Anti-neutrino charged-current reactions on scintillator with low momentum transfer

R. Gran,1 M. Betancourt,2 M. Elkins,1, ∗ P.A. Rodrigues,3, 4, 5 F. Akbar,6 L. Aliaga,7, 8 D.A. Andrade,9

A. Bashyal,10 L. Bellantoni,2 A. Bercellie,5 A. Bodek,5 A. Bravar,11 H. Budd,5 G.F.R. Caceres Vera,12 T. Cai,5

M.F. Carneiro,10 D. Coplowe,3 H. da Motta,12 S.A. Dytman,13 G.A. Dıaz,5, 8 J. Felix,9 L. Fields,2, 14

R. Fine,5 H. Gallagher,15 A. Ghosh,16, 12 H. Haider,6 J.Y. Han,13 D.A. Harris,2 S. Henry,5 D. Jena,2

J. Kleykamp,5 M. Kordosky,7 T. Le,15, 17 J.R. Leistico,1 A. Lovlein,1 X.-G. Lu,3 E. Maher,18 S. Manly,5

W.A. Mann,15 C.M. Marshall,5, † K.S. McFarland,5, 2 A.M. McGowan,5 B. Messerly,13 J. Miller,16

A. Mislivec,5 J.G. Morfın,2 J. Mousseau,19, ‡ D. Naples,13 J.K. Nelson,7 C. Nguyen,19 A. Norrick,7

Nuruzzaman,17, 16 A. Olivier,5 V. Paolone,13 C.E. Patrick,14, § G.N. Perdue,2, 5 M.A. Ramırez,9 R.D. Ransome,17

H. Ray,19 L. Ren,13 D. Rimal,19 D. Ruterbories,5 H. Schellman,10, 14 C.J. Solano Salinas,20 H. Su,13

M. Sultana,5 S. Sanchez Falero,8 E. Valencia,7, 9 J. Wolcott,5, ¶ M. Wospakrik,19 and B. Yaeggy16

(MINERvA Collaboration)1Department of Physics and Astronomy, University of Minnesota – Duluth, Duluth, Minnesota 55812, USA

2Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA3Oxford University, Department of Physics, Oxford, United Kingdom

4University of Mississippi, Oxford, Mississippi 38677, USA5University of Rochester, Rochester, New York 14627 USA

6AMU Campus, Aligarh, Uttar Pradesh 202001, India7Department of Physics, College of William & Mary, Williamsburg, Virginia 23187, USA

8Seccion Fısica, Departamento de Ciencias, Pontificia Universidad Catolica del Peru, Apartado 1761, Lima, Peru9Campus Leon y Campus Guanajuato, Universidad de Guanajuato,

Lascurain de Retana No. 5, Colonia Centro, Guanajuato 36000, Mexico.10Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA

11University of Geneva, 1211 Geneva 4, Switzerland12Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Rio de Janeiro, 22290-180, Brazil

13Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA14Northwestern University, Evanston, Illinois 60208, USA

15Physics Department, Tufts University, Medford, Massachusetts 02155, USA16Departamento de Fısica, Universidad Tecnica Federico Santa Marıa, Avenida Espana 1680 Casilla 110-V, Valparaıso, Chile

17Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA18Massachusetts College of Liberal Arts, 375 Church Street, North Adams, Massachusetts 01247, USA

19University of Florida, Department of Physics, Gainesville, Florida 32611, USA20Universidad Nacional de Ingenierıa, Apartado 31139, Lima, Peru

(Dated: March 28, 2018)

We report on multi-nucleon effects in low momentum transfer (< 0.8 GeV/c) anti-neutrino in-teractions on scintillator. These data are from the 2010-11 anti-neutrino phase of the MINERvAexperiment at Fermilab. The hadronic energy spectrum of this inclusive sample is well-describedwhen a screening effect at low energy transfer and a two-nucleon knockout process are added to arelativistic Fermi gas model of quasi-elastic, ∆ resonance, and higher resonance processes. In thisanalysis, model elements introduced to describe previously published neutrino results have quanti-tatively similar benefits for this anti-neutrino sample. We present the results as a double-differentialcross section to accelerate investigation of alternate models for anti-neutrino scattering off nuclei.

PACS numbers: 13.15.+g, 25.30.Pt

Current and future accelerator-based neutrino oscil-lation experiments analyze flavor oscillations based ondistortions of reconstructed anti-neutrino energy spec-tra. These measurements require models for both thelepton energy and angle, and for the hadronic system.Experiments using calorimetric reconstruction [1, 2] areespecially sensitive to the presence of neutrons in the fi-nal state. To probe for charge-parity (CP) violation inthe lepton sector [3–5], models of anti-neutrino processesrequire similar accuracy to the corresponding neutrino

processes. Otherwise, model uncertainties limit the sen-sitivity to, or possibly mimic, a CP-violating effect.

We present the first anti-neutrino analysis of inclu-sive charged-current reactions to isolate multi-nucleon ef-fects in the quasi-elastic (CCQE) and ∆ resonance kine-matic regions. We reconstruct the hadronic system us-ing calorimetry and obtain an estimate of the three-momentum transfer for each event. The data are sub-divided into six sub-ranges of momentum transfer up to0.8 GeV/c, and within each range we present the ob-

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This document was prepared by [MINERvA Collaboration] using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359

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served hadronic energy in the detector. To describe thesedata, a component of the event rate could be attributedto many-body effects like a two-particle, two-hole (2p2h)process [6–16]. Also, suppression of CCQE interactionsis preferred, such as provided by a random phase approx-imation (RPA) calculation [7, 17–19] applied to a Fermigas model [20].

