The VMC Spectral Function of 4HeIdeally, one should orthogonalize with the single-nucleon overlap

|k0i ⌦ | A�2f i ! |k0i ⌦ | A�2

f i � | A�1f̄

ih A�1f̄

|[|k0i ⌦ | A�2f i]

<latexit sha1_base64="lhHiMtWPf6oF1EzUyCsUl4R/5IY=">AAADRHiclVHLjtMwFHXCayivDizZWFRIbKZKChIdVoPYsCwSnamUZIrj3LSeOnZkO4OqxL/HF7DhH9ghtggnLY8Ow2KuFPnk3HN0LJ+05EybIPji+deu37h5a+92787de/cf9PcfHmtZKQpTKrlUs5Ro4EzA1DDDYVYqIEXK4SRdvWn3J+egNJPivVmXkBRkIVjOKDGOmvc/N6vTuFSsABwrIhbcndK4X42beKLZPD+tXx+M7O+lkfiKlgPc4I6v45SoOre23Yf2j5tvz0tEDY4afLW8BM/7g2A4OhwHzw/xvyAcBt0M0HYm831vFmeSVgUIQznROgqD0iQ1UYZRDrYXVxpKQldkAZGDgrj0pO6e3+KnjslwLpX7hMEd+7ejJoXW6yJ1yoKYpb64a8nLdlFl8nFSM1FWBgTdBOUVx66DtkucMQXU8LUDhCrm7orpkihCjWt8J2UBorvBDtkGGim5tr1eLOAjlUVBROYaOAdqozBxSPKsNUqO60Fo7QXdkphOtyuMHb2RO4Pr4teD4/+D49EwdPjdi8HReNvKHnqMnqBnKEQv0RF6iyZoiqj3yvvgMe/M/+R/9b/53zdS39t6HqGd8X/8BOfLEOE=</latexit><latexit sha1_base64="lhHiMtWPf6oF1EzUyCsUl4R/5IY=">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</latexit><latexit sha1_base64="lhHiMtWPf6oF1EzUyCsUl4R/5IY=">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</latexit><latexit sha1_base64="lhHiMtWPf6oF1EzUyCsUl4R/5IY=">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</latexit>

Inspired by the contact formalism, we put a cut on the relative distance of the pair

Noemi Rocco, Alessandro Lovato, Rocco Schiavilla, Robert B. Wiringa

6

have to be computed. Since the evaluation of the sum rules of the 12C for a single value of q

takes of about 100 seconds (with 32 OMP threads), we decided to split the calculation in such

a way that each ADLB slave calculates the sum rules for a single value of q.

Figure 1. Automatic Dynamic Load Balancing work flow.

• subroutine o_em_wk

Let us concentrate on a particular ADLB energy slave, managing a single configuration.

It enters o_em_wk and immediately puts into the work pool the part of work package

independent on q

call ADLB_Begin_batch_put (rwp%cfl,respon_wp_len_common,ierr)

where rwp%cfl indicates the beginning of the work package, respon_wp_len_common

denotes its size and ierr will get a return code.

Afterwards, the q dependent parts of the work packages are placed in the work pool for

each of the ∼ 60 cases.

call ADLB_PUT(rwp%qh,respon_wp_len_var,-1,myid, adlbwp_respon,i_prior,ierr)

11/26/18, 10)35 AMDeep Underground Neutrino Experiment

Page 1 of 2http://www.dunescience.org/

An international mega-science projectThe Deep Underground Neutrino Experiment (DUNE) is a leading-edge, international experiment for neutrino science and

proton decay studies. Discoveries over the past half-century have put neutrinos, the most abundant matter particles in the

universe, in the spotlight for further research into several fundamental questions about the nature of matter and the evolution

of the universe — questions that DUNE will seek to answer.

DUNE will consist of two neutrino detectors placed in the world’s most intense neutrino beam. One detector will record particle

interactions near the source of the beam, at the Fermi National Accelerator Laboratory in Batavia, Illinois. A second, much

larger, detector will be installed more than a kilometer underground at the Sanford Underground Research Laboratory in Lead,

South Dakota — 1,300 kilometers downstream of the source. These detectors will enable scientists to search for new subatomic

phenomena and potentially transform our understanding of neutrinos and their role in the universe.

DUNE prototype detectors are under construction at the European research center CERN.

The Long-Baseline Neutrino Facility will provide the neutrino beamline and the infrastructure that will support the DUNE

detectors. Groundbreaking for the LBNF excavation and construction at Sanford Lab occurred on July 21, 2017.

