HDOV B idge:
Co espondence be ween unc ional accessibili y ηp
and he dynamical scala ield np(x)
A noldo Wal e Fe nández
[email p o ec ed]
No embe 25, 2025
P ep in — Zenodo
Abs ac
The uni ied HDOV o mula ion in oduces a mas e p opaga ion equa ion o physical
modes Ψin eal media, in which a scala quan i y o unc ional accessibili y
ηp
appea s.
This equa ion desc ibes, wi h p edic i e powe , phenomena as di e se as ul a as cohe en
a enua ion in X- ay sca e ing (TRXS) a em osecond scale, he eme gence o unc ional
p ecu so s a he heliopause (Voyage missions), he damping o g a i a ional ingdown
a e compac me ge s, and cosmological a enua ion o he ype usually a ibu ed o
e ec i e da k ene gy (Fe nandez 2025c). On he o he hand, gene alized HDOV p omo es
unc ional accessibili y o a dynamical scala ield
np
(
x
)wi h an explici Lag angian, a
symme y-b eaking po en ial, acuum expec a ion alue
⟨np⟩
, Yukawa- ype couplings o
e mions and non-minimal coupling o cu a u e. Tha heo y u he p edic s a ligh , neu al
scala exci a ion wi h mass
mnp≈
20
.
5
MeV
, es able in in ensi y- on ie expe imen s and
po en ially con ibu ing o (
g−
2)
µ
(Fe nandez 2025b). This wo k es ablishes he o mal b idge
be ween bo h desc ip ions. We show ha : (i)
ηp
is he e ec i e/mac oscopic limi o he same
ield
np
(
x
); (ii) he uni ied HDOV mas e equa ion a ises as he e ec i e p opaga ion equa ion
o a mode Ψimme sed in an
np
backg ound; (iii) he mul iscale obse a ional signa u es
a ibu ed o
ηp
can be in e p e ed as indi ec measu emen s o
np
; and (i ) di ec sea ches
o he ligh scala pa icle a ound
∼
20
.
5
MeV
p o ide an independen and complemen a y
alsi ica ion channel. Complemen a ily, we inco po a e he holog aphic acuum HDOV
module, in which he same unc ional accessibili y egula es he e ec i e acuum ene gy and
cosmic accele a ion wi hou in oducing a undamen al cosmological cons an (Fe nandez
2025a). In his way, uni ied HDOV s ops being a pu ely phenomenological i and becomes
he obse able e ec i e egime o a conc e e scala ield heo y.
1
Con en s
1 In oduc ion and aim 3
2 Field heo y o np(x)3
3 E ec i e educ ion: om np(x) o he HDOV mas e equa ion 4
4 Mul iscale obse a ional mani es a ions 6
4.1 Ul a as TRXS ( em osecond) ............................ 6
4.2 Heliopause (Voyage 1 and 2) ............................. 6
4.3 G a i a ional ingdown ................................ 6
4.4 Cosmology (da k-ene gy-like a enua ion) ...................... 6
4.5 Quan um acuum and holog aphic cosmic accele a ion ............... 6
5 Complemen a y es channels and alsi iabili y 7
5.1 Labo a o y / low-ene gy channel ........................... 7
5.2 As ophysical / cosmological channel ......................... 7
6 Discussion and conclusions 8
No e on he s a us o he HDOV amewo k 9
Technical no e on he no maliza ion o he po en ial in gene alized HDOV 9
2
1 In oduc ion and aim
The HDOV amewo k (Hypo hesis o Vib a ional Wa e Dispe sion) a ose om he obse a ion
ha e y di e en physical sys ems— om ul a as cohe en dynamics in molecules exci ed
by X ays, h ough plasmas a he bounda y o he Sola Sys em, o he ingdown o black
holes and cosmological a enua ion—sha e he same unc ional law o ampli ude a enua ion
and modula ion (Fe nandez 2025c). This law in ol es a scala quan i y
ηp
, called unc ional
accessibili y, which measu es wha ac ion o he physical mode is uly accessible (obse able,
mani es able, p ojec able) in he local en i onmen .
