Impact of Subleading Corrections on Hadronic B Decays

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by  Kwei-Chou Yang

Impact of Subleading Corrections on Hadronic B Decays
Kwei-Chou Yang
Department of Physics, Chung Yuan Christian University, Chung-Li, Taiwan 320, Republic of China
(Dated: February 1, 2008)
Westudythesubleadingcorrectionsoriginatingfromthe3-parton(qq¯g)Fockstatesoffinal-state
mesons in B decays. The corrections could give significant contributions to decays involving an
ω or η(′) in the final states. Our results indicate the similarity of ωK and ωπ− rates, of order
5×10−6, consistent with the recent measurements. We obtain a (B → J/ψK) ≈ 0.27+0.05i, in
2
good agreement with data. Withoutresorting totheunknownsinglet annihilation effects,3-parton
Fock state contributions can enhancethebranching ratios of Kη′ tothe level above50×10−6.
PACSnumbers: 13.25.Hw,12.39.St,12.38.Lg
4
0 The rare B decays allow us to access the Kobayashi- including the contributionsfromannihilationeffects, the
0 Maskawa(KM)mixinganglesandsearchfornewphysics. pQCDresultsreadBr(B0 ωK0).2 10−6 [9].) This
2 MuchprogressinthestudyofBdecays[1,2,3]hasbeen → × 0
result hints that, to account for the large ωK rate, the
n recently made in QCD-based approaches. In the pertur-
annihilation contributions to the BRs of all B KV
a bative QCD (pQCD) framework, the importance of the →
J modesshouldbeover80%. Ifitwillbetrue,itshouldbe
weak annihilation effects in B Kπ decays was first
4 emphasized by [3], where the an→nihilation contributions easy to observe, for instance, the following simple rela-
1 tion ρ+K−,0 : ρ0K−,0 : ωK−,0 1 : (1/√2)2 : (1/√2)2,
are almost pure imaginary and therefore could lead to ≈
the same as their annihilation ratios squared. Neverthe-
CP asymmetry predictions different from the QCD fac-
2 less,iftheglobalfitisextendedtoallmeasuredB PV
v torization (QCDF) results [1]. Nevertheless, the QCDF modes, a small Kω rate 2 10−6 will be obtain→ed[10]
5 study showedthatthe annihilationeffects mayplayonly ∼ ×
and a reliable best fit cannot be reached. The present
0 a minor role in the enhancement of ππ,πK branching
QCD approachseems hard to offer a coherent picture in
0 ratios (BRs) [1]. A recent QCDF fit to Kπ,ππ rates [2]
dealing with B VP modes.
8 indicatedthateveniftheannihilationcontributionisne- →
0 In this letter we take into account the subleading cor-
glected, one can still get quite good fitting results pro-
3 rections arising from the 3-parton Fock states of final
vided that the strange quark mass is of order 80 MeV.
0 statemesons,asdepictedinFig.1,toQCDFdecaysam-
/ The annihilation effects might be much more impor- plitudes. Wefindthatitcouldgivesignificantcorrections
h tant for VP modes, where P and V denote pseudoscalar
p todecayswithω,orη(′) inthefinalstates. Asimplerule
and vector mesons, respectively. It has been pointed out
- extended to B PP,VP modes is obtained for the ef-
p that in the absence of annihilation effects, the φK BRs fective coefficien→ts aSL with the subleading corrections,
e are 4 10−6 [4] which is too small compared to the i
v:h dωaKta−≈∼ra8t×e×,(61.07−+61.[35, 60]..6)Rec1e0n−t6l,yaBndellωeKob−s/eωrvπe−d a l1a[r5g]e. aS2iL = a2i+[1+(−1)δ3i+δ4i]c2i−1f3/2,
Xi Sizable ωK res−ul1t.s2±are als×o reported in new BaBa∼r mea- aS2iL−1 = a2i−1+(−1)δ3i+δ4ic2if3, (2)
surements [6] with ωK−,0 ωπ− 5 10−6. It is hard
ar to understand the large st∼rength∼of ω×K rates from the wdehfienreedia=tth1e,·s·c·a,le5,µan=d ciΛarme th/2e W1il.s4oGnecVoewffiitchienΛts
theoretical point of view. The ratio ωK0/ωπ− reads h χ B ≃ χ
themomentumoftheemitptedgluonasshowninFig.1(b)
ωK0/ωπ− V /V 2(f /f )2 and
cb ub K π
≈| |
2 √2
×(cid:12)(cid:12)a4−a6rχK +2r2(aa13++ra15a+2 a9/4)+fBfKb3(cid:12)(cid:12) , (1) f3 = m2BfωF1Bπ(m2ω)hωπ−|O1|B−iqqg =0.12 (3)
(cid:12) (cid:12)
(cid:12) (cid:12) inthe SU(3)limit. HereO =sγµ(1 γ )uuγ (1 γ )b,
where(cid:12)r1 = fωF1Bπ/fπAB0ω, r2 = (F1BKfω)/(AB0(cid:12)ωfK), and α is the averaged fr1action of −the5π− mµom−ent5um
rK = 2m2 /[m (m +m )] is the chirally enhanced fac- g
χ K b s u carried by the gluon. For the ωK amplitudes, the term
tor with m being the current quark masses, and b
s,u 3 ≡ a3 + a5, which is originally negligible, is replaced by
b (K,ω) is the annihilation contribution defined in [7].
