven as follows. All the scale-dependent quantities in thescalar sector are evaluated at ¼ 2:1 GeV as perRef. [17]. The ratio of charm-quark mass to the b-quarkmass is taken from Ref. [49]; the resulting value of thecharm-quark mass encompasses the value recommended inRef. [47]. The value of the mixing angle is taken fromRef. [17]. All the other input parameters are taken fromRef. [29]. We have

B1 ¼ �0:54� 0:06; B3 ¼ 0:01� 0:04;

�K1 ¼ 0:3� 0:3; �K2 ¼ 0:1� 0:3;

�B ¼ 0:35� 0:15 GeV; fB ¼ 0:20� 0:03 GeV;

FBK0 ¼ 0:35� 0:03; fK ¼ 0:16 GeV;

�ff0 ¼ 0:460� 0:025 GeV; � ¼ 152:5� 12:5;

mc=mb ¼ 0:27� 0:06; mq=ms ¼ 0:0413: (24)

We note that mq ¼ ðmu þmdÞ=2 and that we neglect the

error in mq=ms. This amounts to neglecting the error in the

light-quark mass mq, since we include the error in the

strange quark mass. This we may safely do as we employmq=ms in the evaluation of rK exclusively; we note that

mass differences of Oðmd �muÞ are tantamount to theinclusion of isospin-breaking effects, which are numeri-cally unimportant to us here. Explicit expressions for the

leading-twist, light-cone distribution amplitudes of thelight mesons are given in Eq. (A1), where we note B1

and B3 are the first and third Gegenbauer moments of thef0ð980Þ and �K1 and �K2 are the first and secondGegenbauer moments of the kaon. Although all theGegenbauer moments are scale dependent in principle,the inclusion of such variations is beyond the accuracy ofthe NLO treatment we effect here, so that we neglect suchrefinements. The first inverse moment of the B mesonlight-cone distribution amplitude, �B, is defined byR10 dx�BðxÞ=x ¼ MB=�B, where �BðxÞ is one of two

twist-2 light-cone distribution amplitudes of the B meson[29,30]. We term fB, F

BK0 , fK, and ff0 the B meson decay

constant, the B! K form factor, the K meson decayconstant, and the scalar decay constant, respectively. Thegiven value of ff0 is specific to the assumed q �q structure of

f0; we recall � is the mixing angle of Eq. (18). For ourinputs, we note that the ratios rM for the K meson and f0meson are

rK ¼ 1:08; �rf0 ¼ 0:402: (25)

Our default value of the B! f0 form factor is

FBfd

0

0 ¼ 0:284; (26)

where we use the CQM model and the parameters m ¼0:3 GeV, � ¼ 0:3 GeV, and �H ¼ 0:4 GeV. Cheng et al.

[17] assert that the FBfd0 form factor should be comparable

in size to the B! � form factor, and we find this to beconsistent with our own numerical estimate. We calculatethe error in the B! f0 form factor by varying � in therange [0.25, 0.35] and �H in the range [0.3, 0.5]. We find

that the variation in FBfd

0

0 with respect to �H is small

compared to the variation with respect to �. That is,

varying �H over our chosen range makes FBfd

0

0 range

over the values [0.28, 0.29]. Varying � over its chosen

range as well, we find that FBfd

0

0 ranges from [0.23, 0.46].

For the random scan we use the range [0.23, 0.46] for our1� scan and triple its range, to yield [0.0048, 0.69] for our3� scan.The coefficients api ðf0KÞ, ap6;8ðKf0Þ, and bi are calcu-

lated at the scale ¼ 2:1 GeV using Eqs. (16) and (17), aswell as the formulas in Ref. [29], to yield

au4 ¼�0:0261� i0:0208; ac4 ¼�0:0340� i0:0110; au6 ¼�0:0581� i0:0185; ac6 ¼�0:0641� i0:00842;

au8 ¼ ð76:6� i0:434Þ � 10�5; ac8 ¼ ð76:5� i0:259Þ � 10�5; au10 ¼ ð�172þ i130Þ � 10�5;

ac10 ¼ ð�172þ i130Þ � 10�5; ap6;8ðKf0Þ ¼ ap6;8ðf0KÞ; b3ðf0KSÞ ¼ �0:0506; b3ðKSf0Þ ¼ 0:0264;

b3;EWðf0KSÞ ¼ �0:00133; b3;EWðKSf0Þ ¼ �0:000768; (27)

Our formulas for b3 and b3EW differ slightly from those given in Refs. [17,29], as we detail in the Appendix, though thenumerical differences are negligible. Now that we have all the input parameters, we can calculate the default value of theAn;su;c amplitudes using Eq. (12). We find

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Anu ¼ ð�8:12þ i0:438Þ � 10�8 GeV;

Anc ¼ ð�7:67� i0:821Þ � 10�8 GeV;

Asu ¼ ð�84:7� i24:2Þ � 10�8 GeV;

Asc ¼ ð�92:5� i11:0Þ � 10�8 GeV:

(28)

If we ignore the annihilation terms from the B! f0KSdecay amplitude, we find

Anu ¼ ð�12:5þ i0:438Þ � 10�8 GeV;

