Zi e bewegung Re isi ed: Insigh s in o Space ime
Damian Piko , Paweł Ku zawski
Abs ac
S anda d physics ea s Zi e bewegung - embling mo ion p edic ed by he
Di ac heo y - as a ma hema ical a i ac . This is because i sugges s an in e nal
s uc u e o he elec on, whe eas expe imen s consis en ly show he elec on o
be a poin pa icle. In his pape , we p opose a e ision o his iew. We pos ula e
ha Zi e bewegung is a eal phenomenon, no as a mo ion o he pa icle’s cha ge,
bu as a pe sis en acuum esonance; an imp in o i s c ea ion. We a gue his
dynamic cons i u es a s able, in e nal phase geome y associa ed wi h he pa icle.
The cen al poin o he model is he s ic sepa a ion o he poin -like pa icle
om his associa ed geome y. We o malize his by in oducing wo dis inc o m
ac o s. The cha ge o m ac o (
FEM
) desc ibes he pa icle i sel and emains
ope a ionally poin -like (
FEM ≡
1), ensu ing ull consis ency wi h sca e ing da a.
Simul aneously, we de ine a new, in e nal obse able – he phase o m ac o (
Fϕ
) –
which desc ibes he cohe ence o his associa ed phase geome y a he Comp on
scale (
¯
λC
). This model is calib a ed agains he anomalous magne ic momen o he
muon (∆
aµ
). This calib a ion hen p edic s he obse ed "silence" o he elec on
anomaly (∆
ae
) and a signi ican , es able anomaly o he au lep on (∆
aτ
). This
app oach esol es he undamen al pa adox, linking he
g−
2anomaly o he Di ac
equa ion’s kinema ic subs uc u e. The model p edic s ha his phase s uc u e
(
Fϕ
), while in isible o sca e ing, is accessible ia cohe ence-sensi i e p o ocols,
such as in e e ome y, whe e i mani es s as a measu able isibili y
V
(
q
)
∝ |Fϕ
(
q2
)
|
linked o CHSH iola ion h esholds.
Key wo ds: Zi e bewegung, Lep on Geome y, Di ac Equa ion, Comp on Scale, Anoma-
lous Magne ic Momen , g-2, SMEFT.
1
1. In oduc ion
The ela i is ic Di ac equa ion is a co ne s one o lep on dynamics, success ully desc ibing
spin and p edic ing an ima e [1-3]. Howe e , i also yields one o physics’ mos endu ing
puzzles: Zi e bewegung (ZBW), an ul a- apid embling o he ee pa icle’s posi ion
ope a o . This mo ion is cha ac e ized by he ZBW equency
ωZ
= 2
mc2/ℏ
(which is
∼
10
20
Hz o he elec on, no
∼
10
15
Hz) and an ampli ude on he o de o he educed
Comp on wa eleng h,
¯
λC
=
ℏ/
(
mc
). This p edic ion is in di ec con lic wi h obse a ion.
High-ene gy sca e ing expe imen s and p ecision QED es s consis en ly cons ain he
lep on’s cha ge adius o be much smalle han i s Comp on wa eleng h,
≪¯
λC
[4, 5].
Because ZBW implies an in e nal s uc u e a he
¯
λC
scale, i is adi ionally dismissed
as an unphysical ma hema ical a i ac [1, 3, 6]. In his wo k, we e ise his iew. We
pos ula e ha he ZBW pa adox a ises om a misin e p e a ion o he phenomenon. Ou
cen al hypo hesis is ha ZBW is a eal, physical p ocess, bu i is no he mo ion o he
pa icle’s cha ge. Ins ead, we posi ha ZBW is a pe sis en , dynamic esonance o he
acuum i sel , imp in ed in he pa icle’s icini y du ing i s c ea ion (e.g., pai p oduc ion).
The pa icle emains poin -like bu is inex icably "d essed" by his associa ed, cohe en
phase-geome y. The key o ou model is a s ic sec o sepa a ion hypo hesis. We
o malize his by de ining wo dis inc o m ac o s o he lep on:
1.
The Cha ge Fo m Fac o (
FEM
(
q2
)): This is he s anda d obse able p obed
in sca e ing expe imen s. In ou model, he cha ge densi y is a poin -like del a
unc ion,
ρEM
(x) =
δ(3)
(x). Consequen ly,
FEM
(
q2
)
≡
1. This "geome ic silence" in
he cha ge sec o ensu es ull consis ency wi h sca e ing da a and p ecision QED
es s [4, 5].
2.
The Phase Fo m Fac o (
Fϕ
(
q2
)): This is a new, in e nal obse able we in oduce.
I does no desc ibe cha ge, bu a he he spa ial cohe ence o he associa ed ZBW
acuum esonance. Unlike
FEM
, his phase- o m ac o possesses a non- i ial
s uc u e, Fϕ(q2)= 1, wi h a cha ac e is ic scale de ined by ¯
λC.
This model esol es he pa adox:
FEM ≡
1explains why sca e ing expe imen s see a
poin pa icle, while
Fϕ
= 1 allows ZBW o be a eal phenomenon wi h i s cha ac e is ic
Comp on scale. Mos impo an ly, his amewo k is no jus a concep ual model; i has
conc e e, alsi iable consequences. We p opose ha his unde lying phase-geome y is he
sou ce o he anomalous magne ic momen (∆
aℓ
) o he lep ons. We use he pe sis en ,
high-p ecision disc epancy in he muon anomalous magne ic momen , ∆
aµ
, as a calib a ion
inpu o ou model. By ancho ing ou geome y o he obse ed ∆
aµ
, he model becomes
p edic i e. I co ec ly "pos -dic s" he obse ed null esul o he elec on anomaly
(∆ae≈0) and p edic s a speci ic, la ge, and po en ially measu able anomaly o he au
lep on, ∆aτ.
