First determination of the one-proton induced Non-Mesonic Weak Decay width of p-shell Λ-Hypernuclei

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

First

determination

of

the

one-proton

induced

Non-Mesonic

Weak

Decay

width

of

p-shell

Λ

-Hypernuclei

The

FINUDA

Collaboration,

M. Agnello

a

,

b

,

L. Benussi

c

,

M. Bertani

c

, H.C. Bhang

d

,

G. Bonomi

e

,

f

, E. Botta

g

,

b

,

∗

,

T. Bressani

g

,

b

,

S. Bufalino

b

, D. Calvo

b

,

P. Camerini

h

,

i

,

B. Dalena

j

,

k

,

1

,

F. De Mori

g

,

b

,

G. D’Erasmo

j

,

k

, A. Feliciello

b

,

A. Filippi

b

,

H. Fujioka

l

,

P. Gianotti

c

,

N. Grion

i

,

V. Lucherini

c

,

S. Marcello

g

,

b

,

T. Nagae

l

, H. Outa

m

, V. Paticchio

j

,

S. Piano

i

, R. Rui

h

,

i

,

G. Simonetti

j

,

k

,

A. Zenoni

e

,

f

a_{DISAT,PolitecnicodiTorino,corsoDucadegliAbruzzi24,Torino,Italy} b_{INFNSezionediTorino,viaP.Giuria1,Torino,Italy}

c_{LaboratoriNazionalidiFrascatidell’INFN,via.E.Fermi 40,Frascati,Italy} d_{DepartmentofPhysics,SeoulNationalUniversity,151-742Seoul,SouthKorea}

e_{DipartimentodiIngegneriaMeccanicaeIndustriale,UniversitàdiBrescia,viaBranze38,Brescia,Italy} f_{INFNSezionediPavia,viaBassi6,Pavia,Italy}

g_{DipartimentodiFisica,UniversitàdiTorino,viaP.Giuria1,Torino,Italy} h_{DipartimentodiFisica,UniversitàdiTrieste,viaValerio2,Trieste,Italy} i_{INFNSezionediTrieste,viaValerio2,Trieste,Italy}

j_{INFNSezionediBari,viaAmendola173,Bari,Italy}

k_{DipartimentodiFisica, UniversitàdiBari,viaAmendola173,Bari,Italy} l_{DepartmentofPhysics,KyotoUniversity,Sakyo-ku,Kyoto, Japan} m_{RIKEN,Wako,Saitama351-0198,Japan}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received23July2014

Receivedinrevisedform3October2014 Accepted9October2014

Availableonline14October2014 Editor:D.F.Geesaman

Keywords:

Λ-Hypernuclei

Two-nucleonandproton-induced Non-MesonicWeakDecay width

PreviousstudiesofprotonandneutronspectrafromNon-Mesonic WeakDecayofeightΛ-Hypernuclei ( A=5–16)havebeenrevisited.Newvaluesoftheratioofthetwo-nucleonandtheone-protoninduced decaywidths,Γ2N/Γp,areobtainedfromsingleprotonspectra,Γ2N/Γp=0.50±0.24,andfromneutron and proton coincidence spectra, Γ2N/Γp=0.36±0.14stat+

0.05sys

−0.04sys, in full agreement with previously

publishedones.Withthesevalues,amethodisdevelopedtoextracttheone-protoninduceddecaywidth inunitsofthefreeΛdecaywidth,Γp/ΓΛ,withoutresortingtoIntraNuclearCascademodelsbutby

exploitingonlyexperimentaldata,undertheassumptionofalineardependenceonA oftheFinalState Interaction contribution.Thisis thefirstsystematicdetermination everdoneand itagreeswithinthe errorswithrecenttheoreticalcalculations.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3.

1. Introduction

Λ

-Hypernuclei (Hypernuclei in the following) decay through Weak Interaction to non-strange nuclear systems following two

modes, the mesonic (MWD) and the non-mesonic (NMWD) one.

TheMWDisfurthersplitintotwo branchescorresponding tothe decaymodesofthe

Λ

infreespace:

*

Corresponding author at: Dipartimento di Fisica, Università di Torino, via P. Giuria 1,Torino,Italy.

E-mailaddress:botta@to.infn.it(E. Botta).

1 _{NowatCEA/SACLAY,DSM/Irfu/SACMF-91191Gif-sur-YvetteFrance.}

A

ΛZ

→

A

(

Z

+

1)

+

π

−

(Γ

π−

)

(1)

A

ΛZ

→

AZ

+

π

0

(Γ

π0

).