The data were taken between November 2010 andFebruary 2011 with the NuMI beam [21] operating inanti-neutrino mode. The primary beam of 120-GeV pro-tons interacts in a graphite target, producing mesons. Apair of magnetic horns focuses negatively-charged mesonstoward a decay pipe where their decay leads to an anti-neutrino spectrum in the MINERvA detector peakingnear 3.0 GeV. We use a GEANT4-based [22, 23] pre-diction for the flux with central value and uncertain-ties adjusted [24] using thin-target hadron productiondata [25–28] and an in situ neutrino-electron scatteringconstraint [29].

A sample of charged-current νµ interactions are se-lected from MINERvA’s 5.3-ton fiducial volume by re-quiring that a muon track leaves the MINERvA detectorand has its positive charge and momentum identified inthe MINOS magnetized iron spectrometer [30] located 2m downstream. The fiducial volume is both an activetracker and calorimeter, built from planes of scintillator(CH) strips with a triangle shape of 3.3-cm base and 1.7-cm height. Alternating and nesting the triangles giveslight-sharing information that improves tracking resolu-tion. Each hexagonal plane contains 127 strips up to245-cm in length. The planes are installed with stripsoriented vertically or rotated ±60◦, ensuring the precisereconstruction of the interaction point and muon trackangle, even when hadronic activity partially obscures themuon. The target mass consists of 8.2%, 88.5%, and2.5% hydrogen, carbon, and oxygen, respectively, plussmall amounts of heavier nuclei.

Particles leaving the active tracking region pass intothe electromagnetic calorimeter (ECAL) where thinsheets of lead are epoxied to each scintillator plane. Far-ther downstream are layers of hadronic calorimetry us-ing alternating planes of scintillator and passive steel.Calorimetric and tracking capabilities of MINERvA areconstrained relative to GEANT4 v.9.4.p2 (with BertiniCascade option) using in-situ [31] and hadron test beammeasurements [32]. With no test beam measurements,the neutron response and its uncertainties come after ad-justing the cross section to match the data from [33] asused by later versions of GEANT4.

The kinematics of each event are reconstructed us-ing the measured muon energy and angle, and measuredenergy deposits attributed to hadrons. The techniqueis nearly identical to [6]. A full simulation of the re-constructed sample with calibrated detector response ismade using the GEANT4 simulation and GENIE ver-sion 2.8.4 neutrino event generator [34]. This simula-

tion is used to obtain a correction [35], as a functionof the calorimetrically measured hadronic energy, to es-timate the energy transfer q0. This correction is ap-plied identically to reconstructed simulation and data.In both cases, the calorimetric neutrino energy estimateis Eν = Eµ + q0, where Eµ includes the muon restmass Mµ. The square of the four-momentum trans-fer is Q2 = 2Eν(Eµ − pµ cos θµ) − M2

µ, and the three-

momentum transfer is simply q3 =√Q2 + q20 . In this

Letter, the kinematics of the analysis sample are limitedto q3 < 0.8 GeV/c. There are no other requirements onreconstructed hadronic topologies for this inclusive sam-ple.

The measured energy deposits are used to form an-other calorimetric estimator, the available energy Eavail

[6]. This is energy due to particles that deposit mostor all of their energy in the detector: proton kinetic en-ergy, charged pion kinetic energy, electrons, positrons,and photons, including those from neutral pion and etadecays. These momentum transfers are too low for pro-duction of heavier mesons and baryons.

When Eavail is formed from a model, it does not in-clude neutrons that leave a small fraction of their energyin the detector or the energy used to unbind nucleons.In the neutrino case [6], where outgoing protons far out-number neutrons, this is a good approximation. In thesimulation of this anti-neutrino subsample, 70% of inter-actions have more than half the energy transfer going toneutrons, including 40% which have neutron-only finalstates. Up to 60% of neutrons at these energies leave re-constructed energy deposits in the detector, so neutronscan contribute significantly to the hadronic energy de-posits. Despite this, reconstructed and generator/modeldistributions of Eavail vs. q3 retain the ability to sepa-rate CCQE and ∆ resonance kinematics, and the regionbetween. Because the analysis is limited to interactionswith little energy in the recoil system, only the energydeposits in the tracker and downstream ECAL regionsare considered. Backgrounds are higher and calorimet-ric resolution is worse for energy deposits in the otherregions, degrading sensitivity to multi-nucleon effects.