Aiming for groundbreaking discoveriesOrigin of Matter

Could neutrinos be the reason that the universe is made of matter rather than antimatter? By

exploring the phenomenon of neutrino oscillations, DUNE seeks to revolutionize our understanding of

neutrinos and their role in the universe.

Unification of Forces

With the world’s largest cryogenic particle detector located deep underground, DUNE can search for

signs of proton decay. This could reveal a relation between the stability of matter and the Grand

Unification of forces, moving us closer to realizing Einstein’s dream.

Black Hole Formation

DUNE’s observation of thousands of neutrinos from a core-collapse supernova in the Milky Way would

allow us to peer inside a newly-formed neutron star and potentially witness the birth of a black hole.

Watch this four-minute animation…

Highlight

CERN and Fermilabannounce big step in DeepUnderground NeutrinoExperiment18 Sept 2018

The first ProtoDUNE detector, the

largest liquid-argon neutrino

detector in the world, has just

recorded its first particle tracks,

signaling the start of a new

chapter in the story of the

international DUNE

experiment… This detector, located

at CERN, took two years to build

and eight weeks to fill with 800

tons of liquid argon. The detector

records traces of particles both

from cosmic rays and a beam

created at CERN’s accelerator

complex. Now that the first tracks

have been seen, scientists will

operate the detector over the next

several months to test the

technology in depth…

Read more…

DUNE and LBNF

Facility for DUNE (LBNF)

LBNF/DUNE Fact Sheet

All Things Neutrino

Frequently Asked Questions

Environmental Assessment

Fermilab (host lab and near site)

Sanford Lab (far site)

For the Media

DUNE at Work

Latest TweetsDUNEScience @DUNEScience

DUNE Science

Search...

Theoretical approaches:• Phenomenological: two body AV18 + three body IL7 • Chiral effective field theory: two- and three-body chiral interactions

Nuclear dynamics is described by:

In electromagnetic processes the current operator is conserved:

= +

The differential cross section for inclusive lepton-nucleus scattering reads

|0i = | A0 i , |fi = | A

f i, | Np , A�1

f i, | ⇡k ,

Np , A�1

f i . . .

The initial and final state are given by

The hadron tensor describes the response of the nucleus

l𝜈

?

Green’s Function Monte Carlo• GFMC algorithms use imaginary-time projection technique to enhance

the ground-state component of a starting (correlated) trial wave function.

| 0i = lim⌧!1

e�(H�E0)⌧ | T i

All the nucleon spin and isospin degrees of freedom are retained, the computational cost grows exponentially with A. To deal with this level of complexity: both MPI and OpenMP parallelism are used

y

R

k

j

9

ԇ ЈЈ(:2o

φ )

oL* R#oL* Rk#�L* R#�L* Rk#L* R#L* Rk#

y

8

Ry

R8

ky

k8

ԇ ֓֓(:2o

φ )

yXy

yXk

yX9

yXe

yX3

RXy

ԇ ֕֕(:2o

φ )

@RXy@yX8yXyyX8RXyRX8kXykX8jXy

ԇ Ј֕(:2o

φ )

@98@9y@j8@jy@k8@ky@R8@Ry@8y

y Ryy kyy jyy 9yy 8yy eyy

ԇ ֓֔(:2o

φ )

ᆂ(J2o)

y

R

k

j

9

ԇ ЈЈ(:2o

φ )

oL* R#oL* Rk#�L* R#�L* Rk#L* R#L* Rk#

y

8

Ry

R8

ky

k8

ԇ ֓֓(:2o

φ )

yXy

yXk

yX9

yXe

yX3

RXyԇ ֕֕(:2

oφ )

@RXy@yX8yXyyX8RXyRX8kXykX8jXy

ԇ Ј֕(:2o

φ )

@98@9y@j8@jy@k8@ky@R8@Ry@8y

y Ryy kyy jyy 9yy 8yy eyy

ԇ ֓֔(:2o

φ )

ᆂ(J2o)

y

R

k

j

ԇ ЈЈ(:2o

φ )

o* R#o* Rk#�* R#�* Rk#** R#** Rk#

y

8

Ry

R8

ky

k8

ԇ ֓֓(:2o

φ )

yXy

yXk

yX9

yXe

yX3

RXy

ԇ ֕֕(:2o

φ )

@RXy@yX8yXyyX8RXyRX8kXykX8jXy

ԇ Ј֕(:2o

φ )

@98@9y@j8@jy@k8@ky@R8@Ry@8y

y Ryy kyy jyy 9yy 8yy eyy

ԇ ֓֔(:2o

φ )

ᆂ(J2o)

