B! K FROM HYBRID SUM RULE. S. Narison. CH Geneva 23. and. Laboratoire de Physique Mathematique. Universite de Montpellier II

Transcription

1 CERN-TH.766/94 PM 94/6 B! K FROM HYBRID SUM RULE S. Narison Theoretial Physis Division, CERN CH - Geneva 3 and Laboratoire de Physique Mathematique Universite de Montpellier II Plae Eugene Bataillon Montpellier Cedex 5 Abstrat HEP-PH Using the hybrid moments-laplae sum rule (HSR), whih iswell-dened for M b!, in ontrast with the popular double Borel (Laplae) sum rule (DLSR), whih blows up in this limit when applied to the heavy-to-light proesses, we show that the form fator of the B! K radiative transition is dominated by the light-quark ondensate for M b!and behaves like p M b. The form fator is found to be F B!K () ' (3:8 :3 3:6 :6) ; where the errors ome respetively from the proedure in the sum rule analysis, the errors in the input and in the SU(3) f -breaking parameters. This result leads to Br(B! K ) ' (4:45 :) 5 in agreement with the reent CLEO data. Parametrization of the M b -dependene of the form fator inluding the SU(3) f -breaking eets is given in (6), whih leads to F B!K ()=F B! () ' (:4 :). CERN-TH.766/94 PM 94/6 February 94

2 Introdution With the advent of the heavy-quark symmetry [], there has been onsiderable interest and progress in the understanding of the semileptoni form fators of the transition of a heavy quark into another heavy quark, sine in this innite mass limit all semileptoni form fators redue to the single Isgur-Wise funtion []. In the ase of heavy-to-light transitions, Isgur and Wise [3] have remarked that, if one assumes that only the upper two omponents of the b quark eld ontribute, there will be relations between various form fators of the semileptoni and rare B-deays. The relations obtained from the stati heavy quark approah are not rigorous as the momentum transfer of e.g, the B! K, whih is of the order of M b =, is large, in suh away that perturbative ontributions related to the hard proess [4] an invalidate the Isgur Wise relations. Burdman and Donoghue [5] have shown that in these proesses, the soft quark model ontributions dominate over the perturbative ones, suh that the results of Isgur-Wise might still be valid and ould be extended to the whole range of q. In this paper, we shall examine the validity of the previous results, using the approah of QCD Spetral Sum Rules (QSSR) in the analysis of the form fator for the B! K radiative proess. The form fator is dened as: <K (p +5 )js q bjb(p)>=i p p F B!K + n (M B M K ) q(p + p ) o F B!K : () In the QSSR evaluation of the form fator, we shall onsider the generi three-point funtion (omitting Lorentz indies): V (p; p ;q)= Z Z d 4 x d 4 yexp(ip x ipy) < jtj q (x)o()j b (y)j >; () whose Lorentz deompositions are analogous to the previous hadroni amplitudes; J q s d is the bilinear quark urrent having the quantum numbers of the K ; J b (M b + m d ) d(i 5 )b is the quark urrent assoiated to the B-meson; O b q s is the weak operator. The vertex funtion obeys the double dispersion relation: V (p ;p ;q )= Z M b ds s p Z ds s p Im V (s; s ;q ); (3) where q p p is the momentum transfer. Choie of the sum rule Widely used in the literature is the double exponential Laplae (Borel) sum rule (DLSR), whih has given suessful preditions in the heavy-to-heavy transitions. This method has been used also for the heavy-to-light transitions [6]{[9]. Though at nite b-quark mass value, the numerial ts of the form fators may be quite good, one an notie that the OPE in the DLSR blows up for M b!, whih invalidates the uses of the DLSR in this

