Geome ic Phase D essing o he S anda d Model Lep on P opaga o :
SMEFT Ma ching, Uni a y Non-Local Regula o , and g−2Anomalies
Damian Piko and Paweł Ku zawski
(Da ed: No embe 27, 2025)
The pe sis en disc epancy in he muon anomalous magne ic momen , ∆
aµ≈
2
.
51
×
10
−9
, sugges s
he exis ence o non- i ial physics a he Comp on scale. In his wo k, we explo e he hypo hesis
ha he obse ed anomalies a ise no om new pa icles, bu om a non-pe u ba i e geome ic
s uc u e o he lep on d essing i sel . We o malize his by in oducing wo dis inc o m ac o s:
(i) a poin -like Cha ge Fo m Fac o ,
FEM ≡
1, consis en wi h high-ene gy sca e ing, and (ii) a
non- i ial Phase Fo m Fac o
Fϕ
(
q2
), encoding a co a ian geome ic d essing o he SM acuum. We
implemen his d essing ia a Gaussian ep esen a i e o he E imo class o en i e unc ion egula o s,
ensu ing pe u ba i e uni a i y and ghos - ee beha io . C ucially, employing a ield ede ini ion
a gumen , we demons a e ha he heo y can be mapped o an e ec i e Weyl geome y whe e he
elec omagne ic cu en emains local in he d essed ame bu mani es s non-local in e ac ions in
he labo a o y ame. This amewo k yields a UV-sa e ealiza ion o dimension-6 SMEFT dipole
ope a o s, mapping he geome ic phase adius
⟨ 2⟩ϕ
o Wilson coe icien s
Ceγ
. Calib a ing o ∆
aµ
,
we de i e a Uni e sal Phase Radius benchma k yielding ∆
ae∼
10
−14
and ∆
aτ∼
7
×
10
−7
, p o iding
a alsi iable p edic ion o u u e p ecision expe imen s.
I. INTRODUCTION
The S anda d Model (SM) has wi hs ood decades o
sc u iny, ye he muon anomalous magne ic momen (
aµ
)
emains a s ubbo n ou lie [
1
–
4
]. While hea y- la o
anomalies sugges uni e sali y iola ion [
9
–
11
], adi ional
explana ions in ol ing new pa icles (e.g.,
Z′
, Lep o-
qua ks) ace inc easing p essu e om high-ene gy collide
bounds (LHC, LEP).
In his wo k, we explo e he hypo hesis ha he ob-
se ed anomalies a ise no om new pa icles, bu om a
non-pe u ba i e geome ic s uc u e o he lep on d ess-
ing i sel . Mo i a ed by he inhe en non-locali y o ield
in e ac ions a he Comp on scale and he necessi y o eg-
ula izing UV di e gences in a uni a y manne , we conside
a co a ian "wo ld-su ace" (Σ) desc ibing a geome ic
phase d essing o he lep on. This pe spec i e aligns wi h
ecen p oposals ea ing elec omagne ism as a pu ely ge-
ome ic heo y, whe e cha ge densi y and Zi e bewegung
a ise di ec ly om he me ic s uc u e o space ime [8].
To econcile his ex ended s uc u e wi h he poin -
like beha io obse ed a high ene gies, we adop he
o malism o Non-Local Quan um Field Theo y (NLQFT)
using E imo -class egula o s [
15
–
18
]. This allows us o
cons uc a heo y ha is:
1.
IR-Rich: Gene a es he equi ed ∆
aµ
ia e ec i e
dipole ope a o s.
2.
UV-Sa e: Remains uni a y and ghos - ee hanks o
he en i e unc ion egula o .
We map his geome ic amewo k di ec ly o he S an-
da d Model E ec i e Field Theo y (SMEFT) [
12
–
14
],
p o iding a igo ous dic iona y be ween geome y and
Wilson coe icien s.