While Eavail is defined assuming neutrons have negli-gible calorimetric response, the actual situation is morecomplex. Interactions that have only neutrons in the fi-nal state are most likely to have reconstructed hadronicenergy between 0 and 10 MeV. Fig. 1 shows the re-constructed energy deposits from the GENIE-producedneutrons exiting the nucleus and simulated by GEANT4with the detector model. The most common outcomesare small (< 10 MeV) energy deposits as the neutronsscatter on hydrogen and carbon in the detector. The re-sponse is mostly uncorrelated with the neutron energy,and not suited for a calorimetric quantity. Larger sec-ondary proton energy deposits become more common asneutron kinetic energy increases. For this analysis and its

r e c o n str u ct e d i s ol at e d e n er g y d e p o sit ( M e V)0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4

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0 < n e utr o n K E < 6 0 M e V

6 0 < n e utr o n K E < 1 2 0 M e V

1 2 0 < n e utr o n K E < 1 8 0 M e V

3

FI G. 1: P r e di c t e d i s ol a t e d e n e r g y d e p o si t s f r o m n e u t r o n si n t h r e e r a n g e s of n e u t r o n ki n e ti c e n e r g y ( K E ). T h e s e a r es el e c t e d f r o m t h e t u n e d M C s a m pl e ( d e s c ri b e d l a t e r ) wi t hq 3 < 0 .8 G e V / c. T h e t h r e e c u r v e s al s o ill u s t r a t e t h e r el a-ti v e a b u n d a n c e of l o w e r ki n e ti c e n e r gi e s i n t h e s el e c t e d M Cs a m pl e. N e u t r o n s i n t hi s e n e r g y r a n g e t y pi c all y l e a v e s m alli s ol a t e d e n e r g y d e p o si t s, u n c o r r el a t e d wi t h t h e n e u t r o n ki-n e ti c e n e r g y.

ki n e m ati c r a n g e, n e ut r o n s wit h t e n s t o a f e w h u n dr e d sof M e V a r e e ff e cti v el y t r e at e d a s bi a si n g r e c o n st r u ct e dE a v ail t o hi g h e r v al u e s t h a n t h e t r u e q u a ntit y.

A n ot h e r c o n s e q u e n c e of t h e n e utr o n r e s p o n s e i n t h eMI N E R v A d et e ct or i s t h at t h e r e s ol uti o n s f o r s o m e r e-c o n st r u ct e d q u a ntiti e s a r e di ff e r e nt t h a n t h e n e utri n oc a s e. T h e t w o h a d r o ni c e n er g y e sti m at or s f o r t h e s e-l e ct e d s a m pl e h a v e si g ni fi c a ntl y w o r s e r e s ol uti o n s. T h esi m ul ati o n i n di c at e s a r o ot- m e a n- s q u a r e ( R M S) r e s ol u-ti o n of 5 8 % f or q 0 c o m p a r e d t o 5 1 % f or t h e n e ut ri n o c a s ei n [ 6]. F o r E a v ail t h e R M S i s al s o 5 8 % w hil e t h e n e ut ri n or e s ol uti o n i s si g ni fi c a ntl y i m pr o v e d t o 4 0 %. W h e n n e u-t r o n s m a y b e t h e o nl y fi n al st at e p arti cl e, t h e a b s ol ut er e si d u al i s a b ett er m et ri c ( s h o w n i n Fi g. 2) t h a n f r a c-ti o n al R M S. T h e n e ut ri n o e n e r g y e sti m at o r i s n e gli gi bl ydi ff e r e nt; b e c a u s e t h e s e e v e nt s h a d s u c h littl e h a dr o ni ce n e r g y t o b e gi n wit h, t h e m u o n e n e r g y d o mi n at e s t h er e s ol uti o n. M u o n e n e r g y a n d a n gl e d ri v e q 3 , it s r e s ol u-ti o n i s b a r el y d e g r a d e d fr o m 2 2 % t o 2 3 % R M S a n d v a ri e slittl e a c r o s s t h e r a n g e of q 3

r e si d u al (r e c o - tr u e M e V)a v aila b s ol ut e E1 0 0− 5 0− 0 5 0 1 0 0 1 5 0 2 0 0

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0. 4 < r e c o q _ 3 < 0. 8 G e V

R M S: 1 0 0 M e V ( d a s h e d)

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.

B e c a u s e t hi s i s a n a n al y si s of a n i n cl u si v e s a m pl e, e v e nts el e cti o n i s mi ni m al. We o nl y c r e at e a b o u n d a r y f o r u n-f ol di n g t h e d at a i nt o a d o u bl e di ff e r e nti al c r o s s s e cti o nt h at c a n b e r e p r o d u c e d b y e xt e r n al e v e nt g e n e r at or s,a n d e x cl u d e r e gi o n s of ki n e m ati c s p a c e t h at d o n ot h a v eg o o d a c c e pt a n c e. T h e m u o n m o m e nt u m i s r e q ui r e d t ob e a b o v e 1. 5 G e V / c a n d a n gl e l e s s t h a n 2 0 d e gr e e s wit hr e s p e ct t o t h e b e a m di r e cti o n. We f u rt h er li mit t h e r e c o n-st r u ct e d a nti- n e ut ri n o e n e r g y t o b et w e e n 2 a n d 6 G e V,w hi c h s p a n s t h e p e a k of t hi s b e a m a n d all o w s dir e ct c o m-p a ri s o n t o t h e n e ut ri n o r e s ult s [ 6]. T h e s e s el e cti o n s ar eu s e d f o r t h e r e c o n st r u ct e d e v e nt s, t h e u nf ol d e d di stri b u-ti o n, a n d t h e t r u e di st ri b uti o n of M C c o m p a r e d t o t h el att e r.

