Physics Letters B 281 (1992) 133-140 PHYSICS LETTERS B North-Holland

Vcb from semi-leptonic B-decays and the reliability of the infinite quark mass limit

Patr ic ia Ball

Institut fiir Theoretische Physik, Universitdt Heidelberg, Philosophenweg 16, W-6900 Heidelberg, FRG

Received 1 March 1992

Using the latest experimental results on semi-leptonic B-decays, we extract I Vcbl by means of both QCD sum rules and sum rules in the infinite quark mass limit. It is shown that corrections from finite quark mass and the analytical structure of the Isgur-Wise function near the normalization point play a crucial part and question the reliability of the limit of infinitely heavy quarks for b --* c decays. From QCD sum rules we obtain I Vcbl (ra/1.32 ps) 1/2 = 0.037+0.002+0.003, where the first error is experimental, and the second comes from uncertainties in the sum rule parameters.

Semi-leptonic decays of B- to D-mesons have been extensively studied since they provide the from the theoretical point of view simplest possibili ty to extract the CKM matrix element IGI from experimental data. Things seemed to become even simpler with the arrival of the effective theory of infinitely heavy quarks (EHQT) (see e.g. ref. [ 1 ] ). In the EHQT, the semi-leptonic decay of a heavy meson into a heavy meson is determined by one single form factor, the universal Isgur-Wise function ~w [2], which does not depend on the heavy quark masses or their ratios at all. The EHQT seemed to apply naturally to b ~ c decays, assuming that both the b- and c-quark be heavy enough to obey the predictions of the EHQT. Various at tempts have been made to extract the Isgur- Wise function and IGI from the experimental data of the differential spectrum of the decay B ~ D*eu [3-5 ], employing the normalizat ion condit ion of the Isgur-Wise function at the kinematical endpoint of the spectrum, ~lw(Y)]>,=l = 1. Using Luke's theorem [6], it was stated in ref. [5] that the kinematical endpoint remains unaffected by 1/mQ corrections. By calculating the fu l l QCD form factors determining the b ---, c decays from QCD sum rules we will show that the higher order mass corrections still contribute sizably to the decay rate, thus strongly questioning the reliabili ty of the l imit of infinitely heavy quarks.

QCD sum rules for the form factors of the semi-leptonic decays of heavy mesons were first calculated in ref. [ 7 ] over the full kinematically allowed range of the invariant lepton mass t. QCD sum rules for the form factors of the B ~ D(*)eu decays at t = 0 were first evaluated in refs. [8,9]. We want to go into details neither about QCD sum rules in general nor about our method to evaluate them, which has been discussed in refs. [ 7,10 ]. We just shortly review the mere bones of the method. We start from the three-point functions

2 2 Ku(pB,pD) = i 2 d4x d4y exp( ipDx-- ipBy)(OITjD(X)Vu(O) j~(y) IO ) = K+(pB,PD, t)(PR +PD)u + . . . (1)

[t = (pB - P D ) 2 ] for the decay B ~ Deu and

~,J (PB,PD) = i 2 f d4x d4y exp(ipDx -- ipsy)(01T jD*, ( x ) j u (0) j~(y) I0)

• 2 2 p a 2 2 = - (PB,PD, t) -- eu, PBpPDJ'v(PB,PD, t) + . . . lglwFo(PB,PD, t) i (pa + PD)uPa.F+ 2 2 (2)

l E-mail address: ACI@DHDURZ1.BITNET.

0370-2693/92/$ 05.00 ~) 1992-Elsevier Science Publishers B.V. All rights reserved 133

Volume 281, number 1,2 PHYSICS LETTERS B 7 May 1992

for the decay B ~ D*eu, respectively. The dots denote further Lorentz invariants that do not contribute to the considered decays. We use the interpolating fields jB = qi75b, jD = Cli75c and jD*~ = qTuc for the B-, D- and D*-meson, respectively, where the q denotes a u- or d-quark. Ju = Vu - Au = cTu (1 - 75)b is the weak current. According to ref. [ 1 1 ], one carries out a non-perturbat ive operator product expansion (OPE) for the three-point functions (1) and (2) in order to factorize the short- from the long-distance behaviour, which is described by vacuum expectation values of gauge-invariant spin-zero operators, the vacuum condensates. The per turbat ive short-distance behaviour is contained in the C-number Wilson coefficients, which depend on the squares of momenta , pB 2, p2 and t as well as on the quark masses mb and mc of the b- and c-quark, respectively. The OPE is matched with the three-point functions expressed in terms of hadronic matrix elements,

