Antineutrino induced Lambda(1405) production off the proton

0.23 MB·English

by  Xiulei Ren

Antineutrino induced Λ(1405) production offtheproton
Xiu-Lei Ren,1,2,3, E. Oset,2, L. Alvarez-Ruso,4, and M. J. Vicente Vacas2,
∗ † ‡ §
1School of Physics and Nuclear Energy Engineering & International Research Center
for Nuclei and Particles in the Cosmos, Beihang University, Beijing 100191, China
2Departamento de F´ısicaTeo´ricaand IFIC,CentroMixto Universidad deValencia-CSIC,
Institutos de Investigacio´n de Paterna, Apartado 22085, 46071 Valencia, Spain
3InstitutdePhysiqueNucle´aire, IN2P3-CNRSandUniversite´ Paris-Sud, F-91406OrsayCedex,France
4Institutode F´ısicaCorpuscular (IFIC),Centro Mixto Universidad de Valencia-CSIC,
Institutos de Investigacio´n de Paterna, Apartado 22085, 46071 Valencia, Spain
Wehavestudiedthestrangenesschangingantineutrinoinducedreactionsν¯lp l+φB,withφB = K−p,
K¯0n, π0Λ, π0Σ0, ηΛ, ηΣ0, π+Σ−, π−Σ+, K+Ξ− andK0Ξ0, using achiral→unitaryapproach. Theseten
5
coupled channels are allowed to interact strongly, using a kernel derived from the chiral Lagrangians. This
1
interactiongeneratestwoΛ(1405)poles,leadingtoaclearsinglepeakintheπΣinvariantmassdistributions.
0
Atbackwardscatteringanglesinthecenterofmassframe,ν¯ p µ+π0Σ0isdominatedbytheΛ(1405)state
2 µ →
ataround1420MeVwhilethelighterstatebecomesrelevantastheangledecreases,leadingtoanasymmetric

r lineshape. Inaddition,therearesubstantialdifferencesintheshapeofπΣinvariantmassdistributionsforthe
p
threechargechannels. Ifobserved,thesedifferenceswouldprovidevaluableinformationonaclaimedisospin
A
I =1,strangenessS = 1baryonicstatearound1400MeV.Integratedcrosssectionshavebeenobtainedfor
theπΣandK¯N channel−s, investigatingtheimpactofunitarizationintheresults. Thenumberofeventswith
5
1 Λ(1405)excitationinν¯µpcollisionsintherecentantineutrinorunattheMINERνAexperimenthasalsobeen
obtained.Wefindthatthisreactionchannelisrelevantenoughtobeinvestigatedexperimentallyandtobetaken

] intoaccountinthesimulationmodelsoffutureexperimentswithantineutrinobeams.
h
p
-
p
e
I. INTRODUCTION
h
[

TheΛ(1405)resonanceisacornerstoneinhadronphysics,challengingthestandardviewofbaryonsmadeofthreequarks.
2
LongagoitwasalreadysuggestedthattheΛ(1405)couldbeakindofmolecularstatearisingfromtheinteractionoftheπΣand
v
3 K¯N channels[1,2]. Thisviewhasbeenrecurrent[3],butonlyaftertheadventofunitarychiralperturbationtheory(UChPT)
7 hasittakenamoreassertivetone[4–9]. Inthisframework,akernel(potential)derivedfromthechiralLagrangiansistheinput
0 intotheBethe-Salpeterequationincoupledchannels.Sometimestheinteractionisstrongenoughtogeneratepoles,denominated
4 asdynamicallygeneratedstates,whichcanbeinterpretedashadronicmoleculeswithcomponentsonthedifferentchannels(see
0
Ref.[10]forareview).
.
1 It came as a surprise that UChPT predicts two Λ(1405) states [6], studied in detail in Ref. [8]. Two poles appear, one
0 around1420MeV with a width of about40 MeV and anotherone around1385MeV with a larger width of about150MeV.
5 These findingshave been reconfirmedin more recentstudies with potentialsthat include higher orderterms of the chiral La-
1
grangians[11–17]. Fromtheexperimentalperspective,theoldexperiments[18,19]producedπΣinvariantmassdistributions
:
v whereasingleΛ(1405)peakisseenaround1405MeV.AccordingtoRef.[20],thissinglepeakresultsfromtheoverlapofthe
i twopolecontributions.IthasalsobeensuggestedthatreactionsinducedbyK ppairsshowapeakaround1420MeVbecause
X −
thepoleat1420MeVcouplesmostlytoK¯N,whiletheoneat1385MeVdoesitmorestronglytoπΣ. Thiswouldbethecase
r
a ofK−p γπΣ[21]andK−p π0π0Σ0. Thelatterone,measuredatCrystalBall[22]andanalyzedinRef.[23],confirmed
→ →
theexistenceofthestateat1420MeV.AnotherreactionthathasproveditsexistenceisK d nπΣ[24],whichwasstudied
−
→
inRef.[25]. TheissuesraisedinRef.[26]wereaddressedindetailinRef.[27]reconfirmingthefindingsofRef.[25].