3 a +a +(c c )f andthe lattergivesthe significantly
Theωπ− ratedependsweaklyontheannihilationeffects. 3 5 4− 6 3
constructive contribution to the rates. It can thus help
Without annihilation, since a and a rK terms in the
4 6 χ understanding the reason for the similarity between Kω
ωK0 amplitude have opposite signs, the ratio ωK0/ωπ− andπ−ω. Onthe other hand, the subleading corrections
shouldbe verysmall. Apossibilitytoexplainthe datais can contribute significantly to the processes with η(′) in
that the annihilation effects may give the dominant con- thefinalstates,forwhichtheterma a alwaysappears
3 5
−
tribution to ωK modes as shown in the QCDF fit [8] for inthedecayamplitudesandbecomesa a +(c +c )f
3 5 4 6 3
−
B PP togetherwithsomeB VP modes. (However, after taking into account the corrections. We also get
→ →
2
Φω Due to G-parity, Φ and Ψ are antisymmetric in inter-
changing α and α for the ω, so that Eq. (6) vanishes.
u¯ u
b
B π In Fig. 1(b), we consider the emitted gluon which be-
comes a parton of the pion. We first take G (vx)
(a) µν
Gµν(0)eivkgπ·x and then adopt the collinear approxima≃-
Φω tionkπ =α p inthefinalstageofthecalculation,where
g g π
α is the averagedfraction of the pion’s momentum car-
b g
B π riedbythegluon. Thecalculationisstraightforwardand
(b) leads to
ωπ− O B−
1 Fig.1(b)
FIG.1: Thecontributionsoftheqq¯g Fockstatesof the(a)ω h | | i
and (b) π− mesons to theB− →ωπ− amplitude. fωmω 1 1
= dv φ (u)
ω
4√2N Z Z
c 0 0
i∂
aSL(J/ψK) 0.27+0.05i which is well consistent with π− d¯γ (1 γ )g G bB− Tr ǫ∗
2 ≈ ×h | µ − 5 s νβ | i∂kπ (cid:26) (cid:20)6 ω
the data. The result resolves the long-standing sign am- gβ
e
bigLueittyuosf sRteu(day2).the subleading corrections originating γν(v 6kgπ +u6pω)γµ γµ(v 6kgπ +u¯6pω)γν
×(cid:18) (vkπ +up )2 − (vkπ+u¯p )2 (cid:19)(cid:21)(cid:27)
fromthe3-partonFockstatesoffinal-statemesons. Tak- g ω g ω
ing the ωπ− mode as an illustration, there are two dif- = 2√2fωpα π− d¯γµγ g G bB− , (7)
ferent types of diagrams shown in Fig. 1. In the follow- ∼−α m2 ωh | 5 s αµ | i
g B
ing calculation, we adopt the conventions Dα = ∂α + where the ω mesons’s asympteotic leading-twist distri-
ig TaAa, G = (1/2)ǫ Gµν, ǫ0123 = 1, and use
s α αβ αβµν − bution amplitude φω(u) = 6uu¯ has been taken and
the Fock-Schwinger gauge to ensure the gauge-invariant
e u¯ = 1 u. We have two unknown parameters
nature of the results pα π− d¯γµ−γ g G bB− and α needed to be deter-
ωh | 5 s αµ | i g
1 mined. First, let us evaluate pα π− d¯γµγ g G bB− .