1 1 K , -

- mB PD - - m2 - - {OIjDID)(DIVuIB)(BIj~IO) + higher resonances,

1 1 p2B 2 2 _ 2 / ,: , ~ \ , ~,, , , , , ,:t, \~OIJD,,ID*,,~IkD*,,%~V_A,ulBjkBIJBIO} higher resonances +

-- mB PD roD* (3)

with the helicity 2 of the D*-meson. Using quark-hadron duality, the contr ibution of higher resonances is modelled by the per turbat ive contr ibut ion to the OPE above certain thresholds s o and s °. The applicat ion of a Borel t ransformation in p2 and p2 suppresses both the cont inuum contr ibution and the one of higher condensates to the OPE, and we arrive at the sum rules

f + ( t ) - f~ fom~rn 2 e x P \ M 2 + M2 j (M~,Mf i , t ) ,

A l ( t ) = mb

mD*mZ fBfD • (mB + mo* ) mD* ~ 2 2 ^ 2 2 exp(m . \ M~ + -7-75-M6 / M ~ M B B Fo ( M~, Mfi, t ) ,

m b ( m B + r n o . ) ( m 2 m 2 . ) 2 ~ ^ A 2 ( t ) = mD. m2fBfD, exp + M ~ M f B F + ( M ~ , M t ~ , t ) ,

mb(mB + mD* ) ( 2 m2D~ ) exp ~ + M~M2[~Fv(M~,Mx~, t )

V ( t ) = 2mD. m2BfBfD. M~ M D ] (4)

with the Borel parameters M~ and MD 2. / )K+ and /~ Fi denote the Borel-transformed expressions of the in- variants in (1) and (2). The full expressions can be found in refs. [8-10] ~1. We have used the form-factor parametr izat ion ,2

(DlVul B) = f + ( t ) ( p B +PD)u + f - ( t ) ( p a - - P D ) u ,

(D*,2IJulB) = - i ( m ~ + mD* )A1 (t)G*, ~;'1

2 V ( t ) ,p~ .(4) iAz(t) (e*(~)PB)(PB +PD)u + iA3( t ) (~*(~)PB)(PB--PD)u + ~ ~ PBpPD~ (5) + mB + mD* m B + mo* mB + mo*

with the polarizat ion vector eft ) of the D*-meson. The fQ are the leptonic decay constants of the mesons, defined by

, l We do not agree with the contribution of the four-quark condensate given in ref. [8], which, however, turns out to be numerically negligible.

,2 Note that in the limit of vanishing lepton masses f_ and A3 do not contribute to the decay rates.

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(01cliysc]D) = JD-~- c , (Olqy.clD*,A) = fD*mD*e~ ~), (6)

and correspondingly for the B-meson. In the numerical evaluation of the sum rules (4) we replace fQ by the appropria te two-point sum rule without radiative corrections (see e.g. ref. [12]) . As discussed in refs. [7,10], this procedure reduces the size of as corrections to the sum rules (4).

The above form factors can be expressed in terms of some invariants ~i more suited for the consideration of the heavy quark l imit (HQL), namely

1 f+(t) = =----~ [(1 + r ) ~ + ( y ) - ( 1 - r ) ~ _ ( y ) ] ,

z v r

rn2B + m~ - t mD Y - 2mamo , r - - m B '

v ~ - m~ + m2. - t r* mD* Al(t) -- 1 + r ~ ( 1 +Y)~A~(Y), Y - - 2rnBmD. ' = mB

Az(t ) - 1 + r* 2x/~- [(1 + r*)~Az(Y) -- (1 -- r*)~As(y)],

V(t) - 1 + r* 2v/~_ @ ( y ) . (7)

Due to the appearance of non-Landau singularities [7], the sum rules (5) can be evaluated only up to/max =

