#### Probing neutrino coupling to a light scalar with coherent neutrino scattering

HJE
Probing neutrino coupling to a light scalar with
Yasaman Farzan 0 1 2
Manfred Lindner 0 1
Werner Rodejohann 0 1
Xun-Jie Xu 0 1
0 Postfach 103980 , D-69029 Heidelberg , Germany
1 P. O. Box 19395-5531, Tehran , Iran
2 Institute for Research in Fundamental Sciences , IPM
Large neutrino event numbers in future experiments measuring coherent elastic neutrino nucleus scattering allow precision measurements of standard and new physics. We analyze the current and prospective limits of a light scalar particle coupling to neutrinos and quarks, using COHERENT and CONUS as examples. Both lepton number conserving and violating interactions are considered. It is shown that current (future) experiments can probe for scalar masses of a few MeV couplings down to the level of 10 4 (10 6). Scalars with masses around the neutrino energy allow to determine their mass via a characteristic spectrum shape distortion. Our present and future limits are compared with constraints from supernova evolution, Big Bang nucleosynthesis and neutrinoless double beta decay. We also outline UV-complete underlying models that include a light scalar with coupling to quarks for both lepton number violating and conserving coupling to neutrinos.
Beyond Standard Model; Neutrino Physics
1 Introduction
2 Light scalar interactions in coherent neutrino-nucleus scattering
2.1
2.2
2.3 From the fundamental couplings to the e ective couplings
3 Existing bounds from particle and astroparticle physics
4 Constraints and future sensitivities from CE NS
4.1
CONUS
4.2 COHERENT
5 Interpretation of the results
6 Summary and concluding remarks
A Suggestions for underlying electroweak symmetric models B Calculation of the cross section 1 2
more experiments will be able to observe CE NS, including CONUS [19], TEXONO [20],
CONNIE [21], MINER [22], GEN [23], Ricochet [24], and -cleus [25].
In the present paper we focus on new neutrino physics caused by a new light scalar.
Such a particle can participate in coherent neutrino-nucleus scattering, and interestingly
modify the nuclear recoil spectrum in a characteristic manner, both for a light scalar as
well as heavy one. Most of our study will focus on the light scalar case. Light new physics
is of course motivated by the lack of signals in collider experiments, and its consequences in
coherent neutrino-nucleus scattering have been mentioned before [14{16, 26{29]. Mostly it
was used that light physics does not su er from limits on neutrino non-standard interactions
{ 1 {
from high energy scattering experiments such as CHARM-II, as stressed in [30, 31].
Discovery limits on light particles in CE N S have been discussed in refs. [14, 26, 29], indicating
already before the observation by COHERENT that CE N S could provide strong constraints
on light mediators. In our work, as mentioned before, we focus on a new scalar particle and
obtain the current limits from COHERENT on its mass and coupling with SM particles. We
also consider explicit future realizations of this experiment and also of CONUS, which will
use reactor antineutrinos to probe coherent scattering. The characteristic distortion of the
spectrum shape for light scalars with masses around the neutrino energy allows to
reconstruct the scalar mass, which we explicitly demonstrate. We also compare our limits with
existing ones from a variety of sources in particle and astroparticle physics. UV-complete
models that may be behind the existence of such light scalar particles are also outlined.
This paper is organized as follows: in section 2, we present the framework of a light
scalar with couplings to neutrinos and quarks, discuss the coherent scattering cross section
and include a discussion of form factors when the coupling to nucleons is considered. In
section 3, we discuss the constraints on such scalar particles from various particle and
astroparticle physics observables. In sections 4 and 5 we discuss and interpret the implications
of various possible observations. In section 6, we summarize our ndings. The calculational
details and outlines for UV-complete gauge invariant models are presented in appendices.
2
Light scalar interactions in coherent neutrino-nucleus scattering
In this section, we rst introduce the possible interactions of neutrinos with quarks (or
nuclei) mediated by a light scalar boson and then discuss the corresponding cross section
of coherent neutrino-nucleus scattering.
2.1
New scalar interactions
We consider a scalar eld denoted by
which couples to neutrinos. There are two
possibilities for such coupling, namely lepton number violating (LNV) and conserving (LNC)
couplings. The latter possibility requires the presence of right-handed neutrinos:
LN
N N
N ;
{ 2 {
LLNC
(C + iD
5)
= y
R L + H:c:;
where y = C
iD . For simplicity, throughout this paper we implicitly assume that
is a real eld but most discussion and overall behavior of the bounds and limits remains
valid for a complex
as well. For real , the hermicity of the Lagrangian implies that C
and D
are real. Note that the couplings have
avor indices, suppressed here for clarity.
The lepton number violating form of the interaction can be written in analogy as
LLNV
y
2
y
2
Lc L + H:c =
LT C L + H:c:
In both Lagrangians y can in general be a complex number.
The same scalar eld can couple to quarks. Since in coherent neutrino scattering we
are concerned with the e ective coupling of
with the whole nucleus N we write the
Lagrangian as:
(2.1)
(2.2)
(2.3)
scattering is
where L
2.2
Cross section
L
L
+ LN
can be either eq. (2.1) or eq. (2.2). The masses of the scalar and nucleus are
respectively denoted by m and M . In appendix A, we present examples in which couplings
to neutrinos and quarks can be embedded in electroweak symmetric models.
The rst thing to notice is that the Yukawa interaction of both LNC and LNV forms
(either eq. (2.1) or eq. (2.2)) leads to chirality- ipping scattering which will not interfere
with chirality-conserving SM weak interactions.2 Thus, one can separate the cross section
into two parts containing the pure SM and the new physics contributions:
d
dT
=
d SM +
dT
d
dT
;
where T denotes the recoil energy. The SM cross section, assuming full coherence is given
by [33]
where
write
N is the Dirac spinor of the nucleus, assuming it is a spin-1/2 particle.1 We can
N
CN + DN i 5:
Again, for real
hermicity of the Lagrangian demands CN and DN to be real numbers.
The conversion from fundamental quark couplings (Cq, Dq) to the e ective coupling (CN ,
DN ) will be discussed later. Note that we here consider both scalar and pseudo-scalar
interactions. In what follows, the latter contribution is usually very much suppressed, and
essentially only the scalar contribution is what matters.