ItissomewhatsurprisingthatthetwopolesemergeinthetheoryevenwhenonlydataonK pscatteringandK patoms[28],
− −
whichareabovetheΛ(1405)polemasses,arefitted. Nevertheless,itisclearthatthebestinformationontheΛ(1405)properties
should come from processes where the Λ(1405) is produced close to its pole masses. In this sense, the abundant Λ(1405)
photoproductiondataobtainedbyCLASwiththeγp K+π+Σ ,K+π0Σ0,K+π Σ+ reactions[29]addmuchinformation
− −
totheearlierdataofRef.[30],bringingnewlightinto→thesubject.AfittothesedataimposingunitarityintheπΣ,K¯N channels
andallowingonlysmallvariationsinthekernelofthechiralLagrangians[31,32]hasreconfirmedtheexistenceofthetwopoles,
in agreementwith theUChPT predictions. The widerangeofenergiesinvestigatedandthe simultaneousmeasurementof the
threeπΣchargedchannelswerethekeytothesolutionsfoundinRefs.[31,32]and,morerecently,inRef.[33].
E-mail:[email protected]
∗
E-mail:oset@ific.uv.es
†
E-mail:alvarez@ific.uv.es
‡
E-mail:vicente@ific.uv.es
§
2
Studiesofpp pK+Λ(1405)performedatANKEshowagainasuperpositionofthecontributionsfromthetwopoles[34],
→
andcanbeexplainedwiththetheoreticalframeworkofUChPT[35].Morerecentmeasurements[36,37]showtheΛ(1405)peak
atalowerenergythanintheANKEexperiment[34]. SomereasonsforthisbehaviorhavebeensuggestedinRef.[37]. Ifmore
dataforthisreactionondifferentconditionsbecameavailable,aglobalanalysisliketheoneofRef.[31,32]forphotoproduction
would be advisable. In between, Λ(1405) electroproduction[38] data [ep eK+Λ(1405)] have unexpectedlyrevealed a
′
→
two-peakstructure,albeitwithlargeuncertainties. Previousmeasurementswithdifferentreactionshaveonlyobservedasingle
peakcomingfromthesuperpositionofthetwopoles,withdifferentshapesdependingontheweightofeitherpole,asdetermined
bythedynamicsofeachprocess.
Lattice QCD simulations have also broughtnew light into the Λ(1405)properties. Using three-quarkinterpolators, a state
associatedwiththeΛ(1405)isproduced[39,40]. ThevanishingstrangequarkcontributiontotheΛ(1405)magneticmoment
forlightquarkmassesclosetothephysicaloneshasbeeninterpreted[41,42]asanevidenceofalargeK¯N componentinthe
wavefunctionoftheΛ(1405). FurtherworkalongtheselineswasreportedinRef.[43]usingsyntheticlatticeresultsfromK¯N
andπΣinterpolators. Theseleadtotherightdescriptionofthemeson-baryonamplitudesinthecontinuumandcontainthetwo
polesinthecomplexplane.
Untilnow,theweakexcitationofΛ(1405)hasneverbeeninvestigated.Itisremarkablethatwhileitsproductioninstrongand
electromagneticprocesseshastoinvolveanextrastrangeparticle(usuallyaK intheinitialstateoraK+inthefinalone),the
−
directexcitationofΛ(1405)inducedbyantineutrinosν¯p l+Λ(1405)isallowedalthoughCabibbosuppressed. Noticethat
l
inΛ(1405)photoandelectroproductiontherearelineshap→edistortionsduetofinalstateinteractionsbetweentheK+ andthe
Λ(1405)decayproducts,whichareabsentintheweakreaction.
Stimulatedbytheprecisionneedsofneutrinooscillationexperiments,thereisa significantongoingeffortaimedata better
understandingofneutrinocrosssectionswith nucleonsandnuclei. Thegoalis to developbetterinteractionmodelsto reduce
systematicerrorsinthedetectionprocess,constrainirreduciblebackgroundsandachieveabetterneutrinoenergydetermination.1
Intherecentpast,severalexperimentshaveproducedvaluablecrosssectionmeasurements(seeRef.[44]foracomprehensive
reviewoftheavailabledata).TheMINERνAexperiment[45,46]atFNAL,fullydedicatedtothestudyofneutrinointeractions
withdifferenttargetmaterialshasrecentlycompleteddatatakingandstartedtoproduceinterestingresults[47–50].
