Bs0→μ+μ−γ from Bs0→μ+μ−

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

B

0

_s

→

μ

+

μ

−

γ

from

B

0

_s

→

μ

+

μ

−

Francesco Dettori

a

,

Diego Guadagnoli

b

,

∗

,

Méril Reboud

b

,

c

a_{EuropeanOrganizationforNuclearResearch(CERN),Geneva,Switzerland}

b_{Laboratoired’Annecy-le-VieuxdePhysiqueThéoriqueUMR5108,UniversitédeSavoieMont-BlancetCNRS,B.P.110,F-74941,Annecy-le-VieuxCedex,France} c_{ÉcoleNormaleSupérieuredeLyon,F-69364,LyonCedex07,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19December2016 Accepted18February2017 Availableonline27February2017 Editor: G.F.Giudice

The B0

s →

μ

+

μ

−

γ

decayofferssensitivityto awiderset ofeffectiveoperators thanitsnon-radiative

counterpartB0

s→

μ

+

μ

−,andasetthatisinterestinginthelightofpresent-daydiscrepanciesinflavour

data. On the otherhand, the directmeasurement of the B0s →

μ

+

μ

−

γ

decayposeschallenges with

respect tothe B0

s →

μ

+

μ

− one. We present anovelstrategy tosearch for B0s →

μ

+

μ

−

γ

decays in

theveryeventsampleselectedfor B0

s→

μ

+

μ

− searches.ThemethodconsistsinextractingtheB0s →

μ

+

μ

−

γ

spectrumasa“contamination”tothe B0

s→

μ

+

μ

−one,asthesignalwindowforthelatteris

extendeddownwardwithrespecttothepeakregion.Weprovideargumentsfortheactualpracticability ofthemethodalreadyonRun 2dataoftheLHC.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

The B0s

→

μ

+

μ

− decay is one of the cleanest low-energy

probes of physics beyond the Standard Model (SM). The pres-ence of new dynamics at scales above the electroweak one can beprobedmostgenerallybyadoptinganeffective-fieldtheory ap-proach, whereby beyond-SM physics manifests itself as shifts to the Wilsoncoefficients of the operators of the b

→

s



effective hamiltonian.Inthiscontext,the B0

s

→

μ

+

μ

− decayissensitiveto

the scalar andpseudoscalar operators

O

(_S)_,P,and to the operator

O

10,definedas(seeforexample

[1]

)

O

S

=

αem

4

π

mb

¯

s PRb

¯ ,

O

P

=

αem

4

π

mb

¯

s PRb

¯

γ5

 ,

O

10

=

αem

4

π

¯

s

γ

μ_P Lb

¯

γ

μ

γ

5

 ,

(1) with

O

_S,P defined from the unprimed counterparts via the re-placements PR

→

PL andmb

→

ms inEq.(1).WithintheSM, to

anexcellentapproximationonlytheoperator

O

10contributes.The goodagreementbetweentheSMprediction

[2]

B

(

B0_s

→

μ

+

μ

−

)

SM

= (

3.65

±

0.23)

×

10−9 (2)

andthecurrentbestmeasurement[3]

B

(

B0_s

→

μ

+

μ

−

)

exp

= (

2.8+₋0₀._.7₆

)

×

10−9

= (

0.76+₋0₀._.20₁₈

)

×

B

(

B0_s

→

μ

+

μ

−

)

SM

,

(3)

*

Correspondingauthor.

E-mailaddress:diego.guadagnoli@lapth.cnrs.fr(D. Guadagnoli).

forces scalar and pseudoscalar contributions to negligible values

[4,5].Ontheotherhand



O(15%)newcontributionstotheWilson coefficientoftheoperator

O

10areallowed bypresenterrors,and actually favoured – provided they are indestructive interference withtheSMcontribution–bytheabout25%toolowcentralvalue inEq.(3).