In summary, the Lagrangian (in addition to the SM) responsible for coherent neutrino
(2.4)
(2.5)
(2.6)
(2.8)
(2.9)
d SM =
dT
G2F M
N
(1
4
4s2W )Z 2
1
T
Tmax
; where Tmax(E ) =
:
(2.7)
2E2
M + 2E
The coherent scattering mediated by the light scalar, independent of whether the new
scalar interaction is of the LNC or LNV form, is (the derivation is given in appendix B):
d
dT
=
M Y 4A2
1The actual spin of the nucleus can take other values but the di erence of the cross section is suppressed
by E2=M 2 | see the appendix in [13].
2More generally, as it has been studied in [32], there is no interference in neutrino scattering between
vector (or axial-vector) form interactions and other forms of interactions, including (pseudo-)scalar and tensor.
in the de nition of Y makes it almost independent of the type of nucleus | cf. eq. (2.19)
and eq. (2.20). The cross section has little dependence3 on DN because the pseudo-scalar
contribution is suppressed by the O T 2=E2 term. This is in analogy to dark matter
direct detection where dark matter nucleon interactions mediated by a pseudo-scalar is
well known to be suppressed.
Obviously scalar interactions lead to a spectral shape di erent from that in the SM
case, see e.g. [13], where an e ective scalar interaction, corresponding to m2
M T , is
considered. For a light scalar under discussion here we see that additional modi cations of
the spectrum are possible, in particular if the scalar mass is of the same order or smaller
than the typical momentum transfer M T
E2.
2.3
From the fundamental couplings to the e ective couplings
The e ective couplings CN and DN originate from fundamental couplings of
with the
quarks. The connection between the e ective couplings and the fundamental couplings has
been well studied in spin-independent dark matter direct detection. Essentially, one needs
to know the scalar form factors of quarks in the nuclei. We refer to [34] for the details and
summarize the relevant results below.
Since the pseudo-scalar coupling DN has no e ect on CE N S [cf. eq. (2.9)], we focus
on the scalar coupling, CN . Taking the fundamental scalar interaction of quarks with
to
be of the form
the e ective coupling CN is related to Cq by
where the couplings to protons and neutrons are
L
X Cqqq ;
q
CN = ZCp + (A
Z)Cn;
"
X Cq
q
p #
fq
mq
Cp = mp
; Cn = mn
"
X Cq
q
fqn #
mq
:
Here mp = 938:3 MeV and mn = 939:6 MeV are masses of proton and neutron; Z and
Z are proton and neutron numbers in the nucleus; mq are quark masses; fqp and fqn are
the scalar form factors in protons and neutrons. According to the updated data for the u
and d quarks from [35, 36] and the data for the s quark from [37], the form factors are:
f
f
f
f
p
c
n
c
dp = 0:0411
dn = 0:0451
f
p
b
f
n
b
f
p
t
f
n
t
0:0028;
0:0027;
2
27
2
27
(1
(1
f
p
d
f
n
d
fup = 0:0208
f
un = 0:0189
p
fs )
n
fs )
f
p
u
f
n
u
0:0015;
0:066;
0:0014;
0:066;
fsp = 0:043
0:011;
f
sn = 0:043
0:011;
3The explicit form of the negligible term O T 2=E2 is actually 1 + DN2 =CN2 2TE22 | see eq. (B.6) in
the appendix.
{ 4 {
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
where fc; b; t are approximately the same for the heavy quarks because their contributions
come from heavy quark loops that couple
to the gluons [38]. The form factors computed
by the e ective eld theory in refs. [39, 40] agree well with the above results. Notice that
despite the di erent quark content, the couplings of proton and neutron to the scalar turn
out to be almost equal: j(Cn
Cp)=Cnj = O(10%). If Cq for the heavy quarks are of the
same order of magnitude as for the light quarks, then the contributions of heavy quarks are
negligible due to suppression by their masses. However, if Cq / mq (as in the case that the
coupling to the quarks comes from the mixing of a scalar singlet with the SM Higgs), the
contribution from all avors will be comparable. Taking the following quark masses [41]:
md = 4:7 MeV;
mc = 1:27 GeV;
mu = 2:2 MeV;
mb = 4:18 GeV;
ms = 96 MeV;
mt = 173:2 GeV;
we obtain
mp
mn
f
p
d ;
md
f n
md
d ;
p
fu
mu
;
f n
mu
u ;
p
fs
ms
;
f n
ms
s ;
p
fc
mc
;
f n
mc
c ;
f
p
b ;
mb
f n
mb
b ;
f
p
t
mt
ftn
mt
(8:2; 8:9; 0:42; 4:9
For Ge and CsI targets, taking the average values of (Z; A) as (32; 72:6) and (54; 130), we
can evaluate the explicit dependence of CN on the Cq:
CN =
(102
103
(6:3 Cd + 6:1 Cu) + (30:5 Cs + 3:5 Cc + 1:1 Cb + 2:5
(1:1 Cd + 1:1 Cu) + (54:7 Cs + 6:3 Cc + 1:9 Cb + 4:7
10 2 Ct) (Ge)
10 2 Ct) (CsI)
:
This means that from the fundamental coupling to the e ective coupling an ampli cation
by a factor of O(102) or O(103) can be present. The de nition of Y in eq. (2.9), in terms
of the fundamental couplings, can be rewritten as
Y
r jCN y j =
A
s
A
Z
A
Z
A
Cn +
Cp y :
The dependence of Y on the types of targets is weak because for heavy nuclei, A Z and
AZ are typically close to 1=2. For example, taking the average values of (Z; A) for Ge and
A
CsI targets, we get
YGe
YCsI
q
q
j(0:56 Cn + 0:44 Cp)y j ;
j(0:58 Cn + 0:42 Cp)y j :
{ 5 {
When comparing the sensitivities of CE N S experiments using di erent targets, we will
ignore the small di erence and assume YGe
YCsI.
(2.17)
(2.18)
(2.19)
(2.20)
Existing bounds from particle and astroparticle physics
In this section we review the relevant bounds on Cn, y or on their product (Cny ) from
various observations and experiments other than coherent scattering. As we shall see, the
bounds on the hadronic couplings are on Cn = R[ n ] (not on Dn) but the bounds on
neutrino couplings are on jy j2 = C2 + D2. The di erence originates from the fact that
while nuclei in the considered setups are non-relativistic, neutrinos are ultra-relativistic.