4

y

k

9

e

3

Ry

� �� �� �� �� �� �� �� �� �� ���

ԓစ�ԓӺ ᇌU��ϵφ V

Ӻᇌ(J2o)

**@k#o@k#�@k#

**@R#o@R#�@R#

FIG. 1. (Color online). The di↵erential rates obtained withone-body (1b) only and both one- and two-body (2b) termsin the vector (V) and axial (A) components of the charge-changing (CC) weak current, and full CC current, are dis-played as function of the ⌫µ-energy in the allowed kinematicalrange. The theoretical uncertainites resulting from combiningstatistical errors in the GFMC calculation with errors associ-ated with the maximum-entropy inversion of the imaginary-time data are shown by the bands. The arrow indicates thekinematically maximum allowed E⌫ , see text for further ex-planations.

body (2b) terms in these currents are listed in Table I,and displayed in Fig 1, separately. Note that the re-sponse function Rxy(E⌫) in Eq. (9) involves interferencebetween the matrix elements of the V and A currents, andtherefore only contributes when both are present. As aconsequence, �(CC) 6=�(V) + �(A); indeed, this V-A in-terference leads to an increase in the �(V) + �(A) resultby ⇡ 10% in both the 1b- and 2b-based calculations.

In the 4He capture, the neutrino energy is in the range0 E⌫ Emax

⌫⇡ 83.6 MeV; however, the distribution,

also on account of the E2⌫-weighing factor present in the

expression for d�/dE⌫ , is skewed towards the high end,confirming the expectation that the energy release in thecapture process is converted primarily in energy for theemitted neutrino [46] with the remaining balance beingabsorbed by the final nuclear system. In the present case,since 4H is not bound, the possible final breakup chan-nels are 3H+n (3+1), 2H+2n (2+2), and 1H+3n (1+3),which have slightly di↵erent thresholds. While the con-tributions of these channels are fully accounted for here,they cannot be individually identified over the allowedE⌫ range—a limitation intrinsic to the present methodand apparent from Eq. (11), which relies on closure toremove the sum over final states. Nevertheless, Caineand Jones [50] estimated the branching ratios into the3+1, 2+2, 1+3 channels to be, respectively, 97.75%, 2%and 0.25%.

A related issue has to do with the behavior of theresponse functions in the threshold region E⌫ . Emax

⌫.

The kinematical constraint that R↵�(E⌫) vanish for E⌫

larger than Emax⌫

is not imposed when performing theinversion (see supplemental material). Even though rel-atively high values of ⌧ ⌧max =0.1 MeV�1 are cal-culated by GFMC, the maximum-entropy procedure weutilize still produces some strength beyond Emax

⌫, as

is apparent from Fig. 1. However, the integrals ofd�/dE⌫ , when evaluated over the whole E⌫-range includ-ing the unphysical region, remain stable to within 1% for⌧max =(0.1, 0.08, 0.05) MeV�1.In Table I we list the results for the 1b and 2b to-

tal rates (indicated as fCC) obtained with a CC weakcurrent in which the term proportional to the inducedpseudoscalar form factor GPS(q2) (in the axial sector)is ignored. The e↵ect is significant: retaining this termreduces the fCC values by ⇡ 15% (14%) in the 1b (2b)calculations. The parametrization for GPS(q2) adoptedhere [44] is consistent with the recent determination ofthis form factor by the MuCap collaboration [16]. It alsoleads, in an accurate ab initio calculation based on es-sentially the same dynamical inputs adopted here [52],to a prediction for the 3He(µ�, ⌫µ)3H total rate that isagreement with the (remarkably precise) measurementof Ref. [53], 1496(4) s�1. Thus, muon capture providesa sensitive test of the GPS(q2) form factor at low mo-mentum transfers. By contrast, this observable is onlyvery marginally a↵ected (at a fraction of a 1% level) bychanges in the parametrization of the nucleon axial formfactor, as we have explicitly verified by calculating howthe total rate changes when the cuto↵ ⇤A is varied by±10% about its central value of ⇤A ⇡ 1 GeV. The reasonis that GA(q2)= gA

