$B\to K^*\ell^+\ell^-$ Forward-backward Asymmetry and New Physics

0.16 MB·English

by  Artyom Hovhannisyan

B → K∗ℓ+ℓ− Forward-backward Asymmetry and New Physics
Artyom Hovhannisyana,∗, Wei-Shu Houb, and Namit Mahajanb
aCenter for Theoretical Sciences, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
bDepartment of Physics, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
(Dated: February 7, 2008)
Theforward-backwardasymmetryAFBinB→K∗ℓ+ℓ−decayisasensitiveprobeofNewPhysics.
Previous studies have focused on the sensitivity in the position of the zero. However, the short
distanceeffectivecouplings arein principle complex,as illustrated byB→ρℓ+ℓ− decay within the
Standard Model. Allowing the effective couplings to be complex, but keeping the B → K∗γ and
K∗ℓ+ℓ− rate constraints, we find the landscape for AFB(B →K∗ℓ+ℓ−) to be far richer than from
entertaining just sign flips, which can be explored by future high statistics experiments.
PACSnumbers: 13.25.Hw,13.20.-v,12.60.-i
7
0
0 It was pointed out 20 years ago [1] that the loop- realandintermsofCKMelementsthatarealreadymea-
2 induced bsZ coupling is enhanced by large m , which sured. Short distance physics, including within SM, are
t
n turns out to dominate b sℓ+ℓ− (B¯ Xsℓ+ℓ−) decay. isolated in the Wilson coefficients C7eff, C9eff and C10.
a The effective bsγ couplin→g gives a low→q2 m2 peak in Eq.(1) canbe useddirectly for inclusiveB decay. For
J the differential rate [2], while Z and γ in≡ducℓeℓd ampli- B K∗ℓ+ℓ−, hadronic matrix elements of quark bilin-
0 tudes interfere across the q2 spectrum. One such effect ear→s give well defined B K∗ form factors. Thus, even
1 is the forward-backward asymmetry [3], AFB, which is forexclusivedecay,theco→efficientsC9eff,C10 andC7eff can
the asymmetry between forward and backward moving be viewed as physical measurables, hence scheme and
2 ℓ+ versus the B meson direction in the ℓ+ℓ− frame. scale independent, up to the definition of form factors.
v
6 The first measurement of FB in exclusive B Indeed, C7eff andC10 in Eq.(1)areatmB scale,with C7
4 K∗ℓ+ℓ− decay was recently repAorted [4] by the Belle e→x- receiving large additive contributions from other Wilson
0 periment, with 3.4σ significance. The results are consis- coefficients through operator mixing [8],
1 tentwiththe StandardModel(SM),rules outthe wrong
0 handed ℓ+ℓ− current, but a sign flip of the bsγ coupling 6
07 is still tolerated by the poor statistics of ∼ 100 signal C7eff =ξ7C7+ξ8C8+XξiCi, (2)
events. However, taking the measured inclusive b sγ i=1
/
h (B¯ X γ) and b sℓ+ℓ− rates together [5], the →latter
p poss→ibilitsy is disfav→ored. where ξi are QCD evolution factors. However,
-
p The relative insensitivity of AFB to hadronic effects C9eff(sˆ)=C9+Y(sˆ), (3)
e makes it an attractive probe for New Physics (NP) in
h the long run. For example, we expect a quantum jump
is alsoa function of the dilepton mass throughY(sˆ), the
:
v inthe number ofeventswiththe adventofLHC in2008. form of which can be found in Ref. [7], and depends on
i A study by the LHCb experiment shows that 7700
X B K∗ℓ+ℓ− events are expected with 2 fb−1 d∼ata [6]. long distance (cc¯) effects.
Within SM, because of the near reality of V in
r → ts
a InthisLetter wepointoutthatthe sensitivityofAFB to the standard phase convention, C7eff, C9 and C10 are
NPisgreaterthanpreviouslythought. Thecomplexityof practically real, with Ceff(sˆ) receiving slight complexity
9
theassociatedeffectiveWilsoncoefficientscanbeprobed
throughY(sˆ). Awidelyinvoked[9]“MinimalFlavorVio-
by d FB/dq2 as early as 2008 at the LHC. lation”(MFV)scenariofurtherasserts(usuallyassuming
A
The quark level decay amplitude is [1, 7],
the operator structure of SM) that there are no further
sources of flavor and CP violation, other than what is
G α
Mb→sℓ+ℓ− = −√F2π Vc∗sVcb (cid:8)C9eff[s¯γµLb][ℓ¯γµℓ] aolfrSeaMd,yspurcehseanstmininSiMma.lInsduepeedr,symmamnyetpriocpSuMlar[e1x0t]eonrsitownos
+C10 [s¯γµLb][ℓ¯γµγ5ℓ] Higgs doublet models [11], tend to follow this pattern.