A (x)= dv vG (vx)xν. (4) e ωh | 5 s αµ | i
µ −Z µν Thematrixelementcanbecalculatedbyconsideringthe
0 e
correlationfunction:
For Fig. 1(a) where the contributions come from the
3-parton Fock states of the ω, because of the V A Πα(p,p+q)=i d4xeipx π−(q)T(j3p(x)jB(0))0
− Z h | | i
structure of the weak interaction vertex, the relevant 3-
m2f 1
parton light-cone distribution amplitudes (LCDA) up to = B B
the twist-4 level are given by [11] mb m2B−(p+q)2
π− d¯γµγ g G bB−(q+p) + , (8)
5 s αµ
ω(p ,λ)u¯(0)γ g G (vx)u(0)0 ×h | | i ···
ω µ s αβ
h f m2| | i where j3p = d¯gseGαµγµγ5b,jB = ¯biγ5u, the ellipses
∼=i ω√2ω Z Dαeipωxvαg(cid:26) pωβgαµ−pωαgβµ Φ(αi) denote contributiones from the higher resonance states,
(cid:0) (cid:1) which can couple to the current j , and the transition
B
1
+ pω(pωx pωx )(Ψ(α ) Φ(α )) , (5) matrix element can be parametrized as
(p x) µ β α− α β i − i (cid:27)
ω π−(q)d¯γµγ g G bB−(p+q)
5 s αµ
h | | i
where Dα = dαu¯dαudαgδ(1 − αu¯ − αu − αg), with =pαf−(p2)+(peα+2qα)f+(p2). (9)
α ,α ,α beingthefractionsoftheωmomentumcarried
u¯ u g
In the deep Euclidean region of (p +q)2, as depicted
by the u¯-quark, u-quark and gluon, respectively. Here Φ
in Fig. 2 the correlation function can be perturbatively
and Ψ are the twist-4 LCDAs. Note that all the compo-
calculated in QCD and expressed in terms of 3-parton
nents of the coordinate x should be taken into account
LCDAs of the pion,
in the calculation before the collinear approximation is
applied. The exponential in Eq. (5) before the collinear 1 du u
approximationisactuallyeikg·xv,wherekg isthegluon’s ΠQαCD = qαZ m2 (p+uq)2 Z dαg
momentum, and the resultant calculation can be eas- 0 b − 0
ily performed in the momentum space with substituting [ 2(p q)f3πφ3π +fπmb(φ˜k 2φ˜⊥)],(10)
× − · −
x (i/v)(∂/∂kα). The result of Fig. 1(a) is found to
α →− g where u = αd +αg, and the 3-parton pion LCDAs are
be
defined by [12, 13]
4√2m2 π(q)d¯(x)g G (vx)σ γ u(0)0
hωπ−|O1|B−iFig.1(a) =fω 3m2 ω h= i|f [q s(qµgν qαβg 5) q| i(q g q g )]
B 3π β µ να ν µα α µ νβ ν µβ
− − −
2Φ(α ) Ψ(α )
×hπ−|d¯6pω(1−γ5)b|B−iZ Dα iα−g i .(6) ×Z Dαφ3πeiqx(αd+vαg), (11)
3
0.07 u 00..88
00..66
00..44
(cid:2) 00..22
2 0.06
Φπ Ω 1.5
m
A 1
u d (cid:1)(cid:3)0.05 f(cid:2)0.5
f
0
00
b 0.04 00..22
p+q p 5 10M1522(cid:1)0G2e5V320(cid:2)35 40 00..ΑΑ44g00..66 0.8 1.
FIG.2: Thediagrammaticillustrationtothecorrelationfunc- FIG. 3: (a) Form factor f−(m2ω) plotted as a function of the
tion, Eq. (8). BorelmasssquaredM2. (b)f−(m2ω)= 01du 0udαgAf− with
M2=10 GeV2. The volume in the ploRt is eqRual to f−(m2ω).