In uni ied HDOV, ηpis used phenomenologically: i is ex ac ed o i ed om obse a ions
and inco po a ed in o a mas e p opaga ion equa ion ha explains he obse ed unc ional
dissipa ion ac oss e y di e en ene gy and ime scales (Fe nandez 2025c). Gene alized HDOV
akes he nex concep ual s ep: i s a es ha unc ional accessibili y is no jus an e ec i e
pa ame e o he medium bu a he he mac oscopic oo p in o a dynamical scala ield
np
(
x
)
ha pe mea es space ime (Fe nandez 2025b). This ield ea u es: (i) a symme y-b eaking
po en ial, (ii) a ini e acuum expec a ion alue
⟨np⟩
, (iii) couplings o e mions ha may
gene a e e ec i e masses, (i ) coupling o cu a u e ha impac s g a i a ional and cosmological
p opaga ion, and ( ) a ligh , neu al scala exci a ion wi h mnp∼20.5 MeV.
The aim o his pape is o make explici ha :
•ηp
in uni ied HDOV is he e ec i e/a e aged e sion o he same
np
(
x
)in gene alized
HDOV.
•
The uni ied HDOV mas e equa ion is de i ed, in he app op ia e egime, om he ac ion
con aining np(x).
•
The mul iscale obse a ions suppo ing uni ied HDOV can be seen as indi ec measu emen s
o np(x).
•
Di ec sea ches o he ligh scala exci a ion p edic ed by
np
(
x
)o e a labo a o y es
channel ha cons ains he same physics om ano he angle.
2 Field heo y o np(x)
Gene alized HDOV pos ula es he exis ence o a eal scala ield
np
(
x
)whose low-ene gy dynamics
may be desc ibed by an e ec i e Lag angian o he o m (Fe nandez 2025b):
L=1
2(∂µnp)(∂µnp)−V(np)−X
i
yinp¯
ψiψi−1
2ξ n2
pR+LSM,massless.(1)
He e:
•V(np)is he scala po en ial,
•yia e Yukawa- ype couplings be ween npand e mions ψi,
• he e m ξn2
pR ep esen s a non-minimal coupling o he Ricci scala cu a u e R,
•LSM,massless
deno es he S anda d Model wi hou explici mass e ms, which in his ame-
wo k may eme ge unc ionally om np.
The po en ial is aken as
V(np) = 1
2m2n2
p+λ
4n4
p, m2<0, λ>0,(2)
3
which implies spon aneous symme y b eaking. The non- i ial minimum is ob ained om
dV
dnp
=m2np+λn3
p= 0 ⇒ ⟨np⟩2=−m2
λ>0,(3)
so ha he ield acqui es a acuum expec a ion alue (VEV)
⟨np⟩= =s−m2
λ.(4)
A ound he minimum, scala exci a ions ha e a mass
m2
np=d2V
dn2
pnp=
=m2+ 3λ 2= 2λ 2=−2m2⇒mnp=√2λ . (5)
Wi h nume ical pa ame e s consis en wi h he phenomenology de eloped in (Fe nandez 2025b),
one inds
m2
np≃420 MeV2, mnp≃20.5 MeV.(6)
A de ailed discussion o he po en ial no maliza ion and o p elimina y e sions is p o ided in
he echnical no e 6.
In summa y, np(x)is no me ely a i pa ame e , bu a physical scala ield wi h:
1. spon aneous symme y b eaking (⟨np⟩ = 0),
2. a ligh , neu al scala exci a ion (∼20.5 MeV),
3. Yukawa couplings yicapable o gene a ing e ec i e e mion masses mi=yi⟨np⟩,
4.
and a g a i a ional coupling
ξn2
pR
able o modi y g a i a ional p opaga ion and cosmological
a enua ion.
Finally, he non-minimal coupling
ξn2
pR
mus obey he s ong cons ain s on scala – enso
heo ies and pos -New onian de ia ions om Gene al Rela i i y de i ed om high-p ecision
es s in he Sola Sys em, bina y pulsa s and g a i a ional wa es (Will 2014). In his wo k we
explici ly assume a weak-coupling egime
ξ 2
M2
Pl ≪1,(7)
whe e
MPl
is he educed Planck mass, so ha scala – enso co ec ions emain below cu en
obse a ional bounds. A sys ema ic explo a ion o he pa ame e space compa ible wi h all hese
cons ain s is le o u u e wo k.