Inthefew-GeVenergyregion,whereseveralofthecurrentandfutureexperimentsoperate,quasielasticscatteringandsingle
pionproductionhavethelargestcrosssectionsbutstrangeparticleproductionisalsorelevant. Thecharged-current∆S = 1
−
quasielastic hyperon(Y = Λ,Σ) productionby antineutrinoshasbeen investigated[51–53] and foundto be a non-negligible
sourceofpionsthroughthe Y Nπ decay[51, 54]. Amongthe inelasticprocesses, associated (∆S = 0)productionof K¯
and Σ or Λ baryonsis the dom→inantone but has a high threshold. Below it, single K (∆S = 1) and single K¯ (∆S = 1)
−
can be producedin chargedcurrentinteractionsinducedby ν and ν¯ respectively. These processeshave beenrecentlystudied
usingSU(3)chiralLagrangiansatleadingorder[55,56]. Theweakhadroniccurrentsandthecorrespondingcrosssectionsat
thresholdareconstrainedbychiralsymmetrywithcouplingsextractedfrompionandhyperonsemileptonicdecays.Asstressed
inRef.[57],whilethederivedK productioncrosssectionisarobustpredictionatthreshold,thesituationcouldbedifferentfor
K¯ productionduetothepresenceoftheΛ(1405)resonancejustbelowtheK¯N threshold.Another,sofarunexplored,∆S = 1
reactionthatcanoccurbelowtheassociatedproductionthreshold,ν¯ p l+Σπ,isboundtogetanimportantcontributionfr−om
l
→
Λ(1405)excitation.
Here we reportthe firststudy ofthe antineutrinoinducedreactionsν¯p l+φB with φB = K p, K¯0n, π0Λ, π0Σ0, ηΛ,
l −
ηΣ0, π+Σ , π Σ+, K+Ξ , K0Ξ0 in coupled channels, paying special a→ttention to the role of the Λ(1405). In Sect. II we
− − −
describethetheoreticalframework.TheresultsarepresentedinSect.IIIfollowedbyourconclusions.
II. THEORETICALFRAMEWORK
A. EffectiveLagrangians
Attreelevel,theprocessν¯p l+φB, withφandB beingthemesonandbaryoninthefinalstate,proceedsasdepictedin
l
→
thediagramsofFig.1. Therearealsobaryon-poleterms(seeFig.1ofRef.[56])whichcontributepredominantlytothep-wave
stateoftheφB system. SinceouraimistogeneratetheΛ(1405),whichappearsinφB s-wave,weneglecttheseterms.
All mechanisms in Fig. 1 consist of a leptonic and a hadronic currents that interact via the exchange of a W boson. The
leptonicpartisprovidedbytheStandardModelLagrangian
= g ψ¯ γ (1 γ )ψWµ+ψ¯γ (1 γ )ψ W µ , (1)
ν µ 5 l l µ 5 ν †
L −2√2h − − i
1Neutrinobeamsarenotmonochromaticsothattheincidentenergyisnotknownforsingleevents. However,oscillationprobabilitiesarefunctionsofthisa
prioriunknownquantity.
3
l+ l+ l+ φ
φ φ
W
ν¯ W ν¯ ν¯ −
l − l l
W
−
K− φ′′
p (a) B p (b) B p (c) B
FIG.1. Feynmandiagramsfortheprocessν¯p l+φB. (a)denotesthekaonpoleterm(KP),(b)representsthecontactterm(CT),and(c)
l
→
standsforthemeson(φ′′)in-flightterm(MF).
whereψ ,ψ andW denotetheneutrino,chargedleptonandgaugebosonW fields,respectively;gisthegaugecoupling,related
ν l
totheFermiconstantbyG =√2g2/(8M2 )=1.16639(1) 10 5GeV 2.
F W × − −
ThehadroniccurrentisderivedfromchiralLagrangians[58–60]atleadingorder. Asmentionedabove,inthisworkweare
onlyconcernedaboutthes-wavecontribution. Inthemesonsector,requiredforCTandMFdiagrams,thelowestorderSU(3)
Lagrangianisgivenby
F2 F2
(2) = 0 D U(DµU) + 0 χU +Uχ , (2)
Lφ 4 h µ †i 4 h † †i
where ... standsforthe tracein flavorspace; F isthe pseudoscalarmesondecayconstantin thechirallimit. Thequantity
0
h i
χ = 2B , with the quark-massmatrix = diag(m ,m ,m ), representsthe explicitbreakingof chiralsymmetry. The
0 u d s
M M
functionU =exp(iφ/F )istheSU(3)representationofthemesonfields
0
π0+ 1 η √2π+ √2K+
√3
φ= √2π π0+ 1 η √2K0  , (3)
− − √3
 √2K− √2K¯0 −√23η 
anditscovariantderivativeD U canbewrittenas
µ
D U =∂ U ir U +iUl , (4)
µ µ µ µ
−
wherel andr correspondtoleft-andright-handedcurrents.Forthechargedcurrentweakinteraction
µ µ
g
r =0, l = (W T +W T ), (5)
µ µ √2 µ† + µ −
with
0 V V 0 0 0
ud us
T = 0 0 0 , T = V 0 0 . (6)
+ ud
−
 0 0 0   Vus 0 0 
   
Here, V are the relevant elements of the Cabibbo-Kobayashi-Maskawa matrix. Their magnitudes are V = cosθ =
ij ud c
| |
0.97425 0.00022and V =sinθ =0.2252 0.0009[61],withθ theCabibboangle.
