Rela i is ic e ec s in wo-pa icle emission o elec on
and neu ino eac ions
I. Ruiz Simo,1C. Albe us,1J. E. Ama o,1M. B. Ba ba o,2J. A. Caballe o,3and T. W. Donnelly4
1Depa amen o de Física A ómica, Molecula y Nuclea , and Ins i u o de Física Teó ica y Compu acional
Ca los I, Uni e sidad de G anada, G anada 18071, Spain
2Dipa imen o di Fisica, Uni e si à di To ino and INFN,
Sezione di To ino, Via Pie o Giu ia 1, 10125 To ino, I aly
3Depa amen o de Física A ómica, Molecula y Nuclea ,
Uni e sidad de Se illa, Apa ado 1065, 41080 Se illa, Spain
4Cen e o Theo e ical Physics, Labo a o y o Nuclea Science and Depa men o Physics,
Massachuse s Ins i u e o Technology, Camb idge, Massachuse s 02139, USA
(Recei ed 17 May 2014; published 18 Augus 2014)
Two-pa icle wo-hole con ibu ions o elec oweak esponse unc ions a e compu ed in a ully
ela i is ic Fe mi gas, assuming ha he elec oweak cu en ma ix elemen s a e independen o he
kinema ics. We analyze he genuine kinema ical and ela i is ic e ec s be o e including a ealis ic meson-
exchange cu en ope a o . This allows one o s udy he ma hema ical p ope ies o he non i ial se en-
dimensional in eg als appea ing in he calcula ion and o design an op imal nume ical p ocedu e o educe
he compu a ion ime. This is equi ed o p ac ical applica ions o cha ged-cu en neu ino sca e ing
expe imen s, in which an addi ional in eg al o e he neu ino lux is pe o med. Finally, we examine he
iabili y o his model o compu e he elec oweak wo-pa icle– wo-hole esponse unc ions.
DOI: 10.1103/PhysRe D.90.033012 PACS numbe s: 13.15.+g, 13.60.Hb, 25.30.Fj, 25.30.P
I. INTRODUCTION
The unde s anding o in e media e-ene gy (0.5–10 GeV)
neu ino-nucleus sca e ing c oss sec ions is an impo an
ing edien o a mosphe ic and accele a o -based neu ino
oscilla ion expe imen s [1–4]. The analysis o hese expe i-
men s equi es ha ing good con ol o nuclea e ec s. The
simple desc ip ion based on a ela i is ic Fe mi gas (RFG)
model does no accu a ely desc ibe he ecen measu e-
men s o quasielas ic neu ino and an ineu ino sca e ing
[5–8]. Mechanisms such as nuclea co ela ions, inal-s a e
in e ac ions, and meson-exchange cu en s (MECs) may
ha e an impac on he inclusi e neu ino cha ged-cu en
(CC) c oss sec ion. In pa icula , explici calcula ions
suppo he heo e ical e idence [9–11] o a signi ican
con ibu ion om mul inucleon knockou o he CC c oss
sec ions ðνμ;μ−Þand ð¯
νμ;μþÞa ound and abo e he qua-
sielas ic (QE) peak egion, de ined by ω¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2þm2
N
p−mN,
whe e ωis he ene gy ans e and qis he 3-momen um
ans e . Recen ab ini io calcula ions [12] o sum ules o
weak neu al-cu en esponse unc ions o 12C ha e also
s essed he impo ance o MECs in neu ino quasielas ic
sca e ing. The size o MEC e ec s is la ge han ha ound
in inclusi e CC neu ino sca e ing om he deu e on [13].
The h ee exis ing mic oscopic models ha ha e p o-
ided p edic ions o mul inucleon knockou e ec s in
quasielas ic neu ino and an ineu ino c oss sec ions om
12C o he expe imen al kinema ical se ings a e hose by
Ma ini [14–19], Nie es [10,20–22], and he supe scaling
analysis (SuSA) model o Re s. [11,23,24].
These h ee models a e based on he Fe mi gas, bu each
one con ains di e en ing edien s and app oxima ions o
ace he p oblem. The Ma ini model is based on he
non ela i is ic model o Re . [25], al hough a emp s o
imp o e i using ela i is ic kinema ics ha e been made.
The model includes MEC and pionic co ela ion diag ams
modi ied o accoun o he e ec i e nuclea in e ac ion.
The in e e ence be ween di ec and exchange diag ams is
neglec ed, in o de o educe he se en-dimensional (7D)
in eg al o e he phase space o a wo-dimensional (2D)
in eg a ion. The Nie es model is simila o Ma ini’s, bu
mos o i is ully ela i is ic. In his model, he momen um
o he ini ial nucleon in he gene ic WNNπ e ex is ixed o
an a e age alue. Unde his app oxima ion, he Lindha d
unc ion can be ac o ized inside he in eg al, lea ing only a
ou -dimensional in eg a ion o e he momen um o one o
he exchanged pions. The di ec -exchange in e e ence is
neglec ed as well. The SuSA model includes all he in e -
e ence e ms a he cos o pe o ming a se en-dimensional
in eg a ion, wi hou any app oxima ion, bu he axial pa o
he MEC is no ye included. I is ob ious ha hese h ee
models should di e nume ically because hey a e di e en .
Bu a quan i a i e e alua ion o hei di e ences has no
been done. Fu he mo e, he accu acy o he app oxima ions
used in hese models only can be de e mined by compa ison
wi h an exac calcula ion o some kinema ics.
Al e na i ely, phenomenological app oaches ha e been
p oposed whe e wo-pa icle wo-hole (2p-2h) e ec s,
es ima ed by a pu e wo-nucleon phase-space model, a e
i ed o he expe imen al c oss sec ion [26,27], while he
PHYSICAL REVIEW D 90, 033012 (2014)
1550-7998=2014=90(3)=033012(23) 033012-1 © 2014 Ame ican Physical Socie y
nucleon ejec ion model o Re . [28] p o ides a phase-
space-based algo i hm o gene a e 2p-2h s a es in a
Mon e Ca lo implemen a ion.
The p esen pape is a i s s ep owa d an ex ension o
he ela i is ic 2p-2h model o Re . [29] o he weak sec o .
We unde ake his p ojec wi h he inal goal o including a
consis en se o weak MEC in he SuSA app oach o CC
neu ino eac ions [11,23]. The model o Re . [29] ully
desc ibed he con ibu ion o 2p-2h s a es o he ans e se
esponse unc ion in elec on sca e ing. Based on he RFG,
he model included all 2p-2h diag ams con aining wo
pionic lines (excep o nucleon co ela ions ha we e
included in Re . [30]), aking in o accoun he quan um
in e e ences be ween di ec and exchange wo-body
ma ix elemen s. P e ious calcula ions o wo-pa icle
emission wi h MEC in ðe; e0Þin ol ed non ela i is ic
models [25,31–36]. The i s a emp s o a ela i is ic
desc ip ion we e made by Dekke [37–39], ollowed by he
model o De Pace e al. [29,40]. The ex ension o his
model o he weak sec o equi es he inclusion o he axial
e ms o MEC. Quasielas ic neu ino sca e ing equi es
one o pe o m an in eg al o e he neu ino lux. This
would conside ably inc ease he compu ing ime o he
nuclea esponse unc ion o Re . [29] in ol ing 7D
in eg als o housands o e ms, al hough imp o emen s
we e made in Re . [30] o pe o m he spin aces nume i-
cally. Thus, in his wo k, we add ess he p oblem om a
di e en pe spec i e, ocusing i s on a ca e ul s udy o
he 7D in eg al o e he 2p-2h phase space as a unc ion o
he momen um and ene gy ans e s. Ou goal is o p o ide a
comp ehensi e desc ip ion o he angula dis ibu ion,
showing ha he e is a di e gence in he in eg and o
some kinema ics and iden i ying ma hema ically he
allowed in eg a ion in e als. A he same ime, we de i e
a p ocedu e o in eg a e he angula dis ibu ion a ound he
di e gence analy ically. This p ocedu e allows us o educe
he CPU ime conside ably. This p og am is ollowed i s
in a pu e phase-space domain, wi hou ye including he
wo-body cu en . We also ske ch he u u e pe spec i es
opened by his gene al o malism applied o he calcula ion
o 2p-2h con ibu ions o elec oweak esponses. In a
o hcoming pape , we will p o ide a ull model o weak
MEC o compu e he comple e se o CC neu ino sca e ing
esponse unc ions.