mˆ With MFV as the prevailing mindset, C7, C9 and C10
−2 sˆb C7eff [s¯iσµνqˆνRb][ℓ¯γµℓ](cid:9), (1) are oftentimes taken as real [12] tacitly, hence the focus
onlyonpossiblesignflipsfromlargeNPeffects. For FB,
A
where s = q2, and we normalize by m , e.g. sˆ= s/m2. therefore,themainprojectionforthefuturehasbeenthe
B B
We factor out V∗V instead of the usual V∗V . Al- sensitivity of the zero to NP [7, 13].
cs cb ts tb
though trivial within SM, it has the advantage of being As a quantum amplitude, however, there is no reason
a priori why C7eff, C9eff and C10 in b→sℓ+ℓ− should be
M
real. Despite the suggested reality from SM and MFV,
whethertheyarerealorcomplexshouldbemeasured ex-
∗OnleavefromYerevanPhysicsInstitute, Yerevan,Armenia. perimentally, and we will be able to do so in just a few
2
years! In fact, currently there are hints [14] for “anoma- cuts that complicate theoretical correspondence.
lies” in time-dependent and direct CP violation (CPV) Having allowed C7eff, C9eff and C10 to be complex, we
measurements of b sq¯q transitions. One possible ex- still need to consider the constraints. Ceff is rather well
7
→
planation is NP in b sq¯q electroweak penguins [15], constrained by b sγ rate measurement. We take a
which are the hadroni→c cousins of b sℓ+ℓ−, but the one sigma experim→ental range [20] for B K∗γ for our
latter is clearly much less plagued by→hadronic effects. exclusive study. Likewise, inclusive b s→ℓ+ℓ− measure-
→
Motivated by possible hints for New Physics CPV in ment (by reconstructing a partial set of Xs states), as
hadronic b s transitions, and in anticipation of ma- well as the exclusive B K∗ℓ+ℓ−, provide constraints
jor experime→ntal progressin near future, we explore how onC7eff,C9eff andC10. At→themoment,measurementsare
much FB can differ from SM by allowing associatedef- notpreciseenough,soweuseonlytheintegratedratefor
A
fective couplings to be complex. Constraints such as de- theexclusivechannel,againwithinonesigmaexperimen-
cayrates,ofcourse,shouldbe respected, andone should tal range. In the future, with high statistics, one could
checkwhether models existwhere C7eff,C9eff andC10 can use the differential dB/dsˆ rate, which is more powerful.
be complex. We find, even without enlarging the oper- We will plot d /dsˆas an illustration.
ator basis, from a theoretical standpoint, MFV may be Our main foBcus is the FB in exclusive B K∗ℓ+ℓ−
A →
too strong an assumption. decay. Assuming the form factors are real, we have
Our insight comes as follows. Part of the impetus for
MFV is the good agreement between theory and experi- dAdsˆFB ∝ nRe(cid:0)C9effC1∗0(cid:1)VA1
mentforinclusiveb sγrate,whichprovidesastringent
constraint on NP. H→owever, while depending on the ex- + mˆsˆb Re(cid:0)C7effC1∗0(cid:1)(cid:2)(VT2)−+(A1T1)+(cid:3)o, (4)
istence of a third generation top quark, the b sγ rate
→
depends very little on the precise value of mt when it is where (VT2)− = VT2(1 mˆK∗), (A1T1)+ = A1T1(1+
−
lraartgeec.hFaonrgmestbiynothnelyrang3e0%of.1I5n0ctoon3tr0a0stG,etVhe,tbhebs→ℓ+ℓs−γ lmˆigKht∗-)c,oannedsuVm, Aru1l,eT(LiCarSeRf)or[2m1]ffaocrtmorsfa[c7t]o.rsWine ouuser nthue-
∼ →
ratedependsverysensitivelyonmt throughtheeffective merical analysis. In Eq. (4) we have exhibited only the
bsZ coupling, aswe statedfrom the beginning, changing dependence on C9eff, C10 and C7eff, since it is customary
by a factor of 4 in the same mt range. to plot d ¯FB/dsˆ which is d FB/dsˆ normalized by the
∼ A A
Suppose there are extra SM-like heavy quarks. These differential rate dΓ/dsˆ. This reduces sensitivity to form
could be the 4th generation, or could be vector-like factormodels. The zero of ¯FB is oftenconsideredquite
A
quarks that mix with the top. Take the 4th generation stable against form factor variations [7, 13].