T h e s el e ct e d i n cl u si v e s a m pl e i s c o m p ar e d t o t h e p r e-

(r e c o-tr u e)/tr u e3

fr a cti o n al r e si d u al q0. 8− 0. 6− 0. 4− 0. 2− 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0

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R M S: 2 4 % ( s oli d)

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R M S: 2 1 % ( d a s h e d)

ar e a n or m ali z e d

FI G. 2: A b s ol u t e r e s ol u ti o n of E a v ail ( t o p ) a n d f r a c ti o n al r e s-ol u ti o n f o r t h r e e- m o m e nt u m t r a n sf e r ( b o t t o m ), f o r t h e M Cs a m pl e wi t h r e c o n s t r u c t e d ( r e c o ) q 3 < 0 .8 G e V / c. T h e d a s h e dli n e i s t h e u p p e r h alf of t h e q 3 r a n g e; t h e s oli d i s t h e l o w e rh alf.

di cti o n of t h e G E NI E e v e nt g e n e r at o r c o m bi n e d wit h aG E A N T 4 si m ul ati o n of t h e o ut g oi n g p a rti cl e s fr o m t h er e a cti o n. G E NI E’ s si m ul ati o n of t h e C C Q E pr o c e s s i sf r o m Ll e w ell y n S mit h [ 3 6] wit h v e ct o r f or m f a ct o r s p a-r a m et eri z e d b y [ 3 7], a n d t h e a xi al f o r m f a ct o r i s t a k e nt o b e a di p ol e wit h a xi al m a s s of 0. 9 9 G e V. F or i nt e r a c-ti o n s o n c a r b o n a n d ot h e r n u cl ei, G E NI E u s e s a Fe r mig a s m o d el [ 2 0]. T h e ∆ a n d hi g h er r e s o n a n c e s u s e R ei na n d S e h g al [ 3 8], wit h a n o n- r e s o n a nt c o m p o n e nt t a k e nf r o m t h e d e e pl y i n el a sti c s c att e ri n g m o d el [ 3 9] a s t h e r e s-o n a n c e s ar e p h a s e d o ut f r o m i n v ari a nt m a s s 1. 4 < W <2. 0 G e V. We a d d t w o mi n or (f or t hi s a n al y si s < 2 % oft ot al r at e) m o di fi c ati o n s t o pi o n p r o d u cti o n. T h e n o n-r e s o n a n c e, si n gl e- pi o n p r o c e s s i s r e d u c e d f oll o wi n g c o m-p a ri s o n of G E NI E t o b u b bl e c h a m b e r e x p e ri m e nt n e u-t ri n o d at a [ 4 0, 4 1]. C o h e r e nt pi o n e v e nt s wit h pi o n ki-n eti c e n e r g y < 0. 4 5 G e V a r e r e d u c e d b y h alf [ 4 2 – 4 4].T hi s b a s e c o m bi n ati o n of m o d el s, c o m p a r e d t o r e c o n-st r u ct e d d at a, h a s di s c r e p a n ci e s i n t h e r e gi o n b et w e e nt h e C C Q E a n d ∆ p r o c e s s a s l ar g e a s a f a ct o r of t w o, a ss h o w n i n t h e t o p p a n el s of Fi g. 3.

We h a v e m o di fi e d t h e d ef a ult G E NI E v er si o n 2. 8. 4 t oi n cl u d e a d v a n c e s i n m o d eli n g t h e i m p o rt a nt p r o c e s s e s.T h e C C Q E pr o c e s s i s m o di fi e d t o i n cl u d e R P A s c r e e ni n gb a s e d o n t h e I FI C V al e n ci a m o d el [ 1 7, 4 5] i m pl e m e nt e db y r e w ei g hti n g G E NI E C C Q E e v e nt s [ 4 6]. A C C Q E-li k et w o- p a rti cl e, t w o- h ol e p r o c e s s “ 2 p 2 h ” f r o m t h e m o d el b yt h e s a m e g r o u p [ 8, 4 5] i s i m pl e m e nt e d i n G E NI E [ 1 5].

T h e I FI C V al e n ci a 2 p 2 h m o d el i n c r e a s e s t h e p r e di ct e de v e nt r at e s, b ut n ot e n o u g h. T hi s p r o c e s s i s i n c r e a s e df u rt h er wit h a n e m pi ri c al e n h a n c e m e nt [ 4 7] b a s e d o nMI N E R v A i n cl u si v e n e utri n o d at a [ 6]. T h e a d diti o n ale v e nt s a r e f r o m w ei g hti n g u p t h e g e n er at e d 2 p 2 h e v e nt sa c c or di n g t o a t w o- di m e n si o n al G a u s si a n i n t r u e q 0 , q3

0. 0 0. 1 0. 2 0. 3

1 0 0 0

2 0 0 0

/ G e V < 0. 43

q0 <

0. 0 0. 1 0. 2 0. 3

/ G e V < 0. 83

q0. 4 <

M C T ot al

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∆M C

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M C T ot al

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R e c o n str u ct e d a v ail a bl e e n er g y ( G e V)