The pape is o ganized as ollows. In Sec. II, we de ine
he ela i is ic 2p-2h esponse and phase space unc ions.
In Sec. III, we e iew he non ela i is ic desc ip ion o
he 2p-2h in eg als, semianaly ical exp essions ha will be
used as a check o he ela i is ic calcula ions, and some
in e es ing p ope ies o he phase-space in eg al, such as
scaling and asymp o ic expansion. In Sec. IV, we add ess
he ela i is ic phase-space unc ion and asymp o ic
expansion and show ha some nume ical p oblems a ise
om a s aigh o wa d calcula ion o high q. In Sec. V,we
desc ibe he 2p-2h angula dis ibu ion in he ozen
nucleon app oxima ion and show ha his dis ibu ion
has a di e gence o some angles. The di e gence is ela ed
o he wo solu ions o he ene gy conse a ion o a ixed
emission angle. We gi e kinema ical and geome ical
explana ions o hese wo solu ions. In Sec. VI, we make
a heo e ical analysis o he angula dis ibu ion and ind
analy ically he bounda ies o he angula in e als. We ge
a o mula, Eq. (95), o he in eg al a ound he di e gen
angles. In Sec. VII, we p esen esul s o he phase-space
unc ion wi h he new in eg a ion me hod. In Sec. VIII,we
discuss how his o malism can be applied o he 2p-2h
esponse unc ions o elec on and neu ino sca e ing.
Finally, in Sec. IX, we p esen ou conclusions.
II. 2P-2H RESPONSE FUNCTIONS
When conside ing a lep on ha sca e s o a nucleus
ans e ing 4-momen um Qμ¼ðω;qÞ, wi h ω he ene gy
ans e and q he momen um ans e , one is in ol ed wi h
he had onic enso
Wμν ¼X
hΨ jJμðQÞjΨiihΨ jJνðQÞjΨiiδðEiþω−E Þ;
ð1Þ
whe e JμðQÞis he elec oweak nuclea cu en ope a o .
In his pape , we ake he ini ial nuclea s a e as he
RFG model g ound s a e, jΨii¼jFi, wi h all s a es wi h
momen a below he Fe mi momen um kFoccupied. The
sum o e inal s a es can be decomposed as he sum o
one-pa icle one-hole (1p-1h) plus 2p-2h exci a ions plus
addi ional channels:
Wμν ¼Wμν
1p1hþWμν
2p2hþ ð2Þ
In he impulse app oxima ion, he 1p-1h channel gi es
he well-known esponse unc ions o he RFG. No ice ha
MEC also con ibu e o hese 1p-1h esponses; howe e ,
he e we ocus on he 2p-2h channel whe e he inal s a es
a e o he ype
jΨ i¼j10;20;1−1;2−1ið3Þ
ji0i¼jp0
is0
i 0
iið4Þ
jii¼jhisi ii;i;i
0¼1;2;ð5Þ
whe e p0
ia e momen a o ela i is ic inal nucleons abo e
he Fe mi sea, p0
i>k
F, wi h 4-momen a P0
i¼ðE0
i;p0
iÞ,
and Hi¼ðEi;hiÞa e he 4-momen a o he hole s a es
wi h hi<k
F. The spin indices a e s0
iand si, and he isospin
is i, 0
i.
In his pape we s udy he 2p-2h channel in a ully
ela i is ic amewo k. The co esponding had onic enso
is gi en by
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-2
Wμν
2p-2h¼V
ð2πÞ9Zd3p0
1d3p0
2d3h1d3h2
m4
N
E1E2E0
1E0
2
× μνðp0
1;p0
2;h1;h2ÞδðE0
1þE0
2−E1−E2−ωÞ
×Θðp0
1;p
0
2;h
1;h
2Þδðp0
1þp0
2−h1−h2−qÞ;
ð6Þ
whe e mNis he nucleon mass, Vis he olume o he
sys em, and we ha e de ined he p oduc o s ep unc ions
Θðp0
1;p0
2;h1;h2Þ¼θðp0
2−kFÞθðp0
1−kFÞθðkF−h1ÞθðkF−h2Þ:
ð7Þ
The unc ion μνðp0
1;p0
2;h1;h2Þis he had onic enso o
he elemen a y ansi ion o a nucleon pai wi h he gi en
ini ial and inal momen a, summed up o e spin and
isospin, gi en schema ically as
μνðp0
1;p0
2;h1;h2Þ¼1
4X
s;
jμð10;20;1;2Þ
Ajνð10;20;1;2ÞA;
ð8Þ
which we w i e in e ms o he an isymme ized wo-body
cu en ma ix elemen jμð10;20;1;2ÞA, o be speci ied. The
ac o 1=4accoun s o he an isymme y o he 2p-2h wa e
unc ion. Finally, no e ha he 2p-2h esponse is p opo -
ional o V, which is ela ed o he numbe o p o ons o
neu ons Z¼N¼A=2by V¼3π2Z=k3
F. In his wo k, we
only conside nuclea a ge s wi h pu e isospin ze o.
In he case o elec ons, he c oss sec ion can be w i en
as a linea combina ion o he longi udinal and ans e se
esponse unc ions de ined by
RL¼W00 ð9Þ
RT¼W11 þW22;ð10Þ
whe eas addi ional esponse unc ions a ise o neu ino
sca e ing, due o he p esence o he axial cu en . The
gene ic esul s coming om he phase-space ob ained he e
a e applicable o all o he esponse unc ions.
In eg a ing o e p0
2using he momen um del a unc ion,
Eq. (6) becomes a nine-dimensional in eg al,
Wμν
2p-2h¼V
ð2πÞ9Zd3p0
1d3h1d3h2
m4
N
E1E2E0
1E0
2
× μνðp0
1;p0
2;h1;h2ÞδðE0
1þE0
2−E1−E2−ωÞ
×Θðp0
1;p
0
2;h
1;h
2Þ;ð11Þ
whe e p0
2¼h1þh2þq−p0
1. A e choosing he qdi ec-
ion along he zaxis, he e is a global o a ion symme y
o e one o he azimu hal angles. We choose ϕ0
1¼0and
mul iply by a ac o 2π. Fu he mo e, he ene gy del a
unc ion enables analy ical in eg a ion o e p0
1, and so he
in eg al is educed o se en dimensions. In gene al, he
calcula ion has o be done nume ically. Unde some
app oxima ions [25,31,32,36], he numbe o dimensions
can be u he educed, bu his canno be done in he ully
ela i is ic calcula ion.
In his pape , we s udy di e en me hods o e alua e
he abo e in eg al nume ically and compa e he ela i is ic
and he non ela i is ic cases. In he non ela i is ic case, we
educe he had onic enso o a wo-dimensional in eg al.
This can be done when he unc ion μν only depends on he
di e ences ki¼p0
i−hi,i¼1,2.
As we wan o concen a e on he nume ical p ocedu e
wi hou u he complica ions de i ed om he momen um
dependence o he cu en s, in his pape , we s a by se ing
he elemen a y unc ion o a cons an μν ¼1. Hence, we
ocus on he genuine kinema ical e ec s coming om he
wo-pa icle– wo-hole phase space alone. In pa icula , he
kinema ical ela i is ic e ec s a ising om he ene gy-
momen um ela ion a e con ained in he ene gy conse a-
ion del a unc ion ha de e mines he analy ical beha io
o he had onic enso , whe e he ene gy-momen um
ela ion is E¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2þm2
N
p, and in he Lo en z con ac ion
coe icien s mN=Ei. Ob iously, he esul s ob ained he e o
cons an μν will be modi ied when including he wo-body
physical cu en . Bu as he inal esul is model dependen ,
i is no possible o disen angle whe he he di e ences
ound a e due o he cu en model employed o o he
app oxima ions ( ela i is ic o no ) used o pe o m
he nume ical e alua ion o he in eg al. In ac all o he
models o 2p-2h esponse unc ions should ag ee a he
le el o he 2p-2h phase-space in eg al Fðq; ωÞde ined as
Fðq; ωÞ≡Zd3p0
1d3h1d3h2
m4
N
E1E2E0
1E0
2
×δðE0
1þE0
2−E1−E2−ωÞΘðp0
1;p
0
2;h
1;h
2Þ;
ð12Þ
wi h p0
2¼h1þh2þq−p0
1. Calcula ion o his unc ion
should be a good s a ing poin o compa e and congeni-
alize di e en nuclea models.