3
E ec i e educ ion: om
np
(
x
) o he HDOV mas e equa ion
In uni ied HDOV, he p opaga ion o a gene ic physical mode Ψ( o example, a cohe en elec o-
magne ic wa e in TRXS, a heliosphe ic plasma pe u ba ion, o he quasi-no mal g a i a ional
mode a e a compac me ge ) is go e ned by a mas e equa ion o he o m (Fe nandez 2025c)
∇µ1+2gcχ(I)ηp∇µΨ+m2
ΨΨ=0.(8)
He e:
•ηpis he local unc ional accessibili y,
4
•χ
(
I
)is a ga e depending on he en i onmen /in a ian s
I
(cu a u e, plasma, magne ic
ield, e c.),
•gccon ols he s eng h o ha e ec i e coupling.
In he WKB egime (quasi-monoch oma ic wa e Ψ =
AeiΘ
), Eq.
(8)
ansla es in o a anspo
law o he ampli ude Ao he o m
dA
ds=−ge ηp(s)A(s), A(s)=A0exp −ge Zs
ηp(s′) ds′,(9)
whe e
s
is he ajec o y pa ame e and
ge
=
gcχ
(
I
). This unc ional a enua ion law
∼
exp[−Rηp]is he one empi ically con i med ac oss e y di e en domains (Fe nandez 2025c).
E ec i e de i a ion om np
The b idge om gene alized HDOV o uni ied HDOV can be ske ched as ollows:
1.
S a ing om he Lag angian
(1)
, which con ains
np
(
x
)and a gene ic mode Ψ( he wa e
p opaga ing in he expe imen ). Fo equencies o ene gies lowe han he as - luc ua ion
scale o np, we decompose
np(x)=⟨np⟩+δn(slow)
p(x)+δn( as )
p(x),(10)
and in oduce a scale sepa a ion
Lcell ≪Lsys
ha allows one o de ine a cell a e age
⟨···⟩cell
o e he as luc ua ions δn( as )
p.
2.
Func ionally in eg a ing ou he as componen s
δn( as )
p
(e.g. ia a cumulan expansion
unca ed a second o de ) yields an e ec i e ac ion o Ψo he o m
Se [Ψ] ≃Zd4x√−gZe (x)∇µΨ∇µΨ−m2
ΨΨ2+···,(11)
whe e he e ec i e kine ic ac o can be w i en as
Ze (x)≡1+2ge ηp(x),(12)
wi h
ge
an e ec i e cons an inco po a ing he en i onmen al ga e
χ
(
I
)and unc ionals o
he co ela ions o
δn(slow)
p
. To leading o de ,
ηp
(
x
)can be in e p e ed as a local unc ional
o
δn(slow)
p
(
x
)and he en i onmen encoded in
I
. In all physical egimes conside ed we
equi e, mo eo e , ha he e ec i e kine ic ac o emain posi i e,
1+2ge ηp(x)>0,(13)
in o de o a oid ghos -like modes and o ensu e ha he ene gy associa ed wi h Ψis well
de ined.
3.
Va ying he e ec i e ac ion
(11)
wi h espec o Ψyields an equa ion o mo ion o he o m
(8), wi h he iden i ica ion gcχ(I)ηp↔ge ηp(x).
The cen al iden i ica ion o he b idge is he e o e
ηp(x)≡Fnp(x)
, I,(14)
whe e
F
is a smoo h, dimensionless unc ional ha encodes he e ec i e accessibili y o he mode
as a unc ion o he a io
np
(
x
)
/
and he en i onmen al in a ian s
I
(including he ga e
χ
(
I
)).
By cons uc ion,
ηp
(
x
) ep esen s a unc ional accessibili y ac ion, dimensionless and physically
bounded in he ange 0≤ηp≤1in he ele an egimes.
Func ional accessibili y
ηp
in uni ied HDOV is he e o e he e ec i e mani es a ion o
np
(
x
)
in gene alized HDOV. They a e no “ wo di e en models” bu wo desc ip ion scales o he
same physical deg ee o eedom.
5
4 Mul iscale obse a ional mani es a ions
Adop ing he iden i ica ion
(14)
, measu emen s o
ηp
in uni ied HDOV can be ein e p e ed as
indi ec measu emen s o np(x)in di e en physical domains.