II. THEORETICAL FRAMEWORK
A. Non-Local Ac ion and Uni a i y
We cons uc a non-local e ec i e ac ion ha p ese es
gauge in a iance and uni a i y. We pos ula e ha he
geome ic d essing modi ies he kine ic e m o he lep on
ia an en i e ope a o E:
LNL =−1
4F2
µν +¯
ψE(D2
µ)(i/
D−m)E(D2
µ)ψ, (1)
whe e
Dµ
=
∂µ
+
ieAµ
is he co a ian de i a i e. To en-
su e loop con e gence and ghos - ee beha io , we choose
he Gaussian egula o o he E imo class:
E D2
µ
Λ2!= exp −D2
µ
2Λ2!.(2)
The egula o scale Λis physically ied o a e e ence
Comp on wa eleng h
¯
λC≡ℏ/
(
m e c
). In he phenomeno-
logical analysis below we ake
m e
=
mµ
and ea
¯
λC
as
a single geome ic leng h scale cha ac e izing he acuum
phase d essing, common o all cha ged lep ons. To eco e
he speci ic o m ac o wid h equi ed by he Gaussian
e ex in Eq. (B3), we iden i y
Λ = √2
¯
λC
.(3)
B. Geome ic Phase Sec o
Be o e analyzing he ield heo e ic consequences, i
is use ul o o mula e he phase sec o a he le el o a
co a ian cohe ence densi y. We dis inguish be ween he
elec omagne ic cha ge dis ibu ion, which we ake o be
s ic ly poin -like:
ρEM(x)=δ(3)(x) =⇒FEM(q2)≡1,(4)
2
and a non- i ial phase cohe ence densi y
ρϕ
(
)associa ed
wi h he geome ic d essing. Assuming spa ial iso opy
in he es ame, he Phase Fo m Fac o is de ined as:
Fϕ(q) = 4πZ∞
0
d 2ρϕ( )sin(q )
q , q ≡ |q|.(5)
The low-momen um expansion de ines he geome ic phase
adius:
Fϕ(q2→0) = 1 −1
6⟨ 2⟩ϕq2+. . . (6)
whe e
⟨ 2⟩ϕ≡ −
6
dFϕ
dq2|0
. To leading o de in
q2
, all IR
obse ables a e "shape-blind" and depend only on
⟨ 2⟩ϕ
.
C. Sec o Sepa a ion ia Field Rede ini ion
A cen al s uc u al p ope y o ou amewo k is he
exac sepa a ion be ween he geome ic d essing (Phase
Sec o ) and he elec omagne ic cha ge (Cha ge Sec o ).
This sepa a ion is implemen ed by an in e ible, gene -
ally non-local ield ede ini ion buil om he co a ian
Laplacian D2
µ≡DµDµ.
Claim. Le
E
(
z
)be an en i e unc ion wi h no ze os on
he eal axis, and de ine he non-local kine ic e m ia
Eq. (1). Then he ield ede ini ion
Ψ≡ E D2
µ
Λ2!ψ, ¯
Ψ≡¯
ψE D2
µ
Λ2!,(7)
maps
LNL
o a local QED Lag angian o Ψ. As shown
in Appendix E, his ans o ma ion induces an e ec i e
Weyl geome y whe e he cu en is local in he in e nal
ame bu gene a es non-local obse ables in he labo a o y
ame.
D. Wa d–Takahashi Iden i y and Absence o Ghos s
The ield ede ini ion in Eq.
(7)
makes he gauge s uc-
u e o he heo y mani es . In he Ψ a iables he ac ion
is ha o local QED, and he s anda d de i a ion o he
Wa d–Takahashi iden i y applies unchanged. The d essed
p opaga o in he ψ-basis eads
SF(p) = iE−2(p2/Λ2)
/
p−m+i0,(8)
and he co esponding d essed e ex Γ
µ
(
p
+
q, p
)(de i ed
in Appendix B) sa is ies
qµΓµ(p+q, p) = S−1
F(p+q)−S−1
F(p),(9)
so ha he Wa d–Takahashi iden i y holds exac ly also
in he non-local ep esen a ion.
Eq.
(8)
shows ha he analy ic s uc u e o he p opa-
ga o is unchanged wi h espec o local QED: he only
pole is a
/
p
=
m
, while
E−2
(
p2/
Λ
2
)is an en i e unc ion
wi h no ze os o poles. Consequen ly, he e a e no ad-
di ional ghos -like poles o nega i e-no m s a es in he
spec um, and he S-ma ix emains uni a y.