III. NONRELATIVISTIC 2P-2H PHASE SPACE
A. Semianaly ical in eg a ion
Fi s , we ecall he semianaly ical me hod o Re . [32]
ha was used la e in Re s. [25,29], o ins ance, o
compu e he non ela i is ic 2p-2h ans e se esponse
unc ion in elec on sca e ing. We shall use his me hod
o check he nume ical 7D quad a u e bo h in he ela i is ic
and non ela i is ic cases.
We s a wi h he 12-dimensional exp ession o he
phase-space unc ion, Eq. (6),
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-3
Fðq; ωÞ¼Zd3p0
1d3p0
2d3h1d3h2
×δðE0
1þE0
2−E1−E2−ωÞ
×Θðp0
1;p
0
2;h
1;h
2Þδðp0
1þp0
2−h1−h2−qÞ:
ð13Þ
The p ocedu e is i s o pe o m he in eg al o e ene gy.
Following Re . [32], we change a iables:
l1¼p0
1−h1
kF
l2¼p0
2−h2
kFð14Þ
x1¼p0
1þh1
2kF
x2¼p0
2þh2
2kF
:ð15Þ
We also de ine he ollowing nondimensional a iables:
qF≡
q
kFð16Þ
ν≡
mNω
k2
F
:ð17Þ
In e ms o hese a iables, he 2p-2h phase-space unc-
ion is
Fðq; ωÞ¼ð2πÞ2k7
FmNZd3l1
l3
1
d3l2
l3
2
×δðl1þl2−qFÞAðl1;l
2;νÞ;ð18Þ
whe e we use he Van O den unc ion de ined as
Aðl1;l2;νÞ¼ l3
1l3
2
ð2πÞ2Zd3x1d3x2δðν−l1·x1−l2·x2Þ
×θ1−
x1−l1
2θ1−
x2−l2
2
×θ
x1þl1
2
−1θ
x2þl2
2
−1:ð19Þ
This unc ion was compu ed analy ically in Re . [32].In
his wo k, we ha e checked ha exp ession because we
ound a ypo in one o he e ms in he o iginal e e ence
( ha ypog aphical e o does no a ec he esul s o he
ci ed e e ence). We gi e in he Appendix he co ec esul
o u u e e e ence.
In eg a ing now o e he momen um l2, we ge
Fðq; ωÞ¼ð2πÞ2k7
FmNZd3l1
l3
1jqF−l1j3Aðl1;jqF−l1j;νÞ:
ð20Þ
The in eg al o e he azimu hal angle ϕ1o l1gi es 2π.
Finally, changing o he a iables
x¼l1;y¼jqF−l1j;ð21Þ
we ob ain
Fðq;ωÞ¼ð2πÞ3k7
FmN
qFZxmax
0
dx
x2ZqFþx
jqF−xj
dy
y2Aðx;y;νÞ;ð22Þ
whe e he maximum alue o x(o k1=kF) is ob ained om
he ene gy conse a ion and momen um s ep unc ions
included implici ly in he unc ion Aðx; y; νÞ. In he
Appendix, we de i e he inequali y
x≤xmax ≡1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1þνÞ
p:ð23Þ
The wo-dimensional in eg al o e he a iables x,yhas
o be pe o med nume ically.
B. Nume ical in eg a ion
The simplici y o he Fe mi gas model used in his pape
allows us o compu e he 2p-2h had onic enso as a 7D
in eg al as shown below. No e ha in a mo e sophis ica ed
model whe e he nuclea dis ibu ion de ails a e aken in o
accoun , like shell models o he spec al unc ion-based
models, some o he nume ical p oblems linked o he
pa icula Jacobian appea ing he e and in he ollowing
sec ion can be a oided, a he p ice o inc easing he
numbe o in eg als o sums o e shell-model s a es, hus
making he calcula ions ha de . The local Fe mi gas used
by Nie es e al. is eally an a e age o di e en Fe mi gases
a di e en densi ies, bu he basic Fe mi gas equa ions a e
he same as he e.
The had onic enso o he elemen a y 2p-2h ansi ion,
Eq. (8), con ains he di ec and exchange ma ix elemen s o
he wo-body cu en ope a o . I one neglec s he in e e -
ence be ween he di ec and exchange e ms, i is possible o
exp ess μν as a unc ion o x,yonly, and one can use he
o malism o he abo e sec ion o educe he calcula ion o
he 2p-2h had onic enso o a 2D in eg al. In he gene al
case, he in e e ence canno be neglec ed. I is hen
necessa y o e alua e a 7D in eg al nume ically. Thus, in
his wo k, we also compu e he phase-space unc ion,
Eq. (12), nume ically. This will allow us i s o check he
nume ical p ocedu es by compa ison wi h he semianaly -
ical me hod o he p e ious sec ion; second, o de e mine he
numbe o in eg a ion poin s needed o ob ain accu a e
esul s, and, hi d, o op imize he compu a ional e o . This
nume ical s udy will be e y use ul when including ac ual
nuclea cu en s.
S a ing wi h Eq. (12), we compu e he in eg and o
ϕ0
1¼0( he azimu hal angle o p0
1) and mul iply by 2π.
Then, we use he δo ene gies o in eg a e o e he a iable
p0
1, o ixed momen a h1and h2and emission angle θ0
1.To
do so, we i s de ine he o al momen um o he wo
pa icles ha is ixed by momen um conse a ion:
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-4
p0¼p0
1þp0
2¼h1þh2þq:ð24Þ
We hen change om a iable p0
1 o a iable E0:
E0¼E0
1þE0
2¼p0
1
2
2mNþðp0−p0
1Þ2
2mN
:ð25Þ
By di e en ia ion wi h espec o p0
1, we ob ain
dp0
1
dE0¼mN
jp0
1−p0
2·ˆ
p0
1j;ð26Þ
whe e ˆ
p0
1¼p0
1=p0
1is he uni ec o in he di ec ion o he
i s pa icle. In eg a ing now o e E0, ene gy conse a ion
is ob ained as
E0¼E1þE2þω:ð27Þ
Subs i u ing Eq. (25) a second deg ee equa ion is ob ained
o p0
1:
2p0
1
2þp02−2p0·p0
1¼2mNE0:ð28Þ
So, we see ha he e can be wo alues o he nucleon
momen um compa ible wi h ene gy conse a ion, o ixed
a emission angle. We deno e he wo solu ions by
p0
1ðÞ ¼1
22
4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2−4p02
2−mNE0
s3
5;ð29Þ
whe e we ha e de ined
≡p0·ˆ
p0
1:ð30Þ
Using his esul , we inally e alua e he phase-space
unc ion as he 7D in eg al
Fðq; ωÞ¼2πZd3h1d3h2dcos θ0
1
×X
α¼
p0
1
2mN
jp0
1−p0
2·ˆ
p0
1jΘðp0
1;p
0
2;h
1;h
2Þjp0
1¼p0
1ðαÞ
;
ð31Þ
whe e he sum inside he in eg al uns o e he wo
solu ions p0
1ðÞ o he ene gy conse a ion equa ion.