4.1 Ul a as TRXS ( em osecond)
In ul a as sca e ing o cohe en X ays (TRXS) on exci onized molecules (e.g. ND
3
), he
cohe en ampli ude decays a e he ini ial exci a ion. Fi s wi h he HDOV anspo law
(9)
ou pe o m pu ely ab ini io s anda d i s when
Q
- esolu ion is ai h ully accoun ed o , wi h
Akaike in o ma ion c i e ion di e ences ∆
AIC ≳
10 in a ou o HDOV (Fe nandez 2025d). In
he language o he b idge, TRXS is p obing he ime a ia ion o
np
(
x
)in a em osecond egime.
4.2 Heliopause (Voyage 1 and 2)
An obse able like
κlocal
(
)ac s as a ace o unc ional accessibili y in he heliosphe ic plasma.
The me ic
κlocal
d ops sha ply be o e he o mal c ossing o he heliopause and ac s as a cohe en
p ecu so in bo h Voyage 1 and Voyage 2 (Fe nandez 2025e). In he b idge language, his means
ha
np
(
x
)changes egime in he heliopause/in e s ella plasma en i onmen be o e classical
diagnos ics ma k he c ossing.
4.3 G a i a ional ingdown
A e he me ge o compac objec s, he emnan black hole ib a es in quasi-no mal modes.
Uni ied HDOV models addi ional damping and/o phase shi s as a unc ional accessibili y
ηp
mod-
ula ed by an en i onmen al ga e
χ
(
I
)depending on s ong cu a u e and elec omagne ic/plasma
en i onmen (Fe nandez 2025c). In e ms o he b idge, his co esponds o he
ξn2
pR
e m in
he Lag angian (1), which makes np(x)a ec he e ec i e g a i a ional p opaga ion.
4.4 Cosmology (da k-ene gy-like a enua ion)
Applying he anspo law
(9)
o ligh p opaga ion on cosmological scales, uni ied HDOV shows
ha pa o he appa en cosmic accele a ion may be unde s ood as accumula ed unc ional
opaci y, wi hou in oking an a bi a y da k luid (Fe nandez 2025c). Wi hin he b idge, his is
ead as a cosmic backg ound o
np
(
x
) ha induces in eg a ed a enua ion along he line o sigh
(Type Ia supe no ae, BAO).
4.5 Quan um acuum and holog aphic cosmic accele a ion
Re e ence (Fe nandez 2025a) shows ha he same HDOV mas e equa ion go e ning he p opa-
ga ion o modes Ψin eal media can be applied o he spec um o acuum modes. Func ional
accessibili y ηpnow en e s as a smoo h spec al weigh Wη(k)on ze o-poin modes,
ρe
ac =ℏc
2π2Z∞
0
k3Wη(k) dk, Wη(k) = exp −k
k0α,(15)
whe e
k0
and
α
=
O
(1) a e ixed by a holog aphic condi ion consis en wi h Bekens ein–Hawking-
ype bounds.
In p ac ice,
Wη
(
k
)is in oduced he e phenomenologically as a smoo h cu o unc ion compa -
ible wi h he no ion o unc ional accessibili y: i assigns lowe weigh o high- equency modes
ha would o he wise sa u a e holog aphic bounds. In his wo k,
Wη
(
k
)is no ye de i ed om
he mic oscopic dynamics o
np
(
x
); a he , he ques ion is explo ed o whe he he e exis s a
egime in which he scales induced by unc ional accessibili y may app oxima e he obse ed
acuum ene gy densi y wi hou eso ing o a undamen al cosmological cons an . A mo e
6
igo ous de i a ion connec ing co ela o s o
np
(
x
) o
Wη
(
k
)is le as a speci ic open p oblem o
u u e wo k.
This cons uc ion p o ides an ope a ional implemen a ion o he idea ha a ini e obse e
can only access a ac ion o he acuum deg ees o eedom be o e iola ing holog aphic limi s.
The esul ing e ec i e acuum ene gy densi y
ρe
ac
is o he o de o he densi y associa ed wi h
he obse ed cosmological cons an , so cosmic accele a ion is in e p e ed as a mani es a ion o
unc ional inaccessibili y ins ead o an ad hoc da k luid. Wi hin he b idge p oposed he e, his
cosmological module is seen as he la ge-scale limi o he same ield
np
(
x
)whose ligh scala
exci a ion is sea ched o in in ensi y- on ie expe imen s.
These i e mani es a ions— em osecond TRXS dynamics, Sola Sys em bounda y, s ong-
g a i y compac me ge s, cosmological unc ional opaci y and holog aphic acuum egula ion—a e
di e en mani es a ions o a single physical en i y np(x).