(2)

A

ΛZ indicates theHypernucleus withmassnumber A and atomic

number Z , A

(

Z

+

1

)

and AZ theresidual nuclearsystem, usually the daughter nucleus in its ground state, and the

Γ

’s stand for thedecaywidths.Sincethemomentumreleasedtothenucleonin MWD (p

∼

100 MeV

/

c,QMWD

∼

37 MeV) ismuchlower than the

Fermi momentum, theMWD is strongly suppressed by the Pauli exclusion principlein all butthe lightest Hypernuclei. In NMWD the Hypernucleus decays through Weak Interaction involving the constituent

Λ

andoneormorecorenucleons.Theimportance of

http://dx.doi.org/10.1016/j.physletb.2014.10.024

0370-2693/©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

500 The FINUDA Collaboration / Physics Letters B 738 (2014) 499–504

suchprocesseswaspointedoutforthefirsttimein[1].Ifthepion emittedintheweakvertex

Λ

→

πN is

virtual,then itcanbe ab-sorbedbythenuclearmediumgivingoriginto:

A ΛZ

→

A−2

(

Z

−

1)

+

n

+

p

(Γ

p

),

(3) A ΛZ

→

A−2Z

+

n

+

n

(Γ

n

),

(4) A ΛZ

→

A−3

(

Z

−

1)

+

n

+

n

+

p

(Γ

2N

).

(5)

The processes (3) and (4) are globally indicated as one-nucleon induceddecays(one-proton

(3)

,one-neutron

(4)

)while

(5)

as two-nucleon induced decay. By neglecting

Λ

weak interactions with nuclearclustersofmorethantwonucleons,thetotalNMWDwidth is:

Γ

NMWD

= Γ

p

+ Γ

n

+ Γ

2N

.

(6)

Thetwo-nucleoninducedmechanism

(5)

wasfirstsuggestedin

[2]

andinterpretedby assumingthat thevirtual pionfromtheweak vertexisabsorbedbyapairofnucleons(np, pp ornn),correlated by the stronginteraction. In (5)we have indicated forsimplicity onlythe mostprobableprocess involving np pairs.Notethat the

NMWD can also be mediated by the exchange of mesons more

massivethanthepion.

TheNMWDmodeispossibleonlyinnuclei;theQ-valueofthe elementary weak reactions driving the decays (3), (4) and (5) is highenough( QNMWD

∼

175 MeV) toavoid anyPauliblocking

ef-fectandthefinalnucleonsthushavealargeprobabilitytoescape from the nucleus. Indeed it is expected that NMWD dominates over MWD for all but the s-shell Hypernuclei andonly for very light systems the two decaymodes are expectedto be competi-tive. The total decay width ofa Hypernucleus,

Γ

T, is thus given

by:

Γ

T

= Γ

π−

+ Γ

π0

+ Γ

p

+ Γ

n

+ Γ

2N

.

(7) In orderto compare data fromdifferent Hypernuclei, partial

Γ

’s are usually given in units of

Γ

Λ, the total decay width of the

free

Λ

.

The NMWD of Hypernuclei has been scarcely studied until a fewyearsago. Experimentallyitisnot onlynecessarytoproduce andto identify Hypernuclei in their ground state by means of a performing magnetic spectrometer, but also to detect in coinci-dencethenucleonsemittedin

(3)

,

(4)

and

(5)

andtomeasuretheir energy.Furthermore,thereisa bigdifficultyinextractingthe ob-servables relatedto theabove mentioned weak processes dueto thestrongdistorsionintroducedbyFinalStateInteraction(FSI)on the spectraof the nucleons emitted in the elementaryprocesses corresponding to (3), (4), (5).The informationon theinitial bare momentamaybe completely lost,andtheir contributionscan be mixed, withpossible additionalquantum-mechanicalinterference effects

[3]

.

Afterthefirstpioneeringexperiments

[4,5]

,theSKS Collabora-tion measured atthe 12GeV KEK PSthe spectra ofprotons and neutronsfromtheNMWD of5_ΛHeand12

ΛC producedby the(

π

+,

K+)reactionat1

.

05 GeV

/

c[6].AttheDA

Φ

NE(e+,e−)colliderthe FINUDA Collaboration measured the spectra of protons and neu-trons from 5

ΛHe andfrom 7 p-shell Hypernuclei. A full account

ofall the papers onthe subjectmaybe found in [7]. Many the-oreticalpapers havealso beenproduced,motivatedbythestrong interestanddiscovery potentialofthestudyofNMWDof Hyper-nuclei;theyarelisted

[7]

aswell.

2. A revisited analysis of the proton spectra from FINUDA

Inan earlypaper [8]proton spectrafromNMWDof5_ΛHe, 7_ΛLi and12

ΛC,measuredbyFINUDA,werepresentedanddiscussed.