Bounds on Cn from neutron nucleus scattering: in the mass range of our
interest, the strongest bounds on Cn come from low energy neutron scattering o
nuclei [42{44], in particular using Pb as target. The e ect of a new scalar would be
to provide a Yukawa-type scattering potential whose e ect can be constrained. Notice
that like the case of CE N S, since in these setups the nuclei are non-relativistic, their
dominant sensitivity is only to Cn (not to Dn).
Bounds on y from meson decay: if the new light boson couples to neutrinos,
it can open new decay modes for mesons such as K+
! l
+
or
+
! l
+
. The
scalar will eventually decay into a neutrino pair appearing as missing energy. From
the absence of a signal for such decay modes, bounds of order 10 3 on (P
and (P
jy j2 )1=2 have been found from di erent modes [45]. Notice that as long
as the mass of
is much smaller than the meson mass the bound is independent of
jy je2 )1=2
m . Moreover, the bound similarly applies for the LNV and LNC cases. In the LNC
case, sensitivity is to the combination jy j2 = C2 + D2.
Bounds from double beta decay: a LNV coupling of form
neutrinoless double beta decay [46] in the form of n+n ! p+p+e +e + , of course
provided that
is lighter than the Q value of the decaying nucleus. From double
beta decay of 136Xe with a Q value of 2.4 MeV the following bound is found [47]:
eT C e can cause
(y )ee < 10 5:
(3.1)
A more recent and a bit weaker bound for m
< 2:03 MeV comes from 76Ge double
beta decay [48].
Supernova bounds and limits: light particles coupled to neutrinos and neutrons
can a ect the dynamics of a proto-neutron star in several ways. Before discussing the
impact of our particular scenario on supernovae, let us very brie y review the overall
structure of the core of a proto-neutron star in the rst few 10 seconds after explosion
when neutrinos are trapped inside the core (i.e., the mean free path of neutrinos is
much smaller than the supernova core radius). For more details, the reader can
consult the textbook [49]. In the rst
10 sec after collapse, the core has a radius of
few 10 km and a matter density of
1014 g cm 3 (comparable to nuclear density).
However, because of the high pressure, most of the nucleons are free. As mentioned
before, neutrinos are trapped inside the core and are thermalized with a temperature
of
10 MeV. The core can be hypothetically divided into the inner core with a radius
{ 6 {
of 10-15 km and the outer core. Within the inner core, the chemical potential of the e
is about 200 MeV which is much larger than their temperature, implying that e are
degenerate. The chemical potentials of
and
are zero and their temperatures are
equal. Thus, inside the inner core n e
n
= n
= n
= n
n e, where n
is
the number density of
. In the outer core, the chemical potential of e decreases and
the number densities of e and e are almost equal. Notice that the average energies
of
,
and their antiparticles throughout the core are given by their temperature
which is of order of few 10 MeV. The energies of e and e in the outer core, where
the chemical potential vanishes, are also of the same order but the energy of e in
the inner core is of order of the chemical potential, 200 MeV. The neutrinos scatter
o nucleons with a cross section of
G2F E2=(4 ). Considering the high density
of nucleons, the mean free path for neutrinos will be of order of
is much smaller than the core radius R
few 10 km. The di usion time, given by
core carry out the binding energy of the star which is of order of 1053 erg.
R(R= ), is therefore of order of O(1 sec) to O(10 sec). Neutrinos di using out of the
' 300 cm, which
Let us now see how new light particles coupled to neutrinos and matter elds can
a ect this picture. First, let us discuss the impact on supernova cooling. If the
interaction of the new particle is very feeble, it cannot be trapped. Thus, if it is produced
inside the core it can exit without hinderance and take energy out of the core leaving
no energy for neutrinos to show up as the observed events of SN1987a. This sets an
upper bound on the coupling of the new particles. On the other hand, if the coupling
is large enough to trap the new particles, the impact on cooling will not be dramatic.
Still, if the new particles are stable, they can di use out and, along with neutrinos,
can contribute to supernova cooling. Considering however the theoretical uncertainty
in the evaluation of total binding energy and the observational uncertainty on the
energy carried away, such contributions can be tolerated. Thus, supernova cooling
consideration, within present uncertainties, can only rule out a range of coupling
between an upper bound and a lower bound. New interactions can also a ect the mean
free path of neutrinos
and therefore the di usion time R2= , which roughly
speaking coincides with the observable duration of neutrino emission from a supernova.
This sets another limit. Finally, if there is a new process that can remove e and/or
convert it to any of e
,
( ) or
( ), it can have profound e ect on the Equation
of State (EoS) in the inner core. For example, if e (with energy of 200 MeV) are
converted to
, the temperature of
will increase dramatically. Conversion of e to
e will lead to the production of e+ which annihilates with electrons inside the core.
In our case the
particles can decay into a neutrino pair. The decay length is
evaluated to be approximately (10 5=jy j)2(E =10 MeV)(5 MeV=m )2 cm, which is much
smaller than the radius of the proto-neutron star (
few 10 km). The consequences
of new interactions on supernova explosions can be categorized into three e ects: (1)
change of equation of state in case of LNV interaction; (2) new cooling modes because
of right-handed (anti)neutrino emission in case of LNC interaction; (3) prolonging
the duration of neutrino emission (R2= ) because of a shorter mean free path . In
{ 7 {
all these three e ects, the neutrino scattering plays a key role. Neglecting
avor
indices, we nd that the neutrino-neutrino scattering cross section is comparable to
the -nucleon scattering cross section if y
Cn. The number density of nucleons is
larger than that of neutrinos, thus the -nucleon scattering should be more
important. The cross section of the scattering due to
both for LNV and LNC cases can
be estimated as
(
( )
+n !
( )
+n) = jy j2Cn2 "log
.
a signi cant fraction of degenerate e in the inner core will convert into antineutrinos,
drastically changing the equation of state. That is, for
p(y )e Cn > 2
200 MeV, the equation of state of the supernova core has to be
reconsidered. In subsequent plots that summarize the limits on our scenario, we call
the associated limit to avoid this feature as \SN core EoS". Since the temperatures
of
and
are expected to be the same and their chemical potentials to be zero,
conversion of
and
to
and
due to non-zero (y ) , (y )
and (y )
will
not change the equation of state.
neutrinos will be trapped. The values of p
When the interaction is LNC, scattering will convert left-handed neutrinos
(righthanded antineutrinos) into sterile right-handed neutrinos (left-handed antineutrinos),
which do not participate in weak interactions. If ( =mn)(10 sec) & 1, a signi cant
fraction of the active (anti)neutrinos will convert into sterile ones. Avoiding this
generates an upper limit on the coupling. If ( =mn)R & 100, the produced sterile
y Cn between these two limits are
therefore excluded by supernova cooling considerations. The limits are denoted in gure 4
as \SN energy loss" and \SN
R trapping", respectively.