3 hannel. This is due to the existene of terms of the type: Mb l ; (4) (Mb p ) k k p whih appear in the suessive evaluation of the Wilson oeients of high-dimension operators. After the double Laplae (Borel) transform, these terms onvert into: M l b k k exp( M b ); (5) where (resp. ) is the sum-rule variable assoiated to the heavy (resp. light) quark. In this limit, it is onvenient tointrodue the non-relativisti sum-rule variable NR M b, where NR is M b -independent. Then, it is lear from (5) that the OPE onverges if and only if k l. This ondition is not fullled in the ase of the B! K proess, as also notied by [], [9], while we have also heked that it is not satised in some of the form fators of the B! and D! K semileptoni proesses analyzed in [6],[7], though this does not eet the numerial estimates of these form fators. In order to restore the good behaviour of the OPE, the authors of Ref. [9] modelize the ondensates in an ad ho way. We do not nd this argument onvining. Moreover, this parametrization annot guarantee the onvergene of the OPE at higher orders in dimensions. An alternative promising diretion is to nd a sum rule, other than DLSR, whih is more appropriate for this hannel. The light-one sum rule has been hosen in Ref. []; but, due to the omplexity (and, perhaps, less understood struture) of the wave funtions entering in this sum rule, we do not have here the simpliity and the transpareny of the original SVZ-expansion, where, in this ase, the non-perturbative eets are simulated by the vauum ondensates. In this paper, we shall present another type of sum rule, whih is in the line of the SVZ-expansion and whih is adequate for the heavy-to-light proess, sine it has a good behaviour when M Q!. This sum rule has been invented in [] for the analysis of the B! D; and semileptoni deays. This sum rule ombines the moment (nite number of derivatives evaluated at p = ) whih is good for the heavy quark as M b (/M b - expansion), and the Laplae one, whih is appropriate for the light quark if p (/p -expansion). We shall hereafter all this sum rule, as in [], the hybrid sum rule (HSR). It has the form: H(n; )! L V(p ; p ; q ) p = = Z Z ds ds exp( s )Im V (s; s ;q ); (6) M b s n+ where L is the Laplae (Borel) exponential operator. As one an obviously notie, the hybrid transform of (4) is: M k (k+n l) b ; (7) whih shows that the onvergene of the OPE for M b!is reahed with the muh weaker ondition k l n than the one k l of the DLSR. Indeed, in the spei ase of the B! K proess, the trunated series with the inlusion of the mixed ondensate already onverges for n =.

4 In order to ome to observables, we insert intermediate states between the eletromagneti and hadroni urrents in (), while we smear the higher-states eets with the disontinuity of the QCD graphs from a threshold t (t ) for the heavy (light) mesons. Therefore, we have the sum rule for the form fator F B!V (q ): F B!V (q ) H res ' C V f B exp ( M V ) ' Z t M b MB n Z t ds s n+ ds exp( s )Im V PT (s; s ;q ) + NPT: (8) PT (NPT) refers to perturbative (non perturbative) ontributions; C V MV =( V ) for light vetor mesons; M V is the light-meson mass. The deay onstants are normalized as: (m q + M Q ) < jq(i 5 )QjP >= p M P f P <jq QjV >= p M V : (9) V 3 HSR estimate of the B! form fator In the following, we shall use the previous sum rule for the estimate of the B! form fator. The QCD expression of the orresponding vertex funtion reads: ImV (s; s ) = 3 8 M b V qq = s M 5 b (s s ) 3 M b < dd > (M b p )( p ) : () We shall use the ontribution of the mixed ondensate obtained by []. Then, we dedue the sum rule: H res ' 3 Z t Z 8 M ds t 5 b ds s s n+ (s s ) exp( s ) 3 < dd > ( ( ) M M n b (n +) 3 + 4M b n +3n+4!) : () We shall also introdue an analogous expression of the deay onstant f B from moments sum rule at the same order [8]: fb (MB) ' 3 Z t n 8 M ds (s M b ) < (!) dd > n(n +) M b : () s n+ s 4 M b M n b For onveniene, we shall work with the non-relativisti energy parameters E and M (b) : s (M b + E) ; M (b) M B M b ; (3) where, as we have seen in the analysis of the two-point orrelator, the ontinuum energy E is [8], [9]: E D ' (:8 :6) GeV; E B ' (:3 :) GeV; E ' (:5 :7) GeV: (4) 3 M b

