Factorization and the Soft-Collinear Effective Theory: Color-Suppressed Decays

0.11 MB·English

by  Sonny Mantry

Factorization and the Soft-Collinear Effective Theory:
Color-Suppressed Decays
Sonny Mantry , Dan Pirjol and Iain W. Stewart1
∗ ∗ ∗
4 ∗CenterforTheoreticalPhysics,LaboratoryforNuclearScienceandDepartmentofPhysics,
0 MassachusettsInstituteforTechnology,Cambridge,MA02139
0
2
Abstract. Wediscussthesoft-collineareffectivetheory(SCET)andkinematicexpansionsinB-decays,focusingonrecent
n
a resultsforcolorsuppressedB D(∗)X decays.InparticularwediscussmodelindependentpredictionsforB¯0 D0p 0and
J B¯0 D∗0p 0,andupdatetheco→mparisonusingnewexperimentaldata.WeshowwhyHQETaloneisinsufficient→togivethese
resu→lts.SCETpredictionsarealsoreviewedforotherBandL decaychannelsthatarenotyettestedbydata.
2 b
1

1 INTRODUCTION cess of disentangling the interactions of hard-collinear-
v softparticlesisknownasfactorization,andissimplified
8
The soft-collinear effective theory [1, 2, 3, 4] (SCET) bytheSCETframework.
5
provides a formalism for systematically investigating Much like any effective theory the basic ingredients
0
1 processes with both energetic and soft hadrons based ofSCETareitsfieldcontent,powercounting,andsym-
0 solely on the underlying structure of QCD. Essentially metries. The Lagrangian and operators, are organized
4 all known methods for simplifying QCD predictions, in a series where only L(0) and O(0) are relevant at
0 without introducing model dependent assumptions, de- LO, an additional L(1) or O(1) is needed at NLO, etc.
h/ pend on exploiting hierarchies of mass scales. For pre- The expansion parameter will be l = L QCD/Q or
p dictions based on SU(3) symmetry we exploit the fact h =L QCD/Q dependingonwhetherthecpollinearfields
- thatm /L 1, andexpectcorrectionsatthe 30% describeanenergeticjetofhadronsoranindividualen-
p u,d,s ≪ ∼
level. In lattice QCD simulations we choose our lattice ergetichadron.Theeffectivetheorywithanexpansionin
e
h spacinga 1/L andvolumeV 1/L 3 sothatwecan l iscalledSCETI,whiletheonewithanexpansioninh
: focusonn≪on-perturbativeeffects≫atscales L .InSCET is called SCET . In processessuch as color-suppressed
v weexpandinL /Q 1,withthelargemo∼mentumofan decays the sepaIIration of scales is Q2 QL L 2 and
i
X energetichadronor≪jetbeing Q.ForBdecayscorrec- thechainQCD–SCET–SCET proves≫tobeu≫seful.The
I II
∼
r tions will be at the 20–30% level depending on the intermediatetheorySCETI providesthedynamicstore-
a ∼
energyscaleQ. arrange soft and collinear quark lines so that they can
Most effective theories that we are familiar with are end up in soft and energetic hadrons. The final theory
designed to separate the physics for hard p2 Q2 and SCET describestheuniversallowenergyhadronicma-
h ≃ II
soft p2 Q2 momenta. Examples include the elec- trix elements. In the case of color-suppressed decays
troweask≪Hamiltonian, chiral perturbation theory, heavy B D(∗)M these are light-cone distribution functions
quark effective theory, and non-relativistic QCD. In f M→(x)whereM=p ,r ,K,orK∗andtwogeneralizedpar-
SCET we incorporate an additional possibility, namely tondistributionfunctionsS(0,8)(k+,k+)fortheB D( )
1 2 → ∗
energetichadronswheretheconstituentshavemomenta transition.
m m
p nearly collinear to a light-like direction n . Both
c
the energetic hadron and its collinear constituents have
n¯ pc Q,wherewehavemadeuseoflight-conecoordi- COLOR-SUPPRESSED DECAYS AND
snta·ittuese∼n(tps+cs,tipl−cl ,hpav⊥ce)=sm(anl·lpocf,fn¯s·hpecl,lnpe⊥cs)s.pT2hecopl2l.inTehaercporno-- SCET
c ∼ s
Color-suppressed decays were investigated in Ref. [5]
using SCET. For B Dp decays the four quark oper-
1 PlenarytalkgivenbyI.S.atthe9’thInternationalConferenceonB →
PhysicsatHadronMachines(Beauty,2003).MIT-CTP3462
p b c
u d B D c D (a) b c (b)
u b b c
u,d u u( s )
b c d,u d B u,d d( s )
B D p d p d d u( s )
u
u,d d u
FIGURE1. B Dp flavortopologies, T,C,andErespec- FIGURE 2. Diagramsin SCET for treelevel matching.