Anc ¼ ð�12:1� i0:821Þ � 10�8 GeV;

Asu ¼ ð�76:4� i24:2Þ � 10�8 GeV;

Asc ¼ ð�84:3� i11:0Þ � 10�8 GeV:

(29)

If we ignore all the 1=mb power-suppressed terms in theB! f0KS decay amplitude, we find

Anu ¼ ð�12:1þ i0:438Þ � 10�8 GeV;

Anc ¼ ð�11:7� i0:821Þ � 10�8 GeV;

Asu ¼ ð�76:4� i24:2Þ � 10�8 GeV;

Asc ¼ ð�84:3� i11:0Þ � 10�8 GeV:

(30)

For our default parameter set, the amplitudes mediated bythe s�s component of the f0ð980Þ are numerically dominant,and the various contributions coming from the power-suppressed terms are small corrections to the leadingpower contributions. However, these power correctionssuffer end point divergences, and their estimate is uncer-tain; they could well affect the value of �Sf0KS . We shallconsider this possibility in detail in Sec. III B. We note thatthe dominance of the penguin contributions associatedwith the strange quark component of the f0 emerges be-cause the a4 and a6 interfere destructively exclusively inthe decay to the nonstrange component of the f0, as we cansee from Eq. (12). Moreover, we note that the contributioncoming from the annihilation amplitude in this case issmaller than the contribution coming from the penguinamplitude, so that the B! f0KS decay amplitude is domi-nated by the s�s component of the f0.

B. Two-body decay results

Using the default values of the An;su;c amplitudes, we caneasily calculate df0KS and thus �Sf0KS and Cf0KS using

Eqs. (6) and (7). We compute the values of �Sf0KS ,

Cf0KS , and the branching ratio (Br) for the two-body decay

as discussed in Sec. II. For our default parameter set wefind

�Sf0KS ¼ 0:0269; Cf0KS ¼ �0:005 61;

Brðf0KSÞ ¼ 13:4� 10�6:(31)

We did the same calculation ignoring the annihilationterms, for which we find

�Sf0KS ¼ 0:0268; Cf0KS ¼ �0:005 84;

Brðf0KSÞ ¼ 12:4� 10�6:(32)

Alternatively, if we neglect all 1=mb suppressed terms wefind

�Sf0KS ¼ 0:0268; Cf0KS ¼ �0:005 87;

Brðf0KSÞ ¼ 12:3� 10�6:(33)

We have done the same calculation at the ¼ mb scale,and our results are very similar to those given in Eqs. (31)–(33). That is, we find the values of �Sf0KS are 0.0269,

0.0269, and 0.0269, respectively, whereas the values ofCf0KS are �0:006 03, �0:006 21, and �0:006 23, respec-

tively. However, the value of the branching ratio doeschange, giving 10:3� 10�6, 9:69� 10�6, and 9:63�10�6, respectively. It is evident from our numerical esti-mates that �Sf0KS and Cf0KS and the two-body branching

ratio receive little contribution from the annihilation termsor, indeed, from any of the 1=mb suppressed terms. That iswhat we expect since the B! f0KS decay amplitude isdriven by the s-quark component of the f0; the leadingpower contributions to the Asu;c amplitudes are much larger

than the contributions coming from the power-suppressedterms, and�Sf0KS andCf0KS depend only on the ratio of the

An;su;c amplitudes df0KS . However, the branching ratio is

controlled by the square of the sum of the amplitudes.Thus any change in the size of the amplitudes themselveswill definitely change the branching ratio; this explainswhy our branching ratios are more sensitive to the valueof . Our default value of the branching ratio is notconsistent with the experimental value of ð5:8� 0:8Þ �10�6 [7] obtained by BABAR and Belle [50,51], so thatthe default set of input parameters we employ cannotexplain the experimental data. This does not falsify ourapproach per se, as it is reasonable to think that we cancompute the ratio of amplitudes df0KS more accurately than

the An;su;c amplitudes alone. We also find that we are able toconfront the empirical branching ratio successfully in ourscan of the theoretical parameter space.The numerical values of�Sf0KS andCf0KS for B! f0KS

decay are calculated by Ref. [17] as well, for which theyfind

�Sf0KS ¼ 0:023; Cf0KS ¼ �0:008: (34)

Our default values of �Sf0KS and Cf0KS for B! f0KSdecay are similar to those in Ref. [17], but they are notexactly same. We note that we also differ in our computedvalues of api ðf0KÞ and ap6;8ðKf0Þ, though this emerges, at

least in part, because they employ a value of �sð ¼2:1 GeVÞ appropriate to Nf ¼ 4.

Our default values of �Sf0KS and Cf0KS are small and

that is what we expect from the SM point of view. In theQCD factorization approach, the uncertainties in the cal-culation of the decay amplitude come from the inputparameters as well as from the 1=mb suppressed terms.