10.5 GeV: (for m b = 4.8 GeV, mc = 1.35 GeV, s ° = 6 GeV: and s o = 34 GeV2), which value, however, nearly meets the kinematical endpoints t = (rob - mD) 2 = 11.6 GeV 2 and t = (mE - mD-)2 = 10.7 GeV 2, respectively, and corresponds to y = 1.06 for the decay B ~ Deu and y = 1.01 for B ~ D*eu. In the HQL, only the universal form factor ~lw survives with

~+ = ~A 1 = 2~A 2 = -2~A3 = ~v = ~IW, ~ _ = 0 . (8)

It has been shown [6,13] that ~+ (1) and ~A~ (1) remain unaffected by 1/mQ corrections. On the other hand, one can deduce a sum rule for the Isgur-Wise function in the EHQT by either taking the

l imit mQ ---+ oo of the sum rules (4) or by recalculating the graphs in the EHQT (cf. refs. [ 14-17] ). Due to hard gluon exchange, ~lw becomes scale dependent. Following the procedure outlined in ref. [ 16 ], we introduce ~iw, which is renormalizat ion-group invariant to one-loop accuracy, by

~lw (y) = ~lw (Y,/z)a~(u) -~o/2~o, (9)

where Y0 = 16/3 [y In (y + X / ~ - 1 ) / V / ~ - 1 - 1 ] is the anomalous dimension of the heavy-heavy current in the EHQT to lowest order [18] and fl0 = 11 - 2 n f is the lowest order coefficient of the QCD beta-function with nf light quark flavours. In contrary to the QCD sum rules now the cont inuum model needed to describe the contr ibution of higher states to the three-point function becomes crucial as we want to cover the whole region of accessible y. We can establish two limiting continuum models characterized by different regions of integration of the perturbat ive double spectral function (see fig. 1 ). Integrating out the symmetric variable co + to' we get the sum rules

l,II fil.ll _y0HH/2fl0 S R 3 p t ( y , AM, As ) ~IW (Y) = as (2AM) S R 2 p t (AM, As) (10)

with Borel parameter A M and cont inuum threshold As, which are related to the QCD sum rule parameters b y

M~ = 2mQAM and s~ = (mQ + As)2. The SRi are given by

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(O'

As

] P-

As m

Fig. 1. Two different continuum models for the three-point function. Light-gray area: model I; light- and dark-gray area: model II.

AS(v) As

SR~p,(y, AIPIAs) I21(I/u2 1 / ~2 1 + y ~ du exp(-u/AM) + 0 As(y)

2 y + 1 -<clq) + ~ ( c w g G q ) ,

As

du u(As - u) exp( -u /AM))

2 y + 1,_ ~ , 12 1 duu2exp(-u/AM - (/lq) + 48--8--~-M-giqcrgtJq), SR~IP ' (y 'AM'As) - jr2 (1 + y ) 2 .

0

As

/ SR2m(AM, As ) = 3 duuZexp(-u/AM) - (qq) + ~ ( / l c r g G q ) (11) 71.2

0

where As (y) = ½As ( 1 + y - ~ 1 ). We have neglected the contribution of the four-quark condensate, which turns out to be negligible. All scale-dependent quantities have to be evaluated at/~ = 2AM. We want to point out that in contradistinction to the EHQT sum rule for the two-point function [ 16], the radiative corrections to (10) are small. We have checked this explicitly by calculating the radiative corrections to the quark condensate which turn out to be smaller than 9% in the whole region of y accessible [10,17]. The sum rule (10) automatically fulfills the normalization condition ~lw (y)ly= 1 = 1 independent of the Borel parameter AM and the cont inuum threshold As. In the numerical evaluation, we take ~iw to be normalized at rh = 2mbmc/(mb + mc) ~ 2 GeV which roughly corresponds to the reduced mass of the system. Actually changing rh by a factor of 2 has no influence on our results due to the smallness of the anomalous dimension Y0 at the y considered.