Adding a photon to the final state, namely considering the B0_s

→

μ

+

μ

−

γ

decay, yields an observable sensitive not only to

O

10, but also to

O

9 and to the electromagnetic-dipole oper-ator

O

7, aswell asto their chirality-flipped counterparts

[6–11]

. (Thesensitivityto

O

7occursforvaluesofthefinal-stateinvariant mass squaredclose to zero; asour discussion will be concerned with the high invariant-mass region, this operator will not be consideredanyfurther.)Increasingthenumberofobservables sen-sitivetotheseoperators,especially

O

9 and

O

10,isveryimportant inthe lightofpresentdata.Infact, theLHCbexperimentaswell asthe B factoriesperformedanumberofmeasurements ofb

→

s transitions, and the overall agreement with the SM is less than perfect.Discrepanciesconcerninparticular:

•

the ratio RK of the branching fractions for B+

→

K+



+



−,

with



=

μ

,

e[12]

RK

≡

B

(

B+

→

K+

μ

+

μ

−

)

B

(

B+

→

K+e+e−

)

,

(4) showinga2

.

6

σ

deficitwithrespecttotheSM

[13–16]

; http://dx.doi.org/10.1016/j.physletb.2017.02.048

0370-2693/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

•

the absolute B+

→

K+

μ

+

μ

− branching ratio [17,18], about 30%lowerthantheSM

[19–21]

;

•

themeasurementof

B(

B0_s

→ φ

μ

+

μ

−

)

[22,23],lowerthanthe SMpredictionbymorethan3

σ

[22];

•

the angular distribution of the B0

_→

_K∗0

_

+

_

− _{decays and,} most notably, the quantity known as P₅ [24], measured by both LHCb [25,26]and Belle

[27]

, whosetheoretical error is, however,stilldebated

[28–32]

.

Remarkably,onecanfindaconsistenttheoreticalinterpretationof all these discrepancies, aswell as of the last equalityin Eq. (3), within an effective-theory approach [5,33–36]. Data can be ac-countedfor atone stroke withnew contributions to C9 only, or jointlytoC9 andC10.Furthermore,theindicationsofnew-physics (NP)couplingspreferringmuonsoverelectrons canbe accommo-datedbyinvokinganeffectiveinteractioncoupleddominantly (be-foreelectroweak-symmetrybreaking) tothird-generationfermions

[37]. This possibility would even allow to relate the mentioned b

→

s discrepancieswithothersexistinginb

→

c transitions

[38]

. Given its sensitivity to C9 and C10 alike, the radiative decay

B0

s

→

μ

+

μ

−

γ

offers anadditionalprobeintophysics beyondthe

SM,andin particularaprobe ofcouplingsthat are interestingin thelightofcurrentdata.However,thedirectmeasurement of ra-diativehadrondecaysisharderwithrespecttotheirnon-radiative counterparts for various reasons. First, the detection and recon-structionefficiency ofa photon is typically smallerthan the one ofchargedtracks.Secondly,theenergybeingsharedwiththe ad-ditionalphotonmakestheotherdaughterparticlessofter,yielding smallertriggerandreconstructionefficiencies.Furthermore,the in-variant massreconstructed indecays with photonshas, at these energies,a worse resolutionthan indecays without.Thisinturn leadstoalargerbackgroundunderthesignalpeak.Theabove con-siderationsholdinparticularforhadron-colliderexperiments,due tothe highoccupancy oftypical events,andforlow-energy pro-cesses such as thoseof interest to flavour physics. Despite these difficulties, rare radiative decays with branching ratios of order 10−6

_÷

₁₀−7 _{have been observed and exploitedfor NP searches} byseveralexperiments,see

[39]

forarecentreview.However,the ratesjust mentioned arestill very‘abundant’ if comparedto the B0

s

→

μ

+

μ

− decayandits radiativecounterpart.The latterposes

thereforeaformidablechallengefordirectdetection. InthispaperweproposeamethodtosearchforB0

s

→

μ

+

μ

−

γ

events in the very same event sample selected for the

B(

B0_s

→

μ

+

μ

−

)

measurement. In one sentence, the method consists in measuring B0

s

→

μ

+

μ

−

γ

as“contamination”to B0s

→

μ

+

μ

−,by

suitably enlarging downward the signal window for the latter search. This possibility requires a number ofqualifications, since the B0_s

→

μ

+

μ

− measurement itselfcomes withsome subtleties asfar as photons are concerned – notably the treatment of soft final-stateradiation.