C. Asymp o ic expansion
I is o in e es o quo e he limi ω→∞because i can
also be used o es ing he nume ical in eg a ion. The mos
use ul case applies o kF,q≪ω, when one can neglec all
momen a compa ed wi h he ene gy ans e ω, because he
phase-space in eg al can be pe o med analy ically. No e
ha o he sca e ing eac ions o in e es , his limi is no
physical (because ω<q, namely, spacelike, o eal
pa icles). I is only a ma hema ical p ope y o he unc ion
F, which is well de ined o all he ω alues no only he
physical ones. We s a w i ing he momen um o he i s
pa icle, Eq. (29),as
p0
1¼
21
2ffiffiffiffi
D
p;ð32Þ
wi h he disc iminan
D¼ 2−2p02þ4mNE0:ð33Þ
The limi ω→∞can be ob ained by no icing ha and p0
do no depend on ωbu only on he momen a h1,h2, and q
and ha E0¼E1þE2þω∼ω. Then,
D∼4mNω;ð34Þ
and he posi i e solu ion o he momen um is
p0
1∼ffiffiffiffiffiffiffiffiffiffi
mNω
p:ð35Þ
Tha is, each nucleon exi s he nucleus aking hal o he
a ailable ene gy.
On he o he hand, using Eq. (29), we no e ha he
denomina o in Eq. (31) can be w i en as
p0
1−p0
2·ˆ
p0
1¼ ffiffiffiffi
D
p∼2ffiffiffiffiffiffiffiffiffiffi
mNω
p:ð36Þ
Then,
Fðq; ωÞ→
ω→∞Faðq; ωÞ
≡2πZd3h1d3h2dcos θ0
1
mN
2ffiffiffiffiffiffiffiffiffiffi
mNω
p
¼4π4
3πk3
F2mN
2ffiffiffiffiffiffiffiffiffiffi
mNω
p:ð37Þ
Thus, o high ene gy, he non ela i is ic phase-space
unc ion inc eases as ffiffiffiffi
ω
p. We shall see in he nex sec ion a
di e en beha io in he ela i is ic case.
D. Non ela i is ic esul s
In Fig. 1, we show he non ela i is ic phase-space
unc ion Fðq; ωÞas a unc ion o ω o h ee ypical alues
o he momen um ans e q¼300, 400, and 500 MeV=c.
The Fe mi momen um is kF¼225 MeV=c. We compa e
he wo compu a ional me hods: he semianaly ical o
Eq. (22) and he nume ical 7D in eg a ion o Eq. (31).
The semianaly ical esul is essen ially exac because we
can choose a e y small in eg a ion s ep o he 2D in eg al
(using s eps o 0.02 o 0.01, he esul s do no change in
he scale o he igu e). Howe e , he 7D in eg al is
compu a ionally ime consuming, and he in eg a ion s ep
canno be e y small. He e, we compu e he in eg al wi h a
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-5
“s aigh o wa d”me hod, as an a e age o e a g id wi h
n o al in eg a ion poin s, uni o mly dis ibu ed. Fo la ge
n, he s aigh o wa d in eg a ion should gi e esul s
simila o he Mon e Ca lo me hods used in p e ious
calcula ions [29,32]. The numbe o poin s chosen o
his calcula ion was 25 o he a iable θ0
1(al hough i can
sa ely be educed o 16) and 16 o each one o he
emaining dimensions. In o al, he numbe o poin s is
n¼0.42 ×109. This is well abo e he maximum numbe
n¼106–107, ypical o p e ious calcula ions [29,32]
pe o med using Mon e Ca lo echniques. Using en
in eg a ion poin s in each dimension gi es e y simila
esul s, excep o some ω egions whe e he nume ical
e o is mani es ed in an appa en ly sligh ly less smoo h
beha io . Inc easing he numbe o poin s would imp o e
he esul s; howe e , his is no p ac ical because he
inclusion o he wo-body cu en would make he calcu-
la ion oo slow. The semianaly ical and nume ical esul s
a e qui e simila , he di e ence be ween hem being o a
ew pe cen . Fo compa ison, we also show he asymp o ic
limi ω→∞, compu ed using he analy ical exp ession in
Eq. (37), which is p opo ional o ffiffiffiffi
ω
p. We see ha o high
ω he unc ion Fðq; ωÞbecomes close o he asymp o ic
alue Faðq; ωÞ.Fo q¼300 MeV=c, he asymp o ic alue
is almos eached a he pho on poin ω¼q. When q
inc eases, so does he dis ance o he asymp o e a he
pho on poin .
IV. RELATIVISTIC 2P-2H PHASE SPACE
Ha ing wo independen calcula ions o he phase-space
unc ion Fðq; ωÞin he non ela i is ic limi , we now
conside he case o he ully ela i is ic calcula ion as
gi en by Eq. (12). This in ol es adding he Lo en z-
con ac ion mN=E ac o s and using ela i is ic kinema ics
in he ene gy δ unc ion. Following he scheme o he
p e ious sec ion, again azimu hal symme y allows one o
ix ϕ0¼0and mul iply by 2π. To in eg a e o e p0
1,we
change o he a iable
E0¼E0
1þE0
2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p0
1
2þm2
N
qþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðp0−p0
1Þ2þm2
N
q;
ð38Þ
whe e again p0¼h1þh2þqis he inal momen um o a
ixed pai o holes. By di e en ia ion, we a i e a he
ollowing Jacobian:
dp0
1
dE0¼
p0
1
E0
1
−p0
2·ˆ
p0
1
E0
2
−1
:ð39Þ
The non ela i is ic Jacobian o Eq. (26) is eco e ed o
low ene gies E0
1≃E0
2≃mN. As be o e, in eg a ion o e E0
gi es E0¼E1þE2þω, and he phase-space unc ion
becomes
Fðq; ωÞ¼2πZd3h1d3h2dθ0
1sin θ0
1
m4
N
E1E2
×X
α¼
p0
1
2
p0
1
E0
1
−p0
2·ˆ
p0
1
E0
2
Θðp0
1;p
0
2;h
1;h
2Þ
E0
1E0
2p0
1¼p0
1ðαÞ
;
ð40Þ
whe e again he sum inside he in eg al uns o e he wo
solu ions p0
1ðÞ o he ene gy conse a ion equa ion
p0
1ðÞ ¼1
~
b~
a~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
~
a2−~
bm2
N
q:ð41Þ
FIG. 1 (colo online). Non ela i is ic phase-space unc ion
calcula ed o ω¼300, 400, 500 MeV, using a nume ical and
a semianaly ical app oach. The numbe o poin s used in wo
nume ical in eg a ions is indica ed in he plo . We also show he
asymp o ic unc ion o compa ison.
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-6
The de ini ions o he quan i ies ~
a,~
b, and ~
a e gi en in
he Appendix. No e ha he e is a di e ence be ween
ou Jacobian in Eq. (40) and ha gi en in Eqs. (15–17)
o Re . [26].
The ela i is ic app oach is mo e in ol ed han he
non ela i is ic one because i equi es aking he squa e
wice in he o iginal equa ion o elimina e he squa ed oo s
in he ene gies. This can in oduce spu ious solu ions o p0
1
depending on he kinema ics, which ha e o be elimina ed
om he abo e sum in he nume ical p ocedu e. This is no
a i ial ask, and de ails a e p o ided in he Appendix. The
appea ance o spu ious solu ions is a di e ence be ween
he ela i is ic and non ela i is ic me hods. A second one
will be discussed below in ela ion o a di e gence o he
in eg and. The e o e, he ela i is ic calcula ion is e y
in ol ed, and i canno be de i ed by simply ex ending he
non ela i is ic code. We de o e he es o his sec ion o
explain in de ail how o ge he ully ela i is ic answe s.