This a icle is o ganized as ollows: Sec ion 2 de ails he o mal, co a ian heo y o he
phase-geome ic wo ld-su ace Σ, i s s abili y, i s gauge compliance (WT/ST iden i ies),
i s de i a ion o spin (Noe he cha ges), and i s ex ension o a cosmological luid. Sec ion
3 p esen s he esul s o ou nume ical analysis, showing how he model is calib a ed on
∆
aµ
and i s p edic ions o ∆
ae
and ∆
aτ
ac oss he UV and Planck-scale pa ame e space,
including i s p ojec ion o had onic (CLN/BGL) sec o s. Sec ion 4 syn hesizes hese
indings, discussing he model’s implica ions, i s connec ion o he .m.s. phase adius
(ϕ)
ms
, and i s alsi iabili y ia in e e ome y. This app oach links he IR slope o
Fϕ
o
2
cohe ence isibili y
V
(
q
)and CHSH iola ion bounds, suppo ed by ep esen a i e (
V, S
)
da a poin s om mode n F anson/emi e and b igh SPDC con igu a ions. De ailed
de i a ions and nume ical p o ocols, including he in e e ome ic p ocedu e (Appendix
F.2), a e p o ided in he Appendices.
2. Theo e ical F amewo k
2.1. Assump ions and Sec o Sepa a ion (WT/ST Compliance)
We in oduce a s ic sepa a ion o obse ables in o a cha ge sec o and a phase-geome ic
sec o . In he es ame, he cha ge densi y is poin -like,
ρEM
(x) =
δ(3)
(x), yielding
FEM(q)≡1[7, 8]. This ensu es ope a ional poin -likeness in sca e ing.
This s ic sepa a ion is he key o gauge compliance. By posi ing
FEM ≡
1, he QED e ex
emains unmodi ied, ensu ing he Wa d-Takahashi (WT) iden i y (
qµ
Γ
µ
=
S−1
(
p′
)
−S−1
(
p
))
is sa is ied by cons uc ion [6]. Simila ly, when ex ended o qua ks, he q-q-g e ex
emains s anda d, p ese ing QCD’s Sla no -Taylo (ST) iden i ies.
All new physics om
Fϕ
is implemen ed ia allowed EFT ope a o s, no by modi ying gauge
e ices. The
Fϕ
ield ( ia i s IR slope) maps o a ca alog o he lowes -dimension, gauge-
in a ian ope a o s ha desc ibe his cohe ence, selec ed by
C, P, T
and
SU
(2)
×U
(1)
symme ies. The p ima y ope a o is he dimension-6 elec omagne ic dipole (see Appendix
E). This e ec i e dipole a ises om he Σ- ield’s in e ac ion wi h he lep on, no om
QED loop co ec ions, hus a oiding double-coun ing wi h s anda d eno maliza ion. The
RGE unning o his ope a o ’s coe icien om he ma ching scale Λ o he lep on mass
mℓ
is equi ed and in oduces a calculable (bu schema-dependen ) unce ain y, ensu ing
he en i e amewo k emains WT/ST-complian [11]. The new sec o possesses a adial
cohe ence densi y
ρϕ
(
)wi h a na u al scale
¯
λC
and an independen phase o m ac o
Fϕ(q2).
We de ine he phase o m ac o in he es ame ia he sphe ical ans o m
Fϕ(q) = 4πZ∞
0
d 2ρϕ( )sin(q )
q
and ea i ela i is ically as a unc ion solely o he in a ian
q2
=
qµqµ
, which ensu es
mani es co a iance. The .m.s. phase adius is de e mined by he s anda d slope- adius
ela ion
⟨ 2⟩ϕ
=
−
6 d
Fϕ/
d
q2|q2=0
, wi h
(ϕ)
ms
=
p⟨ 2⟩ϕ
. This o mally maps he o m- ac o
appa a us on o cohe ence obse ables.
2.2. Co a ian Resonance Wo ld-Su ace (Σ)
The cohe ence ca ie is de ined as a smoo h, wo-dimensional esonance wo ld-su ace
Σ
⊂R1,3
wi h an embedding
Xµ
(
σa
), induced me ic
γab
=
ηµν∂aXµ∂bXν
, wo no mals
nµ
I
, a second undamen al o m
KI
ab
, cu a u es
H
and
KG
, and he Laplace-Bel ami
ope a o ∆Σ. All obse ables a e buil exclusi ely om Poinca é scala s.
The co a ian phase densi y in space ime is de ined dis ibu ionally as
ϱϕ(x;u) = NZΣ
d2σ√−γ δ(4)(x−X(σ)).
3
This is p ojec ed on o he hype plane o hogonal o he ou - eloci y uµ ia
ρϕ(x) = Zdτ ϱϕ(xµ=τuµ+xµ
⊥;u),
which gua an ees a ame-independen de ini ion o he adial densi y
ρϕ
(
). A ske ch o
he p oo o co a iance, including he use o p ojec o s and independence om epa ame-
e iza ion in he IR limi , is p o ided in Appendix A.
2.3. Ac ion Func ional on Σ
The minimal, mani es ly co a ian ac ion unc ional combines su ace ension, cu a u e
elas ici y, and a phase ield
ϑ
on Σ, wi h a "so " esonance condi ion a he Comp on
scale
ℓ∼¯
λC
. The ull ac ion
S
[
X, ϑ
]and he de i a ion o he Hel ich/Willmo e- ype
shape equa ion [9, 10] a e de ailed in Appendix B.
