Eu . Phys. J. C (2023) 83:775
h ps://doi.o g/10.1140/epjc/s10052-023-11891-3
Regula A icle - Expe imen al Physics
Sensi i i y po en ial o a ligh la o -changing scala boson wi h
DUNE and NA64µ
B. Radics1,a, L. Molina-Bueno2, L. Fields3, H. Siebe 4, P. C i elli4
1Depa men o Physics and As onomy, Yo k Uni e si y, To on o, ON, Canada
2Ins i u o de Física Co puscula , Uni e sidad de Valencia and CSIC, Ca e del Ca ed á ic José Bel án Ma inez, 2, 46980 Pa e na,
Valencia, Spain
3Depa men o Physics and As onomy, Uni e si y o No e Dame, No e Dame, IN 46556, USA
4ETH Zü ich, Ins i u e o Pa icle Physics and As ophysics, 8093 Zu ich, Swi ze land
Recei ed: 12 June 2023 / Accep ed: 31 July 2023 / Published online: 4 Sep embe 2023
© The Au ho (s) 2023
Abs ac In his wo k, we epo on he sensi i i y po en-
ial o complemen a y muon-on- a ge expe imen s o new
physics using a scala boson benchma k model associa ed
wi h cha ged lep on la o iola ion. The NA64μexpe i-
men a CERN uses a 160-GeV ene gy muon beam wi h
an ac i e a ge o sea ch o excess e en s wi h missing
ene gy and momen um as a p obe o new physics. A he
same ime, he p o on beam a Fe milab, which is used o
p oduce he neu ino beam o he Deep Unde g ound Neu-
ino Expe imen (DUNE), will also p oduce a high-in ensi y
muon beam dumped in an abso be . Combined wi h he liq-
uid a gon nea de ec o , he sys em could be used o sea ch
o simila scala boson pa icles wi h a lowe -ene gy bu
highe -in ensi y beam. We ind ha bo h NA64μand DUNE
could co e new, unexplo ed pa s o he pa ame e space o
he same benchma k model, p o iding a complemen a y way
o sea ch o new physics.
1 In oduc ion
Obse a ions in cosmology and as ophysics imply he exis-
ence o a da k sec o po en ially con aining new pa icles
ha could weakly couple o S anda d Model (SM) pa i-
cles [1]. Neu ino oscilla ions coupled wi h nonze o neu ino
masses p o ide expe imen al e idence o lep on la o io-
la ion. Fu he mo e, he exis ing disc epancy be ween he
measu ed [2] and expec ed [3,4] alue o he muon anoma-
lous magne ic momen p o ides s ong mo i a ion o new
physics sea ches wi h muons [5]. Inspi ed by hese de elop-
men s, a ce ain class o new heo ies p oposes he sea ch
o cha ged lep on la o iola ion (CLFV), which is hea ily
ae-mail: [email p o ec ed] (co esponding au ho )
supp essed in he SM. In he coming decades, a new gene -
a ion o expe imen s will be conduc ed o sea ch o CLFV
[6–11]. In pa allel, nex -gene a ion long-baseline neu ino
oscilla ion expe imen s [12,13] will s udy neu ino oscilla-
ions while a he same ime p oducing an in ense muon beam
dump wi h i s beam line.
In his wo k, we s udy he sensi i i y po en ial o muon-on-
a ge expe imen s o new physics using a CLFV benchma k
model in oduced in a ecen wo k [14], which uses a ligh ,
scala boson associa ed wi h μ−τcon e sion. While he p e-
ious wo k de i ed cons ain s om da a in exis ing beam-
dump expe imen s (LSND [15], NuTeV [16], CHARM [17])
and o he u u e SHiP expe imen [18], he e we ocus on wo
u he expe imen s: he coming neu ino expe imen , DUNE
[12] deploying a high-in ensi y beam om he Long-Baseline
Neu ino Facili y (LBNF) [19], and a ixed- a ge expe i-
men , NA64μ[20,21], which sea ches o new physics wi h
he high-ene gy muon beam om he CERN Supe P o on
Synch o on (SPS) accele a o . Hence, we s udy new physics
sea ches using wo complemen a y muon beam se ups. E en
hough we ocus on one pa icula benchma k model sce-
na io, he e a e also o he possibili ies o p obe hidden sec-
o s wi h muon beams and using simila echniques [22–26].