A. Rela i is ic asymp o ic expansion
Al hough i is no possible o de i e a semianaly ical
exp ession o Fðq; ωÞas in he non ela i is ic case, i is
s ill possible o ake he limi ω→∞and ob ain an
analy ical esul . As in he non ela i is ic case, we assume
kF,q≪ω. I we add he condi ion mN≪ω, we can
neglec he momen a and ene gies o he wo holes and
w i e
E0∼ωp0∼q:ð42Þ
We can also compu e he quan i ies wi h ildes ha appea
in he solu ion o he ene gy conse a ion (see he
Appendix), ob aining
~
a∼ω
2ð43Þ
~
∼q·ˆ
p0
1
2ωð44Þ
~
b∼1:ð45Þ
Then, he disc iminan o Eq. (41) becomes
~
a2−~
bm2
N∼ω2
4−m2
N∼ω2
4:ð46Þ
The e o e, he allowed solu ion o he ene gy conse a ion
equa ion is
p0
1∼q·ˆ
p0
1
4þω
2∼ω
2:ð47Þ
Thus, in his limi , each nucleon ca ies hal he o al ene gy
and momen um,
E0
1∼p0
1∼E0
2∼p0
2∼ω
2:ð48Þ
Now, he Jacobian, he denomina o in Eq. (40), can be
compu ed as
d≡
p0
1
E0
1
−p0
2·ˆ
p0
1
E0
2¼1−ðp0−p0
1Þ·ˆ
p0
1
E0
2
∼1þp0
1
E0
2
∼2:
ð49Þ
Collec ing Eqs. (47),(48), and (49), he in eg and in
Eq. (40) becomes
p0
1
2
d
m4
N
E1E2E0
1E0
2
∼ω2
8
m4
N
m2
Nω2=4¼m2
N
2:ð50Þ
Finally, pe o ming he in eg al, we ob ain he ollowing
asymp o ic exp ession:
Fðq; ωÞ→
ω→∞Faðq; ωÞ¼4π4
3πk3
F2m2
N
2:ð51Þ
In con as wi h he non ela i is ic beha io , which
inc eases mono onically as ffiffiffiffi
ω
p, he ela i is ic esul
(37) goes o a cons an . The Lo en z con ac ion ac o s
E=mNbalance he ω2beha io coming om he phase
space. This analy ical esul o high ωwill be use ul o
compa ison o he nume ical esul s o high ω.
B. Rela i is ic s aigh o wa d calcula ion
Be o e going o he high-q egion, we i s check he
ela i is ic phase-space unc ion esul s by compa ison wi h
he non ela i is ic coun e pa . Bo h should ag ee o low
ene gy. We p oceed by pe o ming a s aigh o wa d
nume ical in eg a ion o Eq. (40) as in he non ela i is ic
case. In Fig. 2, we show he esul s o his compa ison
o q¼300, 400, and 500 MeV=c. We also show bo h
he nume ical and “exac ”(i.e., using he semianaly ical
o mula) non ela i is ic unc ion Fðq; ωÞ. A uni o m
dis ibu ion wi h en poin s o each dimension is employed
in he 7D in eg a ions. As expec ed, ela i is ic and non-
ela i is ic esul s ag ee a low ene gy. The ela i is ic
e ec s consis o a educ ion o he s eng h a high
ene gy. The amoun o his educ ion is e y small o
q¼300 MeV=c, whe e he non ela i is ic app oxima ion
can be sa ely used, and inc eases wi h q, eaching abou
15% o q¼500 MeV=c. Thus, o low q, we ag ee ha a
numbe o ∼107poin s is adequa e o nume ical in eg a-
ion pu poses. In Fig. 5, he asymp o ic limi Faðq; ωÞ
o he ela i is ic phase space, Eq. (51), is also shown.
Fo hese low q alues, Fðq; ωÞis s ill a below he
asymp o e.
La ge ela i is ic e ec s a e expec ed o in e media e
o la ge momen um ans e . In Fig. 3, we display Fðq; ωÞ
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-7
o q¼700, 1000, and 1500 MeV=c, compa ed wi h he
exac non ela i is ic esul s. Using s aigh o wa d 7D
in eg a ion, we need o inc ease he numbe o poin s o
16 o each dimension in o de o each some s abili y o
he esul s shown in Fig. 3. Howe e , we ind ha ull
con e gence would need mo e poin s. In ac , o
q¼700 MeV=c, a small de ia ion wi h espec o he
exac esul can be no iced a low ω. This de ia ion
inc eases wi h qand u ns in o a p ominen s uc u e wi h
a“shoulde ”shape o q¼1500 MeV=c. One could be
emp ed o a ibu e his e ec o ela i i y. Bu his is no
he case because he same beha io is also p esen in a
non ela i is ic nume ical calcula ion. As we will explain
below, his is jus a consequence o he inadequacy o he
s aigh o wa d in eg a ion me hod a high q. This p oblem
a ec s only he inne in eg al o e θ0
1. Below, we add ess
his issue by a de ailed analysis o he θ0
1dependence o
he in eg and.
V. ANGULAR DISTRIBUTION
A. F ozen phase-space unc ion
We s a ixing a alue o q¼3GeV=c ha is high
enough o ampli y he misbeha io ound abo e and also
allows us o simpli y he analysis ha ollows. In ac , we
no e ha o e y high q≫kF, all o he hole momen a h1,
h2could sa ely be neglec ed inside he in eg al as a i s
app oxima ion. Since his implies ha he ini ial pa icles
a e a es , we deno e his limi he “ ozen nucleon
app oxima ion.”In pa icula , he ene gies o he holes
can be subs i u ed by he nucleon mass in he δ unc ion,
FIG. 3 (colo online). Rela i is ic phase space unc ion o
q¼700, 1000, 1500, calcula ed using s aigh o wa d in eg a-
ion compa ed wi h he non ela i is ic calcula ion using he
semianaly ical app oach.
FIG. 2 (colo online). Rela i is ic phase space unc ion o
q¼300, 400, 500, calcula ed using s aigh o wa d in eg a ion,
compa ed wi h he non ela i is ic calcula ion using he semi-
analy ical app oach.
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-8
Fðq; ωÞ∼Zd3h1d3h2d3p0
1δðE0
1þE0
2−ω−2mNÞ
×Θðp0
1;p
0
2;0;0Þm2
N
E0
1E0
2
;ð52Þ
whe e p0
2¼q−p0
1. Because he in eg and does no depend
on he hole momen a, one can di ec ly in eg a e ou hose
a iables,
Fðq; ωÞ∼4
3πk3
F2Zd3p0
1δðE0
1þE0
2−ω−2mNÞ
×Θðp0
1;p
0
2;0;0Þm2
N
E0
1E0
2
:ð53Þ
Now, he in eg al o e p0
1can be done analy ically as
be o e using he del a unc ion, wi h he same Jacobian
e alua ed o h1¼h2¼0. The in eg al o e ϕ0
1gi es again
a ac o 2π,
Fðq; ωÞ∼2πm2
N4
3πk3
F2Zdθ0
1sin θ0
1
×X
α¼
p0
1
2
p0
1
E0
1
−p0
2·ˆ
p0
1
E0
2
Θðp0
1;p
0
2;0;0Þ
E0
1E0
2p0
1¼p0
1ðαÞ
:ð54Þ
Thus, in his app oxima ion, he phase-space unc ion is
educed o a one-dimensional in eg al o e he emission
angle θ0
1, which has o be pe o med nume ically.
The ozen nucleon app oxima ion ep esen s jus a
pa icula case o he mean- alue heo em o he in eg al
o e h1,h2. We deno e wi h a ba he quan i ies compu ed
by he mean- alue heo em. Thus, we de ine he ba ed
phase-space unc ion
¯
Fðq; ωÞ¼4
3πk3
F2Zd3p0
1δðE0
1þE0
2−ω−E1−E2Þ
×Θðp0
1;p
0
2;h
1;h
2Þm4
N
E1E2E0
1E0
2
;ð55Þ
whe e p0
2¼h1þh2þq−p0
1, and ðh1;h2Þa e a pai o
ixed momen a below he Fe mi sea. Going u he , we will
la e u n o he ques ion o how o choose he a e age
nucleon momen a h1,h2 o low q. Fo high q, we expec
his unc ion no o depend oo much on he chosen alues.
So, a his poin , we es ic ou s udy o ¯
Fðq; ωÞin he
ozen nucleon app oxima ion, i.e., o h1¼h2¼0.
B. Nume ical analysis
We ha e compu ed ¯
Fðq; ωÞin he ozen nucleon
app oxima ion using 100 poin s o pe o m he nume ical
in eg al o e he emission angle θ0
1. Resul s a e shown
in Fig. 4. A misbeha io due o nume ical e o is now
e iden .