Asasecondstep,protonspectrafromNMWDof9_ΛBe,11

ΛB,

13

ΛC,

15

ΛN and 16ΛO were produced andanalyzed [9]; the experimental

energyresolutionwas



E

/

E

=

2% at80 MeV.

Fig. 1

showsallthe experimental spectra;they areup tonowa uniquedatabankfor p-shellHypernucleiwith A

=

5–16,fromwhichseveralinteresting considerationsandconclusionsweredrawn.

In [9]we developeda methodfordisentangling the contribu-tionsfromthedecay

(5)

withoutusingIntraNuclearCascade(INC) calculations,asin

[10,11]

.Thefirststepwastofittheeight exper-imentalspectraof

Fig. 1

above80 MeV(

∼

QNMWD

/

2) toGaussians

withfreecentralvalues,widthsandareas.In

Table 1

thevaluesof theGaussian centers,

μ

0,withtheirstatisticalerrorarereported.

Recently it was outlined in [12] that the values of

μ

0 for 13

ΛC,

15

ΛN and

16

ΛO were significantly larger than those calculated

following the relativistic kinematics with the exact Q-values for the considered eight decays (3), in the hypothesis of a back-to-back emission of the proton–neutron pair with no recoil of the residualnucleusinitsgroundstate;they arereportedinthe sec-ond column of Table 1 and will be labeled

ρ

in the following. The reduced

χ

2 _of

_μ

0 values withrespect to

ρ

ones,

χ

2

/

ndf

=



8

i=1

(

μ

0i

−

ρ

i

)

2

/

16

σ

μ20i,was1.88.

In the present work, we check whetherbetter results can be obtainedbyshiftingdowntheloweredgeofthefittingintervalof the experimentalspectra.The considered

χ

2 _{is minimized inthe} hypothesisofnorecoiloftheresidualnucleus. Ingeneral, the de-cayhappensobeyingtobothenergyandmomentumconservation andthe kinetic energyofthe daughternucleusisnegligible only forhighermasses:indeed,in

[12]

,itwasfoundthatforthelighter nuclei,5_ΛHeand7

ΛLi,arecoilmomentumof

∼

200 MeV

/

c allowsto

reproducetheobtained

μ

0 valueswhilethecorresponding

ρ

val-ues arenot compatiblewithin theerrors. Nevertheless,we chose to minimize thisparticular

χ

2 _{toinvestigatethe highermass} re-gion keepinginmind that inthe lower massregion onlyloosely boundornotboundlightdaughternucleiareproduced.

For70MeV wefindareduced

χ

2 _{of1.33,for60MeVof1.86} and for50 MeV of 3.61. We conclude that the mostappropriate choiceistofitallexperimentalspectrastartingfrom70MeV. We discardstartingfrom50MeVandweconsidertheoptionsof start-ingfrom60and80MeVtoestimatesystematicerrors:theirvalue is

≤

3

.

5% up to 13_ΛC and increases to 4.5% for 15_ΛN and to 7.1% for 16

ΛO, where it is comparableto the statistical error. The new

Gaussian centralvalues,

μ

1,are reportedinthefourthcolumnof

Table 1,whereasinthefifthcolumnthevaluesofthe correspond-ingwidthsandinthesixthcolumntheGaussiancentralvaluesfor fitfrom60MeV,

μ

2,arereported.Thequotederrorsarestatistical only.

Forthe sake of completeness,we observethat itis notuseful to try to fit the proton spectra starting from values higher than 80 MeV becauseonly very fewpoints would be usedandbigger errors would be obtainedon the Gaussiansparameters. In addi-tion, inorder toobtain satisfyingfits it isnecessary to constrain the Gaussian central valuesin quite smallranges,while it isnot necessarytoconstrainthemintheconsideredfitintervals.

We noticethat the widthsfound for5

ΛHe and12ΛCare

consis-tentwiththoseevaluatedtheoreticallyasduetotheFermimotion

[3].ThenewfittingGaussiansarerepresentedbythesolidlinesin

Fig. 1.

Themostrelevantissuefromthisrevisitedanalysisisthatnew valuesfortheareasofthe upperhalf ofthefittingGaussiansare evaluated, withimpacton therelatedphysics itemsthat are dis-cussedinthefollowing.

FINUD A C ollabor a tion / Ph y sics Le tt ers B 738 (2014) 499–504 501

Fig. 1. (Coloronline.) ProtonkineticenergyspectrafromtheNMWDof(fromlefttorightupanddownrows):5

ΛHe, 7 ΛLi, 9 ΛBe, 11 ΛB, 12 ΛC, 13 ΛC, 15 ΛNand 16

ΛO.Thecurvesrepresentthenewanalysisgaussianfitstothespectra:the

solidlinepartindicatestheactualfitregion,thedashedpartindicatestheoneprotoninducedNMWDcontributiontothelowerenergyspectrumpart.Thebluefilledareaisthehigherenergyhalfgaussianarea,wherethe two-nucleoninducedNMWDisnegligible.