⇥1 + 2 q2/⇤2

A+ · · ·

⇤, and q2/⇤2

A⌧ 1

in the allowed kinematical region.In this letter, we have formulated an ab initio QMC

method for calculating inclusive muon-capture rates onlight nuclei (mass number A 12), and have presented,as a first application, a calculation of the total rate in4He. The predicted value is consistent with the lowerrange of available experimental determinations (see Ta-ble I). However, these measurements from bubble cham-ber experiments of the late 60’s have large errors, makingit impossible to establish, at a quantitative level, the va-lidity of the model for the nuclear charge-changing weakcurrent we have adopted here. We hope the present workwill motivate our colleagues to carry out a new experi-ment on 4He.Future plans in this area include (i) the application

of the method to other (light) nuclei, especially in caseswhere more accurate data are known [46], and (ii) itsextension to more fundamental dynamical approachesbased on interactions and electroweak currents derivedfrom chiral e↵ective field theory. The presence of dis-crete states in the final nuclear system substantially com-plicates the calculation of the capture rate, since theimaginary-time response functions E↵�(q, ⌧) would haveto be evaluated at large enough values of ⌧ to reliably

• GFMC calculation of the charged-current electroweak response of 12C at q=700 MeV: one- and two-body currents are includedAccurate calculation of the Euclidean response within GFMC

Maximum Entropy technique is used to obtain the nuclear response function

• GFMC calculation of muon capture on 4He and 3H

n<latexit sha1_base64="YpuTIWIRm4MnQGf9rk1VPmcNIak=">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</latexit><latexit sha1_base64="YpuTIWIRm4MnQGf9rk1VPmcNIak=">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</latexit><latexit sha1_base64="YpuTIWIRm4MnQGf9rk1VPmcNIak=">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</latexit><latexit sha1_base64="YpuTIWIRm4MnQGf9rk1VPmcNIak=">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</latexit>

µ�<latexit sha1_base64="zRFjoyOrRE0xkhITl06LTbsuln8=">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</latexit><latexit sha1_base64="zRFjoyOrRE0xkhITl06LTbsuln8=">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</latexit><latexit sha1_base64="zRFjoyOrRE0xkhITl06LTbsuln8=">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</latexit><latexit sha1_base64="zRFjoyOrRE0xkhITl06LTbsuln8=">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</latexit>

⌫µ<latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit><latexit sha1_base64="P9M/dpXaXxO4WUjCfpE48kLDhyk=">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</latexit>

p<latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">AAACb3icdVFdSxtBFJ1sa9VY69eDDwUZGgr6EnajaHwL9MVHhUYD2UVmZ2+SwflYZu4qYdlf4Kv9cf0Z/Qed3aRgpF4YOJx7DufOvWkuhcMw/N0KPnxc+7S+sdne+rz9ZWd3b//WmcJyGHIjjR2lzIEUGoYoUMIot8BUKuEuffhR9+8ewTph9E+c55AoNtViIjhDT93k97udsBudn/Yv+7QGvX64AGcX0SWNumFTHbKs6/u91ijODC8UaOSSOTeOwhyTklkUXELVjgsHOeMPbApjDzVT4JKymbSi3z2T0Ymx/mmkDfvaUTLl3FylXqkYztzbXk3+rzcucNJPSqHzAkHzRdCkkBQNrb9NM2GBo5x7wLgVflbKZ8wyjn45KylT0M0EK2QdiMZIV7XbsYYnbpRiOivj9BF4NY4Sj4zMaqORtOxEVfVGN2PY6FaFsacXcm/wt/i3cPo+uO11I49vzjqDwfIqG+Qr+UaOSUQuyIBckWsyJJwAeSYv5FfrT3AYHAV0IQ1aS88BWang5C9T0L+v</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit><latexit sha1_base64="Iet7OPTkf1gkRuQASJltGdM+Yxc=">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</latexit>

CC-1B CC-12B exp

Γ(s-1) 265±9 306±9 336±75

��Κ��ϯ��ϵ��φ��Ј

� � � � � � � � �

%_%(

7Kϯ�ϵ )

` (7K)

oJ*:6J*

Variational Monte Carlo Spectral Function• The Spectral Function yields the probability distribution of removing a

nucleon with a given momentum and energy

��Θ��Κ��ϯ��ϵ��φ��Ј��φ��ϵ��ϯ

� � � � � �

ԝ ԚU7K

ϯ V

Ԛ U7KV

ԝϽεͨԝεΑ

Figure 1: VMC mean-field and full momentum distributions of 4He.

ZdE

d3k

(2⇡)3Pn(k, E) =

Zd3k

(2⇡)3nn(k) = A� Z , (4)

where Z is the number of protons and A is the number of nucleons of a givennucleus. This normalization is consistent with the one of the variationalMonte Carlo (VMC) single-nucleon momentum distribution reported in [2].