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We would like to establish the range of �Sf0KS and Cf0KSpossible from SM physics using the effect of the variousuncertainties in the B! f0KS decay amplitude. Thus, it isnecessary to include the corrections arising from the un-certainties associated with all the input parameters, includ-ing those in the scalar meson decay constants, the formfactors, the quark masses, the CKM elements, and theGegenbauer moments of the light-cone distribution ampli-tudes. Since the quark structure of the f0 is not well known,we are particularly interested in whether the uncertaintiesin the B! f0 form factor and the f0 scalar decay constantimpact the value of �Sf0KS in a significant way. Additional

systematic uncertainties in our calculation come from thepower corrections which contain end point divergences;our estimate of these is uncertain. As we have noted, too,the branching ratio we compute from our default parameterset does not confront the experimental branching ratiosuccessfully, though this is not necessary to describe�Sf0KS well.

To see the effect of the above-mentioned uncertaintieson the various observables, we perform a random scan ofthe allowed theoretical parameter space. The central valuesof the inputs and the uncertainties associated with them arein Eqs. (23), (24), and (26). For the parameter scan, wechoose the range of these input parameters by taking 1�and 3� variations from the central values. Once we havethe maximal and minimal values of the inputs in the rangementioned above, we draw random values of the inputparameters in the chosen range. To include the uncertain-ties coming from the hard spectator and the weak annihi-lation terms, we vary �H;A from 0 to 1 and �H;A from 0 to

2�. That is, we draw random values of �H;A in the range 0

to 1 and �H;A in the range 0 to 2�. We use the public

domain random number generator ‘‘rannyu’’ throughout;note Ref. [52] for a discussion of suitable inputs. Eachchosen parameter set corresponds to a theoretical model,and we have plotted the various combination of�Sf0KS and

Cf0KS for 500 000 models for a particular seed, just for

illustration, for parameter ranges fixed at 1� and 3� fromtheir central values in Fig. 1. The boxlike shapes of theresulting regions suggest that there is little correlationbetween �Sf0KS and Cf0KS . That means jdf0KS j is small,

and �Sf0KS and Cf0KS are mainly driven by Reðdf0KSÞ andImðdf0KSÞ, respectively, as we can see from Eqs. (6) and

(7).We did a random scan for two different seeds and for

each seed we use 100 000 and 500 000 different theoreticalmodels. We determine the range in �Sf0KS which results,

or indeed of any observable, by evaluating the extremalvalues which emerge from the parameter scan. This makesthe results sensitive, in principle, to the detailed manner inwhich the scan is effected. The resulting range in �Sf0KSvaries little for different values of the seed and for thedifferent numbers of theoretical models if we choose theinput parameters within 1� of the central values. For

example, the range of �Sf0KS for a 1� scan is found to

be [0.017, 0.034] and [0.016, 0.035] for two different seedsfrom a sample of 500 000 parameter sets. However, therange of �Sf0KS does change significantly with seed if we

choose the input parameters to range within either 2� or3� of the central values. For example, the range of �Sf0KSin the scan over a 3� variation is found to be ½�0:41; 0:29�and ½�0:34; 0:20� for two different seeds from a sample of500 000 parameter sets. It is evident from our scan that,though the range of �Sf0KS is small for 1� variations, it is

large for 3� variations. No color-suppressed tree amplitudecontributes to the B! f0KS decay process, but we havelarge excursions in �Sf0KS over small regions of the pa-

rameter space nevertheless. In these special regions, the Pc

amplitude becomes small and drives the large values of�Sf0KS . However, such small values of Pc always give

theoretical branching ratios which are much too small. Itthus becomes crucial to apply the experimental branchingratio constraint to determine the allowed range in �Sf0KSwithin the SM. We recall that the average branching ratiofor B! f0KS decay process measured by Belle andBABAR [50,51] is ð5:8� 0:8Þ � 10�6. We impose thebranching ratio constraint in such a way that we ignorethose theoretical models which are not compatible within1� of the experimental branching ratio for 1� parameterscans and within 3� of the experimental branching ratiofor the 3� parameter scans, respectively. After we imposethe empirical branching ratio constraint, the seed-averagedrange of �Sf0KS for our scan of models spanning 1� and

3� variations are found to be [0.018, 0.033] and½�0:019; 0:064�, respectively, for the 500 000 point simu-lation. In this context ‘‘seed-averaged’’ means that wereport the extremal values of �Sf0KS which result from

scans using two different seeds. It is interesting to note thatthe range in �Sf0KS once the branching ratio constraint is

FIG. 1 (color online). Range in �Sf0KS and Cf0KS from a scanof 500 000 theoretical models. The lighter interior region corre-sponds to a scan of the parameter space at 1�, where the darker,larger region corresponds to a scan at 3�. Some points in the 3�scan fall outside the window chosen for the illustration.