The s abili y o his ac ion is ensu ed by a ull non-linea s abili y analysis. As de ailed
in Appendix B, he second a ia ion
δ2S
mus be posi i e de ini e. The esonance e m
µ
(2
ℓH −
1)
2
p o ides a posi i e "mass" e m, s abilizing he solu ion agains no mal
luc ua ions wi hin a well-de ined pa ame e space o (
T, κ, ¯κ, µ, h
). This ac ion also de ines
he model’s s ess-ene gy enso ia he s anda d a ia ional p inciple
Tµν
Σ
=
−2
√−g
δSΣ
δgµν
,
which allows he Σ- ield o be ea ed as a cosmological luid (see Sec. 2.11).
2.4. P ojec ion and De ini ion o ρϕ( )
The p ojec ion om
ϱϕ
(
x
;
u
)on o he es ame using he p ojec o
Pµν
=
ηµν−uµuν
leads
o an iso opic adial densi y
ρϕ
(
)sa is ying he no maliza ion 4
πR∞
0
d
2ρϕ
(
) = 1.
This de ines he momen s
⟨ n⟩ϕ
and he sphe ical ans o m
Fϕ
. The p ocedu e is
epa ame iza ion-in a ian , and UV/IR egula o s on Σensu e a unique p ojec ion and
he exis ence o momen s, as is s anda d in analyses o su aces wi h bending ene gy. This
p o ides a di ec in e ace o in e e ome ic measu emen s ia he Fϕ(q2)slope.
2.5. Geome ic Ansa z: "Gab iel’s Ho n"
Fo an axially-symme ic su ace o e olu ion wi h p o ile
(
x
) =
a/x
, he mean cu a u e
H
asymp o ically ealizes he esonance condi ion 2
ℓH ≃
1and selec s he scale
∼ℓ
. A
small-slope app oxima ion
H≃1
2
(
− ′′
+1
/
)wi h a sel -simila ansa z
(
x
) =
ax−α
yields
an ene gy minimum a
α≃
1, jus i ying he in e se p o ile. A e non-dimensionaliza ion
(
ˆ
=
/ℓ, ˆx
=
x/ℓ
), his ancho s he geome y o
¯
λC
. An analysis o he a ia ional
equa ions (Appendix B) con i ms he exis ence and local uniqueness o his ene gy
minimum o he "Gab iel’s Ho n" p o ile, p o ided UV/IR egula o s [
min, max
]a e
applied. The ull de i a ion o he no maliza ion cons an
C−1
is p o ided in Appendix C.
2.6. Spec al Ansa z: Bessel + Localiza ion
Complemen a ily, we employ a spec al ansa z
ρϕ
(
) =
A J2
1
(
k
)
W
(
;
min, max
)wi h
a
C∞
window unc ion
W
, whe e
k
=
c′/ℓ
se s he Comp on scale. The no maliza ion
cons an is
A−1
= 4
πR
d
3J2
1
(
k
)
W
(
). The
J1
(
z
)
∼z/
2beha io a
→
0ensu es
UV egula i y, while he asymp o ic beha io necessi a es he IR-damping window
W
.
4
The calcula ion o
Fϕ
(
q2
)is ealized as a sphe ical Bessel ans o m
j0
(
q
) =
sin
(
q
)
/
(
q
)
using me hods de ailed in Appendix D.
2.7. Low-q2Expansion and "Shape-Blindness"
The expansion o j0(z) = 1 −z2/6+... yields he low-q2expansion o he o m ac o :
Fϕ(q2)=1−q2
6⟨ 2⟩ϕ+(q2)2
120 ⟨ 4⟩ϕ+O((q2)3).
This de ines he slope- adius ela ion
dFϕ/dq2|0
=
−⟨ 2⟩ϕ/
6. The "shape-blindness"
p ope y implies ha a
O
(
q2
),
Fϕ
depends only on he second momen
⟨ 2⟩ϕ
. Any
wo no malized ansä ze (e.g., Geome ic and Spec al) wi h he same
⟨ 2⟩ϕ
will ha e an
iden ical slope and .m.s. adius, wi h di e ences only appea ing a
O
((
q2
)
2
) ia he
⟨ 4⟩ϕ
momen , IR ancho ing wi h
⟨ 2⟩ϕ
= 1
.
5(Comp on uni s) and he demons a ion o
shape-blindness (ag eemen in he slope
O
(
q2
)be ween ansä ze a e ma ching
⟨ 2⟩ϕ
) a e
illus a ed h ough da a and plo s in Appendix F.2.
2.8. Pa ame e Es ima ion and Ancho ing
The ansa z pa ame e s (e.g.,
a, k
) a e de e mined by a uni ied op imiza ion p oblem. The
p ocedu e minimizes he geome ic ac ion
Sgeo
(see Appendix B) subjec o he cons ain s
o no maliza ion (4
πR
d
2ρϕ
(
)=1) and ancho ing o he Comp on scale
ℓ
=
¯
λC
. A
c i ical consis ency es ("slope-ma ch") ensu es ha bo h ansä ze yield he same
⟨ 2⟩ϕ
and hus he same low-q2physics.
2.9. Ansä ze Compa ison: IR Equi alence and UV Di e gence
As es ablished in Sec. 2.7, he Geome ic and Spec al ansä ze a e equi alen in he low-
q2
(IR) egime, p o ided hei second momen s
⟨ 2⟩ϕ
a e ma ched. Thei di e gence occu s a
high-
q2
(UV), which is dic a ed by he la ge-
ails o hei espec i e densi y p o iles and
he choice o egula iza ion window
W
(
). This ensu es ha while he low-
q2
obse ables
(
(ϕ)
ms
,∆
aℓ
) a e obus , he high-
q2
beha io emains dis inc . A nume ical e i ica ion o
his IR equi alence is p o ided in Appendix C.