Spectral function of4He

For clarity, let us deal with the proton spectral function first. The single-nucleon (mean-field) contribution P

MF

p (k, E) corresponds to identifying | A�1

n iwith | 3

H

0i, the ground-state of 3H

PMF

p (k, E) = nMF

p (k)�⇣E � B4He +B3H � k

2

2m3H

⌘. (5)

where B4He ' 28.30 MeV and B3H ' 8.48 MeV are the binding energies of4He and 3H, respectively and m3H is the mass of the recoiling nucleus. In theabove equation we introduced the mean-field proton momentum distribution

nMF

p (k) = |h 4He

0|[|ki ⌦ | 3

H

0i]|2 , (6)

in which h 4He

0|[|ki ⌦ | 3

H

0i is the Fourier transform of the single-nucleon

radial overlap that can be computed within both VMC and Green’s functionMonte Carlo (GFMC) [3].

2

��Ј��φ��ϵ��ϯ��Κ��Θ��ϩ

� ��� � ��� � ��� � ��� �

ԝ ։Ԡ ~ Ԇ����U

7Kϯ V

Ԡ U7K�V

ԝ։Ԡ ~Ԇԝ։Ԡ ~Ԇԡքօ � ���ԝ։Ԡ ~Ԇԡքօ � ���ԝ։Ԡ ~Ԇԡքօ � ���

Lepton-nucleus scattering using factorizationl

𝜈 • In the limit of moderate q, one can factorize the interaction vertex and use a Spectral Function to describe the internal nuclear dynamics

The matrix element of the current can be written in the factorized form

�������

� ���� ��� ���� ��� ���� ��� ���� ��� ����

ԓᅼ�ԓဇ րԓӺ ր(M

`fb`

:2o

)

ᆂ (:2o)

Ӻր4kXky :2o- ᅲր4kyXy ੋ2tT

i?2Q

�������������������������� ��ϩ��� ��ϩ��� ��ϩ��� ��ϩ

� ��� ��� ��� ��� ��� ��� ���

ԓᅼ�ԓဇ ᇋԓӺ ᇋ(M`

fb`

:2o

)

ᆂ (:2o)

Ӻᇌᆡ � � :2o- ᅲᇋ � ��ੋiQi .**

R#k#ၡ.**

1b current 2b current

π

∆

π production

Theory of nuclear electroweak interactions

• Asynchronus Dynamic Load Balancing (ADLB) is a library that handles computational load balancing by accepting work packages from any MPI rank and distributing them to ranks that need work to do

• Distributed Memory (DMEM) carries out memory load balancing by storing large arrays on any node with enough memory and subsequently fetching them when needed.

0 10 20 30 40 50 60 7030.0

50.0

100.

300.

500.

1000

Total number of threads

Tim

e/Sa

mpl

e (s

econ

ds)

1 rank 2 ranks 4 ranks 6 ranks10 ranks 1 rank, BG/QIdeal

A) GFMC Strong OpenMP scalingon KNL node

10 40 100 400 1000 400024

25

26

27

28

29

30

Number of nodes (6 MPI ranks/node, 10 threads/rank)

Tim

e (m

inut

es)

B) GFMC Weak MPI scaling on Theta11 Monte Carlo samples/node

Figure 4: Scaling of GFMC calculations on KNL nodes and Theta

We also have posted [77] many existing QMC results for A 12 nuclei, including one- and two-nucleon coordinate- and momentum-space densities and spectroscopic overlap functions. Thesehave been used by multiple groups to propose and analyze experiments or to test new theoreticalmodels. We will post comparable results for the chiral models as they are generated.

1.f Computational approach and performance of code

The GFMC code has been in continuous production use for more than 20 years. It has steadilyundergone development to take advantage of each new generation of parallel machine at Argonneand has been among the first to deliver new scientific results each time.

The main parts of the code are written in Fortran and compile with modern Intel, gfortran,IBM xlf, and Cray compilers. We make extensive use of Fortran-90 features such as pointers, arraysubscripts, and defined types. We also use older features such as common and equivalence to obtaine�ciency. We use OpenMP to parallelize our Fortran codes across cores within an MPI rank. Weare currently using only OpenMP2 features, but we will be investigating OpenMP4 directives toimprove vectorization.

GFMC relies on the ADLB and DMEM libraries. ADLB [78] handles computational loadbalancing by accepting work packages from any MPI rank and distributing them to ranks that needwork to do. ADLB also routes the work package answers back to the originating rank. DMEMcarries out memory load balancing by storing large arrays on any node with enough memory andsubsequently fetching them when needed. Both libraries are written in extremely portable C anddeveloped locally by coinvestigator Lusk and collaborators, who continue to maintain them for theHPC community under SciDAC grants. These are currently working well on Theta, but we willbe investigating if their performance can be further improved. ADLB and DMEM rely on MPI forinteraddress space communication. Currently the only features of MPI-2 that we need are relatedto thread safety and I/O; the communication is carried out by using original MPI-1 functions.