Inan idealised measurement, the B0

s

→

μ

+

μ

− decayappears

as a peak in the invariant mass squared of the two final-state muons,withnegligibleintrinsicwidth.1Alreadyatthislevel, how-ever, the ‘definition’ of the final-state muons is complicated by the fact that they emit soft bremsstrahlung, giving riseto B0s

→

μ

+

μ

−

+

nγ decays,withthen photonsundetected.Thiseffectis howeverwell known [40–42]. As reappraised in Ref. [43], it can be summed analytically to all orders in the soft-photon approx-imation, yielding a multiplicative correction to the non-radiative rate. This contribution skews downwards the peak region of the

1 _{Theexperimentalresolutioninthemuonmomentagivesthispeakan} approxi-matelyGaussianshape,thewidthbeingforexampleofabout25 MeVfortheLHCb experimentandrangesfrom32to75 MeVfortheCMSexperiment[3].

Fig. 1. Breakup ofthe full B0

s→μ+μ−γ spectrum(solidblue)–calculatedin

Ref.[11],denotedasMNinthelegend–intoitspureISRcomponent(long-dashed blue),FSRone(medium-dashedblue),andISR-FSRinterference(dot-dashedblue). WealsoreporttheB0

s→μ+μ−+nγ spectruminthesoft-photonapproximation

(dottedorange)fromRef.[43],denotedasBGGIinthelegend.(Forinterpretationof thereferencestocolourinthisfigure,thereaderisreferredtothewebversionof thisarticle.)

B0_s

→

μ

+

μ

−distribution,asshownbythedottedorangecurveof

Fig. 1.

In order to compare the measured B0

s

→

μ

+

μ

− rate with

the theoretical one [2], the mentioned soft-radiation tail due to B0_s

→

μ

+

μ

−

+

nγ needstobesubtractedoff.Forexample,a B0_s

→

μ

+

μ

− signalwindowextendingdowntoabout5.3 GeVis equiv-alenttoasingle-photonenergycut Eγ



20

÷

100 MeV, amount-ing to a negative shift of

B(

B0_s

→

μ

+

μ

−

)

as large as 15% [43]. Experimentally, the radiative tail is obtained and taken into ac-count using Monte Carlo B0

s

→

μ

+

μ

− events with full detector

simulation and with bremsstrahlung photon emission modelled through the Photos application [44]. The advantage of this ap-proach over the analytic one [43] is that the correction factor is alreadyadjustedfordetectorefficiencies.

Forsofterandsofterphotons(orequivalentlyfor_mμ+_μ− closer andclosertothe B0

s peakregion),thesingle-photoncomponentin

B(

B0_s

→

μ

+

μ

−

+

nγ

)

isexpected tomatchtheradiative branch-ing ratio

B(

B0_s

→

μ

+

μ

−

γ

)

, ascomputed in Ref. [11] to leading order in

α

em (see also update in Ref. [45]).2 This is indeed the case, asshownbycomparingthe solidblue distributionwiththe dotted orange one in Fig. 1. We can actually go farther in this comparisonby separatingthecontributionsdueto photons emit-tedfromfinal-stateleptons–tobedenotedasfinal-stateradiation (FSR) – with respect to the rest – to be collectively referred to asinitial-stateradiation(ISR)contributions.Thisseparationmakes sensetotheextentthatwecanidentifytworegionsinmμμ where onlyoneofthetwocontributionsisdominant.Thebreakupofthe B0s

→

μ

+

μ

−

γ

spectrum intoitsdifferentcomponentsis likewise

reportedin

Fig. 1

.Aswellknown,theFSRcontributionisdominant for soft photons (or highmμμ), whereas the ISR one dominates forharderandharderphotons,namelyasmμμ decreasesfromthe peakregion.Thecrossoverregionbetweenthetwocontributionsis atmμμ

≈

5

.