E. The Geome ic-SMEFT Dic iona y
Using he Gaussian egula o , we ob ain a geome ic
Phase Radius
⟨ 2⟩ϕ=3
4¯
λ2
C,(10)
whe e
¯
λC
is he single e e ence geome ic leng h scale
de ined in Sec. II A, aken o be ixed by he muon sec o
and applied uni e sally o all cha ged lep ons. To leading
o de in
q2
, he phase d essing shi s he lep on anomalous
magne ic momen by he "Mas e Fo mula":
∆a(ϕ)
ℓ≃m2
ℓ
3⟨ 2⟩ϕ=m2
ℓ
4¯
λ2
C.(11)
Ma ching his on o he Wa saw-basis dipole ope a o
Oeγ
a scale µ=mℓyields he dic iona y:
ℜ[Ceγ,ℓℓ]
Λ2
EFT
=e mℓ
6√2 3
4¯
λ2
C.(12)
I is impo an o emphasize ha unlike s anda d cu o
egula iza ion, he E imo egula o
E
ende s he one-
loop in eg als UV ini e a he han me ely cu o . This
exponen ial supp ession in he deep Euclidean egion
implies ha he dange ous mixing o he dipole ope a o
in o LFV ope a o s unde RGE is na u ally supp essed.
UV Fini eness and Hie a chy S abili y.—Fu he mo e,
he use o an en i e unc ion egula o
E
has p o ound im-
plica ions o he hie a chy p oblem. Unlike polynomial
cu o s o Pauli-Villa s egula iza ion, which ypically
in oduce quad a ic di e gences
δm2
H∝
Λ
2
, he Gaus-
sian o m ac o p o ides exponen ial supp ession in he
Euclidean UV egime.
F. Mac ocausali y and Locali y
A common conce n ega ding non-local heo ies is he
po en ial iola ion o causali y. Howe e , he o mula ion
p esen ed in Sec. II C demons a es ha he non-locali y
can be shi ed en i ely in o he de ini ion o he in e po-
la ing ield
ψ
. In he ee- ield limi , he physical ield
ψ
(
x
)can be iewed as a con olu ion o he local ield Ψ
wi h a Gaussian ke nel o wid h O(1/Λ):
ψ(x) = Zd4z KΛ(x−z) Ψ(z),(13)
Since he commu a o s o Ψ anish exac ly ou side he
ligh -cone, he commu a o o he physical ield
ψ
a space-
like sepa a ions is non-ze o bu exponen ially supp essed.
This ensu es ha he heo y is “mac ocausal” in he sense
o E imo [15,16,18].
3
0 1 2 3 4 5
0
0.5
1
IR Region (g−2)
Momen um T ans e q2[a b. uni s]
Phase Fo m Fac o Fϕ(q2)
IR Shape-Blindness
E imo Regula o (Gaussian)
Gene ic Dipole Fo m
Figu e 1. Illus a ion o IR Shape-Blindness. Di e en UV
comple ions con e ge in he IR egime (shaded) ele an o
g−
2, bu he E imo egula o (blue) ensu es as e supp ession
in he UV.
III. PHENOMENOLOGICAL CONSISTENCY
A. Uni e sal Phase Radius Benchma k
In wha ollows we conside a “Uni e sal Phase Radius”
benchma k. By his, we mean ha he same geome ic
ela ion be ween he egula o scale and he e e ence
leng h ¯
λC,
⟨ 2⟩ϕ=3
4¯
λ2
C,(14)
de i ed in Sec. II E and Appendix B, is applied uni e sally
o all cha ged lep ons. Calib a ing his single geome ic
leng h scale on he obse ed muon anomaly ∆
aµ≃
2
.
51
×
10−9, we ob ain he ollowing p edic ions:
•
Muon (Inpu ): ∆
a(ϕ)
µ
= 2
.
51
×
10
−9
by cons uc ion.
•
Elec on (P edic ion): Scaling as (
me/mµ
)
2
, we p e-
dic ∆
ae≈
5
.
9
×
10
−14
. This is sa ely below cu en
expe imen al sensi i i y (
∼
10
−13
), esol ing he
ension whe e o he models p edic oo la ge an
elec on anomaly.
•
Tau P edic ion and Hea y Fla o Connec ion:
Unde he Uni e sal Phase Radius assump ion, he
anomaly scales as ∆
aℓ∝m2
ℓ
. Fo he au lep on,
his p edic s a subs an ial de ia ion:
∆a(ϕ)
τ= ∆aexp
µmτ
mµ2
≈7.1×10−7.(15)
This alue is well wi hin cu en expe imen al
bounds om LEP and LHC au-pai p oduc ion
[
19
,
20
], bu i is se e al o de s o magni ude la ge
han he S anda d Model expec a ion. F om a phe-
nomenological pe spec i e, such an enhanced au
dipole momen is a ac i e because i can na u ally
co ela e wi h new-physics e ec s in
b→cτν
and
b→sτ+τ−
ansi ions [
11
], sugges ing a common
geome ic o igin o hi d-gene a ion anomalies.