2.10. Noe he Cha ges, Spin, and he Di ac Fac o
The in e nal phase ield
ϑ
on Σ(Sec. 2.3) ac s as a U(1) ield. By imposing a spino ial
holonomy (a 4
π
pe iodici y equi ed o a double co e ing), he associa ed Noe he cha ge
o o a ions on Σyields he in insic angula momen um
S
=
ℏ/
2. This amewo k
geome ically de i es he Di ac " ac o o 2" (
S
=
ℏ/
2and
g
=
−
2) om he wo ld-su ace
opology, a he han pos ula ing i . The ZBW equency
ωZ
= 2
mc2/ℏ
is likewise
eco e ed om he esonance condi ion 2
ℓH ≃
1when he scale is se o
ℓ
=
¯
λC
. This
app oach can be gene alized o spin-3/2 ields by modi ying he holonomy and cons ain s
on Σ, while espec ing he Velo-Zwanzige consis ency condi ions [1].
2.11. Cosmological B idge: The Σ-Fluid
The s ess-ene gy enso
Tµν
Σ
de i ed om
SΣ
(Sec. 2.3) allows he model o ac as
a "cosmological b idge". When popula ed in he ea ly uni e se, he ensemble o Σ-
5
esonances beha es as an e ec i e luid
ρΣ
(
a
). Ou s abili y analysis (Appendix B) and
long-wa eleng h limi (Sec. 2.7) show ha he luid is "cold" (
c2
s,Σ≈
0) and has negligible
aniso opic s ess (
σΣ≈
0). This Σ- luid con ibu es o Ω
m
and mus be consis en wi h
cosmological da a (BBN, CMB, BAO).
This amewo k connec s he lab-measu ed IR-slope
⟨ 2⟩ϕ
( ia
Fϕ
and ∆
aℓ
) o he cos-
mological pa ame e s
ρΣ
and
wΣ
. Admissibili y cu es o
ρΣ
(
a
)show consis ency wi h
BBN and CMB cons ain s. We gene a e a map o (
⟨ 2⟩ϕ,
Ω
Σ
)which shows ha lab-
based me ology o
⟨ 2⟩ϕ
can s ongly cons ain he allowed cosmological pa ame e space,
b eaking degene acies wi h o he sec o s (e.g.,
Ne
o ea ly da k ene gy) in global i s o
Planck and DESI da a [5, 11].
2.12. Consis ency wi h QFT P inciples
The model’s consis ency wi h undamen al QFT p inciples is ensu ed by he
FEM ≡
1
sepa a ion.
•
Heisenbe g Unce ain y P inciple (HUP): The canonical commu a o s [
ˆx, ˆp
]
a e unchanged.
Fϕ
only modula es he cohe ence o a s a e, no he undamen al
algeb a, ully p ese ing ∆x∆p≥ℏ/2.
•
La mo /Un uh Radia ion: As he EM e ex is s anda d, he La mo o mula
(
P∝a2
) and he Un uh empe a u e (
TU∝a
) emain unmodi ied. The Σ- ield does
no couple di ec ly o he pho on ield in his way.
•
EPR S a es (CHSH):
Fϕ
ac s as a "phase-damping" channel. I does no i-
ola e locali y bu modula es he cohe ence ( isibili y
V
) o he en angled s a e.
This co ec ly p edic s a educ ion in
S≈
2
√2V
. To dis inguish his om in-
s umen al noise, a ull open-sys em model (e.g., K aus dephasing) is needed o
sepa a e he in insic
Fϕ
e ec om depola iza ion (We ne /single mixing) and
mul ipai /de ec o noise (
η, κ, p2
). The goal is o isola e
Fϕ
in he
q2≤
0
.
02
−
0
.
04
window, as speci ied in Appendix F.2. We no e ha ep esen a i e (
V, S
)da a om
mode n F anson/emi e se ups and
S
s.
V
cu es om b igh SPDC sou ces
ope a ionally jus i y his phase-law app oach [1, 3].
3.
Nume ical Analysis and Phenomenological P ojec-
ions
3.1. UV G id and Planck-Line Scans
We pe o med a nume ical analysis based on he amewo k de ailed in Sec ion 2 and
Appendix D. We scanned he pa ame e space in wo p ima y ways:
1.
UV G id Scan (54 nodes): We pe o med a Ca esian scan o e he egula o
space (Λ
, p
). This scan con i ms a obus sepa a ion o oles. The muon ancho
∆
aµ
= 2
.
5
×
10
−9
is held cons an by cons uc ion, and he elec on anomaly emains
negligible, wi h a mean
⟨
∆
ae⟩ ≈
1
.
084
×
10
−17
(s d. de . 1
.
93
×
10
−19
). In con as ,
he au anomaly exhibi s a wide, s able pla eau, wi h a mean
⟨
∆
aτ⟩ ≈
2
.
463
×
10
−7
(s d. de . 4.40 ×10−9), as shown in sme _u _summa y.cs .
6
2. Planck-Line T ajec o y (Λ(ℓ) = c/ℓ): We p obed he scale-dependence by sam-
pling a ajec o y along he "Planck-Line." This scan (
sme _planck_summa y.cs
)
con i ms ha he ligh -lep on ancho s (∆
ae,
∆
aµ
) a e in a ian . The au channel,
howe e , exhibi s he cha ac e is ic geome y-con olled c osso e , in e pola ing
om he classical pla eau (max. ∆
aτ≈
2
.
867
×
10
−4
) o a deep comp ession window
(min. ∆aτ≈5.842 ×10−9).