0 GeV.Moreimportantlyforourpurposes,the contri-butionfromtheinterferencetermisalwaysbelow1%ofthetotal spectrum.3 _{ThisholdstruefairlygenerallyalsobeyondtheSM.In} particular,shiftsinC9 andC10 withoppositesignwithrespectto

2 _{Inthisspirit,wewouldalsoexpecttheISRcomponentoftheB}0

s→μ+μ−γ

spectrumcalculatedinRef.[46]tomatch,inthemμ+μ− regionclosetothe end-pointofthis distribution,thecorrespondingspectrumcalculatedinRef.[11].We actuallyfindthat,whilethetwodistributionshaveasimilarshape,thedistribution from[46]is,inthementionedmμ+μ−region,afactorofalmost4abovetheonein [11].BarringanormalisationtypoinRef.[46],weareunabletophysicallyinterpret thisdifference.

Fig. 2. Fractionofthe fullB0

s→μ+μ−γ spectrumasafunctionofthechosen

signal-regionlowerboundmISR

μμ,forthreescenarios, specifiedinthe legend.See

textfordetails.

therespectiveSMcontributions,ashintedatbytherecentb

→

s discrepanciesmentionedearlier,tendtodecreasetheinterference termevenfurther.Asaconsequence,theISRandFSRcontributions canbetreatedastwobasicallyindependentspectra.

Inshort,totheextentthattheFSRcontributioncanbe system-aticallysubtractedoff,asisthecaseforB0

s

→

μ

+

μ

−searches,one

canmeasuretheISRcomponentoftheB_s0

→

μ

+

μ

−

γ

spectrum– andtherebythe B0

s

→

μ

+

μ

−

γ

differentialrate– as

“contamina-tion” of B0_s

→

μ

+

μ

− candidate events as the signal window is enlargeddownwards.Wenotethatsuchcontaminationis,in prin-ciple,alreadypresentinexisting B0

s

→

μ

+

μ

− searches.However,

itisnegligibleinthetypicalwindowof

±

3

÷

5 standarddeviations aroundthe B0

s

→

μ

+

μ

−peak, anditssmooth distributioncanbe

absorbed in other background distributions due, for example, to combinatorialbackgroundorpartiallyreconstructed B decays.For thisreasonitwastypicallynotincludedasseparate componentin recentB0_s

→

μ

+

μ

− decaymeasurements[52–54].

On the other hand, as the signal window is enlarged down-wards,theISRcomponentoftheB0

s

→

μ

+

μ

−

γ

spectrumbecomes

sizable. Fig. 2 shows in more detail how large this contamina-tionisexpectedto be.The figuredisplays thefractionofthe full B0

s

→

μ

+

μ

−

γ

spectrumasafunctionofthechosenvalueformISRμμ fortheSM case,aswellasforthetwoscenariosthat bestfitthe b

→

s anomalies:onewitha V

−

A shifttoC9 andC10,andsuch that

δ

C9

= −

12%CSM₉ , the other with a C9-only shift such that

δ

C9

= −

30%CSM₉ [5].The figurerevealsthat thisfractionislarger withintheSMthanintheconsideredNPscenarios.Forexample,it isabout4.8%intheSM fora B0

s

→

μ

+

μ

− signalwindow

extend-ingdowntomISR_μμ

=

4

.

6 GeV,whereasitisabout4%intheV

−

A scenario.