RUPAK DUTTA AND SUSAN GARDNER PHYSICAL REVIEW D 78, 034021 (2008)

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applied is almost identical to what was found without thebranching ratio constraint in the 1� scan. This underscoresour point that the branching ratio is largely independent ofthe value of �Sf0KS . We find that Cf0KS as well can range

over a wide array of values giving ½�0:47; 0:51� and½�0:55; 0:49� for two different seeds from a sample of500 000 parameter sets within 3� of the default values.However, after applying the branching ratio constraint,such large excursions disappear, giving the seed-averagedrange ½�0:045; 0:051� for 3� and ½�0:013;�0:0024� for1�. We plot the various combinations of �Sf0KS and Cf0KSwith the branching ratio constraint for the seed used inFig. 1, just for illustration, for a random sample of 500 000models in Fig. 2. After applying the branching ratio con-straint, we find that the range in �Sf0KS and in Cf0KSdepends on neither the number of parameter sets nor theparticular seed used in the random scan. To test the accu-racy of our NLO analysis, we also change the renormal-ization scale from mb=2 to mb, and we find very littlevariation in�Sf0KS andCf0KS once the empirical branching

ratio constraint is imposed. That is, the ranges of �Sf0KSand Cf0KS in this case are [0.017, 0.032] and

½�0:012;�0:0032� for 1�, respectively, whereas they are½�0:015; 0:061� and ½�0:044; 0:033� for 3�. The weak dependence we observe follows as the observables involveratios of computed amplitudes in each case.

To study the likelihood of various values of �Sf0KS in

theory space, we have plotted a histogram of�Sf0KS from a

sample of 100 000 and 500 000 theoretical models, afteraveraging over seeds, in Fig. 3. To plot the histogram, wefirst set a bin size and then determine the number of modelswhich fall within each bin. Since we have different num-bers of theoretical models in our random scans, to plot allof them together we divide the number of models falling ineach bin by the total number of theoretical models used and

choose that quantity as the ordinate of each histogram. Ifno points are shown, both here and in later figures, thatmeans that no models whatsoever occupy that region oftheory space. For scans employing parameters within a 1�variation, the range of �Sf0KS varies little if we change the

seed or the number of models used in the scan. However, itdoes vary notably if the parameters chosen range within 2�or 3� variations of their central values. In Fig. 4, we plot ahistogram of �Sf0KS , after averaging over seeds, from a

sample of 100 000 and 500 000 models taking the values ofthe input parameters within 3� of their central values. Itcan be seen from the histogram that, although the actualrange of �Sf0KS varies for different seeds and for different

FIG. 2 (color online). Range in �Sf0KS and Cf0KS from a scanof 500 000 theoretical models with the empirical branching ratioconstraint imposed. The darker points represent the possiblerange of �Sf0KS and Cf0KS within 3�, whereas the lighter points

represent the range for a 1� variation.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.015 0.02 0.025 0.03 0.035 0.04

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 3. Histogram for �Sf0KS for a random scan employingparameters which range over 1� of their central values. We noteþ denotes the scan with 100 000 models and � denotes the scanwith 500 000 models. The range of �Sf0KS is the same for each

case.

0

0.01

0.02

0.03

0.04

0.05

0.06

-0.01 0 0.01 0.02 0.03 0.04 0.05

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 4. Histogram for �Sf0KS for a random scan employingparameters which range over 3� of their central values. We noteþ denotes the scan with 100 000 models and � denotes the scanwith 500 000 models. Some points in the scan fall outside thewindow chosen for the illustration.

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numbers of parameter sets, the shape of the histogram in�Sf0KS does not seem to vary. We also plot the histograms

of �Sf0KS after applying the branching ratio constraint, as

shown in Figs. 5 and 6, and note that the shape of thehistogram changes little. These results are also seed aver-aged. In these figures the total number of points refer to thetotal number of models which satisfy the branching ratioconstraint. In particular, the scale of the ordinate is notsame in Figs. 5 and 6 because the total number of pointswhich satisfy the branching ratio constraint at 1� is smallcompared to those which satisfy the branching ratio con-straint at 3�. Note that the integral of these model spaceresults over �Sf0KS should yield unity in each case.

We have also studied the impact of the hadronic uncer-tainties alone on �Sf0KS for the B! f0KS decay process.

To do this we set the CKM parameters to their defaultvalues and vary all the other input parameters. We includethe variations in the power corrections as well. Similarly, tosee the impact of the CKM parameters, we set all the otherhadronic inputs to their default values and vary only theCKM parameters j�cj, j�u=�cj, �, and �. We include theuncertainties associated with Vij in an uncorrelated way to

find the uncertainties in j�cj and j�u=�cj. We perform arandom scan over these CKM parameters within 1� and3� of their central values as given in Eq. (23) and Ref. [48].The associated histograms of �Sf0KS for the 1� scan are

shown in Figs. 7 and 8. These results are also seed aver-aged. As we see from the figures, the impact of the had-ronic uncertainties is very small compared to the CKMuncertainties in this case. The actual shape and range of�Sf0KS is driven mainly by the CKM parameters. Since

jdf0KS j is small, the second term in the numerator of Eq. (6)

is small compared to the first term—�Sf0KS is mainly

driven by the first term in the numerator. Both terms do

0

0.02

0.04

0.06

0.08

0.1

0.12

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 6. Histogram for �Sf0KS , after that of Fig. 4, once theempirical branching ratio constraint is imposed. Here the totalnumber of points refer to the total number of models whichsurvive the imposed branching ratio constraint.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.01 0.015 0.02 0.025 0.03 0.035 0.04

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 5. Histogram for �Sf0KS , after that of Fig. 3, once theempirical branching ratio constraint is imposed. Here the totalnumber of points refer to the total number of models whichsurvive the imposed branching ratio constraint.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 8. Histogram for �Sf0KS varying only the hadronic pa-rameters over a 1� variation.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.015 0.02 0.025 0.03 0.035

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 7. Histogram for �Sf0KS varying only the CKM parame-ters over a 1� variation.