asanexample,the b sγ rateisnotsensitivetothe ex- The Wilson coefficients are parameterized as,
′ → ∗
iHstoewnecveeor,f“thhaertd”q(usaernksiutinvleestso|hVet′asvVyt′bq|uiasrvkemryalsasr)gaem[1p6li]-. C7 → C7(1+∆7eiφ7), (5)
teuadsielys sauffcehctaesdbb→y fisnℓi+teℓ−V,t∗′BsVst′mb,ixaisngm[t1′7>] emtct. bwyoudledfinbie- CC109 → CC190(1(1++∆∆91e0iφe9iφ)1,0), ((76))
tion. Since Vt∗′sVt′b should be in general complex [18], →
so would C9eff and C10 (and C7eff). With this as an exis- with ∆i = 0 corresponding to SM. These Wilson coeffi-
tence proof,we note further that the three [s¯b][ℓ¯ℓ] terms cientsareevaluatedattheelectroweakscale,thenevolved
in Eq. (1) are 4-fermion operators. The possible under- downtothemB scaletobeusedinEq.(4). Wedonotin-
lying New Physics is precisely what we wish to probe at cludeanycomplexityfromotherWilsoncoefficients. The
B factories and at the LHC. Thus, despite the apparent tree levelC1 andC2 are unchangedby NP,but as asim-
success of MFV, we find the usual assumption of near plifyingassumption,weignorepossibleNPinducedcom-
reality of C7eff, C9eff and C10 unfounded. When sufficient plexities through the gluonic C3−6 and C8 coefficients,
data comes, the experimenters are well advised to keep which enter C7eff and C9eff through operator mixing and
these parameters complex in doing their fit. long distance effects (see Eqs. (2) and (3)). In practice,
We remark that, in fact within SM, B ρℓ+ℓ− decay this should not change our point.
exhibits partially the physics we talk abo→ut. A complex LetusstartwithaSM-likeframework,thatis,viewing
factor δ (V∗ V )/(V∗V ) arises from the u-quark B K∗ℓ+ℓ− as induced by effective bsZ and bsγ cou-
current-ucu≡rrentuodpeurbator,tadswtbellastopquarkintheloop, plin→gs(andbox diagrams). Ifsuchis the case,we expect
makingC9 complex[19]. Wewillusethiscaseattheend C9 and C10 to be approximately the same, i.e.
as an illustration within SM.
In this study we shall keep the operator set as in ∆≡∆9 ∼=∆10, φ≡φ9 ∼=φ10, (8)
SM, since enlarging to include e.g. righthanded currents in Eqs. (6) and (7), and one effectively has the param-
would not be profitably probed in early years of LHC. eters ∆, ∆7, φ and φ7, which covers the usual case of
In the same vein, although inclusive b sℓ+ℓ− (and wrong sign Ceff. The 4th generationalso belongs to this
7
→ ∗
mb →entsaγll)yims tohreeoarcecteicsasilblylecBleanerK, ∗wℓe+fℓo−cu(asnodnBthe eKxp∗eγr)i-. sceWnaeripol,owtidth¯FVBt′/sVdstˆ′babnrdindgin/gdsiˆnicnomFipglse.xi1t(ya.) and (b),
→ → A B
Experimentalstudiesofinclusiveprocessesusuallyapply respectively, for SM and 4th generation model (SM4).
3
*+-*+-HLHL®®ABKllABKllFBFB----00000000........0012122121 a b *+-*+-66HLHL®´®´BRBKll10BRBKll10012012......123123555555 a b Cabdcese −−−∆0004...7528 −−−∆0111...9952 −−−−∆00221....59020 96φ00057◦◦ 2637φ05509◦◦◦ φ6000150◦
--00..33 00
00 00..11 00..22 00..33 00..44 00..55 00..66 00..77 00 00..11 00..22 00..33 00..44 00..55 00..66 00..77
ss`` ss`` TABLE I: Parameter values for Cases b–e in Fig. 2, where
Cases d and e are already ruled out. The SM (Case a) has
FIG. 1: (a) dA¯FB/dsˆand (b)∼dB/dsˆfor B →K∗ℓ+ℓ−. The ∆i = 0. In our numerics [8], we use C7eff ≃ 0.67C7 −0.18,
shadedregionisallowedbyC9 =C10,andCasesa(solid)and with C7≃−0.20, C9≃4.1 and C10 ≃−4.4 at MW scale.
b (dash) are SMand 4th generation model, respectively.