Drawing these gures we have assumed a nominal temperature of 30 MeV for
neutrinos which is the typical energy for all neutrinos in the outer core. In case that the
neutrino-nucleon scattering cross section due to
exchange becomes comparable to
the standard weak cross section G2F E2=(4 ), the di usion time R2=
will be signi
cantly a ected. This limit is shown in gures 4 and 5 as \SN
di usion", and holds
for both the LNC and LNV cases.
{ 8 {
BBN and CMB bounds: the contribution from the LNC and LNV cases to the
additional number of relativistic degrees of freedom ( Ne ) will be quite di erent so
in the following, we address them separately.
i) LNV case: in this case, no R exist so we should only check for the
production.
For m
& 1 MeV,
will be produced at high temperatures but it will decay before
neutrino decoupling without a ecting Ne at the BBN or the CMB era. Thus, within
the present uncertainties, there is no bound from BBN for m
& 1 MeV. Lighter ,
in turn, can contribute to Ne as one scalar degree of freedom if they enter thermal
equilibrium. Taking n
(
!
)H 1
jy j < 5
jT =1 MeV < 1, we nd
10 9 m
MeV
< 1 MeV, although
decays away before the onset
of BBN it still contributes to Ne by warming up the
and
distributions. At
m
= 1 MeV, combining the above bound on y from BBN with that from n-nucleus
scattering, yields Y < 5
10 7. For m
> 1 MeV, this bound does not apply because
decays into neutrinos before neutrino decoupling from the plasma. That is why the
bound denoted \BBN + n scat." appears as a vertical line in gure 5. Our simpli ed
analysis seems to be in excellent agreement with the results of [50] which solves the
full Boltzmann equations.
ii) LNC case: in this case, the production of scalars via
is not possible.
Processes like
process L L !
!
or N !
N
can take place but are suppressed. The t-channel
R R can lead to R and
R production. The L L !
R R process
can also take place but because of cancelation between t and u channel diagrams it
has a smaller cross section. The cross section of the dominant production mode is
!
( L L !
R R) =
jy j4
1
32 m2 x2
2x(1 + x)
1 + 2x
log(1 + 2x) ;
(3.6)
where x = 2E2=m2 , in which E is the energy of colliding neutrinos in the
center-ofmass frame. Taking n
H 1
jT =3 MeV . 0:3 (3 MeV is the temperature at neutrino
decoupling and 0.3 is the bound from CMB on Ne [51]), we nd
y < 1:7
in which m and T are in MeV and x = 2T 2=m2 . Combining this bound with the one
on Cn from neutron-Pb scattering gives the limit denoted \BBN + n scat." in gure 4.
The e ective couplings in eq. (2.1) or in eq. (2.2) can lead to contributions at loop level
to neutrino mass. Before evaluating the contributions, let us notice that the electroweak
symmetric UV-complete models that at low energies give rise to these e ective coupling
can provide mass for neutrinos at tree level, too. In the models summarized in appendix
A, the tree level contribution to neutrino mass turns out to be given by these e ective
{ 9 {
HJEP05(218)6
value to be consistent with the measured neutrino masses. The avor structure of neutrino
couplFingis gtimuesrtehe V1E:V oOfa nnewes-lcoalaoripnthecmoondelt.rTihebVuEVticaon ntaketaonarbnitrearuytrino
mass and y couplings will then have the same pattern. In evaluating the loop contribution
from couplings in eqs. (2.1), (2.2) to neutrino mass, one should bear in mind that these
couplings are valid only at energies below electroweak scale so the natural UV cuto is
cut
100 GeV. Neutrino mass term is a helicity- ipping operator so the loop diagram
providing neutrino mass has to involve an odd number of helicity ipping y couplings. The
direct result of this is that there is no two loop contribution to neutrino mass and at one
loop level, the contribution comes from a tadpole contribution to
(as shown in
gure 1)
which, as usual, can be canceled by a counter-term (see for example sec 11.2 of [52]). At
the three loop level, contributions such as the ones in
gure 2 arise. In these gures, the
lines marked with q and q0 denote quarks and the lines without an arrow attached to
lines can denote R (as in case of eq. (2.1)) or Lc (as in case of eq. (2.2)). The contribution
from the rst diagram can be estimated as
and the other one can be estimated as
m
(Y 2)3
(16 2)3 cut = 10 8eV
m
G2F Y 2
Both these contributions are too small to be discernible at beta decay experiments or (even
in case of LNV coupling) in experiments searching for neutrinoless double beta decay.
Let us now discuss the e ects of forward scattering o nuclei due to the new interaction
on neutrino propagation in matter composed of nuclei, N . The induced e ective mass will
be of chirality ipping form as follows
CN y
m2
mN
T C
or
R L :
ν
φ
φ
ν
φ
ν
•
l
+
q
q′
•
ν
φ
a shift in the mass of neutrinos of order of 3
wihnictheirsacocmtpiolentelyvneergtliigcibelse.compared to
10 12 eV (y =10 5)(Cp=10 5)(5 MeV=m )2,
m2=m . One may wonder why forward
scatasF( Cigpyur=me2 1)(: =Tmph).reTeakilnogofopr ecxaomnptlreitbheudteinosinty tofothne eSuuntarsino= 1m50agscsm. 3
Notice that mN ' Amp ' Amn and CN =mN ' Cp=mp so we can write the e ective mass
T,whee nddots denote
tering due to a possible new gauge boson with similar mass and coupling has such a large
impact on neutrino propagation in matter, while the present case of a scalar does not. The
reason lies in the di erent Lorentz structure of the induced operators. The vectorial
interaction induces a contribution of form
T 0
= y which has to be compared with m2=E .
In the case of scalar, the matter e ects have the operatorial form as the mass themselves
and should be compared to the mass splitting,
m2=m . More detailed discussion can be
found in [53].