The eason o he appea ance o discon inui ies by
nume ical in eg a ion becomes appa en by examining
he angula dependence o he in eg and. We de ine he
angula dis ibu ion unc ion, o ixed alues o ðq; ωÞand
h1,h2,as
Φðθ0
1Þ¼sin θ0
1Zp0
1
2dp0
1δðE1þE2þω−E0
1−E0
2Þ
×Θðp0
1;p
0
2;h
1;h
2Þm4
N
E1E2E0
1E0
2
;
¼X
α¼
m4
Nsin θ0
1p0
1
2Θðp0
1;p
0
2;h
1;h
2Þ
E1E2E0
1E0
2
p0
1
E0
1
−p0
2·ˆ
p0
1
E0
2p0
1¼p0
1ðαÞ
;ð56Þ
whe e once mo e p0
2¼h1þh2þq−p0
1, such ha he
phase-space unc ion is ob ained by in eg a ion o e he
emission angle θ0
1:
¯
Fðq; ωÞ¼4
3πk3
F2
2πZπ
0
dθ0
1Φðθ0
1Þ:ð57Þ
The unc ion Φðθ0
1Þ hus measu es he dis ibu ion o inal
nucleons as a unc ion o he angle θ0
1. This unc ion is
compu ed analy ically, gi en by he in eg and in Eq. (54).
Resul s o Φðθ0
1Þa e shown in Fig. 5 o h1¼h2¼0;
q¼3GeV=c; and o he h ee alues o ω¼1800, 2000,
FIG. 4 (colo online). Phase-space unc ion o q¼3and
0.5GeV=c, compu ed using he ozen nucleon app oxima ion
o ixed hole momen a h1¼h2¼0, using 100 in eg a ion
poin s in an emission angle.
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-9
esul s o all ene gies. Fo high momen um ans e
q¼3GeV=c, hey a e comple ely di e en . The non ela-
i is ic unc ion is pushed owa d highe ene gies due o
he quad a ic momen um dependence o he non ela i is ic
kine ic ene gy. Thus, o q¼1.5GeV=c, he ela i is ic
esul s a e abo e (below) he non ela i is ic ones o low
(high) ene gy. Fo q¼3GeV=c, he ela i is ic esul s a e
abo e o all ω alues allowed.
In all cases, Fðq; ωÞis below he asymp o ic alue,
Eq. (51), also shown in Fig. 11.
In Fig. 12, we ge deepe insigh in o he size o ela i is ic
e ec s. The e, we show esul s o Fðq; ωÞcompu ed using
ela i is ic kinema ics only, bu wi hou including he
ela i is ic Lo en z-con ac ion ac o s mN=E in pa icles
and holes. The esul s inc ease a lo wi h espec o he
non ela i is ic ones. This is ela ed o he ac poin ed ou
a e Eq. (51) o he asymp o ic limi o he ela i is ic
phase-space in eg al. Wi hou he Lo en z ac o s, he unc-
ion Fðq; ωÞwould inc ease as ω2. This seems o indica e
ha in o de o “ ela i ize”a non ela i is ic 2p-2h model,
implemen ing only ela i is ic kinema ics is no su icien ,
since i goes in he w ong di ec ion. In ac , esul s in Fig. 12
show ha he e ec s coming solely om he ela i is ic
kinema ics lead o di e ences e en la ge han he disc ep-
ancy be ween he non ela i is ic and he ully ela i is ic
calcula ions. The e o e, i is essen ial also o include he
Lo en z ac o s mN=E.
No e ha he beha io o ela i is ic e ec s in he 1p-1h
channel goes in he opposi e di ec ion o he one discussed
FIG. 11 (colo online). To al phase-space unc ion o h ee
alues o he momen um ans e . The numbe o in eg a ion
poin s in each dimension in he hole a iables is indica ed by n.
The numbe o in eg a ion poin s o e he emission angle θ0
1is
indica ed as m. We also show he non ela i is ic, exac esul and
he ela i is ic asymp o ic alue.
FIG. 12 (colo online). E ec o implemen ing ela i is ic
kinema ics in a non ela i is ic calcula ion o Fðq; ωÞ. Solid lines:
non ela i is ic esul . Thick do ed lines: ela i is ic kinema ics
only wi hou he ela i is ic ac o s mN=E. Thin dashed lines:
ully ela i is ic esul .
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-16
he e in he 2p-2h channel. In ac , in Re . [41] i was
shown ha implemen ing ela i is ic kinema ics wi hou
he mN=E ac o s in he non ela i is ic 1p-1h esponse
unc ion gi es a esul close o he exac ela i is ic
esponse unc ion (see Fig. 33 o Re . [41]).
In Figs. 13 and 14, we p esen he esul s o a s udy o he
alidi y o he ozen nucleon app oxima ion o compu e
Fðq; ωÞin a ange o momen um ans e s. This app oxi-
ma ion was in oduced o high momen um ans e
q¼3GeV=c, neglec ing he momen a o he wo holes
inside he 7D in eg al, hus educing i o a one-dimensional
(1D) in eg al o e he emission angle θ0
1. In Fig. 14, he
momen um ans e is s ill high, and he ozen nucleon
app oxima ion emains alid. In Fig. 13, he alues o qa e
no so la ge, and one could hink ha he ozen nucleon
app oxima ion is no alid. Howe e , he esul s o
Fig. 13 demons a e ha i is s ill a good app oxima ion
o mode a e momen um ans e excep o e y low
ene gy ans e , whe e he unc ion Fðq; ωÞis small.
This is a p omising esul ; i he ozen nucleon app oxi-
ma ion could be ex ended o he ull esponse unc ions
when including he nuclea cu en , his would mean ha
he 2p-2h c oss sec ion could be app oxima ed by 1D
in eg als o e he emission angle, which would be easy and
as o compu e. In pa icula , calcula ions o his kind
could be implemen ed in exis ing Mon e Ca lo codes.
To illus a e he easons why he ozen nucleon app oxi-
ma ion wo ks o mode a e momen um ans e , we p esen
Figs. 15,16, and 17. We compa e Fðq; ωÞwi h he ba ed
phase-space unc ion ¯
Fðq; ωÞ, de ined in Eq. (55), com-
pu ed o se e al ðh1;h2Þcon igu a ions. The a e age-
momen um app oxima ion is simila o he ozen nucleon
app oxima ion in he sense ha he wo hole momen a
h1,h2a e se o a cons an inside he in eg al. Fo a pai
FIG. 13 (colo online). Rela i is ic phase-space unc ion
Fðq; ωÞcompa ed wi h he ozen nucleon app oxima ion o
low o in e media e momen um ans e .
FIG. 14 (colo online). Rela i is ic phase-space unc ion
Fðq; ωÞcompa ed wi h he ozen nucleon app oxima ion o
high momen um ans e .
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-17
con igu a ion ðh1;h2Þ, he unc ion ¯
Fðq; ωÞgi es he
con ibu ion o such a pai o he phase-space unc ion,
mul iplied by V2
F, whe e VF¼4πk3
F=3is he olume o he
Fe mi sphe e. The o al Fðq; ωÞis he sum o he con-
ibu ions om all o he pai s o , equi alen ly, he a e age
o all o he ba ed unc ions ¯
Fðq; ωÞo e he di e en pai
con igu a ions.
In Fig. 18, we show he geome y o he con igu a ions
used in Figs. 15–17. Fo low alues o he momen a h1,h2,
he ozen nucleon app oxima ion should be a good
app oxima ion o he a e age phase space. Fo la ge alues
o he momen a, we ind pai s o con igu a ions wi h
opposi e o al momen um p¼h1þh2 ha con ibu e
abo e o below he a e age in app oxima ely equal oo ing,
so hey do no change he mean alue e y much.
In he i s example, Fig. 15, we show he con ibu ion
o wo pai s o nucleons wi h he same momen um
h1¼h2¼200 MeV=c, and bo h pa allel, poin ing
upwa d (U) and downwa d (D) wi h espec o he z
axis, ha is, he di ec ion o q. The con ibu ion o he
UU con igu a ion is smalle han a e age, while he DD
is la ge . This is so because in he UU case he o al
momen um p0in he inal s a e is la ge. By momen um
conse a ion, he momen a p0
1and p0
2mus also be la ge.