Wealsonotethattheassociatedeventyieldislarge, compara-bletothatfortheB0s

→

μ

+

μ

−signal,becausetheB0s

→

μ

+

μ

−

γ

rateintegratestoatotalbranchingratioofabout2

×

10−8_[11],an orderofmagnitudeabovetheB0_s

→

μ

+

μ

−one.Theexpectedsize ofthe B0

s

→

μ

+

μ

−

γ

spectrumisdisplayedin

Fig. 3

,by

superim-posingthisspectrum totherecentLHCb B0

s

→

μ

+

μ

−analysisof

Ref.[52].WeshowthecaseofaSM signalaswellastheNPcase mentionedearlier,namely

δ

C9

= −δ

C10

= −

12%CSM9 .Fromthe ab-solutesizeofthesecurveswecanalreadyinferthatNPscenarios withthe B0s

→

μ

+

μ

−

γ

spectrum enhanced by orders of

magni-tudewithrespect to theSM areunlikely in thelight of data: as shownin

Fig. 3

,afactorof10enhancementwouldresultina sub-stantialdistortionofthemeasuredspectrumfrommμμ



5

.

1 GeV downwards.

The B0_s

→

μ

+

μ

−

γ

spectrumshowninFig. 3isobtainedfrom our theoretical calculation, i.e. it is not a fit to existing B0

s

→

Fig. 3. DimuoninvariantmassdistributionfromLHCb’smeasurementofB(B0

s→

μ+μ−) [52]overlayedwith the contributionexpectedfrom B0

s→μ+μ−γ

de-cays(ISRonly).Assumesflatefficiencyversusmμ+μ−.Thelinedenotedas‘B0s→

μ+μ−γ NP’referstotheV−A casewithδC9= −12%CSM9 (seealsoFig. 2).The twofilledcurvesarenotstackedontoeachother.

μ

+

μ

− data. The spectrum assumes that normalisation and effi-ciency be equal to those of the B0s

→

μ

+

μ

− distribution itself.

This is exactly true by definition at the endpoint mμμ

=

m_B0

s,

andincreasingly lessso forlower masses, dueto thevarious se-lectioncriteria. Forexample,typical analyses enforcepointing re-quirementswithrespecttotheprimaryinteractionvertex,andthe latter are lesssatisfied when an additional undetectedphoton is present.TheseissuescanonlybevalidatedinfullMonteCarlo sim-ulationsoftheconsideredexperimentandanalysis.

Withenoughstatistics,onecangobeyondtheintegratedB0

s

→

μ

+

μ

−

γ

branching ratio, and measure the B0_s

→

μ

+

μ

−

γ

spec-trum. This could be within reach of LHC experiments with Run 2data.Infact,shiftstothedifferentialbranchingratioareroughly linearinshiftstoC9orC10.Therefore,foraC9orC10deviationof theorderof15%(ashintedatbytheglobalfitstob

→

s data),the corresponding variation in the spectrum isexpected to be about 15% as well. Then, a fit to data could resolve such shift at one standarddeviationforaneventyieldofabout50.

The above argument is of statistical nature only, i.e.it disre-gards systematic uncertainties. There are two prominent sources of such errors.The first isthe theoretical errorassociated to the B0_s

→

μ

+

μ

−

γ

spectrum prediction

[11]

.The dominantsource of uncertainty in this respect is by far the one associated to the B0_s

→

γ

vector and axial form factors, defined from the rela-tions[11]



γ

(

k

,

)

|¯

s

γ

μ

γ5

b

|

B0_s

(

p

)

=

ie

_ν∗

(

gμνpk

−

pνkμ

)

FA

(

q 2

₎

M_B0 s

,



γ

(

k

,

)

|¯

s

γ

μb

|

B_s0

(

p

)

=

e

_ν∗

μνρσpρkσ FV

(

q2

)

M_B0 s

.