4
Constraints and future sensitivities from CE
NS
To collect large statistics at coherent scattering energies, CE N S experiments require
intensive and low-energy (. 50 MeV) neutrino
uxes. Two types of neutrino sources can be
invoked to carry out CE N S experiments: reactor neutrinos (E
. 8 MeV) and pion decay
at rest (E
. 50 MeV). Two on-going experiments, CONUS [19] and COHERENT [1],
adopt these two sources respectively. In this section, we study the sensitivities of the two
experiments on light scalar bosons.
4.1
CONUS
The CONUS experiment uses a very low threshold Germanium detector setting 17 m away
near a nuclear power plant (3.9 GW thermal power) in Brokdorf, Germany. The total
antineutrino ux is 2:5
1013 s 1 cm 2. Data collection started in 2017 and rst results are
expected soon. To study the sensitivity of CONUS, we compute the event numbers given by
Ni =
Z Ti+ T
Z 8 MeV
dT
Ti
0
dE
(E ) (Tmax(E )
T )
(T; E ) :
(4.1)
d
dT
Here
t is the running time, NGe is the number of Ge nuclei, (Ti; Ti +
T ) is the range of recoil energy in each bin, (E ) is the reactor neutrino ux, and
(Tmax(E )
Heaviside theta function, equal to 0 for Tmax(E )
T < 0 and 1 for Tmax(E )
T ) is the
T > 0.
1.20
The e ective coupling Y is de ned in eq. (2.18) and its dependence on the fundamental quark
couplings is given by eq. (2.17). For comparison, the event numbers of the SM signal are also shown.
It is necessary to insert the
function in (4.1) because the cross section ddT (T; E ) does
not automatically vanish when Tmax(E ) < T . We take
t = 1 year,
T = 0:05 keV
and NGe = 3:32
1025, corresponding to 4 kg natural Ge (with average atomic number
A = 72:6). For (E ), we use a recent theoretical calculation of the ux [54], and normalize
it to meet the total antineutrino ux (2:5
1013 s 1 cm 2) in CONUS.
Using eq. (4.1), we compute the event numbers for several examples (m
= 0:1 MeV,
10 MeV, 30 MeV and 100 GeV) and compare them with the SM value in
gure 3. The
signal strength is quanti ed by the ratio N=N0 where N0 is the SM expectation, and N
contains the additional contributions of light scalar bosons. One can see from the gure
that the shape of the spectrum when we include the scalar contribution can be dramatically
di erent, in particular for low values of m .
To study the sensitivity of CONUS on light scalar bosons, we adopt the following
2 =
i
X [(1 + a)Ni
N 0]2
i
2 2
stat;i + sys;i
+
a
2
2
a
;
stat;i = pNi + Nbkg; i; sys;i = f (Ni + Nbkg; i):
(4.2)
(4.3)
The pull parameter a with an uncertainty of a = 2% takes care of the uncertainty in
the normalization originating from various sources such as the variation of nuclear fuel
supply or the uncertainty of the ducial mass and distance. Other systematic uncertainties
that may change the shape of the event spectrum are parameterized by f in eq. (4.3).
Here we assume they are proportional to the event numbers and take f = 1%. We also
introduce a background in our calculation by adding Nbkg; i to the event number in each
bin. The background in CONUS is about 1 count=(day
keV
kg). The threshold of
ionization energy detection in CONUS is 0.3 keV, which if divided by the quenching factor
(
0:25) corresponds to 1.2 keV recoil energy. The reactor neutrino
ux at E
> 8 MeV
has negligible contributions and also large uncertainties, so we set a cut of E at 8 MeV,
( a, f )
(0.5%, 0.1%)
(0.6%, 0.3%)
(1.0%, 0.5%)
(2.0%, 1.0%)
bounds on Y .
Y (m = 1 MeV) Y (m = 5 MeV) Y (m = 10 MeV) Y (m = 30 MeV)
2:0
2:5
2:8
3:3
which corresponds to about 1.75 keV recoil energy according to eq. (2.7). As a result, in
eq. (4.2) we only sum over the bins from 1.2 keV to 1.75 keV.
The result is shown in gure 4 and gure 5 for the LNC and LNV cases respectively.
Although the constraints of CE N S experiments are independent of the LNC/LNV cases,
the other constraints depend on the nature of the interaction, as explained in section 3.
We therefore present the two cases separately.
We also study future improved sensitivities of CONUS by assuming a 100 kg
Germanium detector as the target and an improved threshold down to 0:1 keV. We assume the
corresponding systematic uncertainties are also reduced to a matching level, ( a; f ) =
(0:5%; 0:1%). This will be possible if the reactor neutrino ux is better understood due to
improved theoretical models and measurements. The forecast for CONUS100 with 5 years
of data taking is also shown in gures 4, 5 with blue dashed lines. If the ux uncertainties
can not be reduced to such an optimistic level, then the constraint should be between the
upcoming bound (blue solid curve) and the optimistic bound (blue dashed curve). In
table 1, we assume several di erent ux uncertainties and compute the corresponding bounds
on Y . Since we care about to what extent the CONUS bound could be improved in the
future, we prefer to present in gure 4 and gure 5 the most optimistic bounds together with
the upcoming (realistic) bounds so that other possibilities fall into the gap between them.
As mentioned above, the shape information for light scalar masses is noteworthy in
the spectrum, see eq. (2.8). It can in fact be used to determine the value of the mass. In
gure 6 we show the potential of CONUS100 for determining the mass and coupling of the
particle assuming two characteristic examples. As long as the mass of the scalar is not
much larger than the neutrino energy or the typical momentum exchange, m2
M T
E2,
reconstruction of the mass is possible.
4.2
The COHERENT experiment uses a CsI scintillator to detect neutrinos produced by
and
+ decay at rest. In its recent groundbreaking publication [1] a 6.7 observation of the
SM coherent scattering was announced. There are three types of neutrinos in the neutrino
+
ux,
,
, and e. The rst is produced in the decay
the third are produced in the subsequent decay
+ ! e+ +
+ !
+ +
while the second and
+ e. Because the rst decay
is a two-body decay and the pion is at rest, the produced neutrinos will be monochromatic
with energy:
E 0 =
m2
m2
2m
29:8 MeV;
where m
= 105:66 MeV and m
= 139:57 MeV are the muon and pion masses, respectively.