5 Complemen a y es channels and alsi iabili y
The uni ied pic u e gene a es wo expe imen al/compu a ional es channels ha ein o ce each
o he bu a e logically independen .
5.1 Labo a o y / low-ene gy channel
The ield
np
(
x
)has a neu al scala exci a ion wi h mass
mnp≃
20
.
5
MeV
(Fe nandez 2025b).
This pa icle may:
•
be p oduced in adia i e p ocesses
e+e−→γ
+
np
a high-luminosi y collide s (e.g. Belle II
(Kou, U quijo, e al. 2019)),
•be p oduced in elec on beams on ixed a ge s (NA64 (Bane jee e al. 2019)),
•decay in o e+e−and γγ, wi h no µ+µ−channel because mnp≪2mµ.
Non-de ec ion a su icien sensi i i y does no au oma ically ule ou uni ied HDOV, bu does
cons ain (and may e en ually alsi y) his speci ic ealiza ion o gene alized HDOV: he conc e e
alues o m2,λ,yiand ξ ha ix mnp≈20.5 MeV and i s isible couplings.
Quali a i ely, in ensi y- on ie expe imen s ha e al eady excluded signi ican egions o he
pa ame e space (
mnp, ye, yµ, . . .
) o ligh scala s in he MeV–GeV ange. The HDOV amewo k
mus i wi hin he su i ing egions, and u u e imp o emen s in sensi i i y may e u e speci ic
pa ame e choices.
5.2 As ophysical / cosmological channel
Independen ly o di ec labo a o y de ec ion, he same amewo k p edic s ha
np
(
x
)( ia
ηp
)
should:
•
keep lea ing i s signa u e o unc ional dissipa ion in nex -gene a ion ul a as TRXS wi h
highe Q esolu ion,
•keep ac ing as a p ecu so in ex eme plasma ansi ions analogous o he heliopause,
•
induce speci ic ex a damping and phase shi s in g a i a ional ingdown (po en ially
measu able by LIGO/Vi go/KAGRA),
•
sus ain a cosmological unc ional opaci y ha can s a is ically compe e wi h s anda d
da k-ene gy models.
7
I hese as ophysical and cosmological signa u es we e o disappea sys ema ically as analyses
become mo e p ecise, uni ied HDOV i sel would be s ongly challenged.
In bo h channels, he non-minimal coupling
ξn2
pR
dese es special a en ion: i e ec i ely
modi ies he Planck mass and, i oo la ge, may con lic wi h p ecision es s o g a i y in he Sola
Sys em and cosmological obse a ions (Will 2014). In his pape we implici ly assume a weak-
coupling egime in which
ξ 2/M2
Pl ≪
1, so ha modi ied-g a i y e ec s emain below cu en
bounds. A quan i a i e analysis o he cons ain s on
ξ
om as ophysical and cosmological da a
is le o u u e wo k.
In summa y, he key alsi iabili y message can be s a ed as ollows:
•
he labo a o y (low-ene gy) channel es s he mic ophysics o
np
(
x
)and i s ligh scala
pa icle;
•
he as ophysical/cosmological channel es s he mac oscopic mani es a ion o
np
(
x
)as
unc ional accessibili y ηpin eal sys ems.
Bo h channels cons ain he same p oposed deg ee o eedom, bu nei he depends en i ely on
he o he o i s exis ence.
6 Discussion and conclusions
We ha e shown ha :
•
The uni ied HDOV mas e equa ion desc ibes he p opaga ion o physical modes Ψin he
p esence o unc ional accessibili y
ηp
, and ep oduces beha iou s obse ed in ul a as
TRXS, a he heliopause, in g a i a ional ingdown and in cosmology (Fe nandez 2025c;
Fe nandez 2025d; Fe nandez 2025e; Fe nandez 2025a).
•
This unc ional accessibili y
ηp
can be iden i ied wi h he e ec i e/ mac oscopic alue o he
dynamical scala ield
np
(
x
)in oduced in gene alized HDOV. The e ec i e educ ion o he
np
(
x
)Lag angian au oma ically gene a es he HDOV mas e equa ion in he app op ia e
egime, so uni ied HDOV is no longe a “s and-alone empi ical law” bu he obse able
mani es a ion o a speci ic ield heo y.