Panel A of Fig. 4 shows GFMC strong OpenMP scaling on a KNL node with 1 to 10 MPI ranksrunning on the node. We see good scaling for six to ten ranks up to using the full 64 cores on thenode. Panel B shows weak MPI scaling on increasing numbers of nodes on Theta. Six ranks with10 threads each were used on each node. The scaling is very good up to 3168 nodes, nearly the

10

⌫ +A ! `� +X

d2�

dE0d⌦0 =1

16⇡2

G2

2L↵�R↵�

R↵�=

X

f

h0|J†↵(q)|fihf |J�(q)|0i�(4)(p0 + q � pf )

H =X

i

p2i

2m+

X

i<j

vij +X

i<j<k

Vijk + . . .

@µJµ = 0 r · J+ i[H, J

0] = 0

Jµ =X

i

jµi +X

i<j

jµij + . . .

Two body currents are necessary to satisfy the continuity equation

Electroweak responses of nuclei

Introduction • Neutrino physics is entering a precision era. The long- and short-baseline programs will provide definitive answers

on neutrino fundamental properties• The success of these experiments greatly relies on the precision with which nuclear structure and electroweak

response functions are calculated

nuclei. This backward peak is a strong signatureof SRC pairs, indicating that the two emittedprotons were largely back-to-back in the initialstate, having a large relative momentum and asmall center-of-mass momentum (8, 9). This is adirect observation of proton-proton (pp) SRCpairs in a nucleus heavier than 12C.Electron scattering fromhigh–missing-momentum

protons is dominated by scattering from protonsin SRC pairs (9). The measured single-protonknockout (e,e′p) cross section (where e denotesthe incoming electron, e′ the measured scatteredelectron, and p the measured knocked-out pro-ton) is sensitive to the number of pp and np SRCpairs in the nucleus, whereas the two-protonknockout (e,e′pp) cross section is only sensitive tothe number of pp-SRC pairs. Very few of thesingle-proton knockout events also contained asecond proton; therefore, there are very fewpp pairs, and the knocked-out protons predom-inantly originated from np pairs.To quantify this, we extracted the [A(e,e′pp)/

A(e,e′p)]/[12C(e,e′pp)/12C(e,e′p)] cross-section dou-ble ratio for nucleus A relative to 12C. The doubleratio is sensitive to the ratio of np-to-pp SRCpairs in the two nuclei (16). Previous measure-ments have shown that in 12C nearly every high-momentum proton (k > 300 MeV/c > kF) has acorrelated partner nucleon, with np pairs out-numbering pp pairs by a factor of ~20 (8, 9).To estimate the effects of final-state interac-

tions (reinteraction of the outgoing nucleons inthe nucleus), we calculated attenuation factorsfor the outgoing protons and the probability ofthe electron scattering from a neutron in an nppair, followed by a neutron-proton single-chargeexchange (SCX) reaction leading to two outgoingprotons. These correction factors are calculatedas in (9) using the Glauber approximation (22)with effective cross sections that reproduce pre-viously measured proton transparencies (23), andusing themeasured SCX cross section of (24).Weextracted the cross-section ratios and deduced therelative pair fractions from the measured yieldsfollowing (21); see (16) for details.Figure 3 shows the extracted fractions of np

and pp SRC pairs from the sum of pp and nppairs in nuclei, including all statistical, systematic,and model uncertainties. Our measurements arenot sensitive to neutron-neutron SRC pairs. How-ever, by a simple combinatoric argument, even in208Pb these would be only (N/Z)2 ~ 2 times thenumber of pp pairs. Thus, np-SRC pairs domi-nate in all measured nuclei, including neutron-rich imbalanced ones.

The observed dominance of np-over-pp pairsimplies that even in heavy nuclei, SRC pairs aredominantly in a spin-triplet state (spin 1, isospin0), a consequence of the tensor part of the nucleon-nucleon interaction (17, 18). It also implies thatthere are as many high-momentum protons asneutrons (Fig. 1) so that the fraction of protonsabove the Fermi momentum is greater than thatof neutrons in neutron-rich nuclei (25).In light imbalanced nuclei (A≤ 12), variational