The e o e, hese s a es need a la ge exci a ion ene gy, and
hey s a o con ibu e o high ω ans e . In he DD
con igu a ion, he o al momen um p0is small, so he inal
momen a p0
1and p0
2can also be small, will small
FIG. 15 (colo online). Rela i is ic phase-space unc ion
Fðq; ωÞcompa ed wi h he a e age-momen um app oxima ion
¯
Fðq; ωÞ o a pai o nucleons wi h momen um 200 MeV=c. In
he UU con igu a ion, bo h nucleons mo e along q(up). In he
DD con igu a ion, bo h mo e opposi e o q(down).
FIG. 16 (colo online). Rela i is ic phase-space unc ion
Fðq; ωÞcompa ed wi h he a e age-momen um app oxima ion
¯
Fðq; ωÞ o a pai o nucleons wi h momen um 200 MeV=c
poin ing in opposi e di ec ions ( o al momen um equal o ze o). In
he UD con igu a ion one mo es along q(U) and he o he
opposi e o q(D). In he T,−Tcon igu a ion, one mo es in he x
di ec ion and he o he in he −xdi ec ion.
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-18
exci a ion ene gy. The e o e, hey s a o con ibu e a
e y low ω.
In he example o Fig. 16, wo an ipa allel con igu a ions
a e shown. In he UD case, one nucleon is mo ing upwa d
and he o he downwa d he zaxis wi h o al momen um
ze o o he pai . This si ua ion is simila o ha o a pai o
highly co ela ed nucleons wi h la ge ela i e momen um
[42]. Since he o al momen um is ze o, he inal 2p-2h s a e
has o al momen um q, exac ly he same ha i would ha e
in he ozen nucleon app oxima ion. The e o e, he con-
ibu ion o his con igu a ion is simila o he a e age.
The same conclusions can be d awn in he case o he
con igu a ion T,−T, wi h one nucleon mo ing along he x
axis ( ans e se di ec ion) and he o he along −xwi h
opposi e momen um. The con ibu ion o his pai is
exac ly he same as ha o he UD con igu a ion in he
o al phase-space unc ion.
Finally, we show in Fig. 17 wo in e media e cases ha
a e nei he pa allel no an ipa allel con igu a ions. They
consis o wo pai s o ans e se nucleons mo ing along
mu ually pe pendicula di ec ions. In he i s case, we
conside a Tnucleon and a second T0nucleon mo ing in
he yaxis ou o he sca e ing plane. The con ibu ion o
he TT0pai is la ge, while he one o he opposi e case, −T,
−T0, is small. On he a e age, hey a e close o he o al
esul .
VIII. PERSPECTIVES ON THE CALCULATION
OF 2P-2H ELECTROWEAK RESPONSE
FUNCTIONS
The nex s ep in ou p ojec o an exac e alua ion o he
ela i is ic 2p-2h elec oweak esponse unc ions in he
Fe mi gas model ini ia ed wi h he app oach in he p esen
pape would be o apply i o a mo e ealis ic si ua ion, i.e.,
elec on and neu ino sca e ing. The 2p-2h s a es can be
exci ed by wo-body MEC ope a o s, in ol ing exchange
o an in e media e meson be ween wo nucleons. A
comple e calcula ion, including all o he MEC diag ams
FIG. 17 (colo online). Rela i is ic phase-space unc ion
Fðq; ωÞcompa ed wi h he a e age-momen um app oxima ion
¯
Fðq; ωÞ o a pai o nucleons wi h momen um 200 MeV=c
poin ing in pe pendicula di ec ions. In he T,T0con igu a ion,
one mo es along x(T) and he o he along y(T0). In he −T,−T0
con igu a ion, hey mo e along he −xand −ydi ec ions,
espec i ely.
FIG. 18 (colo online). Geome y employed o emission o a
pai o nucleons wi h momen a pa allel (cases UU,DD),
an ipa allel (cases UD,T−T), and pe pendicula (cases
TT0,−T−T0).
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-19
wi h one pion exchange, is ou o he scope o he p esen
pape and will be epo ed in a o hcoming publica ion.
Howe e , in his sec ion, we discuss he pe spec i es
opened by he o malism p esen ed he e.
One ques ion can a ise on why he in eg a ion p oblem
ela ed o he di e gence in he angula dis ibu ion, which
is he cen al issue in his wo k, does no appea in o he
app oaches. The models de eloped by Ma ini [9] and
Nie es [10] neglec he di ec -exchange in e e ence e ms
in he had onic enso . In his app oxima ion, in he non-
ela i is ic case, he change o a iables in oduced in
Eqs. (14) and (15) educes he in eg a ion o wo dimensions;
he in eg a ion a iables in his case a e p opo ional o he
magni ude o he ans e ed momen a o he wo nucleons.
In he ela i is ic model o Re . [10], an addi ional app oxi-
ma ion is made o he o al WNN in e ac ion e ex, whe e
he dependence on he ini ial nucleon momen um is
neglec ed by ixing i o an a e age alue o e he Fe mi
sea. This ick allows one o ac o ize he wo Lindha d
unc ions linked o he wo nucleon loops in he many-body
diag ams. Thus, wo app oxima ions a e equi ed in his case
o educe he calcula ion o a ou -dimensional in eg al o e
he 4-momen um o one o he exchanged pions. The exac
calcula ion including all he e ms in he had onic enso
(di ec , exchange, and in e e ence) equi es he comple e
7D in eg al.
Ob iously, a change o a iables can be made o elimina e
he di e gence. One possibili y is o make he change
θ0
1→ffiffiffiffiffiffiffiffiffiffiffi
gðθ0
1Þ
p, whe e gðθ0
1Þis he unc ion de ined in
Eq. (76). This co esponds o he change o a iables made
in Sec. VI B o in eg a e analy ically a ound he di e gence.
The s anda d way o handle his p oblem in he
Mon e Ca lo gene a o s [28] is o compu e he 2p phase-
space angula dis ibu ion in he c.m. sys em o he inal
nucleons, because i is angula independen , al hough Pauli
blocking can o bid some angula egions. A ans o ma-
ion o he labo a o y sys em would gi e exac ly he same
dis ibu ion as conside ed in his pape .
Linked o his, a u he possibili y ha we a e p esen ly
in es iga ing would be o in eg a e o e he c.m. emission
angle ins ead o he Lab one conside ed in his wo k. This
p ocedu e would ha e he ad an age o being ee o he
di e gence coming om he Jacobian bu has he d awback
o equi ing he pe o mance o a boos back o he Lab
sys em o each pai o holes ðh1;h2Þ. One should pe o m
a ull calcula ion wi h bo h app oaches o see he ad an-
ages o each one in e ms o CPU ime.
One o he main p oblems associa ed wi h a comple e,
exac calcula ion o he 2p-2h esponse unc ions is he
compu a ional ime equi ed when he ull cu en is
included. One o he ou comes o his wo k is he possibili y
opened by conside ing wha we called he ozen nucleon
app oxima ion o compu e he in eg al o e he wo holes.
The alidi y o his app oxima ion mus be e i ied in he
comple e calcula ion. I he app oxima ion is ound o be
accu a e enough, hen he calcula ion o he 2p-2h c oss
sec ion could be done wi hou much di icul y and could be
easily implemen ed in Mon e Ca lo gene a o s. The e i i-
ca ion o his app oxima ion is one o he goals o ou u u e
wo k. P elimina y esul s ob ained wi h he seagull dia-
g ams show ha he app oxima ion is alid o his se o
diag ams.
The Mon e Ca lo gene a o s mus no pe o m he
in eg a ion o e he ou going inal s a e bu ins ead mus
keep hese momen a explici ly because one is in e es ed in
gene a ing a ull inal s a e o be p opaga ed. The in eg a-
ion pe o med he e is only needed o he inclusi e 2p-2h
c oss sec ion, which canno be sepa a ed om he measu ed
QE c oss sec ion i he inal nucleons a e no de ec ed.
Wi h ou model, he e is he possibili y o gene a e angula
dis ibu ions o nucleon 2p-2h s a es p oduced by MEC,
ully compa ible wi h he inclusi e 2p-2h c oss sec ions.
They could be use ul o he Mon e Ca lo gene a o s.