We also no e ha o he g oups ha e also pe o med s udies o
(cha ged) lep on la o physics, including he ole o mesons
o bosons wi h di e en spin-pa i y assignmen s [27–32].
2 Fla o -changing scala wi h long li e ime
The model p oposed by [14] conside s a new complex scala
ield, φ, wi h a mass window [mτ−mμ,mτ+mμ] ha cou-
ples o (μ, τ ). Wi h such a mass ange, long li e imes can
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775 Page 2 o 7 Eu . Phys. J. C (2023) 83 :775
be achie ed wi h p opaga ion dis ances on he o de o ens
o kilome e s. The e ec i e Lag angian in e ac ion e ms
desc ibe he coupling o he new scala ield wi h lep ons,
LI=φ¯μ(gV+gAγ5)l+φ∗¯
l(g∗
V−g∗
Aγ5)μ (1)
wi h ec o and axial ec o coupling gV,gA, espec i ely.
This model p oduces a benchma k pa ame e egion explain-
ing he muon g−2 anomaly wi h a ypical alue o |gV|≃
3×10−3, also used in his wo k. The e a e mul iple p oduc-
ion modes o p oduce φa beam dump expe imen s: di ec
elec oweak p ocess, hea y meson decay, and high-ene gy
muons hi ing a ixed a ge . In his wo k, we ocus on he
hi d case ( he so-called μ-on- a ge scena io). In his mode,
muons pass h ough dense ma e ial and can p oduce he φ
boson ia he exchange o a i ual pho on wi h he nuclei o
he a ge , μ(p)N(Pi)→τ(p)φ(k)N(P ). Once c ea ed,
he bosons p oduce a missing ene gy signa u e o p opaga e
long dis ances and may be de ec ed in a de ec o downs eam
om he a ge . When he incoming beam ene gy is much
highe han he pa icle mass, he double-di e en ial c oss
sec ion o he 2 →3 p oduc ion p ocess can be es ima ed
using he equi alen pho on app oxima ion [33],
d2σ(p+Pi→p+k+P )
dEφd cos θφ
=αχ
π
Eμxβφ
1−x
dσ(p+q→p+k)
d(p·k) = min
,(2)
whe e Eμis he ini ial muon ene gy, Eφis he ene gy and
θφ he sca e ing angle o φwi h espec o he ini ial muon
in he lab ame, x=Eφ/Eμ,q=Pi−P , =−q2is he
momen um ans e , α=1/137 is he ine s uc u e cons an ,
βφis he ela i is ic ac o o φ, and χis he e ec i e pho on
lux, de ined as
χ= max
min
d − min
2F2( ), (3)
whe e F( )=Z2/(1+ /d)2is he o m ac o , wi h d=
0.164 GeV2A−2/3, and he limi s a e gi en in he appendix
o [14]. We calcula e he analy ical exp ession o χusing
Ma hema ica [34] and ob ain
χ=Z2 min
+d+ min
d+ +(d+2 min)
d
ln
ln (d+ ) max
min
.
(4)
In Eq. (2), σ(p+q→p+k)is he c oss sec ion o he
2→2 sca e ing p ocess, μ(p)γ (q)→τ(p)φ(k),
dσ(p+q→p+k)
d(p·k)=|¯
A2→2|2
8πs2,(5)
whe e |¯
A|2is he ampli ude squa ed, which we calcula e
using he FeynCalc ools [35] o Ma hema ica. We ob ain
he ollowing exp ession o he (μ, τ) case
|¯
A2→2|2=−e2mμmτ(gAg∗
A−gVg∗
V)
(m2
μ−s)2(m2
τ−u)2
×m4
μ(m2
φ+u)+2m3
μ(m3
τ−mτu)
+m2
μm4
τ−2m2
φs−2m2
τu+u(u−2s)
+2mμmτs(u−m2
τ)+sm2
φs+m4
τ−2m2
τu+u(s+u),
(6)
whe e e=√4πα,mφ,mτ, and mμa e he masses o he
boson φ,τ, and μ, espec i ely, and s,ua e he Mandels am
a iables, which can be e alua ed in he labo a o y ame,
s=(p+q)2≃m2
μ−u−m2
τ
1−x
u=(p−k)2≃−Eμxθ2
φ−1−x
xm2
φ+(1−x)m2
μ.(7)
To calcula e he li e ime o he φboson, τφ, we use Equa-
ions 3.2−3.5 in [14] adap ed o he (μ, τ ) case. We calcula e
he c oss sec ion and decay wid h using he GNU Scien i ic
Lib a y [36]. The p oduc ion c oss sec ion o he φboson as
a unc ion o he incoming lep on ene gy is shown in Fig. 1.