The main difference between the two cont inuum models concerns the analytical structure near y = 1. Whereas ~ v can be expanded around y = 1 as

1̂1 ~ _ p2 ~Iw(Y) 1 (j .... 1) + 0 ( ( 3 ' - 1)2) . (12)

~w starts with a square-root cut:

~ w 0 ' ) = 1 - t72V/y - 1 _ p 2 ( y _ l) d - O ( ( y 1 ) 3 / 2 ) . (13)

According to the discussion of ref. [ 19] this non-analytical structure can be associated with anomalous thresholds

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0.05

0.04

0.03

0.02

0.01

0 1

[Vcb ~Iw(Y) I

1 .I 1.2 1.3 1.4 y

-- -- - EHQT-SR I

. . . . . EHQT-SR II

QCD-SR

• CLEO/ARGUS

Fig. 2. Comparison of the QCD form factors (A~ and ~v to the Isgur-Wise function ~IW (lower curve: (~w, upper curve:

efw .

in principle possible in this channel. Although we know that the lsgur-Wise function is analytical at y = 1, Jaffe [19 ] has shown that a realistic form factor may quite well be reproduced by a function with a singularity which is determined rather by the extension of the meson than by the mass of the neighboured resonance. Since the cont inuum model II does not violate general principles and especially obeys the Bjorken rule [20]

-• V=I 1 ~1w(y) _ _p2 < - ~ , (14)

we see now reason to withdraw it, but take it serious at least as extreme case. We are now in a posit ion to compare the Isgur-Wise function (10) rescaled t o / t = rn according to (9) to

the QCD form factors (4). Fig. 2 shows the QCD form factors ~.41 and ~ . as obtained from (4) with m b = 4.8 GeV, rnc = 1.35 GeV, the cont inuum thresholds s o = 34 GeV 2, s ° = 6 GeV 2 and the Borel parameter Mg = 3 GeV 2 with a fixed ratio M2a/M~ = mb/mc = 3.5. The lsgur-Wise function (10) is displayed for both cont inuum models with A M -- 0.56 GeV and As = 1.1 GeV, which parameters correspond to the ones used in the evaluation of the QCD sum rule. We use the values of the condensates (qq)(1 GeV) = ( - 0 . 2 4 GeV) 3 and (OagGq)(1 GeV) = 0.8 GeV 2 x (qq)(1 GeV). The difference in slope n e a r y = 1 of the two solid curves representing (~w and (IIw, respectively, is clearly visible. No surprise, ~v ( 1 ) violates the normalizat ion condit ion by ,,~ 15%, slightly depending on the Borel parameter. But even for ~.4~ (1), which has no 1/mQ corrections, the normalizat ion condit ion is violated by ~ 10%. This may be at tr ibuted to large 1/rn~ corrections. Remember that mc = 1.3 GeV is small enough to give rise to sizable contributions of that type. Of course this deviat ion from the expected behaviour will influence the determinat ion of ] V~bJ.

The existing experimental data on semi-leptonic b ~ c decays provide various possibilities to extract Vcb. Apart from extracting it from the branching ratios of both the decays B ~ Deu and B ~ D*eu, one can employ the differential branching ratio dB (B ~ D * e u ) / d t with respect to the invariant lepton mass. Since in the HQL dB(B ~ D*ev)/dy is proport ional to IVcb]2~w(y) and the 1/mQ corrections vanish at y = 1, it has been thought suitable to get I Vcb[ directly and model independent by extrapolating the experimental data to y = 1 [5 ]. Unfortunately, the data near the normalizat ion point suffer heavily from phase-space suppression. In addit ion, the extrapolat ion is possible without any problems only in the case that ~lW can be expanded in a rapidly converging Taylor series around y = 1. However, in view of the non-analytici ty of our cont inuum model I, this might be not the case under any circumstances. Figs. 3a and 3b show the experimental data as well as least-squares fits of the sum rules (4) and (10) with the parameters stated in the caption. Fig. 3a displays dB/dt, the differential branching ratio in dependence on the invariant lepton mass, fig. 3b I V~bl~iw (y) according to

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0 . 0 0 8

0.006

0.004

0. 002 F

0 L 0

dB(B->D*ev)/dt [ G e V ~

2 4 6 8 i0

! q

t [GeV 2 ]