Monte Carlo calculations (26) show that this re-sults in a greater average momentum for theminority component (see table S1). The minoritycomponent can also have a greater average mo-mentum in heavy nuclei if the Fermimomenta ofprotons and neutrons are not too dissimilar. Forheavy nuclei, an np-dominance toy model thatquantitatively describes the features of the mo-mentum distribution shown in Fig. 1 shows thatin imbalanced nuclei, the average proton kineticenergy is greater than that of the neutron, up to~20% in 208Pb (16).The observed np-dominance of SRC pairs in

heavy imbalanced nuclei may have wide-rangingimplications. Neutrino scattering from two nu-cleon currents and SRC pairs is important for theanalysis of neutrino-nucleus reactions, which areused to study the nature of the electro-weak in-teraction (27–29). In particle physics, the distribu-tion of quarks in these high-momentum nucleonsin SRC pairs might be modified from that of freenucleons (30, 31). Because each proton has agreater probability to be in a SRC pair than aneutron and the proton has two u quarks foreach d quark, the u-quark distribution modifica-tion could be greater than that of the d quarks(19, 30). This could explain the difference be-tween the weak mixing angle measured on aniron target by the NuTeV experiment and that ofthe Standard Model of particle physics (32–34).In astrophysics, the nuclear symmetry energy

is important for various systems, including neu-tron stars, the neutronization of matter in core-collapse supernovae, and r-process nucleosynthesis(35). The decomposition of the symmetry energyat saturation density (r0 ≈ 0.17 fm−3, the max-imum density of normal nuclei) into its kineticand potential parts and its value at supranucleardensities (r > r0) are notwell constrained, largelybecause of the uncertainties in the tensor com-ponent of the nucleon-nucleon interaction (36–39).Although at supranuclear densities other effectsare relevant, the inclusion of high-momentumtails, dominated by tensor-force–induced np-SRCpairs, can notably soften the nuclear symmetry

energy (36–39). Our measurements of np-SRCpair dominance in heavy imbalanced nuclei canhelp constrain the nuclear aspects of these cal-culations at saturation density.Based on our results in the nuclear system, we

suggest extending the previous measurements ofTan’s contact in balanced ultracold atomic gasesto imbalanced systems in which the number ofatoms in the two spin states is different. Thelarge experimental flexibility of these systems willallow observing dependence of the momentum-sharing inversion on the asymmetry, density,and strength of the short-range interaction.

REFERENCES AND NOTES

1. S. Tan, Ann. Phys. 323, 2952–2970 (2008).2. S. Tan, Ann. Phys. 323, 2971–2986 (2008).3. S. Tan, Ann. Phys. 323, 2987–2990 (2008).4. E. Braaten, in Lecture Notes in Physics (Springer, Berlin, 2012),

vol. 836, p. 193.5. L. Lapikás, Nucl. Phys. A. 553, 297–308 (1993).6. K. I. Blomqvist et al., Phys. Lett. B 421, 71–78 (1998).7. R. Starink et al., Phys. Lett. B 474, 33–40 (2000).8. E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman, J. W. Watson,

Phys. Rev. Lett. 97, 162504 (2006).9. R. Subedi et al., Science 320, 1476–1478 (2008).10. K. Sh. Egiyan et al., Phys. Rev. Lett. 96, 082501 (2006).11. N. Fomin et al., Phys. Rev. Lett. 108, 092502 (2012).12. V. R. Pandharipande, I. Sick, P. K. A. deWitt Huberts,

Rev. Mod. Phys. 69, 981–991 (1997).13. J. Arrington, D. W. Higinbotham, G. Rosner, M. Sargsian,

Prog. Part. Nucl. Phys. 67, 898–938 (2012).14. J. T. Stewart, J. P. Gaebler, T. E. Drake, D. S. Jin, Phys. Rev. Lett.

104, 235301 (2010).15. E. D. Kuhnle et al., Phys. Rev. Lett. 105, 070402 (2010).16. Materials and methods are available as supplementary

materials on Science Online.17. R. Schiavilla, R. B. Wiringa, S. C. Pieper, J. Carlson, Phys. Rev. Lett.

98, 132501 (2007).18. M. M. Sargsian, T. V. Abrahamyan, M. I. Strikman, L. L. Frankfurt,

Phys. Rev. C 71, 044615 (2005).19. L. Frankfurt, M. Strikman, Phys. Rep. 160, 235–427 (1988).20. B. A. Mecking et al., Nucl. Inst. Meth. A 503, 513–553

(2003).21. O. Hen et al., Phys. Lett. B 722, 63–68 (2013).22. I. Mardor, Y. Mardor, E. Piasetsky, J. Alster, M. M. Sargsyan,

Phys. Rev. C 46, 761–767 (1992).23. D. Dutta, K. Hafidi, M. Strikman, Prog. Part. Nucl. Phys. 69, 1–27

(2013).24. J. L. Friedes, H. Palevsky, R. Stearns, R. Sutter, Phys. Rev. Lett.