Assuming mφ≃mτ, he h eshold o he p oduc ion is gi en
by Eμ>[(2mτ+mN)2−m2
μ−m2
N]/2mN≃3.8GeV o
Pb, abo e which he c oss sec ion s eeply ises. In he ol-
lowing, we use he co esponding muon beam lux o each
expe imen (NA64μand DUNE) o e alua e he exp essions
o he φp oduc ion ia he a iables de ined in Eq. (7).
3µ-on- a ge expe imen s
We es ima e he p ojec ed sensi i i y o an expe imen by
inding he pai o pa ame e alues (gV,mφ), whe e Nφ
app op ia e signal e en s a e p oduced ei he di ec ly in he
a ge (NA64μ) o in he de ec o (DUNE), as explained la e ,
a e a gi en exposu e. In his wo k we exploi he φboson
p oduc ion p ocess using di e en and complemen a y μ-
on- a ge expe imen s: NA64μwi h he ac i e dump ech-
nique (using missing ene gy as a signal) compa ed o p o on
beam-dump expe imen s such as he DUNE neu ino expe i-
men using he LBNF p o on beam (di ec de ec ion h ough
in e ac ion as a signal). Al hough he unde lying p oduc ion
mechanism is he same, he wo echniques di e in he lux
o he muon beam, μ(E), and he a ge hickness and ma e-
ials.
In a gene al case, φbosons a e gene a ed by he μ-on-
a ge p ocess, and a e p oduc ion, a ac ion o hem decay
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Eu . Phys. J. C (2023) 83 :775 Page 3 o 7 775
Fig. 1 P oduc ion c oss sec ion o he φboson, wi h mass mφ=mτ
and coupling cons an |gV|=3×10−3, as a unc ion o he incoming
muon ene gy
inside a de ec o olume and can be de ec ed. The numbe o
such signal e en s is
Nφ=dEφφ(Eφ)×lde
γβcτφ
,(8)
whe e γis he ela i is ic Lo en z ac o , lde is he leng h
wi hin which bosons decay, lde /γβcτφis he ac ion o
bosons decaying in ligh o p oduce a signal in he de ec o ,
and φ(Eφ)is he lux o φbosons a he de ec o , es ima ed
as
φ(Eφ)=dEμ(E)E
Emin
dEl
nA
−dE/dl
×θde
0
dθφsin θφ
d2σ(El,Eφ)
dEφd cos θφ
.(9)
He e, μ(E)is he lux o he muon beam as a unc ion
o ene gy, nAis he numbe o a ge a oms pe olume, El
is he muon ene gy a e a eling a leng h lin he a ge
and losing ene gy acco ding o he s opping powe −dE/dl,
Emin is he ene gy o he muon a he end o he a ge , and
θde is he angula accep ance o he de ec o .
In he ollowing subsec ions, we sepa a ely desc ibe he
wo expe imen al scena ios and he assump ions in he
de i ed sensi i i y limi s.
3.1 NA64μscena io
NA64μis a ixed- a ge expe imen a CERN looking o new
pa icles o da k ma e and po al in e ac ions p oduced in
elec omagne ic showe s and coupled o muons. The expe -
imen uses he seconda y 160-GeV muons om he in e ac-
ions o 400-GeV p o ons om he CERN SPS wi h a a ge .