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FI G. 3: R e c o n s t r u c t e d E a v ail di s t ri b u ti o n s c o m p a r e d t o( t o p ) t h e b a s e M o nt e C a rl o si m ul a ti o n ( G E NI E wi t h mi n o rm o di fi c a ti o n s t o pi o n p r o d u c ti o n ) f o r t w o r a n g e s of r e c o n-s t r u c t e d t h r e e m o m e nt u m t r a n sf e r. I n t h e i m p r o v e d si m u-l a ti o n M n v G E NI E- v 1 ( b o t t o m ) t h e r e gi o n b e t w e e n t h e p r e-di c t e d C C Q E p r o c e s s ( d a s h e d ) a n d t h e ∆ ( 1 2 3 2 ) r e s o n a n c e( d o t t e d ) i s fill e d b y e v e nt s g e n e r a t e d f r o m t h e V al e n ci a 2 p 2 hp r o c e s s pl u s a d di ti o n al 2 p 2 h e v e nt s ( d o t- d a s h e d ) i s a si g ni fi-c a nt i m p r o v e m e nt.

w h o s e p a r a m et e r s ar e fit t o t h e n e ut ri n o d at a v e r si o n oft h e s e di st ri b uti o n s. T hi s e n h a n c e m e nt a d d s 5 0 % t o t h ep r e di ct e d 2 p 2 h st r e n gt h, b ut t a r g et s t h e e v e nt r at e i n t h eki n e m ati c r e gi o n b et w e e n t h e C C Q E a n d ∆ p e a k s w h e r et h e r at e d o u bl e s. T h e c oll e cti o n of c h a n g e s i n t hi s a n d t h ep r e c e di n g p a r a g r a p h s a r e r ef e r r e d t o a s “ M n v G E NI E- v 1 ”a n d a r e t h e c e ntr al, t u n e d m o d el f o r m a n y r e c e nt a n al y-s e s [ 4 8 – 5 0].

T h e r e s ulti n g d e s c ri pti o n of t h e a nti- n e ut ri n o d at a i sm u c h i m pr o v e d, a s ill u str at e d i n Fi g. 3 a n d s u m m a ri z e di n T a bl e I u si n g a st a n d a r d χ 2 t e st o n t h e r e c o n st r u ct e ds a m pl e s. T h e s e m o d el s al s o i m pr o v e t h e d e s cri pti o n ofm u o n- o nl y ki n e m ati c di stri b uti o n s of a n o v e rl a p pi n g s u b-s et of t h e s a m e d at a s et [ 5 0] s el e ct e d wit h n o pi o n s i n t h efi n al st at e.

F o r t hi s m o d el c o m p a ri s o n t o r e c o n str u ct e d d at a, t h el a r g e st s y st e m ati c u n c e rt ai nti e s i n cl u d e fl u x, h a d r o n e n-e r g y s c al e, a n d G E NI E r e s o n a n c e i nt er a cti o n a n d fi n al-st at e r e s c att eri n g m o d el u n c e rt ai nti e s. T h e G E NI E u n-c e rt ai nt y o n t h e C C Q E a xi al f o r m f a ct o r i s r e d u c e d t o± 9 % f oll o wi n g t h e a n al y si s of [ 5 1]. A n u n c ert ai nt y o n

t h e R P A C C Q E s u p pr e s si o n [ 4 6, 5 2] i s a d d e d, m o st si g-ni fi c a ntl y fr o m c o m p a ri s o n t o m u o n c a pt u r e d at a. N osi n gl e u n c ert ai nt y d o mi n at e s t h e m o d el p r e di cti o n f o r t h er e c o n st r u ct e d di st ri b uti o n s.

T h e a nti- n e ut ri n o s a m pl e r et ai n s a di s c r e p a n c y j u stb e y o n d t h e e r r o r b a n d i n t h e s e c o n d-l o w e st E a v ail bi n swit hi n t h e r a n g e 0. 3 < q 3 < 0. 8 G e V / c. T h e s e bi n sa r e d o mi n at e d b y e v e nt s wit h n e ut r o n- o nl y fi n al st at e s,i n cl u di n g f e e d- d o w n f r o m hi g h er e n er g y t r a n sf e r C C Q Ea n d 2 p 2 h r e a cti o n s. Li mit e d t o t h e m o d el s a v ail a bl e f o rt hi s a n al y si s, b ot h t h e C C Q E R P A a n d t h e t u n e d 2 p 2 hc o m p o n e nt e a c h h a v e a 1 0 % t o 3 0 % e ff e ct o n t h e s e bi n s.J u d g e d o nl y b y t h e a g g r e g at e d χ 2 v al u e s ( n ot s h o w n),t h e b e n e fit of a d di n g a R P A s u p p r e s si o n i s di ffi c ult t odi s c e r n. It r e d u c e s s o m e bi n s w h e r e t h e M C i s al r e a d yu n d e r p r e di cti n g t h e d at a. F u rt h e r m o r e, t h e χ 2 a c c o u nt sf o r a d diti o n al u n c e rt ai nt y f r o m t h e R P A m o d el. H o w-e v e r, t h e R P A m o d el p r o d u c e s b ett e r a g r e e m e nt i n t h el o w e st E a v ail f o r 0. 0 < q 3 < 0. 3 G e V / c, w hi c h i s al s ow h e r e t h e p r e di ct e d R P A e ff e ct i s m o r e si g ni fi c a nt t h a nt h e pr e di ct e d 2 p 2 h e ff e ct. T h e s e d at a a p p e a r s e n siti v et o s u btl e d et ail s of t h e C C Q E v s. 2 p 2 h p r o c e s s e s n oty et e x p o s e d wit hi n t h e a v ail a bl e m o d el s, s u c h a s t h o s e[ 1 8, 1 9, 5 3] t h at g o b e y o n d t h e Fe r mi g a s.