Remembering that the muon also decays at rest, the neutrino uxes are given by [55]:
(E ) =
(E ) =
e =
0 (E
64E2
0 m3
0
192E2
m3
E 0) ;
3
4
1
2
E
m
E
m
;
;
(4.4)
(4.5)
(4.6)
HJEP05(218)6
Z Ti+ T
Ti
+
Z m =2
0
"
dE
where E should be in the range (0; m =2). The event numbers are computed by
Ni =
tNCs=I
dT
0 (T; Tmax(E 0))
(T; E 0)
d
dT
(E ) +
e (E )
(T; Tmax(E ))
(T; E ) ; (4.7)
d
dT
#
which is similar to eq. (4.1) except that (i) NGe is replaced with NCs=I; (ii) for the
ux,
the delta function in eq. (4.4) has been integrated out. For simplicity, we assume that
Cs and I have approximately the same proton and neutron numbers, (Z; A) = (54; 130).
We also assume that the couplings of neutrinos are avor universal ((y )
= y ), so we
take equal cross section for all neutrino avors. Using eq. (4.7), we also compute the ratio
N=N0 in COHERENT, shown in the right panel of gure 3. Note that there is a kink
around 14 keV in the event spectrum. This is caused by the monochromatic
E = 29:8 MeV, which corresponds to the maximal recoil energy T
2E2=M
beam with
13:7 keV.
Therefore, the monochromatic
beam generates events with T
13:7 keV but does not
contribute to signals with higher recoil energies. Consequently, a kink appears in gure 3
around 14 keV.
In the COHERENT experiment, the recoil energy of the nucleus is converted to
multiple photoelectrons and eventually detected by PMTs. The number of photoelectrons nPE
is approximately proportional to the recoil energy [1]:
nPE
1:17
T
keV
:
(4.8)
For nPE > 20, the signal acceptance fraction is about 70% (cf. gure S9 of [1]). This number
drops down quickly for smaller nPE, and becomes approximately zero for nPE < 5. This
implies that the threshold for T is about 4 keV in COHERENT. Using eqs. (4.7), (4.8) and the
signal acceptance fraction data, we can study the constraint of the COHERENT data (from
gure 3 of [1]) on light scalar bosons. The SM expectation is also provided by [1] which can
be used to compute the total normalization factor. We directly use the relevant
uncertainties provided by [1]. The result is shown in gures 4 and 5 as well. Since reactor neutrinos
provide much larger event numbers, CONUS limits will be better, though of course limited
to the electron-type couplings, whereas COHERENT will also have muon-type neutrinos.
In the future, the COHERENT experiment will further develop the detection of
CE N S with di erent targets,4 including 30 kg liquid argon, 10 kg high purity Ge, and
185 kg NaI crystal. A complete study on the future sensitivities of future COHERENT
including all the di erent targets and di erent detection technology is beyond the scope of
this paper. Considering that the total ducial mass compared to the current value (14.6
kg CsI) will be increased by a factor of
20, plus a prolonged running for few years, at
best the statistics may be increased by a factor of 100, which corresponds to a reduction
of the statistical uncertainties by a factor of 10. It is therefore reasonable to assume that
the uncertainties of future measurement will be reduced by a factor between 1 and 10. To
show the sensitivity of future COHERENT versions on the light scalar coupling, we plot
a black dashed curve in
gure 4 and gure 5 assuming the uncertainties (both
systematical and statistical) are reduced by a factor of 10. For the sake of de niteness, we take
the liberty to denote this potential situation as COHERENT (stat.
100). Again, the
determination of the mass of the scalar particle is possible if its mass lies below the typical
neutrino energy. As seen in
gure 3, the spectral distortion due to the scalar exchange is
less dramatic as for CONUS. This is mostly caused by the larger energy of the neutrinos,
the momentum exchange and nuclear recoil. Figure 6 shows the potential of the assumed
future COHERENT version for determining the mass and coupling of the
particle,
assuming two characteristic examples. Due to the larger energy of COHERENT, and also
because of the smaller statistics, the reconstruction potential is less promising compared
to experiments based on reactor neutrinos.
5
Interpretation of the results
versus the mass of the scalar for lepton number conserving and lepton number violating
interactions, respectively. To draw these lines the coupling of
to neutrinos is taken to
be avor universal. Each limit is however sensitive to a di erent avor structure. Let us
start by discussing the bounds which apply for both lepton number violating and lepton
number conserving interactions. The red-dashed lines show the constraint on Y '
from combining the upper bounds on Cn and y from the n-Pb scattering and meson decay
experiments. As seen from the gures, this bound is relatively weak. The present bound
from COHERENT shown by a solid black line is already well below this combined bound.
The bounds from meson decay are sensitive to P (y )e2 and P (y )2 . Since the beam
p
y Cn
at the COHERENT experiment is composed of
and e
uxes, it will be sensitive
to similar
avor composition. Our forecast for the future bound by the COHERENT
experiment (a factor 10 smaller uncertainties) is shown by dashed black line; the bound
on Y can be improved by a factor of 2. The blue solid and dashed lines are the upper
bounds that CONUS can set with 1 year
4 kg and 5 year 100 kg of data taking, respectively. As seen from these gures, CONUS can improve the bound by one or two
4See: http://webhome.phy.duke.edu/~schol/COHERENT Yue.pdf.
Y
10-4
10-5
10-6
10-3
Y
10-4
10-5
10-6
Meson+n scat.
mϕ (MeV)
interaction, see eq. (2.1). The black, dashed black, blue, and dashed blue curves correspond to
the 95% C.L. constraint of the recent COHERENT data, the sensitivities of future COHERENT,
CONUS 4 kg 1 year, and CONUS 100 kg 5 years (light-blue) respectively. Various other limits
from particle and astroparticle physics are explained in section 3.
and coupling Y in CONUS100 (left panels) and
CO
HERENT (stat.