A se o beam scin illa o s and e o coun e s, low-ma e ial-
budge acke s, and dipole magne s allow us o p ecisely
cons ain he momen um o he incoming 160-GeV muons
impinging on an ac i e a ge . The main de ec o , whe e φ
may be p oduced, consis s o an elec omagne ic calo ime-
e wi h 40 X0 adia ion leng h. Downs eam, he de ec o is
u he equipped wi h e o coun e s and a ∼30-in e ac ion-
leng h had onic calo ime e . New pa icles could be p oduced
by he muon beam sca e ing in he a ge and la e decay o
isible SM pa icles ha could be seen by hei signa u es
in a downs eam de ec o . The cu en wo k is based on he
de ec ion o missing ene gy and momen um ca ied away
by he p oduced hypo he ical, long-li ed φboson, lea ing a
sca e ed muon as expe imen al signa u e (wi h he momen-
um o he sca e ed muon in he ange o 10–80 GeV/c).
The sensi i i y in he sea ch o he φboson is highe wi h
espec o he beam-dump app oach due o he lowe powe
in he coupling s eng h wi hou a decay e ex. Thus, in
NA64μ, only he numbe o e en s a he p oduc ion a ge
needs o be es ima ed. The e o e, he numbe o e en s is
gi en by Nφ=dEφφ(Eφ). Fu he mo e, he p oduc ion
a ge hickness is small, and he muon ene gy loss can be
neglec ed. As a esul , we use he ollowing exp ession o
es ima e he φboson lux
φ(Eφ)=l a ge nAdEμ(E)
×θde
0
dθφsin θφ
d2σ(E,Eφ)
dEφd cos θφ
,(10)
whe e l a ge is he hickness o he a ge .
The e has been ex ensi e s udy o simula ing and e alua -
ing he p oduc ion o da k ma e pa icles wi h a muon beam
a NA64 [26,37,38]. We use he same me hod in his wo k o
es ima e he sensi i i y each o he expe imen . We assume
a muon beam wi h a mean o 160 GeV ene gy and a wid h
o 4.3 GeV hi ing a lead a ge wi h o al da a o ∼3×1013
muons-on- a ge (MOT).
3.2 DUNE/LBNF scena io
The Deep Unde g ound Neu ino Expe imen is a nex -
gene a ion, wide-ene gy-beam long-baseline neu ino expe -
imen a Fe milab. I will use an in ense (an i)neu ino beam
ha passes h ough a nea de ec o a Fe milab and a a de ec-
o 1300km away in Sou h Dako a. The neu ino beam line
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775 Page 4 o 7 Eu . Phys. J. C (2023) 83 :775
o DUNE is a esul o exhaus i e design and op imiza ion
wo k [19]. The beam is p oduced by a 60–120-GeV p o on
beam hi ing a g aphi e a ge , a e which he p oduced pions
and kaons decay o lep ons and neu inos in a ∼220-m-long
decay pipe. A he end o he pipe, a dedica ed ∼30-m-
long s ainless-s eel s uc u e ac s as a beam dump o s op all
muons 300 m ups eam om he nea de ec o . We use he
simula ion o he neu ino beam p oduc ion o ace pa icles
along he beam axis. The muon lux used in he calcula ion is
es ima ed om a dedica ed acking plane, which is loca ed
a he end o he decay pipe in he simula ion. An example o
he ob ained muon ene gy spec um is illus a ed in Fig. 2.
The peak o he spec um is a Eμ≃2.5 GeV; howe e ,
he long high-ene gy ail is esponsible o he majo i y o φ
boson p oduc ion due o he apid inc ease in he c oss sec ion
wi h he incoming lep on ene gy (see Fig. 1). A he lowe -
ene gy egion, he p oduc ion a e is much smalle . F om he
neu ino lux simula ion we es ima e an in eg a ed muon lux
o μ≃5×1019 muons o 1.1×1021 p o ons on a ge
(POT), co esponding o 1 yea o da a- aking.
In DUNE, a signal could be de ec ed in he nea de ec-
o om he decay o he φboson ha was p oduced by
he muons hi ing he s ainless-s eel dump. We conside he
decay channel φ→μ+μ−νμντwi h a b anching a io o
17% [14], mo i a ed by he dimuon esul s om NuTeV [16].