T hi s 2 p 2 h t u n e c o m e s wit h t h r e e ot h er v a ri ati o n s t h att r e at t h e fi n al st at e n u cl e o n c o nt e nt a s u n c ert ai n. I n st e a dof e n h a n ci n g all 2 p 2 h e v e nt s, t h e fir st v ari ati o n e n h a n c e so nl y t h o s e g e n e r at e d f o r p n i niti al st at e n u cl e o n p air s,w hi c h t r a n sl at e s t o p p fi n al st at e f o r t h e n e ut ri n o c a s ei n t h e fit a n d n n f or t h e a nti- n e ut ri n o c a s e w h e r e w ea p pl y t h e t u n e d p a r a m et er s. T h e n e xt v a ri ati o n e n h a n c e sr e a cti o n s t h at a r e n ot o n p n i niti al st at e p ai r s, w hi c h l e a dt o p n fi n al st at e s. Fi n all y, t h e t hi r d v ari ati o n e n h a n c e sC C Q E e v e nt s at t h e s e ki n e m ati c s. I n a d diti o n t o t e sti n gt h e s e v a ri ati o n s a g ai n st t h e r e c o n st r u ct e d d at a, t h e y a r eu s e d a s a n u n c ert ai nt y a p pli e d l at er w h e n p r o d u ci n g ad o u bl e- di ff e r e nti al cr o s s s e cti o n.

T hi s s a m pl e al s o i n cl u d e s a si g ni fi c a nt c o m p o n e nt ata n d b e y o n d t h e ∆ r e s o n a n c e p e a k, w hi c h r e m ai n s p o orl yd e s c ri b e d b y t h e s e m o d el v ari ati o n s. T h e s h o rt c o mi n g oft h e m o d el f o r t h e s e l o w Q 2 = q 2

3 − q 20 ≈ 0 e v e nt s s h o w s

u p o n t h e f a r ri g ht of t h e di stri b uti o n s i n Fi g. 3. Si mil a rmi s m o d eli n g of t h e r e s o n a n c e- r e gi o n r at e h a s b e e n p r e vi-o u sl y r e p ort e d i n m e a s u r e m e nt s o n mi n e r al oil b y Mi ni-B o o N E [ 5 4, 5 5], i n MI N E R v A’ s pi o n fi n al st at e s a m pl e s[ 4 9, 5 6, 5 7], i n t h e n e ut ri n o v e r si o n of t hi s a n al y si s [ 6],a n d i n a r e s o n a n c e- ri c h n e utri n o + Fe s a m pl e f r o m MI N O S[ 5 8]. T h e l att e r u s e d a c al ori m et ri c s a m pl e a s a si d e b a n da n d t u n e d a n a d- h o c, l o w Q 2 s u p p r e s si o n t o t h e d at ai n or d e r t o i m p r o v e t h e e sti m at e of t h e r e s o n a n c e b a c k-g r o u n d i n t h ei r C C Q E a n al y si s. At Q 2 = 0 t h e r at e i s4 0 % of n o mi n al a n d b e c o m e s n o s u p pr e s si o n b y Q 2 = 0 .7( G e V / c) 2 . A p pl yi n g t h e MI N O S p a r a m et e ri z ati o n i m-p r o v e s t h e d e s c ri pti o n of t h e s e MI N E R v A d at a f o r s o m eof t h o s e bi n s at hi g h q 3 , b ut t h e s u p p r e s si o n g o e s t o o f a ra n d p r o d u c e s a m o d el d e fi cit i n t h e hi g h e st e n e r g y bi n s of

5

t h e l o w q 3 p a n el. T h e s e bi n s i n Fi g. 3 w er e al r e a d y w elld e s c ri b e d, a n d t h e χ 2 r e fl e ct s t h at t h e a g r e e m e nt w or s-e n s. Eit h er t h e si n gl e- p a r a m et e r Q 2 w ei g ht o r t h e t u ni n gt o n e ut ri n o + Fe d at a i s n ot a d e q u at e t o d e s c ri b e t h e t w odi m e n si o n al ki n e m ati c s of t h e s e a nti- n e ut ri n o + C H s a m-pl e s.

s a m pl e ν µ ν µ ν µ ν µ

q 3 r a n g e L o w e r U p p e r L o w e r U p p e rd e g r e e s of f r e e d o m 1 9 3 7 2 4 4 1