These scans nume ically alida e he "IR-ancho ed, UV-sensi i e" na u e o he model.
The hea y-lep on channel (
τ
) ac s as a sho -dis ance p obe, while he ligh -lep on channels
(
e, µ
) a e p o ec ed by he low-
q2
"shape-blindness" o he geome y. The unce ain ies on
hese p edic ions (∆
ae
,∆
aτ
) include he ull expe imen al and heo e ical e o budge :
(1) he expe imen al unce ain y on he ∆
aµ
calib a ion inpu , and (2) he heo e ical
unce ain y on he
dFϕ/dq2|0
slope, es ima ed om i s s abili y ac oss he egula o g id
(see Appendix D) and i s dependence on he IR window choice (Appendix F.2). The
p edic ed "pla eau" alue o he au anomaly,
⟨
∆
aτ⟩ ≈
2
.
463
×
10
−7
, is consis en wi h
cu en expe imen al limi s [14, 15], al hough p ecision is no ye su icien o p obe his
alue.
Table 1: Nume ical s abili y o he IR-slope
⟨ 2⟩ϕ
(in Comp on uni s) agains egula o
and g id a ia ions. Me ics (e.g., AIC/BIC) con i m s abili y.
Scan / Window W( )G id Densi y (q)⟨ 2⟩ϕ(Mean) AIC/BIC ( ela i e)
UV-G id (W1) S anda d 1.5002 Re .
UV-G id (W1) 2x Densi y 1.5001 +0.1
UV-G id (W2) S anda d 1.4998 +0.5
Planck-Line (W1) S anda d 1.5000 Re .
3.2. SMEFT P ojec ion and Consis ency
We p ojec hese dipole p edic ions in o he Wa saw basis o he S anda d Model E ec i e
Field Theo y (SMEFT) [11]. We map he anomalies ∆
aℓ
o he elec omagne ic dipole
ope a o coe icien
Cℓeγ
and i s
SU
(2)
L×U
(1)
Y
decomposi ion (
CℓeB, CℓeW
). The ull
con en ions and RGE p ocedu e a e de ailed in Appendix E.
No e on g- ac o and
aℓ
con en ions: We adhe e o he s anda d PDG
con en ion
aℓ
= (
g−
2)
/
2[5]. The de i a ion o
g
=
−
2in Sec. 2.10 e e s
o he base gy omagne ic a io om he Di ac equa ion’s s uc u e, which is
he s a ing poin be o e anomalies (
g≈
2) a e conside ed. All ∆
aℓ
alues
epo ed a e in he PDG con en ion.
This mapping con i ms he esul s o ou scans a he ope a o le el.
•
UV G id:
Cµeγ
is cons an wi h ze o a iance (mean 9
.
610
×
10
−11
, s d. de .
3
.
88
×
10
−26
).
Ceeγ
is ul a-small (mean 1
.
084
×
10
−17
, s d. de . 1
.
93
×
10
−19
).
Cτeγ
shows a s able pla eau (mean 2.463 ×10−7, s d. de . 4.40 ×10−9).
•
Planck-Line:
Cµeγ
and
Ceeγ
emain ixed, while
Cτeγ
spans o de s o magni ude,
om a minimum o 5.842 ×10−9 o a maximum o 2.867 ×10−4.
C i ically, we e i y he low-
q2
consis ency a e e y node by calcula ing he econs uc ion
esiduals (
εℓ
). As shown in Appendix D (and
sme _planck_summa y.cs
), hese esidu-
7
als a e ze o (e.g.,
da _ esid
mean 1
.
07
×
10
−18
, s d. de . 3
.
70
×
10
−18
) o all lep ons
ac oss all scans, ce i ying ha he low-q2slope- adius ela ion is exac ly p ese ed.
When ex ending his amewo k o he qua k sec o , he same gauge-sa e p inciple applies.
The
Fϕ
e ec is no implemen ed by modi ying he q-q-g e ex (p ese ing Sla no -Taylo
iden i ies), bu ia gauge-in a ian ope a o s in he ele an EFT, such as HQET o SCET
[11]. This maps he
Fϕ
cohe ence pa ame e on o he had onic o m ac o s (e.g., in CLN
o BGL pa ame e iza ions) used o desc ibe decays like
B→D(∗)ℓν
. We pe o med a
s abili y es using he same uni a i y p io o bo h CLN and BGL pa ame e iza ions,
con i ming ha he esul s a e obus agains pa ame e iza ion bias. The esul ing e ec
on he di e en ial decay a e [
d
Γ
/dw
]
EFT/
[
d
Γ
/dw
]
SM
is a smoo h,
O
(10
−3
)co ec ion,
consis en wi h EFT powe coun ing and s anda d PDG analysis me hods [5, 11].
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6
0.99
1
1
1.01
1.01
Recoil pa ame e , w
Ra io [dΓ/dw]EFT/[dΓ/dw]SM
Had onic P ojec ion S abili y (CLN s. BGL)
CLN (Uni a i y P io )
BGL (Uni a i y P io )
10−8
10−7
10−6
10−5
10−4
10−3
∆aτLimi
Excluded ∆aτ[14, 15]
Figu e 1: Placeholde : S abili y o
Fϕ
-induced co ec ions in
B→D(∗)ℓν
decays unde
CLN and BGL pa ame e iza ions, using iden ical uni a i y p io s. The g ay shaded egion
( igh axis) indica es he cu en expe imen al exclusion limi s o ∆aτ.
3.3. Neu ino and Z′Po al Consis ency
We pe o med wo uni es s o ensu e ou amewo k is consis en wi h o he phenomeno-
logical cons ain s (de ails in Appendix E).
1.