*+-HL®ABKllFB--0000....01221 ab e c d *+-6HL®´RBKll10012...123555 ce a b d NscBhePeodualdttoKskhe∗eoxγerpptlaotdnhrideesttKafhunel∗clℓep+ga(eℓfnr−oaermrwaienliittstehtyrainnoscfp1eaE,σcqZesc′.aosm(n5sbo)t–dera(feo7ilnsr)et.[,.2W2kI]een)e,dppoeirnenodge-
-0.3 B 0 we→find much richer possibilities than Figs. 1(a)and (b).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
s` s` As plotted in Figs. 2(a) and (b), we illustrate with the
further cases of c, d and e. The SM and SM4 are Cases
FIG.2: (a)dA¯FB/dsˆand(b)dB/dsˆforB→K∗ℓ+ℓ− allow- a and b, respectively, as in Fig. 1. The ∆ and φ values
i i
ing all Wilson coefficients to becomplex as in Eqs. (5)–(7).
are given in Table I.
Case d has wrong sign C10, while Case e has sign
flip in both C7eff and C10 (equivalent to wrong sign C9).
For SM4, we take the CKM parameters which yield the Both are already ruled out [4] by Belle data. The pos-
correctBs-B¯s mixing[17],predictslargetime-dependent sibility of flipping only the sign of Ceff is ruled out by
7
CPVin Bs decay,aswellas accommodating[15] the NP rate constraints [5], hence not plotted. Similar scenar-
hints in CPV in b sq¯q decays. We see that the zero of ios have been considered in the literature, and we give
d ¯FB/dsˆhas shift→ed by a significant amount, with only these cases to illustrate the versatility of Eqs. (5)–(7).
A
a small positive value below the zero. These are due to Though ruled out, Case e is illuminating. From Table I,
theenrichmentof(mostly)theφphase. Forlargersˆ,one ∆7 4.8 overwhelms the SM effect, flipping the sign
Cha7esfflihtatlseddaiffmepreendcaewinayd,Aw¯FhBil/edCsˆ9efrffomandSMC,10asctahrreyeffaelmctoostf owfitCh∼7eaff−m(Euqch. (d2i)m).inAishtetdheCs9a.mHeowtiemvee,r,Cev10enalwsoithflinposcsoigmn-,
the same phase. The general appearance of d /dsˆ for plex phases, the large effects from Case e survives rate
B
SM and SM4 is very similar. constraints, giving a more pronounced low q2 peak for
A broader range is allowed by Eq. (8). Keeping B differential rate, as seen in Fig. 2(a), which lies outside
K∗γ and K∗ℓ+ℓ− rates in 1σ experimental range an→d oftheboundaryoftheshadedregioninFig.1(a). Thisis
exploring ∆, ∆7, φ and φ7 parameter space, the results because Eq. (8) no longer holds. Similar cases may exist
areplottedinFig.1asthe shadedarea,whichillustrates that remain to be probed.
the range of variation allowed by C9 C10. This is An interesting new scenario is illustrated by Case c,
≃
just for illustration purpose and should not be taken as where d /dsˆ and the zero of d ¯FB/dsˆ are hard to dis-
precise boundaries. For instance, we see that below the tinguishBfrom SM, but d ¯FB/dAsˆ above the zero reaches
SM zero d ¯FB/dsˆ could be very small, but the shaded only half the SM value.AThus, a measurement of the
A
rBegionKfo∗rγdBan/ddsˆKba∗sℓi+caℓ−lly. rdefle/cdtsˆs tshheou1lσdcaolnsostrbaeinfitsttoedn zaelrreoaddyoeswintoht 1pinabd−o1wndaCta9.aTthBe sfcaecntoarriieoscaexnpbeectteedstbedy
→ B ∗
in the future, but it depends directly on B K form 2008. Ifsuchphenomenaarediscoveredwith,e.g. LHCb
→
factors, especially the overallscale. data,itwouldimplyNPthatfeedthe(s¯γ Lb)(ℓ¯γµℓ)and
µ
We illustrate the power of early LHC data with the (s¯γµLb)(ℓ¯γµγ5ℓ) operators differently.