G E NI E 2. 8. 4 + pi o n [ 3 4, 4 0 – 4 3] 2 3 9 1 6 7 3 9 4 2 8 1

+ Q E R P A [ 4 6, 5 2] 2 6 1 1 4 0 2 6 5 2 5 3

+ 2 p 2 h [ 8, 4 5] 1 0 5 1 0 8 1 4 9 2 9 4

+ t u n e [ 6, 4 7] 6 9 8 0 7 7 1 5 0

t u n e o nl y p n i ni ti al s t a t e 6 5 8 6 7 6 1 6 0

t u n e n o t p n i ni ti al s t a t e 7 1 7 4 8 4 1 6 3

t u n e C C Q E r e a c ti o n s 5 9 1 2 3 1 0 8 1 6 6

+ MI N O S r e s o n a n c e t u n e [ 5 8] 1 5 1 4 5 1 1 4 1 4 1

T A B L E I: C o m p a ri s o n of t h e m o d el s t o r e c o n s t r u c t e d d a t as h o wi n g t h e e v ol u ti o n of t h e χ 2 wi t h e a c h m o d el c h a n g e. T h er e c o n s t r u c t e d d a t a a n d t h e b a s e m o d el a r e a s i n t h e t o p p a n-el s, a n d t h e “ + t u n e ” m o d el a r e a s i n t h e l o w e r p a n el s Fi g. 3.T h e c al c ul a ti o n a c t u all y u s e s t h e r e s ol u ti o n- d ri v e n si x bi n s ofq 3 { 0. 0, 0. 2, 0. 3, 0. 4, 0. 5, 0. 6, 0. 8 }

0. 0 0. 2 0. 40

1

2

3/ G e V < 0. 5

3q0. 4 <

2 0. 1 5

1 5. 5 5

0

1

2

3/ G e V < 0. 2

3q0. 0 <

0. 0 0. 2 0. 4

/ G e V < 0. 63

q0. 5 <

1 9. 5 5

1 6. 0 7/ G e V < 0. 3

3q0. 2 <

1 2. 2 5

9. 2 5

0. 0 0. 2 0. 4

/ G e V < 0. 83

q0. 6 <

D at aM n v G E NI E- v 1:Q ED elt a2 p 2 h

T ot al

1 5. 8 5

1 2. 9 6/ G e V < 0. 4

3q0. 3 <

p ot L E A nti n e utri n o2 0

1 0×1. 0 2

1 6. 8 4

1 4. 0 9

A v ail a bl e e n er g y ( G e V)

)2

/Ge

V2

c

m-42

(10

3q

davail

E/

dσ

2d

G e V / c f o r b e s t s e n si ti vi t y, a n dt h e y a r e s u m m e d i nt o t h e s a m e t w o r a n g e s s h o w n i n Fi g. 3.T h e ri g ht- m o s t c ol u m n s a r e m a d e u si n g t h e n e u t ri n o d a t a [ 6]t h o u g h t h e m o d el s b ei n g t e s t e d i n t hi s L e t t e r h a v e a d v a n c e dsi n c e t h a t e a rli e r p u bli c a ti o n.

FI G. 4: U nf ol d e d d 2 σ / d E a v ail d q 3 c r o s s s e c ti o n p e r n u cl e o nc o m p a r e d t o t h e m o d el wi t h R P A a n d t u n e d 2 p 2 h c o m p o-n e nt s. T h e b r e a k d o w n of t h e p r e di c t e d Q E ( d a s h e d ), 2 p 2 h( d o t- d a s h e d ), a n d ∆ r e s o n a n c e ( d o t t e d ) p o r ti o n s a r e s h o w n.T o s h o w t h e d e t ail, t h e d a t a a n d m o d el p r e di c ti o n f o r t h e fi r s tbi n ( d o mi n a t e d b y n e u t r o n- o nl y fi n al s t a t e s ) a r e n o t s h o w n;t h e y a r e f a r o ff t h e t o p of t h e pl o t wi t h v al u e s b e t w e e n 9 a n d1 7 × 1 0 − 4 2 c m − 2 / G e V 2 p e r n u cl e o n.

T o all o w d e v el o p m e nt a n d t e sti n g of i m p r o v e d m o d el s,t hi s di stri b uti o n i s u nf ol d e d t o pr o d u c e a d o u bl e di ff e r-e nti al c r o s s s e cti o n d 2 σ / d E a v ail d q 3 , s h o w n i n Fi g. 4 a n d

t a b ul at e d i n t h e s u p pl e m e nt. T h e r e s ol uti o n f o r q 3 i nFi g. 2 i s w ell b e h a v e d wit h a n R M S n e a r 2 3 % t hr o u g h-o ut. T h e r e c o n st r u ct e d a v ail a bl e e n er g y i s t h e s u m of ac o m p o n e nt f r o m c h a r g e d h a dr o n a n d el e ct r o m a g n eti c e n-e r g y d e p o sit s wit h a c e nt r al p e a k of 3 0 % r e s ol uti o n b utR M S of 4 0 % a s i n t h e n e ut ri n o c a s e [ 6]. T h e n t h e r a n-d o m t e n s of M e V e n e r g y fr o m a b o ut h alf t h e fi n al st at en e ut r o n s f u rt h e r d e g r a d e t h e r e s ol uti o n t o Fi g. 2. T h et w e nt y- fi v e E a v ail , q 3 bi n s w e r e c h o s e n b a s e d o n t h e s er e s ol uti o n s.