us c c
± | | ±
ThelowestorderchiraleffectiveLagrangiandescribingtheinteractionbetweentheoctetofpseudoscalarmesonsandtheoctet
ofbaryonscanbewrittenas
D F
(1) = B¯(iD/ M )B + B¯γµγ u ,B + B¯γµγ [u ,B] , (7)
LφB h − B i 2h 5{ µ }i 2h 5 µ i
withthebaryonfieldsarrangedinthematrix
1 Σ0+ 1 Λ Σ+ p
√2 √6
B = Σ 1 Σ0+ 1 Λ n  ; (8)
− −√2 √6
 Ξ− Ξ0 −√26Λ 
4
M denotesthebaryonoctetmassinthechirallimit;D =0.804andF =0.463aretheaxial-vectorcouplingconstants,which
B
aredeterminedfromthebaryonsemi-leptonicdecays[62]. Thecovariantderivativeofthebaryonfieldisdefinedas
D B =∂ B+[Γ ,B], (9)
µ µ µ
1
Γ = u (∂ ir )u+u(∂ il )u , (10)
µ † µ µ µ µ †
2 − −
(cid:8) (cid:9)
andu isgivenby
µ
u =i u (∂ ir )u u(∂ il )u , (11)
µ † µ µ µ µ †
− − −
(cid:8) (cid:9)
whereu=√U.
B. ChiralUnitaryTheory
Asdiscussedintheintroduction,theΛ(1405)isdynamicallygeneratedbytheinteractionofS = 1s-wavemeson-baryon
−
pairsincoupledchannels.ThiscanbeachievedbysolvingtheBethe-Salpeterequationwiththeinteractionpotentialprovidedby
thechiralLagrangianofEq.(7).InthediagramsofFig.1,theoutgoingmesonandbaryoncaninteractproducingtheresonance.
Therefore,onemustconsiderthe diagramsdepictedin Fig.2. The solidsquarein thefiguresrepresentsthedifferentT
ij φB
amplitudes, where the pair of indices ij = K p, K¯0n, π0Λ, π0Σ0, ηΛ, ηΣ0, π+Σ , π Σ+, K+Ξ , K0Ξ0 denote an→y of
− − − −
thetenallowedchannels.
l+ l+ l+
φ φ φ
φ
ν¯l W− ν¯l W φ′ ν¯l W− ′
−
K− B φ′′ B
′ ′
p (a) B p (b) B p (c) B
FIG.2.Iteratedloopdiagramsforν¯ p µ+φB.ThesolidboxesrepresenttheT matrixofthetencoupledchannels.
µ
→
FollowingtheapproachofRef.[5]forthestronginteractionintheS = 1sector,
−
T =V +VGT =[1 VG] 1V , (12)
−
−
wherethelowest-orderinteractionamplitudeV,extractedfromthelowestorderchiralLagrangian (1),isgivenby
LφB
1
V = C (k0+k 0) (13)
ij − ij4F2 ′
φ
after a nonrelativisticreduction. Here, k0 and k 0 are the energiesof the incomingand outgoingmesonsin the φB center of
′
mass(CM)frame;F hasbeenreplacedbytheaveragevalueofthephysicaldecayconstantsF =1.15f withf = 93MeV
0 φ π π
asinRef.[5]. The10 10matrixofcoefficientsC canbefoundinTable1ofRef.[5].
ij
×
Themeson-baryonloopfunctionG isgivenby
ij
d4q M 1 1
j
G =i ,
ij Z (2π)4E (q~)k0+p0 q0 E (q~)+iǫq2 m2+iǫ
j − − j − i
d3q 1 M 1
= j , (14)
Z (2π)32ω (q~)E (q~)p0+k0 ω (q~) E (q~)+iǫ
j j i j
− −
wherem ,M arethephysicalmesonandbaryonmassesoftheijstatewhileω =(m2+~q2)1/2,E =(M2+~q2)1/2arethe
i j i i j j
correspondingenergies. ItisafunctionoftheCMenergyM = p0+k0. InRef.[5],theloopfunctionisregularizedwitha
inv
cutoffq =630MeV.
max
5
C. Crosssection
Thereactionunderconsiderationis
ν¯(k )+p(p) l+(k )+φ(k )+B(p), (15)
l ν¯ l ′ ′
→
wherek =(k0,~k )[k =(k0,~k )]isthe4-momentumoftheincomingneutrino[outgoingchargedlepton]whilep=(E ,p~),
ν¯ ν¯ ν¯ l l l p
p = (E ,~p )andk = (ω ,~k )denotethemomentaoftheinitialproton,finalbaryonandfinalmeson,inthisorder. Itscross
′ B ′ ′ φ ′
sectionisgivenby
2M m d3k m d3k 1 d3p M
σ = p ν¯ l l ′ ′ B(2π)4δ4(p+k k k p) t2, (16)
λ1/2(s,m2,M2)Z (2π)3 k0 Z (2π)32ω Z (2π)3 E ν¯− l− ′− ′ | |
ν¯ p l φ B X
whereλ(x,y,z)=x2+y2+z2 2xy 2xz 2yzands=(p+k )2; denotesthesumoverfinalstatepolarizationsand
ν¯
− − −
averageovertheinitialones. ItisconvenienttoperformtheintegralsoverPp~ and~k intheφB CMframe,takingadvantageof
′ ′
the fact that the amplitudeis projectedontothe s-wave state of the φB pair. The last integrationover~k is carried outin the
l
global(ν¯p)CMframe.Weobtain
σ = 2 mν¯mlMpMB √s−ml dM +1dcosθ~k ~k t2, (17)
(2π)3 √s(s M2) Z invZ | l|ν¯p| ′|φB | |
− p mφ+MB −1 X
whereθistheanglebetween~k and~k intheν¯pCMframe.InEq.(17)
l ν¯
λ1/2(s,m2,M2 ) λ1/2(M2 ,m2,M2)
~k = l inv , ~k = inv φ B (18)
| l|ν¯lp 2√s | ′|φB 2M
inv
arethecharged-leptonmomentumintheν¯pCMframeandthemesonmomentumintheφB CMframe,respectively.