I
→ ⊗
tively.Forthecolor-suppresseddecaysonlyCandEcontribute. denotestheoperatorQ(0,8)andthedotsareinsertionsofL(1).
j x q
Thesolidlinesanddoublesolidlinescarrymomenta pm L
∼
andformtheBandD.Thedashedlinesareenergeticcollinear
atorswhichcontributeare[G =V A] quarksthatformthelightmesonM.
−
G V V
HW = F cb u∗d C1(c¯b)G (d¯u)G +C2(c¯ibj)G (d¯jui)G , (1)
√2 from 1 are explained by an SCET power correction.
(cid:2) (cid:3)
Other mechanisms for testing factorization for color-
with flavor contractions shown by the Fig. 1 diagrams.
allowed decays include using multibody states to make
For the amplitudes we use A = A(B¯0 D+p ),
A =A(B D0p ),andA +=−A(B¯0 D→0p 0).W−rit- testsasafunctionofq2orwmax [12],lookingfordecays
te0n−interms−o→fisospi−namplitu0d0es → which do not occur in naive-factorization [13], or tests
usinginclusiveB D( )Xspectraortheequalityofrates
∗
→
forparticularmultibodyfinalstatesX [14].
1 2
A = T+E= A + A , Thecolor-suppressedamplitudeA hascontributions
+− √3 3/2 r3 1/2 fromC and E, but not T. With larg0e0N very little can
c
A = T+C=√3A , (2) besaidabouttheCandE contributions,besidesthefact
0 3/2
− that we expect A <A A . In SCET the ampli-
A00 = C√−2E =r23A3/2−√13A1/2. tDuedsepsitCe tahnisdpEowae0r0er ssuupppp+rr−eesss∼sieodn,0b−pyreLd/icEtipvereploatwiveer itsorTe-.
tained since only a single type of SCET time ordered
TheamplitudesfordecaystoB D( )r aredefinedina I
→ ∗ product contributes to give the proper quark rearrange-
similarfashion.
ment, T(Q(0,8)(0),iL(1)(x),iL(1)(y)) [5]. This combi-
In the large Nc limitC/T E/T 1/Nc (where we j x q x q
∼ ∼ nation contributes to both C and E as shown in Fig. 2.
takeC 1 andC 1/N ). The color-allowed ampli-
1 2 c
∼ ∼
tudesA andA aredescribedbyafactorizationthe-
+ 0
orem[6,−7,8],pro−venwithSCET[9] WhenmatchedontoSCETII thetime-orderedproduct
(0,8)
givesaproductofsoftO andcollinearO operators.
s c
A(∗)=N(∗)x (wmax)Z01dxT(∗)(x,mc/mb)f p (x)+..., (3) sTohfutsohpDer(a∗t)opr|O[Ps(L0,8=)O(1c|Big=5)/hD2](∗)|Os(0,8)|Bihp |Oc|0i.The
−
where x (wmax) is the Isgur-Wise function at maximum O(0) = (h¯(c)S)n/P (S†h(b))(d¯S) n/P (S†u) , (5)
recoil, f p (x) is the light-cone distribution function for s v′ L v k1+ L k2+
thepion,T =1+O(a s)isthehardscatteringkernel,and whileO(8) isidenticalbutwithcolorstructureTA TA.
N(∗)= G√F2VcbVu∗dEp fp √mD(∗)mB(1+mB/mD(∗)).Theel- Inadditisonthereareoperatorsencoding“long”dis⊗tance
lipsesinEq.(3)denotetermssuppressedbyL /Qwhere contributionsinSCET thatarethesameorderinL /Q.