Possible backg ounds leading o a dimuon signa u e include
deep inelas ic sca e ing (DIS) and esonance p oduc ion o
mesons in cha ged-cu en (CC) muon–neu ino (νμ) in e -
ac ions wi h a a ge nucleus. The mesons could decay in
semi-lep onic mode, p oducing an ex a muon. In o de o
analyze such po en ial backg ound p ocesses, we pe o med
simula ions wi h he GENIE [39] Mon e Ca lo (MC) e en
gene a o ha p o ides a comp ehensi e neu ino in e ac ion
modeling in he Eν∼100 MeV − ew-hund ed GeV neu-
ino ene gy egion, including quasi-elas ic, esonance, and
DIS p ocesses. Unlike he φboson decay, e en s wi h DIS o
esonan meson p oduc ion a e accompanied by addi ional
ac i i y in he inal s a e. Simila ly o p e ious indings o
NuTeV, a e disc imina ing o low-mul iplici y e en s wi h
wo muons, we ound ha hese backg ounds could be com-
ple ely supp essed in an MC simula ion o 400 million e en s
o neu ino in e ac ions on an a gon a ge (co esponding
o ∼5 yea s o ope a ion o DUNE a he nominal in en-
si y). Howe e , u he s udies a e planned wi h a ull de ec-
o simula ion o gain a de ailed unde s anding o he possible
bounds on he backg ound ejec ion.
4 Resul s
We illus a e he sensi i i y po en ial o he benchma k
CLFV scena io wi h he complemen a y muon beams a
NA64μand DUNE in Fig 3. Fo bo h expe imen s, he
Fig. 2 Ene gy spec um o muons a he end o he decay pipe om
he ull DUNE neu ino beam line simula ion [19], see explana ion in
he ex
double-di e en ial c oss sec ion in Eqs. (9)o (10) is e al-
ua ed gi en he ene gy spec a o each expe imen , μ(E),
and he kinema ic limi s on he inal-s a e φ-boson ac ional
ene gy, x, which is cons ained by he masses o he boson φ
and he τ.
In he case o he NA64μexpe imen , a o al in eg a ed
muon lux o ∼3×1013 MOT is achie able [20]. The ime
needed o accumula e he assumed o al MOT is es ima ed
o be ∼100 days. This conse a i e es ima ion is based on
he CERN SPS deli e ing on a e age 3500 spills pe day
and 2 ×108muons pe spill. Bene i ing om he unique
combina ion o a 160-GeV muon beam wi h missing-ene gy
and momen um sea ch, NA64μwould be able o pe o m a
compe i i e sea ch o such a CLFV signal. Fu he mo e, he
sensi i i y each c i ically depends on he leng h o he ac i e
a ge . We ind ha a 1-m-long a ge would al eady be able
o explo e a la ge pa o he pa ame e space, gV≥6×10−3.
Howe e , he expe imen is highly modula , and a possible
op imiza ion o he se up could u he enhance i s po en ial:
a easible op ion is o inc ease he a ge leng h o 5m which
would make i possible o comple ely co e he (gμ−2)
p e e ed egion and o p obe a a ie y o o he new physics
scena ios in ol ing muons. We also ind ha , al eady wi h
a10
12 MOT and a ∼3-m-long a ge , he gV≥×10−2
pa ame e egion can be co e ed. A de ailed Mon e Ca lo
simula ion o an op imized se up will ollow as a nex s ep.
Fo he DUNE expe imen , he φ-boson p oduc ion is
d i en by he high-end ail o he muon lux. Compa ed o
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Eu . Phys. J. C (2023) 83 :775 Page 5 o 7 775
NA64μ, he lowe muon ene gies a DUNE and hus lowe
p oduc ion c oss sec ion a e compensa ed by he mo e in ense
muon lux. Assuming 10–20 yea s o ope a ion a nominal
in ensi y, DUNE would be able o explo e a signi ican pa
o he pa ame e space, eaching in o he gV≃10−2 egion
and po en ially imp o ing he cons ain s om NuTeV. How-
e e , an op imiza ion o he neu ino beam line could u -
he enhance he con ibu ion o high-ene gy muons in he
lux and subsequen ly app oach he gV≤10−2benchma k
egion. This scena io is pa ially mo i a ed by a ecen wo k
explo ing an al e na i e beam-dump ope a ion mode o p obe
new physics wi h DUNE [40].