(5)

To the authors’knowledge, no first-principle calculation of these formfactorsexists,forexamplewithinlatticeQCD.Theform-factor predictionsusedinthisworkareobtainedfromtherecentanalysis

[48] of heavy-meson transition form factors, based on the rela-tivistic constituentquark model[49,50]. The analytic expressions forthe formfactorsfrom theconstituentquark modelreproduce theknownresultsfromQCDforheavy-to-heavyandheavy-to-light form factors [51]. Form-factor predictions within this model are therebyattachedan uncertaintyofabout10%,implyinga20% un-certainty on the branching-ratio prediction. It is clear that such levelofaccuracyisnotsufficienttoclearly resolvetheeffects ex-pected from new physics (see legend of Fig. 2). However, what is needed for the proposed method are the form factors in the

high-q2 _{range closeto the kinematic endpoint.This range isthe} preferredoneforlattice-QCDsimulations.

The second potential source of systematicuncertainty forour methodisofexperimental nature.The impact ofthisuncertainty dependsontheactualpossibilitytowellconstraintheother back-ground components populating the signal window as it is en-largedtowardslowervalues.Thispartofthespectrum,inaddition to combinatorialbackground,consistsmainly of semileptonic de-cays in the form B

→

h±

μ

∓

ν

(

+

X

)

, where h is a pion or kaon misidentified as muon and X can be any other possible hadron (not reconstructed), and rare decayssuch as B0,+

_→

_h0,+

_μ

+

_μ

−_, whichdo not need anymisidentification. While thesemileptonic decays do not represent a problem as they can be constrained from control channels directly in data, the rare decays need to be estimated witha combination ofexperimental measurements andtheoreticalinputs;asanexampletheB0

→

π

0

_μ

+

_μ

−_decayis not yetobserved experimentally and iscurrentlyconstrained us-ingthespectral shapemeasured fromthe B+

→

π

+

μ

+

μ

− decay andtheoretical estimates of theratio of the two branching frac-tions[3,52].Thespecificdetailsonhowtotreatthesinglesources of backgrounds will have to be addressed by the single experi-mentsdepending ontheexperimental capabilities,butwedonot foreseethesetobeirreduciblebackgrounds.

Weemphasizethatourproposedmethodispotentially applica-bletoseveralotherdecays– inprincipletheradiativecounterpart of any two-body decay whereby the initial-state meson mass is completelyreconstructible.Straightforwardexamplesareprovided by alltheother Bq

→ 

+



−

γ

modes,forwhich theonlyexisting

limitsconcernB0

_→

_e+_e−

_γ

_or

_μ

+

_μ

−

_γ

_{withatechniquebasedon} explicitphotonreconstruction

[55]

.Seriousconsiderationofthese decays will be timely when mature measurements of the corre-spondingnon-radiativedecayswillbecomeavailable.

Inconclusion,wepresenteda novelmethodforthe extraction ofthe B0_s

→

μ

+

μ

−

γ

spectrum athighm2_μμ.The methodavoids thedrawbacksofexplicitphotonreconstruction,andtakes advan-tage of the fact that this spectrum inevitably contaminates the B0s

→

μ

+

μ

−eventsampleasthem2μμ signal windowisenlarged downward.

Fig. 3

showsthatorder-of-magnitudeenhancementsof the B0_s

→

μ

+

μ

−

γ

decayrateare unlikely,alreadyin thelight of existingdatabelowmμμ



5

.

1 GeV.Morelikely,themeasurement willinvolveadedicatedfitby experiments,andthisiswhereour methodmaymakethedifference.Thismethodcanrealisticallybe applicableinLHCRun2data,andwouldtherebyallowtosetthe firstlimitfor

B(

B0s

→

μ

+

μ

−

γ

)

,orprovidethefirstmeasurement

thereof.

Acknowledgements

The work of DG is partially supported by the CNRS grant PICS07229.TheauthorsareindebtedtoDmitriMelikhovformany clarificationsonRef.[11]andrelatedwork.Theauthorsalsothank GinoIsidoriforcommentsonthemanuscript, andMikolajMisiak fordiscussions.

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