D. Invariantamplitude
Inthe(k k )2 q2 M2 limit,theamplitudecanbecastas
l− ν¯ ≡ ≪ W
it=2G V LµH , (19)
F us µ
−
wheretheleptoniccurrentis
Lµ =v¯(k )γµ(1 γ )v(k ), (20)
ν¯ 5 l
−
whilethehadroniccurrent
Hµ =u¯(p′)Γµu(p) (21)
isdeterminedbythesumofthefollowingcontributions
KP(vector)
•
1 q
ΓKµP =−2Fφq2−m2Kµ− +iǫTK−p→φB. (22)
NotethatinFig.2(a),thesumovertheintermediatestatesφB producestheK p φB t-matrixelementbyvirtueof
′ ′ −
→
Eq.12.
CT(vectorplusaxial)
•
1
ΓµCT(V) =−4Fφ Cφ(VB)γµ+φX′B′Cφ(V′B)′γµGφ′B′Tφ′B′→φB, (23)
 
1
ΓCµT(A) =−4Fφ Cφ(AB)γµγ5+φX′B′Cφ(A′B)′γµγ5Gφ′B′Tφ′B′→φB. (24)
 
6
ThecoefficientsC(V)andC(A) aretabulatedinTableIandTableII,respectively.Theloopfunctionisgivenby
φB φB
d4l 1 1
Gφ′B′ =iZ (2π)4l2−m2φ′ +iǫ/p+/q−/l −MB′ +iǫ. (25)
MF(axial)
•
1 (2k q) (k q) γνγ5
ΓMµF = 4√2Fφ Xφ′′ Cφ′′φCφ′′B (k′′−−q)µ2−′m−2φ′′ν+iǫ +φ′Xφ′′B′Cφ′′φ′Cφ′′B′Gµφ′φ′′B′Tφ′B′→φB, (26)
 
whereφ denotestheinternalmesoninthetreeleveldiagram(c)ofFig.1. Inmostcases,onlyonetypeofmesoncanbe
′′
exchangedbutithappensthatbothπ0andηareallowedintermediatestates. TheGµ functionisgivenby
φ′φ′′B′
d4l 1 1 1
Gµ =i (2l q)µ(l q)ν γ γ5 . (27)
φ′φ′′B′ Z (2π)4 − − ν l2−m2φ′ +iǫ(l−q)2−m2φ′′ +iǫ/p−/l +/q−MB′ +iǫ
Finally,coefficientsC andC aretabulatedinTableIIIandTableIV,respectively.
φ1φ2 φB
C(V) p n Λ Σ0 Σ+
φB
K− 2 0 0 0 0
K¯0 0 1 0 0 0
π0 0 0 √3 1 0
2 2
η 0 0 3 √3 0
2 2
π− 0 0 0 0 1
TABLEI.CoefficientsC(V)appearingintheCTcontributiontothehadroniccurrent[Eq.(23)].
φB
C(A) p n Λ Σ0 Σ+
φB
K− 2F 0 0 0 0
−
K¯0 0 (D+F) 0 0 0
−
π0 0 0 1 (D+3F) 1(D F) 0
−2√3 2 −
η 0 0 1(D+3F) √3(D F) 0
−2 2 −
π− 0 0 0 0 D F
−
TABLEII.CoefficientsC(A)appearingintheCTcontributiontothehadroniccurrent[Eq.(24)].
φB
Cφ1φ2 K− K¯0 π0 η π−
π0 1 0 0 0 0
−√2
η 3 0 0 0 0
−q2
π+ 0 1 0 0 0
−
K+ 0 0 1 3 0
√2 q2
K0 0 0 0 0 1
TABLEIII.CoefficientsC appearingintheMFcontributiontothehadroniccurrent[Eq.(26)].
φ1φ2
Thehadroniccurrentpresentedabovedoesnottakeintoaccounttheq2dependenceoftheweakinteractionvertices,whichis
poorlyknown.FollowingRef.[56],wehaveparametrizedthisdependencewithaglobaldipoleformfactor
q2 −2
F(q2)= 1 (28)
(cid:18) − M2(cid:19)
F
thatmultipliesallthetermsinH . UptoSU(3)breakingeffects,thevalueoftheaxialmassM shouldbesimilartotheone
µ F
inelectromagneticandaxialnucleonformfactors. Therefore,asinRefs.[55,56]wehaveadoptedM 1GeV,acceptingan
F
≃
uncertaintyofaround10%.