RUPAK DUTTA AND SUSAN GARDNER PHYSICAL REVIEW D 78, 034021 (2008)

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tend to zero, however, as � becomes small. Thus, weexpect to get very small values of �Sf0KS for sufficiently

small values of �. As we can see from the histograms,variations in � and, indeed, the CKM parameters impactthe likely values of �Sf0KS . Since � is never negative and

the jdf0KS j2 term is negligible, �S can only be negative if

Reðdf0KSÞ< 0, which is an exceptionally rare occurrence.

When we repeat this analysis for a scan in which theparameters are allowed to range over 3�, as shown inFigs. 9 and 10, we find that the hadronic uncertaintiesdominate over the CKM uncertainties. This emerges, de-spite the detailed shapes in Figs. 9 and 10, because we usethe extremal values found in the simulation to define therange. The range of �Sf0KS is thus found to be

½�0:34; 0:25� for the hadronic uncertainties and[0.000 76, 0.039] for the CKM uncertainties. However,after applying the branching ratio constraint the �Sf0KSrange is found to be small, giving ½�0:018; 0:048� for thehadronic uncertainties and [0.000 80, 0.038] for the CKMuncertainties. In the 3� scan the range of �Sf0KS is large,

so that jdf0KS j is no longer small, for certain values of the

parameter set. It is the small value of the Pc amplitudewhich is responsible for these large values of jdf0KS j. For ¼ mb, the ranges of �Sf0KS are quite similar giving

½�0:0073; 0:043� for the hadronic uncertainties and[0.000 75, 0.039] for the CKM uncertainties once thebranching ratio constraint is imposed.

We have also studied the impact of the individual had-ronic parameters on the value of �Sf0KS for the B! f0KSdecay process. To do this we set all the other input pa-rameters to their default values except the one which wewant to study. Note that we include the variations in thepower corrections as well throughout. We also use thesame seed used to generate Figs. 1 and 2. We find thatfor any hadronic input with values in the 1� range, therange of �Sf0KS is very small. For the input parameters

with values within the 3� range, the largest excursions in�Sf0KS come from the first inverse moment of the Bmeson

distribution amplitude �B, which enters the B! f0KSdecay amplitude through the hard spectator interactionterms. However, after applying the branching ratio con-straint, the�Sf0KS range is found to be small giving [0.026,

0.034] if we employ a renormalization scale of ¼ mb=2and giving [0.026, 0.034] for ¼ mb. The negative valuesof �Sf0KS found in the scan with inputs which range up to

3� of their central values are driven by the uncertaintiesassociated with �B, XA, and XH. To see the impact of theseparameters alone on �Sf0KS , we set all the other input

parameters to their default values and vary �B, XA, andXH within 3� from their default values. The range of�Sf0KS is quite large giving ½�0:25; 0:19�. However, wefind �Sf0KS to be small and positive once we impose the

branching ratio constraint, giving the range [0.024, 0.036]and [0.025, 0.036] for the two different renormalizationscales we employ, ¼ mb=2 and ¼ mb, respectively.Fixing the values of �B, XA, and XH to their default values,we perform a random scan over the other input parameterswithin 3� and find the range of �Sf0KS to be

½�0:0015; 0:047�. Once we impose the branching ratioconstraint, we find the range of �Sf0KS to be [0.000 54,

0.046] and [0.000 37, 0.046] for ¼ mb=2 and ¼ mb,respectively. It is interesting to note that the range of�Sf0KS we find by fixing the values of �B, XA, and XH to

their default values, and imposing the branching ratioconstraints, captures most of the range we find once weperform the scan with all the parameters within 3�.However, we note that the extension of those scan resultsinto negative values come from the variations associatedwith �B, XA, and XH as well.In principle, �Sf0KS can have large excursions either by

having a small Pc amplitude or a large Pu amplitude. The

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 10. Histogram for �Sf0KS varying only the hadronicparameters over a 3� variation. More points exist beyond thewindow chosen for the illustration.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

No.

of o

ccur

renc

es/to

tal n

o. o

f poi

nts

∆Sf0 KS

FIG. 9. Histogram for �Sf0KS varying only the CKM parame-ters over a 3� variation.