15, 38–41 (1965).25. M. M. Sargsian, Phys. Rev. C 89, 034305 (2014).26. R. B. Wiringa, R. Schiavilla, S. C. Pieper, J. Carlson, Phys. Rev. C

89, 024305 (2014).27. L. Fields et al., Phys. Rev. Lett. 111, 022501 (2013).28. G. A. Fiorentini et al., Phys. Rev. Lett. 111, 022502 (2013).29. Neutrino-Nucleus Interactions for Current and Next Generation

Neutrino Oscillation Experiments, Institute for NuclearTheory (INT) workshop INT-13-54W, University of Washington,Seattle, WA, 3 to 13 December 2013.

30. O. Hen, D. W. Higinbotham, G. A. Miller, E. Piasetzky,L. B. Weinstein, Int. J. Mod. Phys. E 22, 133017 (2013).

31. L. B. Weinstein et al., Phys. Rev. Lett. 106, 052301 (2011).32. G. P. Zeller et al., Phys. Rev. Lett. 88, 091802 (2002).33. G. P. Zeller et al., Phys. Rev. Lett. 90, 239902 (2003).34. I. C. Cloët, W. Bentz, A. W. Thomas, Phys. Rev. Lett. 102,

252301 (2009).35. J. M. Lattimer, Y. Lim, Astrophys. J. 771, 51 (2013).36. A. Carbone, A. Polls, A. Rios, Europhys. Lett. 97, 22001 (2012).37. I. Vidana, A. Polls, C. Providencia, Phys. Rev. C 84, 062801(R)

(2011).38. C. Xu, A. Li, B. A. Li, J. Phys. Conf. Ser. 420, 012090 (2013).39. B.-A. Li, L.-W. Chen, C. M. Ko, Phys. Rep. 464, 113–281

(2008).

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy (DOE)and the National Science Foundation, the Israel ScienceFoundation, the Chilean Comisión Nacional de InvestigaciónCientífica y Tecnológica, the French Centre National de la

616 31 OCTOBER 2014 • VOL 346 ISSUE 6209 sciencemag.org SCIENCE

Fig. 3. The extractedfractions of np (top)and pp (bottom) SRCpairs from the sum ofpp and np pairs innuclei.The green andyellow bands reflect68 and 95% confidencelevels (CLs), respec-tively (9). np-SRC pairs dominate over pp-SRC pairs in all measured nuclei.

SR

C P

air

fract

ion

(%)

100

50

010 50 100 A

C Al Fe Pb

68% C.L.

95% C.L.

np fraction

pp fraction

RESEARCH | REPORTS

on June 17, 2019

http://science.sciencemag.org/

Dow

nloaded from

• Electron- and proton-nucleus scattering experiments allow to study the high momentum component of the nuclear wave function. This is dominated by the presence of short range correlated pairs of nucleons

• Observed dominance of np-over-pp pairs for a variety of nuclei is ascribed to the tensor part of the nuclear force

• An accurate description of short-range correlated pairs appears to be relevant for the understanding of the EMC effect

✐ Subedi et al., Science 320, 1476 (2008)

Muons can be captured by the nucleus: inverse process of charge current neutrino scattering

✤ Total (left) and differential (right) capture rate of 4He obtained within GFMC including one-and two-body currents. The measured rate is taken from Nuovo Cimento 33, 1497 (1964).✤ Scaling of GFMC calculations on KNL nodes and Theta

✤ The Mean Field part (left) is obtained from VMC estimates of single-nucleon overlaps. The VMC two-nucleon momentum distributions are utilized to describe the short-distance and high-momentum component of the nuclear wave-function

✤ Total spectral function of 4He (left) and single nucleon momentum distribution, obtained integrating over E (right). Both the total and MF contribution are shown

✤ Electron-4He (top left) differential cross section obtained using the VMC spectral function. Only one-body current is included

✤ Charge current neutrino-12C (bottom right) differential cross section. We included one- and two-body current, π production

leading to a simplified expression for the nuclear cross section

given in terms of the elementary one d�A =X

N

ZdEd3k PN (k, E)d�N

h0|J↵(q)|fi !X

k

h A0 |[| N

k i ⌦ | A�1f i]h N

k |X

i

ji↵(q)| N 0

p i

E↵�(�,q) =

Zd!K(�,!)R↵�(!,q) = h A

0 |J†↵(q)K(�, H � E0)J�(q)| A

0 i

PN (k, E) = PMFN (k, E) + P corr

N (k, E)