- - -- - EHQT-SR I: Vcb = 0.040

. . . . . EHQT-SR II: Vcb = 0.033

- - QCD-SR: Vcb = 0.038

• CLEO/ARGUS

1.2

1

0.8

0.6

0.4

0.2

0

. . . . ~AI

. . . . . . . ~V

- - ~ I W

L ~ , I ~ , , J J , , , , I , ~ , J I , , J ,

i.i 1.2 1.3 1.4 y

Fig. 3. (a) Experimentally and theoretically determined spectra of the decay B ~ D*eu in dependence on the invariant lepton mass. The EHQT sum rules are calculated at AM = 0.56 GeV and As = 1.1 GeV, which corresponds to the parameters used in the QCD sum rules, m b = 4.8 GeV, mc = 1.35 GeV, s o = 34 GeV 2, s ° = 6 GeV 2 and M 2 = 3 GeV 2 with

2 2 M ~ / M o = 3.5. (b) ]Vcb~lW(y) ] according to eq. (15) with the same data as displayed in (a).

d B ( B ---* D * e u ) d B ( B --* D * e u ) 2mBmo* = =

dt dy 2 2

_ G F [ V c b [ . 2 ~ 3 / ~ r * = , B ~ t r t B r t ~ D * k / y " - 1(1 + y ) [ ( 1 -- )2(1 + 5) ' ) -- 8 y r * ( y - 1 ) ] ~ 2 w ( y ) . (15)

A l t h o u g h in all cases the da ta can be f i t ted very well, the e x t r a p o l a t i o n to y = 1 reveals the p r o b l e m s s ta ted

a b o v e a n d yields qu i t e d i f fe ren t va lues o f ]Vcb 1.

In t ab le 1 we give the resul ts for [V~b] as o b t a i n e d f r o m b o t h the Q C D a n d the E H Q T sum rules us ing the

e x p e r i m e n t a l resul t s o f the b r a n c h i n g ra t ios B ( B ---+ D e u ) = (1.6 + 0 .3 )% a n d B ( B ~ D * e u ) = (3.7 i 0 . 5 ) %

as well as the d i f f e ren t i a l decay ra te o f the decay B --+ D*eu, all ave raged ove r the C L E O a n d A R G U S resul ts

[21 ,22] . N o t e t h a t we have i n c l u d e d the p r e l i m i n a r y resul t o f CLEO, B ( D * ~ D~r) = 66% [21 ,23] , wh ich

dev ia t e s f r o m the va lue g iven in ref. [24] a n d lowers the va lue o f B (B --* D * e u ) by 17%. The B- l i fe t ime is set

to the average va lue o b t a i n e d at LEP (cf. ref. [25] ): rB = (1.32 + 0 .07) ps ~3 . All q u o t e d d e t e r m i n a t i o n s o f

Vcb are resca led a c c o r d i n g to these values. In e v a l u a t i n g the Q C D sum rules, we h a v e v a r i e d the q u a r k masses

in the wide range m b = 4 .6 - 4 .8 G e V a n d mc = 1 .3-1 .35 GeV. The c o n t i n u u m t h r e s h o l d s were set to be s o =

3 4 - 3 6 G e V 2 a n d s ° = 6 G e V 2, wh ich va lues are d e t e r m i n e d by r equ i r i ng s tab i l i ty o f the t w o - p o i n t sum rules

#3 Note that this value is a lifetime averaged over all B-hadron species produced in e+e - annihilations, weighted by the product of their semi-leptonic branching ratios and their production cross section.

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Table 1 [Vcb I based on the average B-lifetime obtained at LEP and the latest results on branching ratios and differential decay rates, averaged over the values measured by ARGUS [22] and CLEO [21]. The first error is experimental, the second from uncertainties in the sum rule parameters. Neubert [5] uses various parametrizations of the Isgur-Wise function, Stone [21] averages over various quark model calculations.

I Vcbl (zB/1.32 ps)U2

from B(B ~ Deu) from B(B ~ D*ev) fit to spectrum

QCD sum rules EHQT sum rule I

II

0.036 + 0.002 4- 0.003 0.044 4- 0.004 4- 0.002 0.035 ± 0.003 4- 0.001

IVcbl(rB/1.32 pS)U2

0.037 ± 0.003 4- 0.004 0.038 4- 0.003 4- 0.002 0.032 4- 0.002 4- 0.001

0.038 + 0.005 4- 0.003 0.040 4- 0.006 4- 0.002 0.033 4- 0.005 4- 0.001

result of this paper result of ref. [5] result of ref. [21]

0.037 + 0.002 ± 0.003 0.039 ± 0.006 0.035 4- 0.002 -1- 0.003

Table 2 The ratio of the decay rates F (*) = F (B ---, D(* ) eu ) and the asymmetry c~ for the experimental cuts on the lepton momentum employed by CLEO and ARGUS, respectively.