Neu ino Uni Tes : We de ined an e ec i e neu ino magne ic momen p oxy,
ξν
, and de e mined he maximum allowed alue consis en wi h expe imen al limi s.
Ac oss he 54-node UV g id (
sme _nu_summa y.cs
), his alue is uni o mly small
and s able:
ξmax
ν
has a mean o 6
.
940
×
10
−11
wi h a s anda d de ia ion o only
6
.
347
×
10
−13
(min 6
.
531
×
10
−11
, max 6
.
959
×
10
−11
). This con i ms ha ou
model induces no ension wi h neu ino cons ain s, demons a ing obus sec o
sepa a ion.
2.
Lep ophilic
Z′
Po al: We con i med ha a modula
Z′
po al can be in oduced
o accoun o he
g−
2anomaly. A scan o e (
mZ′, g′
)shows iable pa ame e
8
space (e.g.,
mZ′≃
0
.
261
GeV
,
g′≃
1
.
78
×
10
−3
) ha ma ches he ∆
aµ
a ge while
passing a p oxy " iden " sc een [12, 13]. This demons a es ha he
Z′
can be
ea ed as an o hogonal e ec o , lea ing he co e ZBW-d i en ∆
ae/
∆
aτ
p edic ions
in ac .
4. Discussion
We ein e p e Zi e bewegung (ZBW) as a phase cohe ence esou ce encoded in a gauge-
sa e o m ac o ,
Fϕ
(
q2
). This o m ac o is no malized as
Fϕ
(0) = 1 and de ined by i s
in a ed (IR) slope
⟨ 2⟩ϕ
. I is undamen ally sepa a ed om cha ge sca e ing, o which
he o m ac o emains FEM ≡1.
This sepa a ion ele a es ZBW om a ma hema ical a i ac o an ope a ional obse able.
I is measu able ia in e e ome ic isibili y,
V
(
q
)
∝ |Fϕ
(
q2
)
|
, and is linked o CHSH
nonlocali y h ough he app oxima ion
S≈
2
√2V
. This holds in he dominan phase-
dephasing egime, wi h a iola ion h eshold o
V>
1
/√2
. This ela ionship has nume ous
di ec p oxies in exis ing expe imen s. F anson/emi e sys ems and b igh SPDC sou ces
bo h demons a e ha
S
deg ades as isibili y
V
dec eases—a p ocess go e ned by
pa ame e s like pu e dephasing (γd/Γ) o mul ipai p obabili y.
Me ology, Noise Models, and IR S abili y
This "lab- o-
Fϕ
" mapping is ealized by a join i
V
(
q2
) =
V0
(1 +
c1q2
) o ex ac
⟨ 2⟩ϕ=−6c1. In p ac ice, he S(q)p edic ion equi es a hie a chical noise model:
S=2√2ηV0|Fϕ(q2)|
1+κp2
He e,
η
(depola iza ion) is es ima ed omog aphically on complemen a y bases, and
p2
(mul ipai p obabili y) is cons ained by b igh ness and second-o de coincidences.
This model ep oduces he obse ed dependencies in F anson/emi e and b igh SPDC
expe imen s, calib a ing
|Fϕ|
as an in insic cohe ence me ic independen o ins umen al
deg ada ion. Fo me ological obus ness, we de ine wo IR windows: a p ima y window
whe e
q2
linea esiduals a e whi e, and a seconda y window o con ol
O
(
q4
) e ms. We
u ilize he e oscedas ic weigh s and an op ional Gaussian P ocess (GP) o phase d i s
(calib a ed by Allan a iance), manda ing he epo o Du bin-Wa son s a is ics o GP
hype pa ame e s. A nominal Fishe /CRLB analysis o a
q2
=
{
0
,
0
.
01
,
0
.
02
}
g id (IR
uni s) a SNR=2 yields
σc1≈
3
×
10
−3
and
σ
(
⟨ 2⟩ϕ
)
≈
0
.
018. This se s he baseline
equi emen s o in eg a ion ime and phase s abili y needed o measu e
⟨ 2⟩ϕ
and he
S(q)cu e.
QFT Rigo : WT/ST Compliance
The model’s QFT igo is ensu ed by cons uc ion. The elec omagne ic (
FEM ≡
1)
and QCD (
q−q−g
) e ices emain unmodi ied. Consequen ly, he **Wa d-Takahashi
iden i y** (
qµ
Γ
µ
=
S−1
(
p′
)
−S−1
(
p
)) and he non-abelian **Sla no -Taylo iden i ies**
a e p ese ed, main aining eno malizabili y and BRST symme y. All
Fϕ
e ec s en e
exclusi ely as pe missible ope a o s wi hin EFT cu en s, hus sepa a ing cha ge sca e ing
om cohe ence obse ables. Key diagnos ic isks a e explici ly de ined:
9
•ξmax
ν(min): 6.531 ×10−11
•ξmax
ν(max): 6.959 ×10−11
•ξmax
ν(mean): 6.940 ×10−11
•ξmax
ν(s d. de .): 6.347 ×10−13
The ex eme s abili y and smallness o his alue con i ms he "neu ino silence" o he
model.
E.3. Z′Po al Scan
The
Z′
scan (Sec 3.3) was pe o med o e a g id spanning
mZ′∈
[0
.
1
,
1
.
0]
GeV
and
g′∈
[10
−4,
10
−2
]. The accep ance mask
ok_all = T ue
equi es passing bo h he ∆
aµ
a ge window (∆
a a ge
µ
= 2
.
5
×
10
−9
,
δ ol
= 0
.