7
C p n Λ Σ0 Σ+
φB
π0 D+F 0 0 0 0
η - 1 (D 3F) 0 0 0 0
√3 −
π+ 0 √2(D+F) 0 0 0
K+ 0 0 1 (D+3F) D F 0
−√3 −
K0 0 0 0 0 √2(D F)
−
TABLEIV.CoefficientsC appearingintheMFcontributiontothehadroniccurrent[Eq.(26)].
φB
E. Non-relativisticreductionoftheinvariantamplitude
BecauseweonlyfocusonthesmallmomentaoftheφB componentscreatingtheΛ(1405),wecanperformanonrelativistic
reduction,whichwasalsousedinthedescriptionoftheφB amplitudeincoupledchannelsofRef.[5]. FortheCTweget
1
−itCT(V) =−4Fφ(2GFVus)L0Cφ(VB)+φX′B′Cφ(V′B)′G′φ′B′Tφ′B′→φB,
 
1
−itCT(A) =+4Fφ(2GFVus)(L~ ·~σ)Cφ(AB)+φX′B′Cφ(A′B)′G′φ′B′Tφ′B′→φB, (29)
 
wheretheloopfunction,afterremovingthebaryonnegativeenergypart,becomes
G′φ′B′ =Z (2dπ3l)32ωφ1′(~l)EMB′B(~′l)Minv ωφ′(~l)1 EB′(~l)+iǫ. (30)
− −
Afterthenonrelativisticreduction,theMFcontributionscanbewrittenas
1 L0(2k q)0+L~ ~q
−itMF = 4√2Fφ(2GFVus)Xφ′′ Cφ′′φCφ′′B ~σ·~q (k′−q′)−2−m2φ′′ +·iǫ

+φ′Xφ′′B′Cφ′′φ′Cφ′′B′hL~ ·~σG(φ1′)φ′′B′ +(L~ ·~q)(~σ·~q)G(φ2′)φ′′B′i , (31)
wheretheloopfunctionsare 
G(φ1φ)′B′ =Z (2dπ3l)3ωφ′(~l)ω1φ(~l ~q˜)EMB′B(~′l)~l32 (cid:26)hωφ(~l−~q˜)+ωφ′(~l)i2
−
+ ωφ(~l ~q˜)+ωφ′(~l) EB′(~l) p˜0 q˜0ωφ′(~l)
h − ih − i− o
1 1
×Minv EB′(~l) ωφ′(~l)+iǫp˜0 EB′(~l) ωφ(~l ~q˜)+iǫ
− − − − −
1 1
, (32)
×q˜0+ωφ(~l ~q˜)+ωφ′(~l) iǫωφ′(~l) q˜0+ωφ(~l ~q˜) iǫ
− − − − −
and
G(φ2φ)′B′ =Z (2dπ3l)32ωφ′(~l)ω1φ(~l ~q˜)EMB′B(~′l)(cid:26)hωφ(~l−~q˜)+ωφ′(~l)i2
−
+ ωφ(~l ~q˜)+ωφ′(~l) EB′(~l) p˜0 q˜0ωφ′(~l)
h − ih − i− o
1 1
×Minv EB′(~l) ωφ′(~l)+iǫp˜0 EB′(~l) ωφ(~l ~q˜)+iǫ
− − − − −
1 1
. (33)
×q˜0+ωφ(~l ~q˜)+ωφ′(~l) iǫωφ′(~l) q˜0+ωφ(~l ~q˜) iǫ
− − − − −
ThequantitieswithtildearedefinedintheφB CMframe.
8
III. RESULTS
Throughoutthissection, theresultsarepresentedforthemuonflavorl = µ. TheΛ(1405)canbeobservedintheinvariant
massdistributionofπΣpairsthathasitsthresholdbelowthepeakoftheΛ(1405)states. ThecleanestsignalforI =0Λ(1405)
production appears in the π0Σ0 channel because I = 1 is not allowed. In Fig. 3, we show dσ/dM for π0Σ0 production
inv
atthreedifferentlaboratoryenergies,E = 900,1100,and1300MeV. We canclearlysee theresonantshapeoftheΛ(1405)
ν¯
at all the energies. Note that, in spite of the two poles, there is a single peak. This is common to all the reactions, with the
exceptionofelectroproduction[38],wherethedataarestillrelativelypoor.Onlythedifferentweightofthetwopolesmakesthe
peakappearatdifferentenergiesindifferentprocesses. Inthepresentcasethedistributionpeaksaround1420MeVindicating
thatthere is moreweightfromthe pole at 1420MeV or, in other words, thatthe Λ(1405)productioninducedby the K p is
−
dominant. TogainfurtherinsightintotheinterplayofthetwopolesoftheΛ(1405)resonanceinthisreaction,wehavelooked
10
V] ν-µ p → µ+ π0 Σ0
e
G 8
2/ Eν-µ = 0.9 GeV
m Eν- = 1.1 GeV
41 c 6 Eν-µµ = 1.3 GeV
-0
1
[ 4
v
n
Mi
2
d
σ/
d
0
1.3 1.35 1.4 1.45 1.5
M [GeV]
inv
FIG.3. (coloronline). Differentialcrosssectionforthereactionν¯ p µ+π0Σ0asafunctionoftheinvariantmassM ofthefinalmeson
µ inv
→
baryonsystemforthreedifferentincidentantineutrinoenergies.