2 fb−1 study of LHCb, where 7700 reconstructed We mark the simulated errors from the 2 fb−1 study
B K∗ℓ+ℓ− events are expecte∼d. We take the simu- at LHCb as before on Fig. 2(a), illustrating its power.
→
lated errors [6] (with signal events generated according The actual possibilities are far richer. The shaded area
to SM)ford ¯FB/dsˆfromthree bins,onearoundthe SM of Fig. 1(a) illustrates that, even with Eq. (8) imposed,
A
zero, one below and one above, and plot in Fig. 1(a) to a broad range is allowed for sˆ< 0.2. With the full free-
guide the eye. It shouldbe clear that our suggestioncan domofEqs.(5)–(7),theregionallowedbyrateconstraint
be tested early on in the LHC era would likely cover a large part of Fig. 2(a), which is up
The narrow, long “tail” at sˆ& 0.3 for d ¯FB/dsˆ indi- to experiment to explore. One should keep the effective
A
cates that Eq. (8) is probably too strong an assumption. Wilson coefficients of Eq. (1) complex and use the gen-
Even if we keep the operator basis as in Eq. (1), treat- eral parametrizationof Eqs. (5)–(7) to fit for ∆ and φ .
i i
ing these as 4-fermion interactions arising from possible Finite φ implies violation of MFV.
i
4
starting with Eq. (1), can be easily employed and fol-
0.2 lowed. Third, while the 4th generationmodel provides a
B®K*l+l-
0.1 goodexample,ourapproachisfullygeneralandaimedat
AFB 0 the experimental study, and does not depend on specific
-0.1 models. In fact, the 4th generation provides a good ex-
-0.2 B®Ρl+l- ample against the prevailing MFV prejudice that limits
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 the perspective for flavorand CP violation expectations
s` in b s transitions. It is our opinion that the study of
→
FIG. 3: dA¯FB/dsˆfor B→K∗ℓ+ℓ− and ρℓ+ℓ− within SM. b → s transitions is still in its infancy, and is the least
constrained. Imposing MFV may be overstretching our
experiencefromotherareasofflavorviolation. Itisupto
experiment to reveal what may be in store for us in the
OursuggestionofkeepingtheWilsoncoefficientscom- excellent probe of FB. Finally, the 1 ab−1 final data at
A
B factories would only give limited improvement on the
plex is not just for NP. Even within SM, for the CKM
suppressed decay B ρℓ+ℓ−, one already expects [19] existing result. The next round of major improvement
→ would come from LHC. The Super B factory upgrade
complexityineffectivecouplings,arisingfromboththeu-
quark andtop contributions. We plot d ¯FB/dsˆin Fig. 3 in the future could bring back competitiveness of e+e−
for B K∗ℓ+ℓ− and ρℓ+ℓ− in SM. AltAhough the differ- machines, especially for inclusive studies.
ence is→not so great, especially since with 2 fb−1 data at Insummary,wehaveexploredtheCP conservingcon-
LHCb one expects less than 200B ρℓ+ℓ− events with sequencesofcomplex Wilsoncoefficients onthe forward-
largerbackground,thedifferentbeh→aviorind ¯FB/dsˆcan backward asymmetry FB in B K∗ℓ+ℓ− decay. The
A A →
be tested with a larger dataset. possibilities are much broader than the usual considera-
Weoffersomeremarksbeforeclosing. Wehavefocused tion of sign flips under minimal flavor violation frame-
on FB for exclusive B K∗ℓ+ℓ−, mostly because of work. In view of hints of CP violation anomalies in
A →
ease of experimental access, and with upgrade of statis- b sqq¯ decays, the large increase in statistics with
→
ticsimminent. Second,itisusuallystressedthatthezero the advent of LHC would make FB one of the clean-
A
of FB is insensitive to form factors. With NP sensitivi- est probes for New Physics in the near future.
A
ties now going beyondthe zero,form factor issues would
have to be considered. One would have to combine the We thank A. Buras, B. Grinstein, P. Koppenburg,
progress from form factor models, lattice, as well as ex- Z. Ligeti, and I. Stewart for discussions. This work is
perimental studies of B ρℓν. One can of course try supported in part by NSC 95-2119-M-002-037,NSC 95-
→
to measure FB in inclusive mode, and our discussion, 2811-M-002-031,and NCTS of Taiwan.
A
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