B. Consis ency wi h Elec oweak P ecision Da a
A na u al ques ion a ises ega ding he compa ibili y
o he geome ic d essing wi h he Higgs sec o . Since he
mechanism o mass gene a ion ia Spon aneous Symme y
B eaking occu s in he acuum con igu a ion whe e he
Higgs ield ca ies ze o momen um ans e (
q2
= 0),
he ele an o m ac o is e alua ed a he s a ic limi
Fϕ
(0) = 1. Consequen ly, he physical mass gene a ion
mechanism emains s ic ly S anda d Model-like:
mphys
µ=yµ
√2Fϕ(0) ≡yµ
√2,(16)
p ese ing he s anda d ela ion be ween he Yukawa
coupling and he lep on mass.
C. In e e ome ic Falsi iabili y
The model p edic s a momen um-dependen loss o
cohe ence in elec on wa epacke s. The inge isibili y
V(q) ollows:
V(q)≈1−1
6⟨ 2⟩ϕq2.(17)
Fo calib a ed alues, his implies a slope
s≈ −
0
.
25 in
dimensionless uni s. Expe imen al Reach: To dis inguish
his slope om uni y (
V
= 1) a a momen um ans e o
q∼
10 keV, an expe imen al p ecision o ∆
V/V ∼
10
−3
is equi ed.
IV. CONCLUSION
We ha e p esen ed a phenomenologically obus Geo-
me ic Phase D essing model. By in eg a ing Comp on-
scale acuum cohe ence wi h he ma hema ics o E imo
non-local QFT, we cons uc ed a heo y ha : 1. Explains
∆
aµ
ia SMEFT dipole ope a o s. 2. Ensu es UV sa e y
ia a uni a y Gaussian egula o . 3. Remains es able ia
p ecise in e e ome y.
We pos ula e ha he obus ness o his phase agains
en i onmen al decohe ence may s em om a opological
o igin, akin o a Be y phase acqui ed by he acuum
s a e along he e mion’s wo ldline.
Finally, we ema k on he heo e ical o igin o he en i e
ope a o
E
(
D2
). Fo m ac o s o he exponen ial ype
e−D2
a e no a bi a y; hey eme ge na u ally in S ing
Field Theo y (SFT) in e ac ions. This p o ides a com-
pelling UV comple ion pa hway, linking he low-ene gy
muon anomaly o undamen al space ime geome y.
4
This amewo k sugges s ha he "missing physics"
may be a signa u e o he geome ic cohe ence o he
lep on acuum.
Appendix A: BRST Symme y and Gauge In a iance
In his appendix, we p o e he BRST in a iance o he
heo y using he ield ede ini ion a gumen .
1. Cons uc ion
We s a wi h he gauge-in a ian ma e Lag angian
(Eq. 1) and add he s anda d
Rξ
gauge ixing and ghos
e ms:
Lg =−1
2ξ(∂µAµ)2,(A1)
Lgh = ¯c□c. (A2)
The BRST ope a o sis de ined s anda dly:
sAµ=∂µc, sψ =iecψ, (A3)
s¯
ψ=−ie ¯
ψc, sc = 0.(A4)
2. P oo
The gauge- ixing sec o is
s
-exac and hus in a ian .
Fo he ma e sec o , we le e age he ield ede ini ion
Ψ=Eψ. The d essed ield Ψ ans o ms exac ly like he
o iginal ield:
sΨ = s(Eψ)=iecΨ.(A5)
Consequen ly, he ans o med Lag angian
¯
Ψ
(
i/
D−m
)Ψ
is mani es ly BRST in a ian .
Appendix B: S-Ma ix and Feynman Rules
He e we de i e he Feynman ules in he In e ac ion
Pic u e.
1. Decomposi ion and P opaga o s
We decompose
L o
=
L ee
+
Lin
. Expanding
E
(
D2
µ/
Λ
2
) o ze o h o de in
Aµ
, we iden i y he kine ic
ope a o . The p opaga o is he in e se o he kine ic
ope a o :
SF(p)=iexp(−p2/Λ2)
/
p−m+i0.(B1)
This exhibi s Gaussian supp ession in he UV (
p2→ ∞
),
ensu ing loop con e gence.