at the line shapes of the doubledifferentialcross section d2σ/(dM dcosθ) for differentvaluesof the θ angle between the
inv
initialν¯ andthefinalµ+ intheν¯pCMframe(Fig.4). Whenθ increases,sodoes q2 ,andtheformfactorcausesareduction
µ
| |
inthecrosssection. Tocomparetheshapeswehavenormalizedallcurvestothesameareabymultiplyingthecrosssectionat
cosθ =0( 1)by3.4(14).Inthebackwarddirection,thedistributionclearlyresemblesasingleBreit-Wignerwithamassanda
−
widthremarkablyclosetothevaluesoftheheavierpoleoftheΛ(1405).Itisthispolethatappearsdominantatthiskinematics.
Asθdecreases,thepresenceofthelighterstatebecomesmoreevidentwithlargerstrengthaccumulatingbelowthepeak,which
isshiftedtowardssmallerinvariantmasses. Thelineshapebecomesasymmetricbutthesecondstatenevershowsupasapeak
inthecrosssection.
Itis also veryinterestingto considerdσ/dM for the threechargedchannelsπ0Σ0, π+Σ andπ Σ+. Thisis shown in
inv − −
Fig.5. Thepeakpositionforthedifferentreactionsisslightlyshifted,butthelargestdifferencesarepresentbelowthemaxima.
ThisisduetothecontributionofanI =1amplitudewhichaddsconstructivelyordestructivelydependingonthechannel[32].
ItwasalsoshowninRef.[32]thatΛ(1405)photoproductiondatahinttoapossibleI =1statearound1400MeV,whichappears
insomeapproaches[6]butisataborderlineinothers[8]. IntheworkofRefs. [63,64], theexistenceofsuchI = 1state is
claimed from the study of the K p Λπ π+ reaction. The large differences seen in the cross sections for the three πΣ
− −
→
channelsinthepresentreactionindicatethattheyareindeedrathersensitivetotheI =1amplitudeand,thus,thereisapotential
fortheextractionofinformationonthepossibleI =1state.
InFig.6, weshownowtheintegratedcrosssectionsforπ0Σ0, π Σ+, andπ+Σ production. Weobserveasteadygrowth
− −
ofthecrosssectionswiththeantineutrinoenergy. ThesecrosssectionsarelargelydrivenbytheΛ(1405)resonance. Indeed,in
Fig.6,bothtreelevelandfullmodelcrosssectionsareshown.Weobservethatthecontributionofthemeson-baryonrescattering
hasa drasticeffectinthe results. Thecase ofthe π+Σ channelisthemostspectacularbecausethetreelevelcontributionis
−
exactlyzero.
WehavealsoinvestigatedtheK¯-nucleonproductionreactions. Notethatinthiscasethethresholdenergies,√s = mK− +
Mp =1430MeVandmK¯0+Mn =1437MeV,arealreadyabovetheΛ(1405)peak.Thus,wedonotplotdσ/dMinvinthiscase
andshowonlytheintegratedcrosssectionasafunctionofenergy. TheseareshowninFig.7forK pandinFig.8forK¯0n.
−
AscanbeseenintherightpanelsofFigs.7,8,unliketheπΣproductioncase,thecrosssectionisnotincreasedbytheresonance.
Onthecontrary,thefastfalldownofdσ/dM closetotheK pthreshold,seeninFig.3forπΣ,reflectsthesimilartrendof
inv −
9
7
]
nits 6 ν-µ p -> µ+ Σ0 π0
u
rary 5 E−νµ = 1 GeV
[arbit 4 ccoossθθ == 10
θ) cosθ = -1
os 3
c
d
nv2
Mi
d
( 1
σ/
2
d
0
1.3 1.35 1.4 1.45 1.5 1.55 1.6
M [GeV]
inv
FIG.4.(coloronline).Areanormalizeddoubledifferentialcrosssectionforν¯ p µ+π0Σ0atE =1GeV,asafunctionofM forthree
µ → ν¯µ inv
differentvaluesoftheangle(θ)betweentheincomingneutrinoandtheoutgoingmuoninthereactionCMframe.
5
GeV] 4 νν--µµ pp →→ µµ++ ππ0- ΣΣ+0 Eν-µ = 1.0 GeV
2/ ν-µ p → µ+ π+ Σ-
m
c 3
1
4
-0
1
[ 2
v
n
Mi
1
d
σ/
d
0
1.3 1.35 1.4 1.45 1.5
M [GeV]
inv
FIG.5.(coloronline).Invariantmassdistributionforthethreechargechannels:π0Σ0(solidline),π−Σ+(dashedline)andπ+Σ−(dot-dashed
line).TheincidentantineutrinoenergyisE =1GeV.