5 Using (8), (){(4), we alulate the form fator F B () in Fig. for dierent values of n,, E and t.we use the following values of the QCD parameters for 5 avours [, 3]: M = (:8 :) GeV ; = (75 4) MeV; M b = (4:59 :5) GeV; M =(:47 :5) GeV; < dd > ( ) = (89 MeV) 3 log = =3 ; (5) and the experimental value =(:55 :6) from the -meson eletroni width. Values of and t at whih this experimental number is reprodued from the sum rule are []: ' (:6 :) GeV and t ' (:7 :3) GeV. As an be notied from Fig., the values of orresponding to the stability are onsistent with the previous ones. Inreasing values of n tend to destroy the existene of the stability points due to the inrease of the anomalously large values of the =Mb ontributions. The inlusion of higher-dimension ondensates in the OPE should restore the stability for larger values of n. In our present trunated series, the dierent eets remain still orretions to the < qq > ondensate ones, for n. Moving the values of E B and t within the previous ranges modies the shape of the urves but aets only slightly the value of the stability point; and M b introdue eah an error of 4 and %. We do not expet that radiative orretions will aet this result in a sensible way from dierent experienes of alulating similar observables in the heavy-to-heavy transition form fators. Indeed, large radiative orretions due to the oulombi-likeinterations anel out in the ratio of the three- over the two-point funtion sum rules while the radiative orretions due to the light quark ondensate is known to be small in the ases of heavy and of light quark proesses. In these ases, the total eet due to the radiative orretions is about 4-5%, whih we onsider as another soure of errors. Taking into aount these dierent soures of errors, we obtain from the HSR: F B! () ' (7: : 3:) ; (6) where the rst error omes from the sum-rule proedure and the seond one from the input parameters. Extending this analysis to the D mass, we get, for the hypothetial D! proess: F D! () ' (6: :) : (7) 4 M b -dependene of the B! form fator In order to understand the meaning of the previous results, let us study analytially the sum rule at large values of M b. Sine we shall work with the full theory of QCD, the pseudosalar quark urrent assoiated to the B meson does not aquire any anomalous dimension. Using [8] f B ' E B 3 M b MB M b < dd > (E B ) 3 n (! 3 ( (n +) E B + 3 (n + 3)(n +)+! E B M b 5 4 M b!) n(n +) M ; (8) 4 Mb 4

6 from (), we an dedue from (): F B! () ' p Mb <! dd > exp(m 4(E) B 3= C ) + () + () + () ; (9) M b Mb with: () = (n +) M () = 3 4 (n +)EB + () = I < dd > M 4 n+ <qq > (E B ) 3 M (b) +n(n+) 4! <qq > + 3 E B (E B ) 3 64 (83n + 3n + 63); () where: I 3 Z E B 8 de + E n+ B M b Z t ds s + E Mb s! 3 exp( s ): () We have heked that this approximate expression gives a slightly lower (about %) value of F B! (). We evaluate numerially the oeients of the =M b and =Mb terms at the values n ' and ' :5 :7 GeV, where the HSR optimizes, from the full non-expanded expression of the three-point funtion. We use the expression of f B given by (8), whih naturally has the expeted large M b behaviour. Then, we dedue the interpolating formula in units of GeV: q F B! < dd () ' :5 GeV > ( ( ) M b + :5: + 6:3: ) ; () (E B ) 3= M b Mb where eah oeient ompares reasonably well with that of the expanded expression. We have absorbed the error due to E B into the errors in the orretions. One should understand in the previous formula that: The overall fator.5 is xed in suh away that the interpolating formula reprodues, with the entral values of the numbers in (), the numerial estimate in (6). This fator also absorbs in it the eet of the mixed ondensate M as given by ().However, if we assume that the fatorisation works for the high-dimension ondensates (however, it is known to be largely violated by a fator to 3 []), we ould resum all light-quark-like ondensates eets ( idea behind the notion of non-loal ondensate) by replaing < dd > with < dd >exp( M =3), whih onverges perfetly for M b!ontrary to the ase of the DLSR mentioned by []. The =M b orretion is mainly due to f B (one should ompare this oeient with the one of f B inluding the =Mb term (see e.g. []))and to the meson-quark mass-dierene M (b) (see ()). The oeient of the =Mb term omes, partly from an =M b -expansion of f B, whih is known to give a quite good approximation of f B even at the quark mass. However, 5