IX. CONCLUSIONS
We ha e pe o med a de ailed s udy o he wo-pa icle–
wo-hole phase-space unc ion, which is p opo ional o he
nuclea wo-pa icle emission esponse unc ion o con-
s an cu en ma ix elemen s. To ob ain physically mean-
ing ul esul s, one should include a model o he wo-body
cu en s inside he in eg al. Howe e , he knowledge
gained he e by dis ega ding he ope a o and ocusing
on he pu ely kinema ical p ope ies has been o g ea help
in op imizing he compu a ion o he 7D in eg al appea ing
in he 2p-2h esponse unc ions o elec on and neu ino
sca e ing. The ozen nucleon app oxima ion, ha is,
neglec ing he momen a o he ini ial nucleons o high
momen um ans e , has allowed us o ocus on he angula
dis ibu ion unc ion. We ha e ound ha his unc ion has
di e gences o some angles. Ou main goal has been o
ind he allowed angula egions and o in eg a e analy i-
cally a ound he di e gen poin s. The CPU ime o he 7D
in eg al has been educed signi ican ly. The ela i is ic
esul s con e ge o he non ela i is ic ones o low ene gy
ans e . We a e p esen ly wo king on an implemen a ion o
he p esen me hod wi h a comple e model o he MEC
ope a o s.
ACKNOWLEDGMENTS
This wo k was suppo ed by DGI (Spain), G an s
No. FIS2011-24149 and No. FIS2011-28738-C02-01; by
he Jun a de Andalucía (G an s No. FQM-225 and
No. FQM-160); by he Spanish Consolide -Ingenio 2010
p og ammed CPAN, in pa (M. B. B.) by he INFN p ojec
MANYBODYand in pa (T. W. D.) by U.S. Depa men o
Ene gy unde Coope a i e Ag eemen No. DE-FC02-
94ER40818. C. A. is suppo ed by a CPAN pos doc o al
con ac .
I. RUIZ SIMO e al. PHYSICAL REVIEW D 90, 033012 (2014)
033012-20
APPENDIX A: CALCULATION o xmax
He e, we de i e he uppe limi o he in eg al o e x
in Eq. (22). We i s no e ha he unc ion Aðx; y; νÞinside
he in eg al con ains he ene gy del a unc ion δðωþE1þ
E2−E0
1−E0
2Þand he s ep unc ion θðkF−h1ÞθðkF−h2Þ.
This implies ha
E0
1≤E0
1þE0
2¼ωþE1þE2≤ωþ2EF:ðA1Þ
The e o e,
p0
1
2
2mN
≤ωþ2k2
F
2mN
:ðA2Þ
Taking he squa e oo and ea anging, one has
p0
1≤kFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2þ2mNω=k2
F
q:ðA3Þ
Recalling now he de ini ion o he nondimensional a i-
able ν¼mNω=k2
F, we ha e
p0
1
kF
≤ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1þνÞ
p:ðA4Þ
Finally, using his inequali y in he de ini ion o he x
a iable, one inds ha
x¼
p0
1−h1
kF
≤p0
1þh1
kF
≤p0
1þkF
kF
≤1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1þνÞ
p:
ðA5Þ
APPENDIX B: FUNCTION Aðl1;l2;νÞ
The unc ion Aðl1;l
2;νÞwas compu ed analy ically in
Re . [32]. In his wo k, we ha e epea ed he analy ical
calcula ion, and we ha e ound a ypog aphical e o (a
minus sign) in ha e e ence. Al hough he demons a ion
and nume ical esul s o Re . [32] a e co ec , aking in o
accoun he gi en e o is essen ial. Fo comple eness, and
because he e o can mislead he eade , we w i e in his
Appendix he co ec inal exp ession wi h he sligh ly
di e en no a ion used by us. We w i e he unc ion Aas he
sum o 16 e ms,
Aðl1;l
2;νÞ¼X
4
i¼1X
4
j¼1
Aijðl1;l
2;νÞ;ðB1Þ
whe e he Aij unc ions ha e he symme y
Aijðl1;l
2;νÞ¼Ajiðl2;l
1;νÞ:ðB2Þ
Thus, we only need o gi e he analy ical exp essions o
he diagonal and he uppe -hal o -diagonal ij elemen s:
A11 ¼l1l2
C3
11
3! þðl1þl2ÞC4
11
4! þC5
11
5! θðC11Þ
C11 ≡ν−l2
1
2−l1−l2
2
2−l2
A12 ¼l1l2
C3
12
3! þðl2−l1ÞC4
12
4! −C5
12
5! θðC12Þθðl2−2Þ
C12 ≡ν−l2
1
2−l1−l2
2
2þl2
A13 ¼−l1l2
C3
13
3! þðl1þl2ÞC4
13
4! þC5
13
5! θðC13Þθð2−l2Þ
C13 ≡ν−l2
1
2−l1þl2
2
2−l2
A14 ¼l1
C3
14
3! þC4
14
4! l2
2θðC14Þθð2−l2Þ
C14 ≡ν−l2
1
2−l1
A22 ¼l1l2
C3
22
3! −ðl1þl2ÞC4
22
4! þC5
22
5!
×θðC22Þθðl1−2Þθðl2−2Þ
C22 ≡ν−l2
1
2þl1−l2
2
2þl2
A23 ¼−l1l2
C3
23
3! þðl1−l2ÞC4
23
4! −C5
23
5!
×θðC23Þθðl1−2Þθð2−l2Þ
C23 ≡ν−l2
1
2þl1þl2
2
2−l2
A24 ¼l1
C3
24
3! −C4
24
4! l2
2θðC24Þθðl1−2Þθð2−l2Þ
C24 ≡ν−l2
1
2þl1
A33 ¼l1l2
C3
33
3! þðl1þl2ÞC4
33
4! þC5
33
5!
×θðC33Þθð2−l1Þθð2−l2Þ
C33 ≡νþl2
1
2−l1þl2
2
2−l2
A34 ¼−l1
C3
34
3! þC4
34
4! l2
2θðC34Þθð2−l1Þθð2−l2Þ
C34 ≡νþl2
1
2−l1
A44 ¼l2
1l2
2
ν3
3! θðνÞθð2−l1Þθð2−l2Þ:ðB3Þ
No e ha he equi alen unc ion A13 in Re . [32]
(deno ed F3) has a missing global minus sign.
RELATIVISTIC EFFECTS IN TWO-PARTICLE EMISSION …PHYSICAL REVIEW D 90, 033012 (2014)
033012-21
APPENDIX C: SOLUTIONS OF RELATIVISTIC
ENERGY CONSERVATION
Gi en q,ωand ixing he momen a o he wo holes h1,
h2, he o al ene gy and momen um o he wo pa icles is
also ixed by
E0¼E1þE2þω
p0¼h1þh2þq:
Fo ixed emission angles o he i s pa icle, ϕ0
1¼0and
θ0
1, he alue o p0
1is es ic ed by momen um conse a ion
p0
2¼p0−p0
1and ene gy conse a ion, E0
2¼E0−E0
1.In
ac , aking he squa e o he las equa ion, we should sol e
E0
2
2¼ðE0−E0
1Þ2:ðC1Þ
Ha ing squa ed, we ha e in oduced spu ious solu ions
wi h E0−E0
1<0, which should be h own away.
Expanding he igh -hand side, using he ene gy-
momen um ela ion in he squa ed ene gies, and ea ang-
ing e ms, we a i e a he equi alen equa ion
E0
1¼~
aþ~
p0
1;ðC2Þ
whe e we ha e de ined
~
a¼E02−p02
2E0ðC3Þ
~
¼p0·ˆ
p0
1
E0:ðC4Þ
Taking he squa e o Eq. (C2) and again using he ene gy-
momen um ela ion, we a i e a he second-deg ee
equa ion o p0
1,
~
bp0
1
2−2~
a~
p
0
1þðm2
N−~
a2Þ¼0;ðC5Þ
whe e we ha e de ined
~
b¼1−~
2:ðC6Þ
No e ha Eq. (C2) p o ides an al e na i e way o compu e
he ene gy E0
1once p0
1is known. I also is alid as a check
o he solu ion. Howe e , cau ion is needed because aking
he squa e o Eq. (C2) in oduces spu ious solu ions wi h
~
aþ~
p0
1<0 ha should be dis ega ded.
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