ν¯µ
the t matrix which is commonto all the channels. This affects the K¯-nucleonproductioncross sections, most noticeablyfor
K¯0n,thechannelwithalargerthreshold.TheseunitarizationeffectswereabsentinthecalculationsreportedinRef.[56]. There
areotherdifferencesbetweenthepresentstudyandtheoneofRef.[56]. First,herewehaveusedtheaverageF =1.15f ,for
φ π
consistencywiththevaluetakeninthestudyofφB scattering[5](seeSec.IIB),insteadofF = f inRef.[56]. Thisleads
φ π
tolittlesmallercrosssectionwithrespecttothoseofRef.[56]. Furthermore,thep-wavecontributionsconsideredinRef.[56]
butnotheremakethecrosssectionsbiggerasonedepartsfromthreshold. Finally,thenonrelativisticapproximationbecomes
poorerforthehigherenergyandmomentumtransfersthatcanbeprobedasthereactionenergyincreases. Asanexample,the
CTcontributionhereisabout30%lowerthaninRef.[56]atE = 1200MeVandabout40-45%smalleratE = 2000MeV
ν¯ ν¯
(aftercorrectingforF ). Forbetterprecision,oneshouldrestricttosmallerantineutrinoenergiesorimplementkinematiccuts
φ
tokeepq0and ~q smallcomparedtothenucleonmass.
| |
In the K p channel, the largest contribution arises from the CT mechanism (left panel of Fig. 7), in line with Fig. 3 of
−
Ref.[56]. IntheK¯0nchannel,instead,theMFcontributionbecomesincreasinglylargerthantheCTaboveE = 1200MeV
ν¯
(leftpanelofFig.8),invariancewithFig.5ofRef.[56]. Nevertheless,itshouldbementionedthatourpredictionsforKP,CT
andMFtermsconvergetothoseofRef.[56]intheheavy-nucleonlimit.
10
1.2
ν-µ p → µ+ π0 Σ0
1 ν-µ p → µ+ π- Σ+
2m] 0.8 ν-µ p → µ+ π+ Σ- Full Model
c
1
4 0.6
-0
1
σ [ 0.4
0.2 Tree Level
0
0.8 1 1.2 1.4 1.6 1.8 2
Eν- [GeV]
µ
FIG.6. (coloronline). Crosssectionsasafunctionoftheantineutrinoenergyforthethreeν¯ p µ+πΣreactionchannels. Thethreeupper
µ
curveshavebeenobtainedwiththefullmodelwhilethetwoloweroneswithtreelevelcontribut→ionsalone. Thelaterisabsentfortheπ+Σ−
channel.
1.6 1.6
ν- p → µ+ K- p ν- p → µ+ K- p
1.4 µ 1.4 µ
] 1.2 Full Model ] 1.2 Full Model
2 2
m Contact m Tree Level
1 1
1 c Meson in flight 1 c
4 0.8 Kaon Pole 4 0.8
-0 -0
1 0.6 1 0.6
[ [
σ 0.4 σ 0.4
0.2 0.2
(a) (b)
0 0
0.8 1 1.2 1.4 1.6 1.8 2 0.8 1 1.2 1.4 1.6 1.8 2
Eν- [GeV] Eν- [GeV]
µ µ
FIG. 7. (color online). Integrated cross section for the ν¯µp µ+K−p reaction as a function of the antineutrino energy. Left panel:
→
contributionofthedifferenttermstothefullmodelresult.TheKPcontributionisnegligibleandcannotbediscernedintheplot.Rightpanel:
comparisonbetweenthefullmodelandtreelevelcalculations.
A. Λ(1405)productionatMINERνA
OneofthegoalsoftheMINERνAexperimentistostudyweakstrangenessproduction[46]. Itisthereforeimportanttoobtain
the numberof eventsin which the Λ(1405)resonance is primarilyproducedduringthe antineutrinorun. Let us consider the
processν¯ p µ+πΣ. ThenumberofeventsforagiveninvariantmasoftheπΣpairis
µ
→
dN dσ
=N fMN dE φ(E ) πΣ (E ). (34)
POT A ν¯ ν¯ ν¯
dM Z dM
inv inv
The differentialcross section is averagedoverthe antineutrinoflux φ(E ). The flux prediction, in unitsof ν¯/cm2/POT, for
ν¯
thelow-energyconfigurationistakenfromTableV ofRef.[50]. Thepresentestimatecorrespondstoanumberofprotonson
targetofN =2.01 1020inν¯mode,neglectingthesmallν¯ componentinthebeamofmuonantineutrinos.Althoughthe
POT e
×
MINERνAdetectorismadeofdifferentmaterials,hereweconsideronlythescintillator(CH).Inthiscasetheprotonfraction
f =(1+6)/(1+12). OneshouldrecallthatπΣpairscanalsobeproducedonneutronsbut,inthiscase,thepairhasnegative
charge, not leading to Λ(1405) excitation. The scintillator mass is M = 0.45M + 0.55M , with M = 2.84 106 and
1 2 1
M = 5.47 106 grams,totakeintoaccountthat45%oftheν¯dataweretakenduringtheconstructiontime,using×areduced
2
×
fiducialvolume[50]. Finally,N denotestheAvogadronumber.
A