502 The FINUDA Collaboration / Physics Letters B 738 (2014) 499–504

Table 1

Kinematicsand Gaussianfitparameters.Firstcolumn:Hypernucleus;second col-umn:protonkineticenergy,ρ,froma2-bodykinematicsofoneprotoninduced NMWD, with no daughternucleus recoil (see the text for more details); third column:Gaussianfit meanvalue from [9], μ0;fourthcolumn:present analysis Gaussianfitmeanvalue,μ1;fifthcolumn:presentanalysisGaussianfitstandard deviation,σ1;sixthcolumn:Gaussianfit meanvalue startingform 60MeV, μ2. Statisticalerrorsonlyarequoted.

ρ (MeV) μ0 (MeV) μ1 (MeV) σ1 (MeV) μ2 (MeV) 5 ΛHe 76.65 68.5±4.1 66.9±11.8 22.3±9.9 65.0±16.9 7 ΛLi 82.99 76.7±5.2 74.9±3.8 18.0±2.1 77.7±2.9 9 ΛBe 76.48 78.2±6.2 77.7±9.1 20.8±10.8 77.3±3.8 11 ΛB 79.72 75.1±5.0 71.7±10.8 23.8±5.5 70.0±6.3 12 ΛC 78.36 80.2±2.1 77.3±2.9 22.0±2.1 79.9±2.2 13 ΛC 74.44 83.9±12.8 81.6±5.8 22.6±3.5 82.8±3.1 15 ΛN 77.55 88.1±6.2 84.2±4.5 18.6±2.8 80.6±3.3 16 ΛO 78.25 93.1±6.2 85.0±6.8 21.9±3.5 81.0±5.7 3. A refined determination of Γ2N

/Γ

NMWD

In

[9]

atechniquewas devisedto disentanglethe contribution comingfrom the2Ninduced decays (5)fromthe one-proton in-duceddecaysaffectedby FSIby exploitingthesystematicsinthe massrangeA

=

5–16.Eachspectrumof

Fig. 1

wasdividedintotwo parts, one below the value

μ

0,with area Alow, the other above,

withareaAhigh.Sincewefindthatthenewcurves,centeredat

μ

1, provideabetterdescriptionoftheexperimentalspectra,we calcu-latethe newvaluesof Alow and Ahigh (bluefilledareasin

Fig. 1

).

Newvaluesoftheratio R

=

Alow

/(

Alow

+

Ahigh

)

arefoundandwe

repeatthen exactlythe sameprocedureasin

[9]

.Finally wefind thenewvalues:

Γ

2N

/Γ

p

=

0.50

±

0.24stat

±

0.04sys (8)

and

Γ

2N

/Γ

NMWD

=

0.25

±

0.12stat

±

0.02sys (9)

by using the value

Γ

n

/Γ

p

=

0

.

48

±

0

.

08, weighted average (w.a.

fromnowon)takenfromthedatain

[13]

.Werecallthatthemain assumptions in the above procedure are a linear dependence of theFSI contributionon A andtheconstancy ofboth

Γ2N

/Γ

NMWD

and

Γ

n

/Γ

p forHypernucleiintherange A

=

5–16 under

consider-ation, as discussed in [14]. We remark that the new values are fully consistent with the previous ones (

Γ2N/Γ

p

=

0

.

43

±

0

.

25,

Γ2N

/Γ

NMWD

=

0

.

24

±

0

.

10);the smallerrelative erroron

Γ2N/Γ

p

isduetothelarger Ahighintegralobtainedwiththenewfitsfrom

70 MeV,whiletheerroron

Γ2N/Γ

NMWDisdominatedbytheerror

on the w.a. from [13]. The systematicerror quoted in (8) refers to the maximum difference obtained considering the fits from 60 MeV,

Γ2N/Γ

p

=

0

.

42

±

0

.

21,and80MeV,

Γ2N/Γ

p

=

0

.

43

±

0

.

25 [9]too;in

(9)

thesystematicerroriscalculatedbypropagatingthe previousone.

Inasecondapproach

[15]

we determined

Γ2N

/Γ

NMWD by

con-sidering both protons and neutrons emitted in coincidence with the

π

− fromtheformationreactionofHypernuclei.Werepeatthe sameprocedureanddefineforeachHypernucleustheratioR1 as:

R1

≡

Nn

(

Ep

≤ (

μ

1

−

20 MeV),cos

θ (

np

)

≥ −

0.8)

Np

(

Ep

>

μ

1

)

(10) where Nn

(

Ep

≤ (

μ

1

−

20 MeV

),

cos

θ (

np

)

≥ −

0

.