CP ASYMMETRIES IN B! f0KS DECAYS PHYSICAL REVIEW D 78, 034021 (2008)

034021-11

first possibility can be controlled by demanding that thetheoretical model confront the empirical branching ratiosuccessfully. As we have already noted, such small Pc

amplitudes give theoretical branching ratios which aremuch too small, so that they can be excluded by imposingthe experimental branching ratio as a constraint. We nowwish to consider whether it is possible to have a large Pu

amplitude without enhancing Pc as well. If it were pos-sible, one could have a large value of �Sf0KS without

necessarily violating the branching ratio constraint. In thecase of the default parameter set, the aq4 , a

q6 , and a

q10—aq8 is

so small as to play no role—terms interfere destructivelygiving small Anu;c amplitudes. In this case, the hard specta-

tor terms, which enter through the aq4 and aq10 terms, are

similar in magnitude to the vertex terms in aq6 . However,once we do a random scan over the complete theoreticalparameter space, the hard spectator terms play a dominantrole in enhancing the aq4 and aq10 terms, which, in turn,

enhance the Anu;c amplitudes. The real and imaginary parts

of aq4 � 0:5aq10, noting Eq. (12), e.g., can both be very

large, so that the cancellation of these terms with aq6 is

no longer possible in these regions of parameter space.However, these effects act to enhance both Pu and Pc at thesame time—it is not possible to have a large Pu amplitudewithout having a large Pc amplitude as well. Since both Anuand Anc amplitudes are large,�Sf0KS is not enhanced. Thus,

large excursions in the hard scattering terms cannot pro-duce large excursions in the value of �Sf0KS .

Our primary motivation in conducting our current analy-sis is to determine whether it is possible to control therange of �Sf0KS irrespective of the structure of the f0resonance. In Sec. II we noted that the structure of the f0impacts both the B! f0 form factor and f0 decay con-stant. Thus, the impact of these parameters alone on�Sf0KSgives us insight as to whether we can constrain the range of�Sf0KS to small values no matter the structure of the f0. We

perform a random scan varying only these two parameterswithin 1� and 3� from their central values, both with andwithout the branching ratio constraint, for the particularseed used to generate Figs. 1 and 2. Without the branchingratio constraint, the range of �Sf0KS is [0.026, 0.028] for

1� and [0.025, 0.028] for 3�, respectively. Once we im-pose the branching ratio within 3� of the experimentalvalue as a constraint, we find the range to be [0.025, 0.028]for the scan in which the B! f0 form factor and f0 decayconstant are varied within 3� of the central value. For ¼mb, the range of �Sf0KS in this case is also very small,

giving [0.026, 0.028]. It is evident from our analysis thatthe impact of the B! f0 form factor and f0 decay constanton �Sf0KS is small irrespective of the branching ratio

constraint. Apparently, the strange quark content of thef0 is so important in determining�Sf0KS that the particular

value of the ratio of the B! f0 form factor and f0 decayconstant is quite unimportant.

IV. CONCLUSIONS

In this paper, we have analyzed the two-body decay ofthe B meson to one scalar meson and one pseudoscalarmeson, namely, to the f0ð980ÞKS final state. The empiricalstudy of this mode measures sinð2�Þ through a decayprocess controlled by the b! s penguin amplitude. Thisis important for several reasons. First, the study of suchmodes is complementary to the study of the b! s tran-sition in Bs mixing. The hint that the empirical weak phasedetermined from Bs mixing is larger than the SM predic-tion [53], if borne out, would require new sources of CPviolation beyond that contained in the CKM paradigm.This makes the confirmation of, or constraints on, sucheffects in the b! s penguin modes crucial. Generally, wewould like to assess the possible deviation of SfðtÞ, thetime-dependent CP asymmetry, from SM physics in arobust way, so that we can determine what new physics,if any, exists in these decay modes. To that end, thef0ð980ÞKS final state itself is of intrinsic interest. Thetime-dependent CP asymmetry in this case has no color-suppressed tree contribution; it is intrinsically subject toless theoretical uncertainty than modes in which suchcontributions are present. Moreover, its empirical studycan be complementary to that of another theoretically cleanprocess: B! �Ks, as both modes occupy the B! KKKsDalitz plot. Indeed, B! f0Ks and B! �Ks, assumingthe � to be ideally mixed, are the only known b! spenguin modes which have no color-suppressed treecontributions.We have investigated the size of �Sf0KS , i.e., the devia-

tion of SfðtÞ from sinð2�Þ, in the B! f0KS decay process

in the SM using the QCD factorization approach, assumingthe f0 to be a q �q state. We employ a parameter scan toprobe a broad range of possible theoretical models, explor-ing variations in the inputs at the 3� level and ill-knownOð�QCD=MBÞ corrections with 100% uncertainty. The

value of �S in the B! f0KS decay mode has been studiedby other authors within the QCD factorization approach[15–17,54]. We note in passing that work in the kT facto-rization approach exists as well [55–57], though suchleading-order treatments in �s do not permit a quantitativeassessment of �S. Our calculation differs most signifi-cantly from the other work in QCD factorization in itstreatment of the B! f0 form factor, for which, for con-creteness, we employ the CQM approach of Refs. [41–44].The earlier work simply asserts that the B! f0 formfactor ought be comparable to that of B! �0. Our nu-merical studies support this, though we assign a largeuncertainty to our assessment. The parameter scan tech-nique we employ borrows heavily from the work of Beneke[14], in which the deviations to SfðtÞ from sinð2�Þ werestudied for the B! ð�0; �0; �; �0; �ÞKS decay modes.That work eschewed the B! f0Ks decay mode due tothe uncertain q �q structure of the f0 [58]; thus we havestudied the possible numerical uncertainty incurred by this

RUPAK DUTTA AND SUSAN GARDNER PHYSICAL REVIEW D 78, 034021 (2008)

034021-12

with great care. Our work also differs from Beneke’s in thatwe explore the theoretical model space for input parame-ters which vary within 3� of their default values, as well asidentify the parameter morphologies which give the largestexcursions in �Sf0KS .