Hereu=αd+αg.
π(q)d¯(x)γ g G˜ (vx)u(0)0 =if (q g q g )
µ s αβ π α βµ β αµ
h | | i −
q
×Z Dαφ˜⊥eiqx(αd+vαg)−ifπqxµ(qαxβ −qβxα) f(9± p2r0e)dGicetVio2n. Tbyheardeosputltinisgdse0pi≃cted37inGFeiVg.23(aan)d. TMhe2r≈e-
×Z Dα(φ˜k+φ˜⊥)eiqx(αd+vαg). (12) sthu−eltiunngcveartluaeinitsyfc∓o(mme2ωs)f=rom±(0th.0e57su±m0.r0u0l5e)aGneaVly2s,isw.hWeree
then get
Here φ is a twist-3 DA, φ˜ and φ˜ are all of twist-4,
3π ⊥ k
pα π− d¯γµγ g G bB−
1 ωh | 5 s αµ | i
φ3π(αi) = 360αdαu¯α2gh1+ω1,02(7αg−3) =−f−(m2ω)×(em2B −m2π−m2ω)≃−1.6 GeV4. (16)
+ω2,0(2−4αdαu¯−8αg+8α2g) Next, we determine the value of αg. For illus-
tration, we plot in Fig. 3(b) the amplitude A of
+ω (3α α 2α +3α2) , f−
1,1 d u¯− g g i the f−(m2ω) sum rule versus αg and u(= αd + αg)
1 by adopting M2=10 GeV2, where A satisfies f =
φ˜ (α ) = 30δ2α2(1 α )[ +2ε(1 2α )] , f− −
⊥ i g − g 3 − g 1du udα A , i.e. the volume in the plot is equal to
0 0 g f−
φ˜k(αi) = −120δ2αdαu¯αg[31 +ε(1−3αg)] . (13) rfRe−g(imon2ωR)w.hTerheeure&sul6ta0n%t,fαorm.fa3c0t%or. isTdhoemaivneartaegdedbyfrtahce-
g
tion of the pion momentum carried by the gluon is then
Since the quark’s momentum after emitting the gluon is
1 u
estimated to be α =( du dα α A )/f 0.23.
roughly of order mB/2 and the emitted gluon’s momen- g 0 0 g g f− − ≃
ttuhme scisalΛeχfo∼r thαegpsπep(aαrgatwioinllobfethdeispcuerstsuedrbbateilvoewa),ndwenosnet- anWd ef3the=refo0re.12obwtahiinRchhωgπiRv−e|sO1t|hBe−icFoirgr.1e(cbt)ion≃to0.1a3i
perturbative parts at µh = ΛχmB/2 1.4 GeV. The as defined in Eq. (2). We list ai without and
corresponding parameters apt the scale ≃µh read: f3π = with the subleading corrections in Table I, where
δ02.00=320G.1e9VG2,eVω12,,0ε==−20..6435,[ω123,]0. =To9.6c2a,lcωu1la,1te=f±−,1.t0h5e, tphαωehπa−p|dp¯γroµxγi5mgasGtioαnµb|−B√−2i/ipFαπ1Bhωπ|u¯hγaµsgsbGeeeαnµb|mBa−die/.AB0Nωot≃e
contributionsofhigherresonancesin Eq.(8)areapprox- that a and a do not receive subleading corrections.
6 8
e
imated by Inthefollowinganalysis,theLCsumruleformfactors
andm =80MeVareused. Wewillinsteaduseasmaller
1 ∞ ImΠQαCD ds, (14) AB0ρ =s0.28whichispreferredbytheωπ−data. Theωπ−
π Zs0 s−(p+q)2 mode is ideal for extracting AB0ρ since its rate is insen-
sitive to annihilation effects. We find that the spectator
wEqhse.re(8s)0aisndth(e10th)raenshdomldaokfinhgigthheerBreosroenlatnracenss.foErmquaattiionng: parameter XH = lnmΛhB(1+ρHeiφH) is consistent with
zero in the analysis. The reasonis that since the specta-
[m2 (p + q)2]−1 = exp( m2/M2), we obtain the
B B − − B tor interaction with a gluon exchange between the emit-
light-cone sum rule
tedmesonandtherecoiledpseudoscalarmesonoftwist-3
LCDA Φ is end-point divergent in the collinear expan-
f−(p2)= 2mm2bf Z 1duZ udαg e(mM2B2−mu2b−Mu¯2p2) sion, theσvertex of the gluon and spectator quark should
B B 0 ∆ be consideredinside the pion wave function, i.e., for this
m2 p2 m situation the pion itself is at a 3-parton Fock state. An-
f b − φ f b(φ˜ 2φ˜ ) , (15)
×(cid:20) 3π u2 3π − π u k− ⊥ (cid:21) nihilationeffectshavebeenemphasizedinφK studies[4].