•
The ield
np
(
x
)has a ligh , neu al scala exci a ion wi h
mnp∼
20
.
5
MeV
and Yukawa
couplings
yi
ha may gene a e e ec i e e mion masses and con ibu e o (
g−
2)
µ
(Fe nandez
2025b). This exci a ion can be sea ched o in in ensi y- on ie expe imen s (Belle II,
NA64).
•
On he o he hand, mul iscale unc ional accessibili y signa u es— em osecond TRXS,
Voyage heliopause, damping in g a i a ional ingdown, da k-ene gy-like cosmological a en-
ua ion and holog aphic egula ion o acuum ene gy—cons i u e independen as ophysical
and cosmological e idence ha he e exis s a deg ee o eedom modula ing he physical
mani es a ion o modes Ψin di e en en i onmen s.
Taken oge he , hese esul s es ablish ha :
1.
Uni ied HDOV and gene alized HDOV a e no wo compe ing heo ies bu wo (mac oscopic
and mic oscopic) desc ip ions o he same physical deg ee o eedom.
2.
The e a e wo complemen a y e i ica ion/ alsi ica ion ou es: (i) he di ec sea ch o
he ligh scala exci a ion a
∼
20
.
5
MeV
in in ensi y- on ie expe imen s, and (ii) he
pe sis ence (o absence) o unc ional accessibili y signa u es in mul iscale eal da a.
In his sense, unc ional accessibili y ceases o be jus a use ul phenomenological ool and
becomes an ope a ional window on o new scala physics in he
O
(10
MeV
) ange, accessible bo h
in labo a o ies and in as ophysical and cosmological obse a ions.
8
Acknowledgmen s
The au ho hanks colleagues and semina pa icipan s o discussions and commen s on p elimi-
na y e sions o his wo k.
No e on he s a us o he HDOV amewo k
The HDOV amewo k p esen ed in his wo k should be unde s ood as an explo a o y p oposal
in de elopmen . The quan i a i e esul s summa ized he e a e based on a se ies o p ep in s
deposi ed in open-access eposi o ies ( o example, zenodo.o g), which a he ime o his e sion
ha e no ye unde gone pee e iew in specialized jou nals.
In o de o a ou anspa ency and independen assessmen , each o hese wo ks is published
oge he wi h he da ase s, sc ip s and igu es needed o ep oduce he nume ical analyses. This
allows hi d pa ies o:
• e i y s ep by s ep he i ing p ocedu es and e idence me ics;
•c i icize he me hodology and p opose a ian s;
•
e u e o imp o e he conclusions p esen ed in he di e en domains (TRXS, heliopause,
geodynamics, cosmology and g a i a ional wa es).
The pu pose o his b idge a icle is, he e o e, o o e a cohe en global iew o he HDOV
amewo k and i s cu en unc ional alida ions, while making explici ha i s s a us is ha o an
open esea ch p og am, subjec o empi ical con on a ion and c i ical e iew by he communi y.
Technical no e on he no maliza ion o he po en ial in gene alized
HDOV
In p elimina y e sions o gene alized HDOV an inco ec nume ical no maliza ion o he scala
po en ial was used, incompa ible wi h he phenomenological alues adop ed in his wo k. To ix
he no a ion used he e, ecall ha a sel -in e ac ion po en ial can be w i en as
V(φ)=−1
2µ2φ2+λ
4φ4, µ2>0, m2≡ −µ2<0,(16)
whe e
φ
is a gene ic eal scala which, in he HDOV con ex , can be iden i ied wi h he adial
mode o he ield np(x). F om (16) one ob ains
V′(φ)=−µ2φ+λφ3= 0 ⇒ 2=µ2
λ,(17)
and he mass o he scala exci a ion a ound he minimum is
m2
φ=V′′( ) = −µ2+ 3λ 2= 2µ2= 2λ 2.(18)
Wi h nume ical alues consis en wi h he phenomenology de eloped in (Fe nandez 2025b),
one has
≈32.4 MeV, λ ≈0.20 ⇒mφ≈√2λ ≈20.5 MeV,(19)
and
m2=−210 MeV2(no −105 MeV2), m2
φ= 2λ 2.(20)
Yukawa couplings can be pa ame ized compac ly as
y =κ
m
,0< κ ≤1,(21)
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