8

)

is the number of neutrons in coincidence with a proton of energy lower than

μ

1

−

20 MeV andformingananglewiththeprotondirectionsuch ascos

θ (

np

)

≥ −

0

.

8,while Np isthenumberofprotonswith

en-ergylargerthan

μ

1(blueareasin

Fig. 1

).Theseevents

[15]

should

Fig. 2. (Coloronline.) R1=Nn(Ep≤ (μ1−20MeV),cosθ (np)≥ −0.8)/Np(Ep>μ1) valuesasafunctionofA for5ΛHe,

7 ΛLi, 9 ΛBe, 11 ΛB, 12 ΛC, 13 ΛC, 15 ΛNand 16 ΛOfromthe present analysis.Thebluelineisalinearfit tothedata; seethetext for more details.

correspond mainlytotheprocess

(5)

plusa notnegligible contri-butionduetoFSI.

In Fig. 2 the newexperimental values of R1 foreach Hyper-nucleus are plotted asa function of A. By a simple linear fit to

(

a

+

b A

)

(blueline inFig. 2) we findthe valuesa

=

0

.

58

±

0

.

23, b

= −

0

.

017

±

0

.

090 with

χ

2

_/

_ndf

₌

₁

_.

₀₄₅

_/

_{6 and then, following} theapproximationsadoptedin

[15]

:

Γ

2N

Γ

p

=

[

R1

(

A

)

−

b A

]

1.6

=

a 1.6

=

0.36

±

0.14stat +0.05sys −0.04sys

.

(11) Furthermore

Γ

2N

/Γ

NMWD

=

0.20

±

0.08stat₋+₀0._.04₀₃sys_sys

.

(12)

The new estimations (11) and (12) agree well with the

pre-vious ones [15] (

Γ2N/Γ

p

=

0

.

39

±

0

.

16stat+ 0.04sys

−0.03sys,

Γ2N

/Γ

NMWD

=

0

.

21

±

0

.

07stat+ 0.03sys

−0.02sys) andare inagreementwithrecent

theoreti-cal predictions [16] andthe resultfromKEK [10] (

Γ2N/Γ

NMWD

=

0

.

29

±

0

.

13). The systematic error quoted in (11) contains also thecontribution,

±

0

.

015,duetothemaximumdifferenceobtained consideringthefitsfrom60MeV,

Γ2N

/Γ

p

=

0

.

37

±

0

.

14stat+

0.04sys

−0.03sys,

and80MeV,

Γ2N

/Γ

p

=

0

.

39

±

0

.

16stat+ 0.04sys

−0.03sys [15]too;in

(12)

the

systematicerroriscalculatedbypropagatingthepreviousone. Wefinallyrecallthattheanalysisoftheneutron–proton coinci-dencesallowedustofindthreecandidateeventsforthe2N decay with full reconstruction of thefinal state particles

(

n

,

n

,

p

)

kine-matics

[17]

.

4. First determination of Γp

/Γ

Λfor eight hypernuclei (A

=

5–16)

The previous studies [9,15] demonstrated that the higher en-ergypartofthebumpobservedaround80MeVforallthe exam-inedHypernucleiisduetothedecay

(3)

,eventhoughsignificantly distortedby FSI.Inordertoquantify

Γ2N

/Γ

p,theimportant

con-tributionduetoFSIwasparametrizedbyresortingtothe measure-mentofrelativequantities,theratiosR and R1.

On the contrary, in order to deduce the absolute values of

Γ

p

/Γ

Λ fromthe measured spectrait isnecessary to calculateas

accuratelyaspossibletheeffectiveinfluenceoftheFSIeffect.More indetail:

a) therealnumberofprimary protonsduetodecays

(3)

and

(5)

b) thereisanincreaseofthenumberofprotonsduenotonlyto FSIofprotonsathigherenergyinthespectrum,butalsotoFSI ofhigherenergyneutronsfrom

(4)

;

c) quantum-mechanical interference effects may occur among protons of the same energy from the different sources (pri-maryfrom

(3)

and

(5)

,secondaryfromFSI).

Alltheseeffectsmaybeevaluated byappropriateandpreciseINC calculations,asdonein

[3]

andin

[11]

.

We tryto evaluate the effect ofthe FSI on our spectra with-out using INCcalculations butexploiting only experimental data andsimple hypotheses. If we consider the portions ofthe spec-tra abovethe

μ

1 values(blue areasin Fig. 1), the importanceof theeffect b) maybesafelyneglected,following

[3]

.The contribu-tionofthedecay

(5)

above70MeVisnotlargerthan5%of

Γ

NMWD [3], and, consideringour determination

(11)

, thetotal amountof primary protons from (5) wouldnot be larger than 2% of those from(3).Thenalsotheinterferenceeffectc) maybeneglected.