7 the main ontribution to the =Mb term in () omes from the perturbative vertex diagram. One should understand that the extration of this eet omes from the exat non-expanded expression of the Wilson oeient without any approximation related to the large value of M b. The appearane of the =Mb term is only due to the analytial struture of the perturbative ontribution, and its numerial oeient absorbs in it all the eets of the perturbative graph. From this feature, the dominane of this term at the quark mass does not mean that the formula given in () annot be used at this sale. One should understand () as an interpolating formula. This sum rule expliitly indiates that the M b behaviour is dominated by the soft proess term < qq > instead of the perturbative hard diagram for M b!. This is a peuliar feature of this heavy-to-light transition proess, whih is not the ase of the heavy-toheavy one. We also obtain a similar behaviour in the ase of the semileptoni B! e form fator [4]. However, the authors of Refs.[], [5] who work with the light-one sum rule do not have this dominant behaviour in M b, sine in their ase the behaviour M 3= b is similar to that oming from the perturbative graph, whih is non-leading in M b, in our approah based on the SVZ-expansion. A lariation of this disrepany needs a better understanding of the struture of the meson wave funtions used in this analysis for M b!. The M b dependene obtained here at q = is the same as the one obtained by Isgur-Wise at qmax, whih might be in line with the Isgur-Wise onjeture that the relations among form fators an be valid at any q values if the heavy-to-light transition form fators are dominated by the soft proess: in suh a ase, the heavy quark stays almost on its mass shell. The dominane of the soft proess obtained by Burdman-Donoghue [5] is onrmed, in a ompletely independent way, by our analysis. However, at the real value of M b, the agreement of the dierent previous sum-rule preditions for the B and D meson form fators is enouraging, despite the fat that the sum rule used in [9] is not well-dened for M b!(however, trunating their QCD series at the level of the quark ondensate, one an notie that their result has the same M b -behaviour than ours (see their equation (4)), while the results of [] have a similar M b -behaviour than our non-leading perturbative ontribution. The agreement between dierent numerial estimates might be due to the large numerial value of the M 3= b - term, whih an also invalidate the nave extrapolation of the result from the D to the B when only the leading M b behaviour of the form fator is used. Using the previous interpolating formula at the D mass, we obtain: F D! () ' (6:5 3:5) ; (3) in aordane with the previous result in (7) from a numerial t. 5 SU(3) f -breakings and the B! K proess We shall onsider in this setion the expliit SU(3) f -breakings on the form fator of the B! K, proess, due to the s quark mass and to the < ss > ondensate, whih have the values [, 3]: m s ( ) ' 5 MeV < ss > < dd > ' (:6 :): (4) 6

8 For this proess, the QCD expression of the SU(3) f breaking parts of the vertex funtion reads to leading order in s : ImV (s; s ) SU(3)! 3 8 M b m s s Mb s (s s ) < qq > SU(3)! m s M b < dd>: (5) One an notie that the only eet of the quark ondensate whih survives after the sum rule proedure is the one from the d quark line of the B-meson urrent, suh that only the < dd > ondensate ontributes. Using an analytial approximate evaluation of the SU(3) f -breaking eets, we dedue the K version of the interpolating formula in (): q < dd F B!K () ' :3 GeV > ( ) M b (E B ( )3= + m s + :5: + 6:3: M b M b Mb + m se B t!) ; (6) where we have used K ' (:8 :3) from deay. One should notie that the SU(3) f -breakings due to the meson mass and oupling give an eet of +7.7 %, whih is inluded in the overall fator. The expliit breaking due to the s quark mass from the Wilson oeient of the ondensate ontributes as +3.%. The SU(3) breaking from the perturbative diagram is about 3% of the perturbative ontribution and might explain the large SU(3) f -breakings obtained in []. One an also note that the SU(3) f -breakings vanish for M b!, ontrary to the ase of f Bs where these orretions remain onstant []. We dedue from the previous formula: F B!K ()=F B! () ' (:4 :); F D!K ()=F D! () ' (: :4); (7) where it is lear that the systemati errors in the evaluation of the oeients of the =M b and =Mb anel out in the ratio. The quoted error in (7) is mainly due to SU(3) f - breaking from the perturbative graph. The SU(3) f -breaking orretions are smaller than the ones in Ref. [] as explained before. Combining this result with the one in (6), we dedue: F B!K ' (3:8 :3 3:6 :6) ; (8) where the last error is due to (7). This implies the branhing ratio: Br(B! K ) ' (4:45 :) 5 ; (9) in agreement with the CLEO data of (4:5:5:9) 5 [6]. Our result for F B!K () is in agreement with the one obtained from an eetive lagrangian approah [7]. It also agrees with the value (:3 :5) obtained from the light-one sum rule [], whih, a priori, is an approah quite dierent from ours, as indeed, the two approahes do not provide the same large M b -behaviour of the form fator. A omparison with the value (:35 :5) obtained in [9] from the DLSR is not very informative as this number omes for the uses of inonsistent sets of input parameters as notied by [] (we agree with these ritiisms). Moreover, the analysis of [9] also suers from the bad behaviour of the DLSR whih blows up for large M b. 7