II
Q= mb,mc,Ep .Intheheavyquarklimit,Eq.(3)pre- These come from the region of momentum space for
dicts{A=A∗, so}Br(B¯0 D+p −)=Br(B¯0 D∗+p −) Fig. 2wherethe gluonstill has p2 QL , butthe quark
and Br(B− D0p −)=→Br(B− D∗0p −). T→his agrees propagatorhas p2 L 2. ∼
wellwithth→eexperimentalresult→s[10,11],whichyield Usingheavyqua∼rksymmetryonecanprove
Br(B¯0→D∗+p −) = 1.03 0.14, hD(∗)|Os(0,8)|Bi=SL(0,8)(k1+,k2+), (6)
Br(B¯0 D+p ) ±
→ − so that the matrix elements for B¯0 D0p 0 and B¯0
Br(B−→D∗0p −) = 0.93 0.11. (4) D∗0p 0 arethesame[5].Furthermore→,SL(0,8) arecompl→ex
Br(B D0p ) ± from their dependence on nm , the direction of the light
−→ −
meson,andencodeanon-perturbativestrongphaseshift.
Eq.(3)alsopredictsA =A ,howeverexperimentally
+ 0 Thisleadstothepredictions
A /A = 0.77 0−.05 for−Dp and 0.81 0.05 for
0 +
D| ∗p−. We−w| ill see t±hat the reduction of thes±e numbers d (D∗p ) = d (Dp ), (7)
0.8 =Tr[H¯v(c)g m PLHv(b)Xm ] (10)
p ′
0.6 == DD*p +m1cTr[H¯v(′c)is ab 1+2v/′g m PLHv(b)Rmab ]+...
where Hv and Hv are HQET superfields and Xm and
′
0.4 Rmab are the most general tensor functions compati-
R 3A00 ble with the symmetries of QCD. Here the Rmab term
0.2 I 2A0_ hoanstahechcrhoamrmomqauganrekt.icUospuearlalytoirninHseQrtEioTn,thh¯ev′sRabmab Gabterhmv′
d f would be suppressed relative to the Xm term. However,
m m
0 in Eq. (10) the pion momentum pp = Ep n is an al-
0 0.2 0.4 0.6 0.8 1 lowed four-vectorin Rmab . Since Ep /mc 1.5 the two
≃
termsarethesamesize(andthiswillalsobethecasefor
FIGURE3. ExperimentaldatafortheDp andD p isospin all other terms in the 1/m heavy quark expansion, ie.
∗ c
triangles.ThisfigureupdatestheoneinRef.[5]toincludethe the expansion does not converge). Since Ep 2.3GeV
recentBaBardata. the “soft” gluons are carrying hard momen≃ta. Terms
like Rmab break the heavy quark spin symmetry and
Br(B¯0→D∗0p 0) = Br(B¯0→D0p 0), gSiCvEeTA(wB¯e0c→anDex∗0ppan0)d6=inAL (/B¯E0p→anDd0fpac0t)o.rIinzecaowntaryastth,eweinth-
where d = arg(A A ) is the strong phase shift be- ergetic pion. Thus the matrix elementin Eq. (6) has no
tweenisospinamp1l/i2tud∗3e/s2.ThepredictionsinEq.(7)have Ep dependenceandispartofaconvergentexpansion.
corrections at O(a (Q)) and O(L /Q). For M =p 0,r 0 The SCET analysis also gives predictionsfor several
s
thelongdistanceamplitudeissuppressedbya (Q).The channelswherethedataisnotyetavailable.Forinstance,
s theanalysisabovealsoappliesforther ,predicting
current experimental data [15, 16, 17] gives the world
averages[branchingratiosbelowareinunitsof10−3] d (D∗r ) = d (Dr ), (11)
Br(D0p 0)=0.29 0.03, d (Dp )=30.4 4.8◦, (8) Br(B¯0→D∗0r 0) = Br(B¯0→D0r 0).
± ±
Br(D∗0p 0)=0.26±0.05, d (D∗p )=31.0±5.0◦, AsimilarpredictioncanbemadefordecaystoD(s∗)K(∗)
except in this case the long distance contributions to
showinggoodagreementwithEq.(7).Iffurtherdatain-
theamplitudesarenotsuppressed.Thismeansthatboth
dicatesagreementoftheanglesbeyondthecurrent17%
longitudinalandperpendicularpolarizationsoccuratthe
level then this would be an indication that L /Q correc-
sameorder.TheanalogofEq.(11)istherefore:
tionstoEq.(7)aresmallerthanexpected(orperhapsab-
sent),andthesameappliesfortheratiosinEq.(4).The Br(B¯0 D K ) = Br(B¯0 D K ), (12)
→ ∗s − → s −
agreementcanalsobeshowngraphically.Theisospinre- Br(B¯0 D K¯ ) = Br(B¯0 D K¯ ),
lationbetweenamplitudesimpliesthat → ∗s ∗− → s ∗−
k k
where these color-suppressed decays are not part of an
3 A
1=R + 00 , (9) isospintriangle.Cabbibosuppresseddecaystokaonsare
I
√2A0 moreanalogoustoDp andDr ,exceptthattheyalsohave
−
longdistancecontributionswhicharenotsuppressed.In
where R = A /(√2A ) = (A A /√2)/A .