The assumed quark structure of the f0ð980Þ does enter inthe evaluation of the hadronic matrix elements and in theassessment of the f0 decay constant and the B! f0 formfactor, in particular. We have investigated the range of�Sf0KS which results from varying either the B! f0form factor and the f0 scalar decay constant, after impos-ing the branching ratio constraint, giving [0.026, 0.028] forvariations within 1� and [0.025, 0.028] for variationswithin 3�. The value of �Sf0KS is simply not sensitive to

the value of the FBf0 form factor and the f0 decay constant.Although we can accommodate the experimental branch-ing ratio in a q �q model of the f0, we cannot draw con-clusions concerning the structure of the f0 on the basis ofour analysis.

Let us summarize our results for the range of�Sf0KS and

Cf0KS , which emerge from a scan of the complete theoreti-

cal parameter space, allowing for uncertainties in theinfrared-cutoff dependent Oð�QCD=MBÞ corrections,

which we characterize by the parameters XA and XH. Inlimited regions of the parameter space, the Pc amplitudecan become small, driving large excursions in �Sf0KS .

Such excursions are removed, however, by demandingthat the theoretical model confront the empirical branchingratio up to some tolerance, that is, up to 1� for the 1� scansand up to 3� for the 3� scans. Once we demand that ourtheoretical models satisfy the branching ratio constraint,the possible seed-averaged range in �Sf0KS is greatly

reduced, giving [0.018,0.033] and ½�0:019; 0:064� for thescans in which the parameters are allowed to range within1� and 3� of their central values, respectively. In com-parison, we find that Cf0KS ranges over ½�0:013;�0:0024�and ½�0:045; 0:051�, respectively. Generally, we find df0KSto be sufficiently small that the values of �Sf0KS and Cf0KSare uncorrelated, note Fig. 1. It is worth emphasizing thatwe estimate the ranges using the extremal values of ourparameter scans. This is a conservative approach. If wewere to define the range of �Sf0KS to capture, e.g., 95% ofthe models about its most likely value in theory space, wewould have much smaller ranges. Retaining our currentapproach, we could also sharpen our estimates once im-proved measurements of the branching ratio becomeavailable.

Nevertheless, let us proceed to compare our range for�Sf0KS with the empirical result. The time-dependent CP

asymmetry induced by the interference between B �B mix-ing and direct decay, as well as the direct CP asymmetry,measured by Belle and BABAR [7,59,60] are 0:85� 0:07and 0:08� 0:12, respectively. The value of sinð2�Þ mea-sured by Belle and BABAR [7] in B! J= KS decays andrelated charmonium modes is 0:668� 0:026, so that the

empirical value of�Sf0KS is 0:18� 0:10 for the B! f0KSdecay mode. Its error is substantially larger than our largestestimated value of �Sf0KS , so that further refinement of the

experimental results is warranted. If we compare our re-sults with those of Ref. [14] for the B!ð�0; �0; �; �0; �ÞKS decay modes, we see that the B!f0KS mode compares favorably: it yields small values of�Sf, as do the B! �Ks and B! �0Ks modes. Since both

B! f0Ks and B! �Ks decays lack color-suppressedtree contributions, the estimation of �Sf should be par-

ticularly reliable, so that significant differences in theempirical values of �Sf in these modes could truly signal

new physics.We find that the largest single variation in our predic-

tions of the decay amplitudes come from the uncertaintyassociated with the inverse moment of B meson distribu-tion amplitude �B which enters the decay amplitudethrough the hard scattering terms. In our scans for whichthe parameters range over 3� of their central values, thehard scattering terms can dominate over all other terms inthe decay amplitude and produce large values of the Anu;camplitudes. The cancellation of aq4 with a

q6 reflected in the

default values of the parameter set is no longer possible forsuch large values of the hard scattering amplitude. Thismechanism generalizes to any other mode where aq4 and a

q6

interfere destructively. However, these large excursions inthe Anu;c amplitudes do not generate large excursions in

�Sf0KS , as they also act to enhance the charm penguin

amplitude. Nevertheless, the extremal values of �Sf0KS ,

and its negative values in particular, do come from theuncertainties associated with �B, XA, and XH. One expectsthis observation to be relevant to other b! smode as well,though if there is a color-suppressed tree amplitude, thereis more freedom for �S to be large.In future studies we would like to consider how the finite

width of the f0ð980Þ resonance can impact the value of�Sf0KS determined from B! f0KS ! �þ��KS and B!f0KS ! KþK�KS decay processes. In particular, we wishto consider the role of the f0 ! �þ�� and f0 ! KþK�scalar form factors, as well as that of possible OZI-violating effects, in the determination of �Sf0KS in a later

publication [28].