We parametrizethen the effect a) by means ofthe following relationship:

Γ

p

Γ

Λ

=

Γ

T

Γ

Λ BR

(

p

)

=

Γ

T

Γ

Λ 2(Np

−

N2N

)

+

α

(

Np

−

N2N

)

Nhyp (13) whereBR

(

p

)

isthebranchingratioof

(3)

,Np isthenumberof

pro-tonsinthehigherenergyhalfpartofthefittingGaussian,N2N the numberofprotonsfrom

(5)

(about2%), Nhyp thenumberof

pro-ducedHypernuclei,the factor2 takesinto account thetotal area ofthe Gaussians and

α

isa coefficient to be determined, which accountsforthenumberofprotons moved below

μ

1 duetoFSI. Moreprecisely

α

/(

2

+

α

)

isthefractionofprotonsaffectedbyFSI. To calculate

α

for the considered Hypernuclei,

Γ

p

/Γ

Λ values

for5_ΛHeand12_ΛCareconsideredandalinearscalinglawwith A is assumedfortheFSI contribution,andconsequentlyfor

α

.

Γ

p

/Γ

Λ

for5_ΛHeand12_ΛCcanbeevaluatedfrom

(7)

,explicitly:

Γ

T

Γ

Λ

=

Γ

_π−

Γ

Λ

+

Γ

_π0

Γ

Λ

+

Γ

p

Γ

Λ

+

Γ

n

Γ

p

·

Γ

p

Γ

Λ

+

Γ

2N

Γ

p

·

Γ

p

Γ

Λ

,

(14)

by means of the value of

Γ2N

/Γ

p given by (11) and other

ex-perimental values existing in the literature. More precisely for 5

ΛHe,by substituting theexperimental values of

Γ

T

/Γ

Λ

=

0

.

96

±

0

.

03 (w.a. of [4,18]),

Γ

n

/Γ

p

=

0

.

45

±

0

.

11

±

0

.

03[19],

Γ2N/Γ

p

=

0

.

36

±

0

.

14stat+ 0.05sys

−0.04sys (11),

Γ

π−

/Γ

Λ

=

0

.

34

±

0

.

02 (w.a. of [4,18, 21]),

Γ

_π0

/Γ

_Λ

=

0

.

20

±

0

.

01 (w.a. of [4,22]), we obtain

Γ

p

/Γ

Λ

=

0

.

22

±

0

.

03,tobecomparedwith0

.

21

±

0

.

07 givenin

[4]

. For12_ΛCweuse

Γ

T

/Γ

Λ

=

1

.

22

±

0

.

04 (w.a.of

[18,23]

),

Γ

n

/Γ

p

=

0

.

51

±

0

.

13

±

0

.

05 [10],

Γ2N

/Γ

p

=

0

.

36

±

0

.

14stat+₋0_0..₀₄05sys_sys (11),

Γ

π−

/Γ

Λ

=

0

.

12

±

0

.

01 (w.a.of

[4,5,24,25]

),

Γ

π0

/Γ

Λ

=

0

.

17

±

0

.

01

(w.a.of

[22,26]

), andwe obtain

Γ

p

/Γ

Λ

=

0

.

49

±

0

.

06,tobe

com-pared withthe values 0

.

31

±

0

.

07 given in [5] and 0

.

45

±

0

.

10 givenin

[10,25]

.

Forthe sake ofclarity, it mustbe observed that the available experimentaldeterminationsof

Γ

p

/Γ

Λ for5ΛHe

[4]

and12ΛC

[5,10,

25]were not used directlyin

(13)

tocalculate theFSI correction factor

α

:thosevalues,infact,wereobtainedtreatingFSIwiththe helpofINCcalculationsorsimulationsandcouldincreasethe sys-tematicerrorsinourFSIeffectevaluation.

Withtheindirectvaluesof

Γ

p

/Γ

Λfor5_ΛHeand12_ΛCwemay

ob-tainfrom

(13)

twoevaluationsfor

α

,byusingtheabovereported valuesof

Γ

T

/Γ

Λ andtheexperimental valuesof Np (70MeV fit)

andNhyp.

We find

α

5(5_ΛHe

)

=

1

.

15

±

0

.

26 for 5_ΛHe (indicated as sub-script) from5

ΛHe measurements (indicated betweenparentheses)

Table 2

Firstcolumn:hypernucleus;secondcolumn:totaldecaywidthΓT inunitsofthe

freeΛdecaywidthΓΛ;thirdcolumn:αfactor;fourthcolumn:presentevaluation ofΓp/ΓΛ;fifthcolumn:previousmeasurements;sixthcolumn:recenttheoretical calculationofΓp/ΓΛ[27].