Eq. (9) cIan be1r/e2presente3d/2by a tria+n−gl−e in00the comp0le−x thiscasetheanalogofEq.(11)is
plane. The current world averages for Dp and D∗p are d (D∗K¯0) = d (DK¯0), (13)
shown in Fig. 3, where the overlap of the 1-s regions Br(B¯0 D 0K¯0) = Br(B¯0 D0K¯0),
∗
indicatestheagreement. → →
d (D K¯ 0) = d (DK¯ 0),
ItisusefultonotethatEq.(7)providesasensitivetest ∗ ∗ ∗
k k
ofSCET-factorization,andnotjustheavyquarksymme- Br(B¯0 D 0K¯ 0) = Br(B¯0 D0K¯ 0).
∗ ∗ ∗
try.Thebasicreasonisthatapriori“soft”gluonexchange → k → k
betweentheborcandthelightquarksinthepionspoils ThepredictionsinEqs.(11),(12),and(13)willbetested
theprediction.Toseethismoreclearlywecanconsider once data on B¯0 → D∗0r 0, B¯0 →D∗sK(∗)−, and B¯0 →
usingjust HQETwith fullQCD forthe lightquarks.In D 0K¯( )0 becomeavailable.Thesignificanceofthelong
∗ ∗
thiscasetheamplitudewouldbe distance terms will be tested by comparing Br(B¯0
D K¯ ) to Br(B¯0 D K¯ ) or Br(B¯0 D 0K¯ 0)→to
hD(∗)0p 0|(h¯v(c′)g m PLh(vb))(d¯gm PLu)|B0i/√mBmD Br∗sk(B¯0k∗−→D∗0K¯∗0).→ ∗s⊥ ⊥∗− → ∗k k∗
⊥ ⊥
The full factorization theorem for color suppressed p u,d
decaystakestheform[5] u d L b b cLScc,
d,u d
AD(∗)M = NM dxdzdk+dk+T(i) (z)J(i)(z,x,k+,k+) b c u
00 0 Z 1 2 L∓R 1 2 L b d d L c p
u u
×S(i)(k1+,k2+)f M(x)+ADlo(n∗g)M. (14) L c,
GwFheVrcebVuT∗dL(∓fi)MR√amreBmhDa(rd)/2s.caTthteerinnogn-kpeerrntuerlbsatainveddNyn0Mam=- L b db cqqS c LLbb d ub dc qqLScc,
ics is contained in f ∗, the light-cone distribution func- u d
M p p
tionformesonM,andS(i),i=0,8,ageneralizedparton
distributionfunctionfortheB D( ) transitionwithk+
andk+ beingmomentumfract→ionso∗fthelightspectato1r FIGURE4. Classesof diagramsforL b decays, giving am-
2 plitudesT (tree) andC (color-commensurate) forthetop two
quarks.Finally,thejetfunctionJ(i)issensitivetophysics
diagrams, and E (exchange), and B (bow-tie) for the bottom
at the m 2 EML scale and is responsible for the quark two.Bow-tiediagramsareuniquetobaryondecays[24].
∼
rearrangement.
Thepredictionsdiscussedaboveareallvalidindepen-
dentoftheformofJ(i),meaningtoallordersina s(m 0) or in other words that the Dp and Dr triangles (as in
patowtheersinoteframse(dmi0a)tethsecnaleth,ims02in∼troEdMuLce.sIfadwdeitieoxnpaalnudnJceirn- Fcaigse. 3th)ewnililtbweosuilmdilianrd.iIcfattehitshatutrnthseroeutarneotsutobsbtaenttihael
tainty, butgivesfurtherpredictions.At lowest orderwe a 2(m ) corrections to J(0,8). This would mean that the
s 0
have subset of predictions that follow from Eq. (15), which
AD00(∗)M = N0MCL(0)16pa 9s(m 0)seff(m 0)hx−1iM, (15) bdeeptreunsdteodn.Parpeedritcutriobnatsivfoerecxoplaonrs-siounppforersJs(e0d,8d),eschaoyusludsninogt
othermethodshavebeendiscussedinRefs.[23,18,19,
where CL(0) = C1 +C2/3, hx−1iM = dx/x f M(x), 20,21,22].