F*/F c~(pg > 1.0 GeV) o~(pe > 1.4 GeV)

CLEO 2 2 +13 0.65 ± 0.66 -t- 0.25 " - 0 . 8

ARGUS 9 a+ L2 0.7 :k 0.9 - ' - - 0 . 6

QCD sum rules 2 -~+0.2 0.42 ± 0.03 0.16 + 0.03 "~-0A EHQT sum rules I, I1 2 R+0.2 0.47 ± 0.01 0.15 + 0.01 • v_0.1

for fB and fro*), respectively. Actual ly the sum rules do not strongly depend on these values and the errors f rom

paramete r uncer ta in t ies g iven in the table are mainly de te rmined by the var ia t ion o f the Borel pa ramete r wi th in

M 2 = 2 - 4 G e V z for the three-point sum rule (cf. ref. [16] ). The Q C D sum rules yield nearly co inc id ing values

for I Vcbl f rom both the de t e rmina t ion f rom the branching ratios and the fit to dB/dy. As m e n t i o n e d above, the

large exper imenta l errors o f the spec t rum data result in a rather large er ror o f b Vcbl as compared to the pa ramete r

uncertaint ies . The E H Q T sum rules are faced with the p rob lem of ext rapola t ion and yield qui te di f ferent central

values IVcbL = 0.040 for mode l I and IVcbl = 0.033 for mode l II f rom the ' f i t to the spectrum. The difference

o f the values ob ta ined f rom the branching ratios f rom those obta ined f rom the fit shows the impor tance of

1/rnQ correct ions to ~w. We have var ied the E H Q T sum rule parameters wi th in A M = 0 .5-1 .0 G e V a n d As =

1.09-1.37 GeVas suggested by the analysis o f the two-point sum rule [16]. Al though the fit o f the Q C D sum

rule and the E H Q T - I I sum rule in fig. 3b extrapolate to nearly the same value at y = 1, the values o f IV~bl do

not agree because o f ~A 1 ( l ) ~ 1.

As a cross check for the val id i ty o f our sum rules, we give in table 2 the ratio o f the decay rates F ( B ---*

D * e v ) / F (B ~ D e u ) and the asymmet ry pa ramete r a = 2FL/Fr - 1, which are both independen t on rs and V~b.

FL,T denotes the decay rate where the D*-meson is longi tudinal ly or t ransversely polarized. In the calcula t ion we

have taken into account the exper imenta l lower cut on the lepton m o m e n t u m Pt, employed to enhance the fract ion

o f tagging leptons or iginat ing direct ly f rom the decaying b-quark. Al though the exper imenta l de t e rmina t ion o f a

is not that accurate, we find good agreement . The ratios o f branching ratios compare very well to the exper imenta l

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results, slightly favouring the QCD sum rules.

Taken altogether, we conclude that the EHQT sum rules are not suited to determine ]V~bl from the full branching ratios due to the non-negligible influence of 1/mQ corrections. In addition, the possible effects of anomalous thresholds on the extrapolation of the differential rate of the decay B ~ D*eu as well as remaining 1/m~ corrections at the normalization point prevent the EHQT from being more accurate than the full QCD sum rules and quark models. The sum rule analysis of both the two- [ 16 ] and the three-point function indicates that charmed and to some less extent even beauty mesons are not heavy enough for the HQL to be reliable. As for the

b --* c decays, the limited statistics of the experimental data favours the extraction of I Vcb] from the full rather than from the differential branching ratios. Averaging over the QCD sum rules, we finally obtain

IVcbl = 0 . 0 3 7 + 0 . 0 0 2 + 0 . 0 0 3 rB , (16)

which compares very well to the results obtained from quark models (cf. ref. [21]).

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