100) (right panels) assuming the presence of a scalar boson, with the true values
indicated by the green stars.
orders of magnitudes. Since CONUS is a reactor neutrino experiment, it can only probe
Cn1=2(P
jy je2 )1=4. Since the uncertainties of CONUS are mainly limited by statistics, the
bound that it can set on the cross section
/ Y 4, the bound on Y will scale as t 1=8.
scales as t 1=2 with data taking time. Since
The violet curve in
gure 5 up to 2.4 MeV is the combined bound from the n-Pb
scattering on Cn and from double beta decay on (y )ee. CONUS with only one year of
data taking can provide a stronger bound. The dashed green lines in gures 4 and 5 denoted
\SN
di usion" show the limits resulting essentially from a neutrino-nucleon scattering
cross section due to
exchange being equal to that in the SM. As we discussed before, in
the vicinity of this line supernova evolution and emitted neutrino ux will be dramatically
a ected. As seen from the gures, the CONUS experiment with 1 year of data taking can
already probe all this range. The green area between solid and dotted-dashed green lines
in gure 5 is ruled out by supernova cooling and R trapping considerations.
In the LNC case, the orange line in gure 4 denoted by \BBN + n scat." shows the
combined bound from n-Pb scattering and BBN. As seen from the gure, for m
> 3 MeV,
the bound from COHERENT is already stronger. Both for LNC and LNV cases, a
particle with m
2 (1:5 3) MeV and y
the gures this mass range can be probed by CE NS experiments. For pCnjy je above the
dotted-dashed green line in gure 5, the equation of state in supernova inner core will
drasfew
10 5 can signi cantly a ect BBN. As seen from
tically change because of e + n !
+ n scattering. As seen from the gure, a signi cant
part of the parameter space above this line can be probed by CONUS. We can therefore
deduce that coherent scattering results may have dramatic impact on SN and BBN physics.
and COHERENT, assuming true values of (m ; Y ) are (10 MeV; 10 5) or (60 MeV; 5
10 5) for CONUS100, and (10 MeV; 7:5
10 5) or (60 MeV; 10 4) for COHERENT (stat.
100). Here for comparison, we choose the same masses for the two experiments. However,
the couplings in COHERENT (stat.
100) are set to larger values to lead to similar
precision as CONUS100 (cf. gures 4 and 5). Even with larger couplings, COHERENT
still cannot measure (m ; Y ) as good as CONUS100. As shown in gure 6, for true values
m
= 10 MeV and Y = 10 5, the mass and coupling can be determined with better than
10% accuracy by CONUS100. In comparison, COHERENT (stat.
100) loses its capability
to determine m
but it has still reasonable precision in determining Y provided that Y is
large enough (close to its present bound). This is understandable because m
= 10 MeV
is larger than the typical energy-momentum transfer in the CONUS100 (m2 & M T
but is smaller than the energy-momentum transfer in COHERENT (m2 . M T
E2)
E2).
Another reason is the better statistics in CONUS. If the mass is raised to 60 MeV, then
both lose their ability to determine the mass and the coupling separately. They however
maintain their sensitivity to Y =m . In principle, there could be a scenario where the future
COHERENT experiment is able to determine the mass. For example, as we have checked,
if (Y; m ) = (10 4; 10 MeV), then the coupling and the mass can be determined separately
by the future COHERENT experiment. However, such a large coupling has already been
excluded by the current COHERENT data. To
nd a scenario which is not excluded
by the current COHERENT data but still within the sensitivity of future COHERENT,
we can only choose (Y; m ) in the band between the black solid and dashed curves in
gure 4. Since the band is too narrow, we cannot nd such a scenario where the mass can
be determined. Besides, the di erence between s-channel and t-channel signals is also a
reason. The new scalar boson causes a t-channel process. In the s-channel, it is crucial
to have the energy match the new boson mass to get the resonance, which provides useful
information on the mass and the coupling. For example, the masses of the Z boson or Higgs
can be precisely measured from the resonances observed in colliders. The measurement of
masses in the t-channel, however, depends rather on the high statistics and low thresholds,
which is the strength of CONUS.
Discovering a positive signal for the e ects of
by CONUS will have drastic
consequences for the analysis of supernova evolution. If a value of Y below the green solid line
in gure 4 is found, the supernova cooling bounds tell us that interaction cannot involve R
so the interaction should be of lepton number violating form. If
turns out to have a mass
around 2 MeV, it will be more intriguing as it may be discovered at double beta
experiments. If, however, double beta decay searches fail to discover
with expected mass and
HJEP05(218)6
coupling, we may draw a conclusion that (y )ee
(y )e ; (y )e . In any case, in analyzing
BBN, e ects of such light m with sizeable Y has to be taken into account. Comparing gures 4 and 5, we conclude that because of the BBN bounds, discovery of m < 1 MeV will
indicate LNC interaction with light right-handed neutrinos with immediate consequences
for supernova evolution.
If CONUS and/or COHERENT
nds Y
10 5, the chances of nding a signal for
in meson decay experiments as well as in n-Pb scattering experiments increase. If CONUS
nds a signal for Y > 5
10 5, this means that signals for K+ ! e+
and for + ! e+
will be within reach of next generation [56], and new n-Pb measurements would be very
interesting. If CONUS
nds Y < 10 5 and if meson decay experiments nd y
& 10 4,
we would conclude Cn < 10 6, making it di cult to see an e ect on n-Pb scattering
experiments. Similarly Y < 10 5 and Cn
10 4 (close to the present bound) would imply
y < 10 6
.
We conclude this section by mentioning some possibilities on inferring the avor
structure or type of interaction that arise due to the complementarity of the various sources and
limits. If the e couplings are large enough to be within the reach of COHERENT (that
is if Y > 5
10 5), the CONUS experiment will easily determine P
j(y )e j2CN2 , where
runs over all active avors for the LNV case (over all light right-handed species for LNC
case). The information by COHERENT can then determine P
j(y ) j2CN2 . If
COHERENT alone would be able to distinguish the
avor content of the events (e.g. by timing
cuts), information on avor structure of y (i.e. on P
j(y ) j2= P
j(y )e j2) could be
extracted. In the special case that j(y )e j
j(y ) j, it may be possible that COHERENT
will discover new e ect but CONUS will report null results for new physics discovery.
6
Summary and concluding remarks
Coherent elastic neutrino-nucleus scattering can probe both new light as well as heavy
physics. Focussing here on the light case we demonstrated the discovery potential of current
and future coherent scattering experiments on the mass and coupling of scalar particles
interacting with neutrinos and quarks. The shape of the nuclear recoil spectrum is distorted
by the scalar interaction, and allows even to determine the mass of the scalar, if its mass
is around the energy of the scattered neutrinos. Even current limits by COHERENT
are competitive with a combination of bounds from BBN and from various terrestrial
experiments such as meson decay and neutron-scattering experiments. Moreover, these
bounds probe areas in parameter space that can have important consequences for BBN
and supernova evolution, in particular for lepton number violating interactions. Future
versions of the experiment or upcoming reactor experiments such as CONUS will reach not
yet explored areas in parameter space.