and s = s(0) + C /(4C + 4/3CR )s(8) with
eff 2 1 2
−
s(0,8) = dk+dk+/(k+k+)S(0,8)(k+,k+). Correctionsto
1 2 1 2 1 2
Eq. (15)Rare O(a s(m 0)) and O(L /Q). At this order the BARYONDECAYS
strong phase f in Fig. 3 is generated by s and so is
eff
independentofwhetherM=p orM=r .Therefore,we Recently, the authors in Ref. [24] have used SCET to
predictthatf isuniversalforD( )p andD( )r . make model-independent factorization predictions for
∗ ∗
For the ratio of charged amplitudes Eq. (15) can be baryon decays. The main results for L L p , L r ,
b c c
→
usedtopredicttheleadingpowercorrection, S (c∗)p , S (c∗)r , and X c(′,∗)K are briefly summarized here.
RD(∗)M = A(B¯0→D(∗)+M−) (16) ThTehneoetaleticotnroiwseS ack=HSacm(2il4to5n5i)ananfdorS t∗che=seS bc(a2ry5o2n0)d.ecays
c A(B D( )0M )
−→ ∗ − is in Eq. (1). The diagrams for the flavor contractions
= 1−91(m6paB+smmDD((∗)))hxx−(w1i0M) EseMff . dTihffeedrefcraoymsttoheL mc egseotncodnetcriabyustiaonndsfarroemshTo,wCn,iEn,Faingd.B4.,
Avalueseff ≃(430MeV)ei44◦ give∗s|RDcp |≃0.8,fitting ddeeccaayyssttooXS c(c∗o)nhlayvferocmontthriebuEtiaonndsBfroammpCli,tuEd,easn.dB,and
theexperimentalvaluesgivenbelowEq.(4)withparam- InthelargeN limitC/T E/T N0,whileB/T
etersofnaturalsize.Innaivefactorizationthecorrection N .Thus,evenL c L p de∼caysdo∼notcfactorizeinth∼e
c b c
terminR woulddependonthedecayconstant f ,how- →
c M large N limit. The extra factors of N arise from the
c c
everforthetruefactorizationtheoremthatgivesEq.(16) choice of which of the N quarks in the L and/or L
c b c
thisturnsoutnottobethecase.Theobservedsimilarity
participateintheweakinteraction.
ibnegtwhexe−n1iRp c≃fohrx−D1pir aannddDisr nocatnspboeileedxpblayintheedfbaycthtahva-t SCEExTpaonndeinfigndisnCL/T/Q wEh/eTre QL /=Q{amnbd,mB/c,TE<p }L 2u/sQin2g.
fr /fp ≃1.6.Experimentally[11] For L b L cp the le∼ading o∼rder result is fro∼m T and
→
RDp gives
| c | = 0.96 0.13. (17)
Withthisapprox|RimDcra|teequalityan±dthef p =f r predic- AL cp = NLz (wmax)Z01dxTL(x,mc/mb)f p (x) (18)
tion,wewouldexpectthatthestrongphased Dr d Dp ,
≃
+ NRz (wmax)Z01dxTR(x,mc/mb)f p (x)+..., tB¯h0ep iDsr(∗e)p0lpac0ewdebyupadratoedrkthaoenexwpeerreimalesnotdalisccoumsspeadr.iFsoonr
→
inFig.3andEq.(8)totakeintoaccountthenewBaBar
where z (w ) is the L L Isgur-Wise function at
max b c results[17].
→
maximum recoil, T are hard scattering kernels, and
L,R
NL,R=√2GFVcbVu∗dEp fp √mL cmL b u¯(v′)n/PL,Ru(v)with ACKNOWLEDGMENTS
u¯(v)u(v)=2 and the states normalized as in the PDG.
The other factors are the analogs of those in Eq. (3).
I.S. would like to thank A. Leibovich, Z. Ligeti, and
Thevalueofz (w )willbedeterminedbyq2spectrum
max M.Wiseforcollaborationonthebaryonresultsdiscussed
measurements of L L ℓn¯ which are not yet avail-
able.Ifz (w )issbim→ilarctoxℓ(w )thenEq.(18)pre- here.ThisworkwassupportedinpartbytheDepartment
max max ofEnergyunderthecooperativeresearchagreementDF-
dictsthatBr(L L p ) 2Br(B¯0 D+p )inagree-
b→ c ∼ → − FC02-94ER40818. I.S. was also supported by a DOE
mentwiththemeasurementsfromCDF[25].
OutstandingJuniorInvestigatoraward.
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→ →