Fo compa ison, we also show he p ojec ed sensi i i y o
he same μ-on- a ge mode calcula ed by [14] o CHARM,
NuTeV, and SHiP. He e, we do no show he cons ain s o
p ojec ed sensi i i y limi s de i ed o he di ec elec oweak
and hea y meson decay p ocess, which was included in he
p e ious wo k [14], since we only ocus on he μ-on- a ge
scena io. In he case o SHiP, he assumed o al da a co -
esponds o 2 ×1020 p o ons on a ge (POT). The wo se
sensi i i y in DUNE ela i e o SHiP s ems om he lowe
muon ene gies in he lux [41]. I is also no ed ha he e a e
di e ences in he beam in ensi y be ween NA64μand SHiP.
We assume 1012 POT pe spill in ensi y a he CERN SPS in
he case o he muon beam line used by NA64μ, while o
SHiP he p o on beam in ensi y is gene ally expec ed o be
an o de o magni ude highe .
We no e ha a simila se up o NA64μis capable o
sea ching o o he new scala pa icle candida es using he
same muon beam [26]. In addi ion, he p oposed Muon Miss-
ing Momen um (M3) expe imen a Fe milab [42] plans o
p obe new physics wi h a dedica ed muon beam. Finally,
a numbe o expe imen s also ha e he po en ial o sea ch
o hidden-sec o scala pa icles, such as SHADOWS [43],
HIKE [44], and ATLAS [45].
5 Conclusions
In summa y, we p esen he sensi i i y po en ial o wo μ-on-
a ge expe imen s, NA64μand DUNE, as complemen a y
modes o sea ching o new physics wi h muon beams. We
ind ha bo h NA64μand DUNE ha e he po en ial o co e a
signi ican po ion o he benchma k model pa ame e space,
(mφ,gV). NA64μwi h an op imized se up could p obe he
coupling pa ame e down o gV≃3×10−3, comple ely
co e ing he muon gμ−2 p e e ed egion and hus p o id-
ing a simila p ojec ed each as SHiP. DUNE will also be
able o co e unexplo ed pa s o he pa ame e space, po en-
ially imp o ing on he ob ained cons ain s om NuTeV. An
op imiza ion o he neu ino beam line, inc easing he con-
ibu ion om he high-ene gy ail, could allow us o u he
enhance he sensi i i y o DUNE.
Fig. 3 Cons ain s om CHARM, NuTeV [16,17], and p ojec ed sen-
si i i y cu es o SHiP [14,18] (dash-do ed, magen a line), DUNE
(solid b own and blue lines o 10 and 20 yea s o ope a ion, espec-
i ely), and NA64μ(dashed b own and black lines) in he μ-on- a ge
p oduc ion mode (excep o CHARM). In he case o NA64μ, he op i-
mized se up wi h 1-m-long (b own dashed line) and 5-m-long (black
dashed line) a ge s a e also shown sepa a ely. The ligh blue egion
shows he benchma k pa ame e ange explaining he muon gμ−2
anomaly
Al hough we use a gi en CLFV model as a benchma k
in his wo k, we no e ha simila echniques can be used
o s udy he sensi i i y po en ial o expe imen s wi h muon
beams o o he physics scena ios. Beyond he gμ−2 disc ep-
ancy and neu ino la o oscilla ions, sea ches o da k sec o
pa icles a e also mo i a ed by he ma e –an ima e asym-
me y, he known da k ma e abundance om as ophysical
and cosmological obse a ions, o by heo e ical mo i a ions
s ongly sugges ing he exis ence o addi ional gauge g oups
weakly coupling o SM ields [1,46].
Acknowledgemen s We g a e ully acknowledge con e sa ions wi h
D. Ha is. The wo k o LMB is suppo ed by SNSF G an no. 186158
(Swi ze land), RyC-030551-I, and PID2021-123955NA-100 unded by
MCIN/AEI/ 10.13039/501100011033/FEDER, UE (Spain).
Da a A ailabili y S a emen This manusc ip has no associa ed da a
o he da a will no be deposi ed. [Au ho s’ commen : The simula ions
and calcula ions ha suppo he indings o his s udy a e a ailable
om he au ho s upon eques ].
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