Acknowledgments
We thank Giorgio Arcadi, Tommy Ohlsson, Kate Scholberg and Stefan Vogl for many
helpful discussions. YF thanks MPIK Heidelberg where a part of this work was done for their
hospitality. This project has received funding from the European Union's Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant agreement
No 674896 and No 690575. YF is also grateful to ICTP associate o ce for partial nancial
support. WR is supported by the DFG with grant RO 2516/6-1 in the Heisenberg program.
A
Suggestions for underlying electroweak symmetric models
In this section we show how one can build a toy model symmetric under SU(2)
U(1) that can give rise to e ective coupling of form
q qq
q(Cq + i 5Dq)q
as well as the ones shown in eqs. (2.1), (2.2). The coupling of
to nuclei should of course
arise from its coupling to quarks. The latter can originate from the mixing of the singlet
scalar
with an electroweak doublet, . Taking complex couplings of form
YuuR
T CQ + H:c: and YddR yQ + H:c:
and a mixing of
between
and the neutral component of , we nd the coupling of
to the u and d quarks respectively to be given by
u = sin Y
u and
d = sin Yd , or
equivalently
Cq = R[Yq] sin
and
Dq = I[Yq] sin :
Br(H !
case
Notice that taking Yq to be real, the coupling will be parity invariant and therefore Dq = 0.
The most economic solution is to identify
with the SM Higgs. Remember that the
of
couplings of the SM Higgs to quarks of rst generation are O(10 5). Moreover, the mixing
with SM Higgs cannot exceed O(10 2), otherwise the rate of invisible decay mode
) will exceed the experimental limits. Combining these, we conclude that in
is taken to be the SM Higgs, j q j
10 2 mq=hH0i. As we have seen in section 2.3,
the contributions from quarks of di erent generations to the coupling of a nucleus to
will be of the same order and too small to lead to discernable e ects on current CE NS
(A.1)
(A.2)
experiments.
as O(1). Taking
Taking thus to be a new doublet, its coupling to quarks can in principle be as large V ( ; ) = 2
m21 2 + m22j j2 + (A Hy
+ H:c:);
we obtain sin
= AhHi=(m22
m21), m
' m2 and m
'
for ne-tuned cancelation m sin
m
1 TeV, this implies sin
should be smaller than O(m ). For m
5 MeV and
10 5. For u
d
10 5, naturalness (i.e., absence of
ne-tuned cancelation) requires the mass of
to be within the reach of the LHC and its
coupling to u and d quarks to be O(1) which in turn promises a rich phenomenology at the LHC.
In what regards neutrinos,5 considering rst the LNC interaction, a mechanism similar
to the one described above can provide a lepton number conserving interaction of
m2 sin2 . To avoid a need
q
m21
through mixing of
violating coupling
with neutral component of the
doublet. For a lepton number
T C , two scenarios can be realized:
5A model in which a scalar couples both to neutrinos and charged leptons has been considered in [57].
T C coupling can be obtained by the mixing of
with a neutral component of
an electroweak triplet
which couples to the left-handed lepton doublet as LT C
The quantum numbers of
should be the same as the triplet scalar whose tiny
vacuum expectation value is responsible for neutrino mass in the type II seesaw
mechanism. Identifying these two, the avor structure of the
coupling to neutrinos
will be determined by that of neutrino mass: (C ) ; (D )
/ (m ) . Naturalness
(i.e., m
1 MeV
m
TeV without ne tuned cancelation) again implies that
is not much heavier than TeV and its coupling to leptons are of order of 1 which
promises rich phenomenology at the LHC such as production of
++ and its decay
into a same sign pair of charged leptons.
Another scenario that can provide coupling of form
T C is suggested in [58]. The
scenario is very similar to the inverse seesaw mechanism for generating mass for
neutrinos and requires a Dirac fermion singlet
with the following Lagrangian
L = M
+ y RHT CL + y0
TRC
R:
(A.3)
When M
m , we can integrate out
and arrive at C
shown in [58], C even as large as 10 3 can be obtained by this mechanism. Moreover,
if
develops a vacuum expectation value, an inverse seesaw mechanism for neutrino
mass generation will emerge and the avor structure of C and neutrino mass matrix
will be similar. In this scenario, the SM Higgs will have an invisible decay mode
= (yhHi=M )2y0. As
H !
L governed by y2.
The discovery of a scalar eld in coherent scattering experiments will therefore, at least
in the models outlined here, hint towards rich new collider phenomenology.
B
Calculation of the cross section
In this appendix, we give the analytic calculation of N and
N cross sections, assuming a lepton number conserving interaction with . The cross section for the lepton number { 21 {
violating case is identical. Let us rst focus on the N case. The initial and nal momenta
are denoted in the way shown in gure 7. The scattering amplitude of this diagram is
iM
= vs(p1)PR (i
) vs0 (k1) q2
i
m2
ur0 (k2) (i N ) ur(p2);
where
C + D i 5 and
q = p1
k1; q2 =
2M T:
We have inserted a right-handed projector PR = (1 + 5)=2 in eq. (B.1) because the initial
antineutrino produced by the charged current interaction should be right-handed. Because
of PR, if the initial antineutrino is left-handed, the amplitude automatically vanishes. We
can therefore sum over all the spins and apply the trace technology:
1
2
1
1
jiM j2 =
(2M T + m2 )2 tr [ p1PR
k1
PL]
tr [(
k2 + M ) N ( p2 + M ) N ] :
p1PR
k1(C2 + D2)PL 2M CN2 (2M + T ) + DN2 T : (B.4)
For N scattering, one needs to change eq. (B.1) to
iM
= us(p1)PL (i
) us0 (k1) q2
i
m2
ur0 (k2) (i N ) ur(p2):
Consequently, one has to interchange PR $ PL in eq. (B.3). From eq. (B.4) we can see
that PR $ PL does not change the result so the cross sections are equal for
N and
N
scattering. From eq. (B.4) we obtain
d
dT
4 (2M T + m2 )2
CN2 MET2 + CN2 + DN2
T 2
2E2 ;
which is the cross section for both neutrino and antineutrino scattering.
Open Access.
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