T h e l ar g e st f r a cti o n al u n c e rt ai nti e s i n h alf t h e bi n s,u p t o 1 4 %, c o m e f r o m v a ri ati o n s o n t h e 2 p 2 h e n h a n c e-m e nt u s e d i n t h e u nf ol di n g m o d el. W h e n t h e e n h a n c e-m e nt i s f o r m e d o nl y f r o m e v e nt s wit h p n i niti al st at ep ai r s ( p r ef er e nti all y n n fi n al st at e s i n t h e a nti- n e utri n oc a s e), t h e mi g r ati o n m at ri x h a s hi g h e r p r o b a bilit y t o p ute v e nt s i n t h e l o w E a v ail bi n s. T h e o p p o sit e i s t r u e w h e nt h e e n h a n c e m e nt o nl y a d d s p p i niti al st at e p air s. T h eu n c e rt ai nt y a s si g n e d t o G E NI E’ s i ntr a n u cl e a r r e s c att er-i n g m o d el i s al s o l a r g e b e c a u s e it m o di fi e s t h e u nf ol di n gm o d el i n t hi s st e e p r e gi o n of t h e c r o s s s e cti o n. T h e s eu n c e rt ai nti e s a r e of si mil a r si z e t o t h e fl u x u n c e rt ai nt y,s u g g e sti n g a f ut ur e c y cl e of c r o s s s e cti o n m o d el i m p r o v e-m e nt s c o ul d yi el d a n e v e n m o r e p r e ci s e cr o s s s e cti o n. T h eb r e a k d o w n of u n c e rt ai nti e s a n d t h e f ull c o v ari a n c e m atri xa r e pr e s e nt e d i n t h e s u p pl e m e nt ar y m at e ri al.

I n c o n cl u si o n, t h e h a d r o ni c e n e r g y s p e ct r u m f r o m as a m pl e of l o w m o m e nt u m tr a n sf e r a nti- n e ut ri n o i nt e r a c-ti o n s s u g g e st s t h e n e e d f o r a R P A-li k e s u p pr e s si o n [ 1 7]of q u a si- el a sti c e v e nt s, r el ati v e t o a Fer mi g a s m o d el. I na d diti o n, a n e n h a n c e m e nt o n t o p of t h e I FI C V al e n ci a2 p 2 h c o m p o n e nt [ 8, 4 5] i s e s s e nti al t o s u p pl y t h e o b-s e r v e d e v e nt r at e i n t h e r e gi o n b et w e e n t h e C C Q E a n d ∆p e a k s. We a d d t o t h e e vi d e n c e f or a l o w Q 2 s u p p r e s si o nof r e s o n a n c e e v e nt s b y d e m o n str ati n g t h e MI N O S p a-r a m et eri z ati o n [ 5 8] o ff er s s o m e i m p r o v e m e nt t o t h e χ 2 .T h e m o d el el e m e nt s a b o v e w er e t e st e d or fit t o d e s c ri b el e pt o n a n d h a d r o ni c c o m p o n e nt s of n e ut ri n o d at a. C rit-i c al f o r o s cill ati o n e x p e ri m e nt s i n t hi s n e ut ri n o e n er g yr a n g e, t h e y o ff er si mil arl y g o o d d e s c ri pti o n of t h e s e a nti-n e ut ri n o d at a.

T hi s d o c u c m e nt w a s p r e p a r e d b y m e m b e r s of t h e MI N-E R v A c oll a b o r ati o n u si n g t h e r e s o u r c e s of t h e Fe r miN ati o n al A c c el e r at o r L a b o r at o r y ( Fer mil a b), a U. S. D e-p a rt e nt of E n e r g y, O ffi c e of S ci e n c e, H E P U s e r F a cil-it y. Fe r mil a b i s m a n a g e d b y Fe r mi R e s e a r c h Alli a n c e,L L C ( F R A), a cti n g u n d er C o nt r a ct N o. D E- A C 0 2-0 7 C H 1 1 3 5 9. T h e s e r e s o u r c e s i n cl u d e d s u p p o rt f or t h eMI N E R v A c o n st r u cti o n pr oj e ct, a n d s u p p o rt f o r c o n-st r u cti o n w a s al s o g r a nt e d b y t h e U nit e d St at e s N a-ti o n al S ci e n c e F o u n d ati o n u n d e r a w ar d P H Y- 0 6 1 9 7 2 7a n d b y t h e U ni v e r sit y of R o c h e st e r. S u p p o rt f o r p a r-ti ci p ati n g s ci e nti st s w a s p r o vi d e d b y N S F a n d D O E( U S A) b y C A P E S a n d C N P q ( Br a zil), b y C o N a C y T( M e xi c o), b y Pr o y e ct o B a s al F B 0 8 2 1, C O NI C Y T PI AA C T 1 4 1 3, F o n d e c yt 3 1 7 0 8 4 5 a n d 1 1 1 3 0 1 3 3 ( C hil e), b y

6

CONCYTEC, DGI-PUCP, and UDI/IGI-UNI (Peru),and by the Latin American Center for Physics (CLAF).We thank the MINOS Collabortion for use of its neardetector data. Finally, we thank the staff of Fermilab forsupport of the beamline, the detector, and computinginfrastructure.

∗ Now at Iowa State University, Ames, IA 50011, USA† now at Lawrence Berkeley National Laboratory, Berke-

ley, CA 94720, USA‡ now at University of Michigan, Ann Arbor, MI 48109,

USA§ Now at University College London, London WC1E 6BT,

UK¶ now at Tufts University, Medford, MA 02155, USA

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