9 6 Conlusions We have estimated the B! form fator in (6). The value of the hypothetial D! form fator is given in (7). The value of the B! K form fator inluding SU(3) f -breakings is given in (8) and leads to the branhing ratio in (9). We have performed our analysis with the so-alled hybrid sum rule (HSR) in (6), whih is well-dened for M b!, ontrary to the ase of the double exponential Laplae sum rule (DLSR) whih blows up in this peuliar proess. Our numerial estimates are in agreement with the previous results from light-one sum rule [], though the two results do not have the same large M b -behaviour. Indeed, we have also studied the M b -dependene of the previous transition form fator whih an be parametrized with the interpolating formula in (4). It expliitly shows that, in the large-mass limit, this form fator is dominated by the light quark ondensate and behaves like p M b, though at low M b mass the perturbative ontribution is numerially important. This dominane of the soft < dd > ontribution, whih is in line with the results of Isgur-Wise [3] and Donoghue-Burdman [5], might allow the extension, of the result obtained at qmax, to the whole range of q -values. In this large mass limit, the light-one sum rule result has a M 3= b behaviour, whih isvery similar to the one from the perturbative diagram, in our approah within the SVZ-expansion. A lariation of this disrepany between our result with the one from the light-one, needs a better understanding of the meaning of the < qq > ondensate in the language of the wave funtions. Finally, wehave extrated an analyti expression of the SU(3) f -breaking terms due to the s quark mass in (6) and (7). It shows, that the SU(3) f -breakings tend to zero for M b!in ontrast with the ase of f Bs []. Our numerial value in (7) is smaller than the one in []. Aknowledgements I have enjoyed useful onversations with Ahmed Ali, Vladimir Braun and Antonio Masiero. Figure aptions and n dependenes of the form fator F B! () of the B! proess. 8

10 Referenes [] For reviews see e.g. H. Georgi, Proeedings of TASI-9 (World Sienti, Singapore, 99), edited by R.K. Ellis et al.; N. Isgur and M. Wise, Proeedings of Heavy Flavours (World Sienti, Singapore, 99), edited by A. Buras and M. Lindner; T. Mannel, Talk given at the 5th International Symposium on Heavy Flavours, Montreal, Canada, 6-th June 993, CERN preprint TH-75/93 (993). [] N. Isgur and M.B. Wise, Phys. Lett. B33 (989) 3. [3] N. Isgur and M.B. Wise, Phys. Rev. D4 (99) 388. [4] S.J. Brodsky and G.P. Lepage, Phys. Lett. B87 (979) 959; Phys. Rev. D (98) 57. [5] G. Burdman and J.F. Donoghue, Phys. Lett. B7 (99) 55. [6] P. Ball, V.M. Braun and H.G. Dosh, Phys. Lett. B73 (99) 36; Phys. Rev. D48 (993). [7] P. Ball, Phys. Rev. D48 (993) 39. [8] T.M. Aliev, A.A. Ovhinnikov and V.A. Slobodenyuk, Phys. Lett. B37 (99) 569; [9] P. Colangelo, C.A. Dominguez, G. Nardulli and N. Paver, Phys. Lett. B37 (993) 83. [] A. Ali, V.M. Braun and H. Simma, CERN preprint TH-78/93 (993). [] S. Narison, Z. Phys. C55 (99) 55; Phys. Lett. B83 (99) 384. [] S. Narison, Leture Notes in Physis, Vol. 6, QCD Spetral Sum Rules (World Sienti, Singapore, 989) and referenes therein. [3] S. Narison, Phys. Lett. B6 (989) 9; B97 (987) 45. [4] S. Narison (to be published). [5] V.M. Belyaev, A. Khodjamirian and R. Rukl, Z. Phys. C6 (993) 349. [6] CLEO ollaboration: R. Ammar et al, Phys. Rev. Lett. 7 (993) 674. [7] R. Casalbuoni et al., Phys. Lett. B3 (993) 35. [8] S. Narison, Phys. Lett. B98 (987) 4; Phys. Lett. B38 (993) 365 and Talk given at the Third Cf Workshop, -6 June 993, Marbella, Spain, CERN preprint TH-74/93 (993) and referenes therein. [9] S. Narison and K. Zalewski, Phys. Lett. B3 (994) 369. [] S. Narison, Phys. Lett. B3 (994) 47. 9