5
×
10
−9
) and a p oxy iden p oduc ion
sc een. Figu e 3 shows he accep ed pa ame e space co ido , which ollows he expec ed
g′∝mZ′scaling.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−4
10−3
10−2
Z’ Mass, mZ′(GeV)
Coupling, g′
Z′Scan Accep ance Region o ∆aµ≈2.5×10−9
Accep ed Region (ok_all = T ue)
Example Poin (0.261, 1.78e-3)
Figu e 3: The phenomenologically accep ed pa ame e space o he
Z′
po al. The blue
band ep esen s he "na ow s ipe" o (
mZ′, g′
) alues ha simul aneously sa is y he ∆
aµ
a ge window and pass iden cons ain s [12, 13]. The ed do highligh s he speci ic
benchma k poin ci ed in he ex .
E.4. Li e a u e and Limi a ions
The ∆
aτ
obse able is bounded by expe imen al limi s [14, 15] and global SMEFT i s [11].
The
Z′
cons ain s om iden p oduc ion a e aken om [12, 13]. The model’s p ima y
limi a ion is he sensi i i y o high-
q2
obse ables (like he ∆
aτ
pla eau) o he choice o
UV egula o (W( )o Λ, p), hough his does no a ec he low-q2 (ϕ)
ms p edic ion.
16
F. Expe imen al P o ocol o (ϕ)
ms Measu emen
The key p edic ion,
(ϕ)
ms ≈p3/2¯
λC
, is no accessible ia sca e ing bu can be measu ed
using cohe ence-sensi i e p o ocols.
F.1. In e e ome ic P o ocol
A p ac ical measu emen can be achie ed ia elec on in e e ome y (e.g., in a Ramsey-
Bo dé o Mach-Zehnde se up) by measu ing he cohe ence ac o ( inge isibili y)
V
as
a unc ion o momen um ans e
q
. The isibili y is di ec ly p opo ional o he modulus
o he phase o m ac o , V(q)∝ |Fϕ(q2)|.
P ocedu e:
1.
P epa e a cohe en elec on beam (e.g., om a cold-a om sou ce o a high-b igh ness
ield-emission ip).
2. Use a beam spli e (e.g., a nano ab ica ed g a ing) o sepa a e he wa e packe .
3. Impa a di e en ial momen um kick q o one a m o he in e e ome e .
4. Recombine he beams and measu e he inge con as V(q)a he de ec o .
5. Plo V(q) s. q2in he low-q2 egime (q≪1/¯
λC).
6. Fi he da a o he low-q2expansion: V(q2)≈ V01−q2
6⟨ 2⟩ϕ.
7. The linea slope c1=−V0⟨ 2⟩ϕ/6di ec ly yields ⟨ 2⟩ϕ, and hus (ϕ)
ms.
Sys ema ics: The p ima y unce ain y budge will be domina ed by he ini ial beam
cohe ence V0, he p ecision o he momen um kick q, and decohe ence om s ay ields.
F.2. Al e na i e P o ocols
Al e na i ely, he ZBW phase geome y can be p obed by:
•
Spin-Echo P o ocols: Using Ramsey- ype pulse sequences (spin-echo o echo-
Ramsey in e e ome y) o measu e he dephasing caused by he ZBW dynamics.
•
In e e ome ic Phase Es ima ion: The ZBW equency
ωZ
= 2
mc2/ℏ
is
∼
10
20
Hz, a beyond elecommunica ion o THz anges. This elimina es "bea -no e"
spec oscopy as a iable di ec measu emen . The only iable app oach, de ailed in
Appendix F.2, elies on in e e ome y in he
q≪
1
/¯
λC
egime (e.g.,
q2≤
0
.
02
−
0
.
04
in IR uni s) o es ima e small,
q2
-dependen phase shi s, pa ame e ized by
V
(
q2
)
≃
V0(1+c1q2), a he han demodula ing ωZ.
17
Appendix G: In e e ome ic IR es o he phase o m
ac o
G.1. Goal and se up
This appendix de ails a di ec , low-
q
in e e ome ic es o he phase o m ac o
Fϕ
ha
is independen o cha ge sca e ing. In he IR, he sphe ical ans o m implies
Fϕ(q2)=1−q2
6⟨ 2⟩ϕ+O(q4),
so he slope a
q2→
0 eco e s
⟨ 2⟩ϕ
and
ms
(
ϕ
) =
p⟨ 2⟩ϕ
. We measu e he isibili y
V
(
q
)
∝ |Fϕ
(
q2
)
|
in a small window o
q
, and i
V
(
q2
)
≈V0
(1+
c1q2
), yielding
⟨ 2⟩ϕ
=
−
6
c1
.
This app oach is necessa y as he ZBW equency (
∼
10
20
Hz) is inaccessible o di ec
"bea -no e" demodula ion.
G.2. IR ancho ing and shape-blindness
We en o ce he IR ancho
⟨ 2⟩ϕ
= 1
.
5(Comp on uni s), i.e.,
ms
(
ϕ
) =
p3/2¯
λC
. Two
independen ansä ze (GH and spec al) a e ma ched o his a ge . A e he ma ch:
•
The IR slopes coincide o nume ical p ecision, ∆
c1≃
3
×
10
−5
a
q2
= 0
.
05,
demons a ing IR shape-blindness (ag eemen in o de q2).
•
The no maliza ion is co ec ,
Fϕ
(0)
≈
1, and IR esiduals
2
slope − 2
momen
a e
nume ically small ac oss UV-g id and along he Planck-line a e IR ema ch,
con i ming s abili y o he IR me ic.
G.3. Da a p oduc s and plo s
We p o ide Planck-line and UV-g id low-
q
isibili y se ies
V
(
q
)a e
⟨ 2⟩ϕ
= 1
.