We adopt the annihilation parameters ρ 0.9,φ 0
A A
≃ ≃
andf (p2)= f (p2),where∆=u (m2 p2)/(s p2). which give Br(B− φK−) 8.5 10−6, consistent
+ − − − b− 0− → ≃ ×
Using the above parameters for LCDAs, m = (4.7 with the current data. Here the annihilation parameter
b
0.1) GeV, and fB = 180 MeV, we obtain the stab±le ofVP modesis definedasXAVP =lnmΛhB(1+ρAeiφA)[1]
4
TABLE I: Values for ai for charmless B decay processes without (first row) and with (second row) 3-parton Fock state
contributions of final state mesons, where a3−10 are in units of 10−4 and theannihilation effects are not included.
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
1.02+0.014i 0.10−0.08i 26+26i −328−91i 1.2−30i −487−72i 0.7+0.3i 4.5+0.6i −89−0.1i −5.9+7i
0.974+0.014i 0.25−0.08i −55+26i −291−91i 112−30i −487−72i −0.3+0.3i 4.5+0.6i −88−0.1i −18+7i
(cid:2)K10. Η(cid:6)75. result for aS2L is well consistent with that extracted from
Ω (cid:1)a(cid:2) (cid:5)K (cid:2)b(cid:6) data. This solves the long-standing sign ambiguity of
(cid:5)Π, ’,Η50. a2(J/ψK)whichturnsouttobepositiveforitsrealpart.
Ω 5. K Note that if Re(a ) were negative, f would have to be
(cid:2) (cid:3)25. 2 3
0.3 which in turns would lead to φK 20 10−6
B B ∼ − ∼ ×
(cid:1)Br 0 Br(cid:2) 000 and ωπ−,ωK ∼1×10−6!
0o 90o 180o 270o 360o 0o 90o 180o 270o 360o The subleading corrections could give significant con-
Γ Γ
tributions to the decays with η(′) in the final states be-
cause these decay amplitudes always contain the singlet
FIG. 4: (a) Dash, solid, and dot-dash for B →
factor a a . We plot the BRs of Kη′,K∗η modes
Kωπ−−η,′ω,KK0−η′a,Knd∗−ωηK,0a;n(dbK) s∗o0lηid.,Bdrostas,redoint-udnaisths,oafn1d0−d6a.sh for versus γ3i−n F5ig. 4(b), where XAPP ≈ 0, the annihila-
tion parameter for PP modes, has been used as it could
give good fit results for Kπ,ππ rates [2]. We do not
Ρ(cid:5)15. (cid:2)a(cid:5) (cid:7)(cid:2)(cid:9)(cid:10)Η(cid:10)15. (cid:2)b(cid:10) mcoondsiedserbetchaeusseinigtleits astninllihhialartdiotnocdoertreercmtioinne[a1t5]pirnesKenηt′.