ΓT/ΓΛ αA Γp/ΓΛ Γp/ΓΛ Γp/ΓΛ

this work previous works [27] 5 ΛHe 0.96±0.03 1.08±0.16 0.22±0.05 0.21±0.07 0.237 [4,18] [4] 7 ΛLi 1.12±0.12 1.51±0.22 0.28±0.07 0.297 9 ΛBe 1.15±0.13 1.94±0.28 0.30±0.07 0.401 11 ΛB 1.28±0.10 2.37±0.34 0.47±0.11 0.30±0.07 0.444 [5] 12 ΛC 1.242±0.042 2.58±0.37 0.65±0.19 0.31±0.07 0.535 [18,23] [5] 0.45±0.10 [25] 13 ΛC 1.21±0.16 2.80±0.40 0.60±0.14 0.495 15 ΛN 1.26±0.18 3.23±0.47 0.49±0.11 0.555 16 ΛO 1.28±0.19 3.44±0.50 0.44±0.12 0.586

and

α

12(12_ΛC

)

=

2

.

48

±

0

.

46 for12_ΛCfrom12_ΛCmeasurements.By as-sumingthat

α

scaleslinearlywithA,itisstraightforwardtoobtain the crossed evaluations:

α

5(12_ΛC

)

=

1

.

04

±

0

.

19 and

α

12(5_ΛHe

)

=

2

.

77

±

0

.

63.Thew.a. ofthetwoevaluationsare

α

5

=

1

.

08

±

0

.

16 and

α

12

=

2

.

58

±

0

.

37. We adopt finally the general expression for

α

A:

α

A

=

α

5 5

·

A

=

α

12 12

·

A

= (

0.215

±

0.031)

·

A (15) where the statisticalerrorcomes from the errors onthe quanti-ties used to evaluate

α

. A systematic error can be evaluated by takinginto accountthe difference between

α

5(_Λ5He

)

and

α

5(12_ΛC

)

for 5_ΛHe and between

α

12(12_ΛC

)

and

α

12(5_ΛHe

)

for 12_ΛC: this er-ror amounts to 6%. It is also worth to observe that in [9] and

[15] the assumption

Γ2N

 Γnp

was made, which gives a

sys-tematicunderestimation of

Γ2N

/Γ

p of

∼

16% (muchsmallerthan

the experimental errors): we remind, indeed,that following [20]

Γ

np

: Γpp

: Γ

nn

=

0

.

83

:

0

.

12

:

0

.

04.Ifthissystematiceffectistaken

into account in the calculation of

α

from (14) and (13), a de-creaseof

(

9–5

)

% ariseswhichgives afurthersystematicerror on

α

(

A

=

5–16

)

,foratotalof

(

10–7

)

%.

We remark that the hypothesis that FSI effects are to a first approximation proportional to A was already adopted in [9,15]. With

(15)

we findthat 35% ofthe primary protonsfrom NMWD arelost(movedbelow

μ

1)forFSIin5_ΛHeand63%in16_ΛO.

We are thus able to determine with Eq. (13) the values of

Γ

p

/Γ

Λfor5ΛHeandallstudiedp-shellHypernuclei.Theyaregiven

in

Table 2

,which reportsalso theexperimental valuesof

Γ

T

/Γ

Λ

we used,when available;fortheother Hypernucleiwe adoptthe parametrization

Γ

T

/Γ

Λ

(

A

)

= (

0

.

990

±

0

.

094

)

+ (

0

.

018

±

0

.

010

)

·

A

proposed in [21].The errors on

Γ

p

/Γ

Λ arecalculated by

consid-ering statistical errors for Np and Nhyp, the errors reported in

columntwofor

Γ

T

/Γ

Λ

(

A

)

andthe statisticalerrorfor

α

A,which

isthelargest one(15%).

Table 2

reports,inthe sixthcolumn, the theoreticalvaluesof

Γ

p

/Γ

Λcalculatedrecentlyin

[27]

.