5, sui able
o linea i s:
•Planck-line se ies: zmg_cl_i 150_planck_V_se ies.cs (columns: idx, α,q,V).
•UV-g id se ies: zmg_cl_i 150_u 54_V_se ies.cs (columns: node, q,V).
•
Summa y able:
zmg_cl_i 150_summa y.cs
o
GHscaled
and
SPma ched
showing
2
momen ≈1.5, 2
slope ≈1.495,F(0) −1≈0, and ∆c1in he IR window.
Illus a i e plo s (p o ided in he eposi o y) include:
•V(q) s q2wi h linea i V0(1+c1q2).
•
"
F
(
q
) s
q2
" in he IR, using
V≈ |Fϕ|
, wi h he e e ence cu e 1
−q2
6⟨ 2⟩ϕ
a
⟨ 2⟩ϕ= 1.5.
G.4. Robus ness ac oss egula o s
To demons a e egula o -independen IR:
•
UV-g id (54 nodes): a e IR ema ch, esiduals
2
slope −
1
.
5and
F
(0)
−
1 emain
nume ically small o all nodes, con i ming IR in a iance o he obse able slope.
18
•
Planck-line: scanning
α
wi h ema ched scale p ese es he IR slope and no maliza-
ion along he ajec o y, as quan i ied by he same esidual diagnos ics.
G.5. P ac ical SNR and e o budge
A pilo SNR scan on ideal (noise- ee) syn he ic se ies e u ns ex emely small pe -poin
σ
(SNR
∼
10
9
), an a i ac o dense, clean da a. Fo ealis ic planning, we ecommend
he “SNR 2.0” budge :
•
Decompose
σ
in o s a is ically and sys ema ically mo i a ed componen s:
σs a ∼
1/N,σsys (phase/V0s abili y a 10−3–10−2), and σcal (q-axis calib a ion).
•De ine he a ge IR window: q2≤0.02 −0.04 (in IR uni s).
•
Implemen phase d i con ol (e.g., ia Gaussian P ocess (GP) p io s o Allan
a iance diagnos ics, epo ing GP hype pa ame e s o Du bin-Wa son s a is ics).
•
Fi
V0
join ly wi h
c1
. The a ge p ecision, based on nominal CRLB calcula ions,
is σ(c1)∼10−3, implying a a ge o σ(⟨ 2⟩ϕ)∼10−2 o con i m he slope.
•
C oss- alida e i s ac oss mul iple IR windows (e.g.,
q2≤
0
.
02
,
0
.
03
,
0
.
04) and quo e
he wo s -case as he equi emen .
Fo CHSH es s (Sec. 2.12), his
S≈
2
√2V
app oxima ion mus be e ined using an
open quan um sys em app oach (e.g., K aus dephasing) o dis inguish he
Fϕ
signal om
depola iza ion and mul ipai noise (
η, κ, p2
). As an o de -o -magni ude (OOM) example
o CHSH: wi h
q2
= 0
.
02 (IR uni s),
V0
= 0
.
95,
η
= 0
.
9,
p2
= 10
−3
, and
κ≈
1, we ind
S≈2√2·η·V0·|Fϕ(0.02)|/(1+κp2)≈2.2, which is clea ly abo e he classical bound.
G.6. Summa y
The IR in e e ome ic p o ocol di ec ly measu es
⟨ 2⟩ϕ
om he isibili y slope, eco e s
ms
(
ϕ
) =
p3/2¯
λC
, and exhibi s IR shape-blindness a e
⟨ 2⟩ϕ
ma ching. S abili y ac oss
egula o amilies (UV-g id, Planck-line) and a p ac ical SNR oadmap oge he p o ide
a clea , alsi iable pa h o expe imen al alida ion o he phase-geome y sec o wi hou
elying on cha ge sca e ing.
Funding
This esea ch ecei ed no ex e nal unding.
Con lic s o In e es
The au ho s decla e no con lic o in e es .
Da a and Code A ailabili y
All analysis code and nume ical a i ac s suppo ing his s udy a e publicly a ailable. The
analysis sc ip (n10.py, n5.py) and he esul da ase s:
19
•sme _u _wilsons_ es .cs
•sme _planck_wilsons_ es .cs
•sme _u _summa y.cs
•sme _planck_summa y.cs
•sme _nu_summa y.cs
•zmg_cl_i 150_planck_i .cs
•zmg_cl_i 150_planck_V_se ies.cs
•zmg_cl_i 150_u 54_i
•zmg_cl_i 150_summa y.cs
These will be deposi ed as ancilla y iles wi h he submission and mi o ed in he public
eposi o y (see Appendix D).
Re e ences
[1]
A. O. Ba u and A. J. B acken, "Zi e bewegung and he in e nal geome y o he
elec on," Phys. Re . D 23, 2454 (1981).
[2]
P. K eko a, Q. Su, and R. G obe, "Di ac-equa ion-based dynamics o elec on’s
Zi e bewegung," Phys. Re . A 63, 032107 (2001).
[3]
D. Hes enes, "The zi e bewegung in e p e a ion o quan um mechanics," Found.
Phys. 20, 1213 (1990).
[4]
S. J. B odsky e al., "P ecision es s o quan um ch omodynamics and he s anda d
model," Phys. Re . D 69, 076001 (2004).
[5]
Pa icle Da a G oup (PDG), "Re iew o Pa icle Physics," P og. Theo . Exp. Phys.
2022, 083C01 (2022).
[6]
J. Schwinge , "Quan um elec odynamics. III. The elec omagne ic p ope ies o he
elec on," Phys. Re . 76, 790 (1949).
[7] G. B. Wes , "The s uc u e o he p o on," Phys. Rep. 18, 263 (1975).
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