K(cid:3)10. ,ΠΡ10. With (without) the subleading corrections, we see that
BrB(cid:2) 5. (cid:4)K(cid:3) 5. Kin−uηn′it&s oKf 10η0−′ 6≈. T55h(e3c5o),rraenctdioKns∗−giηve&70K%∗0aηnd≈2254%(2e0n)-,
B
00o 90o 180o 270o 360o Br(cid:2) 00o 90o 180o 270o 360o thhaantcewmitehnts(wtiothKouηt′)atnhde Kco∗rηrecrtaitoenss,,rKes∗pηe′ct.ive1ly. N10o−te6
Γ Γ ×
(&4 10−6)andKη.1 10−6 (&1 10−6). InpQCD
× × ×
FIG. 5: (a) Solid, dash, dot-dash, and dots for B → calculation[16],itseemsthatK0η′(=41 10−6)wasun-
×
K0ρ0,K−ρ+,K0ρ−,andK−ρ0;(b)solid,dot-dash,dash,and derestimatedwhileK0η(=7 10−6)overestimatedcom-
dotsfor B→K∗−π+,K∗0π0,ρ−η and ρ−η′. Brs are in units pared to the data[5, 6]. Wit×hin the QCDF framework,
of 10−6. by only considering the two-parton LCDAs of mesons,
BenekeandNeubert[15]obtainedK∗η 13 10−6,just
∼ ×
half of the experimental value, but with a huge error.
and its the imaginary part is neglected since the BRs
Forfurthercomparisonwithothercalculations,weplot
are insensitive to it. In Fig. 4(a) we plot the BRs of Kρ,K∗π,ρ−η(′) modes in Fig. 5. The subleading con-
γωπ≈−(a6n0d−ω1K20)m◦oadreesinvegrosoudsaγg(r≡eemaregnVtu∗wb)i.thTdhaetrae.suAlttsγfo=r thraivbeutKio0nρs0,tKo −thρe+s,eKB0Rρ−s,aKre−ρ.0 =156%,.7,7A,t2(γ =10−960)◦,,awnde
90◦, it gives ωπ−,ωK−,ωK0 to be 5.5,4.5,4.3, respec- K∗−π+,K∗0π0,ρ−η,ρ−η′ = 9,2,6,4 ( 10−×6). In [7],
tively, in units of 10−6. Without the contributions from ×
to fit φK,ωK rates, the annihilation effects dominate
3-parton Fock states of mesons, φK−,ωπ−,ωK−,ωK0 the decay amplitudes and the form factors FBK,ABπ
will become 11,3.9,3.1,2.9 (in units of 10−6). The 3- are rather small, such that it will lead to ρ1+K−,00 :
parton Fock state effects give constructive contributions ρ0K−,0 : ωK−,0 1 : (1/√2)2 : (1/√2)2, while the
to ωπ,ωK modes, but destructive one to the φK mode. pQCD results for r≈ates are K0ρ0,K−ρ+,K0ρ−,K−ρ0 =
The corrections from 3-partonFock states of the kaon
2.5,5.4,3.0,2.2( 10−6) [9].
also give a definite answer to the longstanding problem ×
for a (J/ψK). In the earlier study, to account for the In conclusion, we have calculated the contributions
2
experimental value a , the parameter ρ has to be & arising from 3-parton Fock states of mesons in B de-
2 H
1.5 [14]. As emphas|ize|d in passing, without fine-tuning cays. We find that the contributions could give signif-
ρ , we calculate the amplitudes from the 3-parton Fock icant corrections to decays with an ω or η(′) in the fi-
H
states of the kaon. With the same procedure as shown nal states. Our main results for γ (60 110)◦ are
≈ −
above,we obtain f (m2 ) 0.08, f 0.14and thus ωπ−,ωK−,ωK0 6.0,(6 5),5.1, respectively, in units
∓ J/ψ ≃± 3 ≃ ≈ ∼
aSL =at2+c (µ )f 0.10+0.05i+c (µ )0.14=0.27+ of10−6,whilethepreviousQCDFglobalfittoVP modes
0.205i, w2here1at2his3d≃etermined up to1thhe twist-2 order and the pQCD results gave smaller ωK BRs of order
andtheSU(3)2approximationforf hasbeenmade. The . 3 10−6 [9, 10]. We predict that K0ρ0 ωK and
3
× ∼
5
K0ρ0/K−ρ0 3. Including the corrections we obtain Kη′ to the level above 5 10−5.
≈ ×
a (J/ψK) 0.27+0.05i which is well consistent with
2
≈
the data. The sign of Re(a ) turns out to be positive. This work was supported in part by the grant from
2
Without resorting to the unknown singlet annihilation NSC91-2112-M-033-013. IamgratefultoH.Y.Chengfor
effects, 3-parton Fock state contributions can enhance acriticalreadingofthemanuscriptandusefulcomments.
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