Itis interestingto notethat thenewevaluation,following the determination of

α

A given by (15), and the former indirect

cal-culation for both 5

ΛHe and 12ΛC from (14) are compatible within

the error: thus it appears that the method used to evaluate

α

A

doesnot introduceany systematicerror inthe

Γ

p

/Γ

Λ value. On

theotherhand,asystematicerroron

Γ

p

/Γ

Λ canbeestimatedby

repeating all the previous procedure with the fits from 60 MeV andfrom80MeV:itcorresponds to5–6% for5_ΛHe, 7_ΛLi,9_ΛBe, 16_ΛO and 9–10% for 11

504 The FINUDA Collaboration / Physics Letters B 738 (2014) 499–504

Fig. 3. (Coloronline.) a)Γp/ΓΛ valuesasafunctionofA for5ΛHe, 7 ΛLi, 9 ΛBe, 11 ΛB, 12

ΛC,13ΛC,15ΛNand16ΛOfromthepresentanalysis(bluestars,errorsfromTable 2). TheoreticalcalculationsofΓp/ΓΛ[27](violetsquares)arealsoshown.Γp/ΓΛfrom

[4]for 5

ΛHe(brownfullcircle),from[5] forΛ11Band12ΛC(greenfullcircles)and from[25]for 12

ΛC(orange fullcircle) havealso beenplotted.b) Γπ−/ΓΛ values asafunctionofA for5 ΛHe, 7 ΛLi, 9 ΛBe, 11 ΛBand 15

ΛN(redstars)from[21]andfor 12

ΛC(orangecross)from[25].TheoreticalcalculationsofΓπ−/ΓΛfor5ΛHe(grayup triangle)from[28],for7

ΛLi, 9 ΛBe, 11 ΛB, 12 ΛCand 15

ΛN(downviolettriangles)from[27] andfor5 ΛHe, 7 ΛLi, 9 ΛBe, 11 ΛBand 15

ΛN(cyandiamonds)from[29]arealsoreported.

on

Γ

p

/Γ

Λ can be obtained from the systematic error on

α

(

A

)

:

itamountsto

(

3–3

.

5%

)

only;thetotalsystematicerroramountsto 6–7% for5_ΛHe,7_ΛLi,9_ΛBe,16_ΛOand10–11% for11_ΛB,12_ΛC,13_ΛCand15_ΛN andissmallcomparedtothestatisticalone.

Fig. 3a)showsthecomparisonamongthevaluesfromthis ex-periment (blue stars), previous data [4,5,25] (brown full circle, greenfull circles andorangefull circlerespectively) and theoret-icalvalues

[27]

(violetsquares);inthefigurestatisticalerrorsonly areindicated onthepresentresultstomatchwithprevious data. Ageneralagreementbetweenourdataandthetheoreticalonesis evident,eventhoughtheexperimentalerrorsarequitelarge,with the exception of9

ΛBe, which is lower by 1

.

5

σ

andof 16ΛO which

islowerby 1

.

3

σ

.

Fig. 3

b)showstheexperimentaldeterminations of

Γ

π−

/Γ

Λ from

[21]

(red stars) for5_ΛHe, 7_ΛLi, 9_ΛBe, 11_ΛB and15_ΛN

and from [25] (orange cross) for 12

ΛC; theoretical calculations of

Γ

π−

/Γ

Λfrom

[28]

(grayuptriangle)for5_ΛHe,from

[27]

(down

vi-olettriangles)for7

ΛLi, 9ΛBe, 11ΛB,Λ12Cand15ΛNandfrom

[29]

(cyan

diamonds)for5_ΛHe, 7_ΛLi,9_ΛBe,11_ΛBand15_ΛNarealsoindicated.The

purposeisto showthefirstexperimental verification,atleastfor the

Γ

’s relative to charged particles, ofthe long-time advocated

complementary behavior ofMWD andNMWD of Hypernuclei in

therelevant A range(5–16).

5. Conclusions

Wehavedeterminedthepartialdecaywidthsoftheone-proton inducedNMWDfrommeasuredprotonspectraforeight Hypernu-clei ( A

=

5–16):itisthefirstsystematicdeterminationeverdone for p-shell

Λ

-Hypernuclei.The measured values, though affected by errors ranging from20% to 30%, agree reasonablywith those predictedbyarecentprecisetheoreticalcalculation

[27]

.

To make a better comparison smallerexperimental errors are necessary, at a 10% level or less, on both

Γ

T

/Γ

Λ and Np (Nhyp)

(see (13)). At present, the Laboratory equipped with the beams and detector’sarrays necessary forthiskind of measurements is J-PARC. It is also necessary to develop a simple INC calculation, without two-nucleon induced contributions,to verifythe validity ofthehypothesisoflinearitywithA oftheFSIcorrection.

With all theserequirements satisfied, it could then be possi-ble to try to face the problem of the experimental study of the

Λ

N

→

N N WeakInteraction fromtheHypernucleardata.Indeed, aprecisestudyoftheNMWDofHypernucleiwillbetheonlyway to get information on the four-baryon weak process

Λ

N

→

N N andtorealizethentheideaofusinganuclearsystemasa “Labo-ratory”forthestudyofinteractionsbetweenelementaryparticles nototherwiseaccessibleinvacuo.

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