ACKNOWLEDGMENTS

We thank Timo A. Lahde, Ulf-G. Meißner, and JoseOller for discussions concerning the scalar form factor inthe f0ð980Þ mass region and Martin Beneke for a discus-sion concerning the modeling of power corrections withinQCD factorization. We also thank A.D. Polosa for usefuldiscussions regarding the CQM model and the determina-tion of the various form factors within this model. Wethank the Institute for Nuclear Theory (INT) for gracioushospitality and acknowledge partial support from the U.S.Department of Energy under Contract No. DE–FG02–

CP ASYMMETRIES IN B! f0KS DECAYS PHYSICAL REVIEW D 78, 034021 (2008)

034021-13

96ER40989. S. G. thanks the SLAC theory group for gra-cious hospitality as well, and R.D. acknowledges theUniversity of Kentucky in generous support of his visit tothe INT. S. G. also acknowledges partial support from theNational Science Foundation under Grant No. NSFPHY05-51164 at its completion.

APPENDIX: ANNIHILATION AMPLITUDES INQCD FACTORIZATION

The weak annihilation contributions b3 and b3;EW in the

B! f0KS decay processes are expressed as linear combi-

nations of the annihilation amplitudes Ai;f1;2;3. Explicit ex-

pressions for these amplitudes in terms of the light-conedistribution amplitudes for scalar and pseudoscalar mesonsare given in [17,29]. The general expressions for theleading-twist, light-cone distribution amplitudes for pseu-doscalar and scalar mesons are written as

�Pðx;Þ ¼ 6xð1� xÞ�1þ X1

n¼1

�Pn ðÞC3=2n ð2x� 1Þ

�

�Sðx;Þ ¼ �fS6xð1� xÞ�B0 þ

X1m¼1

BmðÞC3=2m ð2x� 1Þ

�:

(A1)

Here B0 is zero, as are all the even Gegenbauer moments,in the SU(3) limit. In order to present simple expressions

for the Ai;f1;2;3 amplitudes, we truncate the expansion of the

pseudoscalar meson after the second Gegenbauer polyno-mial and that of the scalar meson after the thirdGegenbauer polynomial.We perform the integration over the light-cone distribu-

tion amplitudes in such a way that we always have Ixy ¼Iyx, where

Ixy ¼Z 1��

0dx

Z 1

�dyfðx; yÞ;

Iyx ¼Z 1

�dy

Z 1��

0dxfðx; yÞ:

(A2)

We retain finite � throughout the calculation and put �! 0only at the end to avoid dropping finite contributions. Thisamounts to a model of the power corrections which differsslightly from that employed previously [17,30], givingslightly different results for Ai3ðPSÞ and Ai3ðSPÞ. Insteadof the �2=3 found in Refs. [17,30], we get �2=6. That is,the annihilation amplitudes can be written as

Ai1ðPSÞ ¼ ��sfK �fsf0½B1½ð180� 18�2Þ þ �K1 ð3726� 378�2Þ þ �K2 ð27 720� 2808�2Þ�þ B3½ð593� 60�2Þ þ �K1 ð37 305� 3780�2Þ þ �K2 ð714 168� 72 360�2Þ�þ ð3�K2 � 3�K1 þ 3Þ½B1ð18XA � 36Þ þ B3ð60XA � 190Þ� � 2�rSr

KX

2A�; (A3)

Ai1ðSPÞ ¼ ��sfK �fuf0½B1½ð�540þ 54�2Þ þ �K1 ð3726� 378�2Þ þ �K2 ð�13 860þ 1404�2Þ�þ B3½ð�5930þ 600�2Þ þ �K1 ð124 350� 12 600�2Þ þ �K2 ð�1 190 280þ 120 600�2Þ�þ ð�3B3 � 3B1Þ½6ðXA � 1Þ þ 18�K1 ðXA � 2Þ þ 12�K2 ð3XA � 8Þ� þ 2�rSr

KX

2A�; (A4)

Ai3ðPSÞ ¼ ��sfK �fsf0

�rK

�18B1

�X2A � 4XA þ 4þ �2

6

�þ 60B3

�X2A �

19

3XA þ 191

18þ �2

6

��

þ �rS

�6

�X2A � 2XA þ �2

6

�� 18�K1

�X2A � 4XA þ 4þ �2

6

�þ 36�K2

�X2A �

16

3XA þ 15

2þ �2

6

���; (A5)

Ai3ðSPÞ ¼ ��sfK �fuf0

�rK

��18B1

�X2A � 4XA þ 4þ �2

6

�� 60B3

�X2A �

19

3XA þ 191

18þ �2

6

��

� �rS

�6

�X2A � 2XA þ �2

6

�þ 18�K1

�X2A � 4XA þ 4þ �2

6

�þ 36�K2

�X2A �

16

3XA þ 15

2þ �2

6

���; (A6)

Af3ðPSÞ ¼ ��sfK �fsf0XA½rK ½6B1ð6XA � 11Þ þ B3ð120XA � 374Þ�� �rS½6ð2XA � 1Þ � 6�K1 ð6XA � 11Þ þ �K2 ð72XA � 186Þ��; (A7)

Af3ðSPÞ ¼ ��sfK �fuf0XA½rK ½6B1ð6XA � 11Þ þ B3ð120XA � 374Þ�� �rS½6ð2XA � 1Þ þ 6�K1 ð6XA � 11Þ þ �K2 ð72XA � 186Þ��: (A8)

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