2. In e ac ion Ve ex
The in e ac ion e ices a ise om he expansion o
E
(
D2
µ
)in powe s o he coupling
e
. The 1-pho on e ex
Γ
µ
(
p′, p
)is ob ained by collec ing e ms linea in
Aµ
. Fo
he Gaussian egula o , his yields:
Γµ(p′, p) = −ieγµexp −(p′−p)2
4Λ2.(B2)
Iden i ying he momen um ans e
q
=
p′−p
, we eco e
he Phase Fo m Fac o used in he main ex :
Fϕ(q2) = exp −q2
4Λ2.(B3)
Appendix C: In e e ome ic P obes
We conside a Kapi za-Di ac-Talbo -Lau in e e ome e .
The momen um ans e
q
is de e mined by he g a ing
pe iodici y
d
and o de
n
:
q
=
n2πℏ
d
. Fo elec ons, he
isibili y
V
is educed by he o m ac o . The ela i e
con as loss is
δV ≈1
6⟨ 2⟩ϕq2
. To dis inguish he ge-
ome ic slope
s≈ −
0
.
25 om he poin -like p edic ion
(
V
= 1) a a momen um ans e o
q∼
10 keV, a ela i e
con as p ecision o δV/V ∼10−3is equi ed.
Appendix D: Co a ian Wo ld-Su ace Σ
We model he d essing as a dis ibu ion on a wo ld-
su ace Σde ined by embedding
Xµ
(
ξ
). The co a ian
cohe ence densi y ρϕ( )is cons uc ed such ha :
ρϕ( ) = ZΣ
d2ξ√−h δ(3)(x−X(ξ)).(D1)
Fo he Gaussian egula o choice, Σis e ec i ely a uzzy
mani old ep esen ing he quan um delocaliza ion o he
phase d essing.
Appendix E: Non-Local Field Rede ini ion and
E ec i e Weyl Geome y
Fo comple eness, we collec he e he pa h-in eg al
de i a ion o he ield ede ini ion used in Sec. II C and
cla i y i s geome ic in e p e a ion. We s a om he
gene a ing unc ional in he
ψ
-basis. We use
wide ex
o accommoda e he long pa h in eg al exp ession:
5
Z[η, ¯η] = ZDψD¯
ψexp (iZd4x¯
ψE(D2
µ)(i/
D−m)E(D2
µ)ψ+ ¯ηψ +¯
ψη).(E1)
wi h E(z)an en i e unc ion as in Eq. (2). Unde he linea change o a iables
Ψ = E(D2
µ/Λ2)ψ, ¯
Ψ = ¯
ψE(D2
µ/Λ2),(E2)
he gene a ing unc ional becomes
Z[η, ¯η] = ZDΨD¯
Ψ exp (iZd4x¯
Ψ(i/
D−m)Ψ + ¯ηE−1Ψ + ¯
ΨE−1η).(E3)
While his esembles he s anda d local QED unc ional
o Ψ, he coupling o he sou ces
η, ¯η
in ol es he in-
e se egula o
E−1
, which depends on he gauge ield
Aµ
h ough
Dµ
. This esul equi es a ca e ul physical in e -
p e a ion. In he basis o he d essed ield Ψ, he cu en
Jµ
=
¯
Ψγµ
Ψappea s poin -like. Howe e , he ede ini ion
Ψ =
E
(
D2
µ
)
ψ
induces a momen um-dependen scaling o
he in e ac ion s eng h. As a gued in ecen geome ic
app oaches o elec omagne ism [
8
], such ans o ma ions
can be in e p e ed as a ansi ion o an e ec i e Weyl
geome y whe e he co a ian de i a i e o he me ic
does no anish (semime ici y).
In his pic u e, he appa en con lic be ween he local
cu en in he in e nal basis and he non-local sca e -
ing o m ac o in he labo a o y ame is esol ed: he
"poin -like" na u e is an a i ac o he Weyl ame, while
he physical obse able
Fϕ
(
q2
)a ises om he geome ic
d essing o he acuum i sel . This is equi alen o a
"ligh -speed ci cula ion" o Zi e bewegung o he cha ge,
which smea s he e ec i e in e ac ion e ex o e a egion
o size
∼
1
/
Λ[
8
]. Thus, he ield ede ini ion p o es he
uni a i y o he heo y (absence o ghos s) while he e -
ec i e geome y ensu es he phenomenologically equi ed
o m ac o .
Appendix F: Ma ching o Ceγ o ∆aℓ
In his appendix we gi e a ully explici ma ching be-
ween he SMEFT dipole ope a o
Ceγ,ℓℓ
and he anoma-
lous magne ic momen ∆
aℓ
, e i ying ha he geome ic
dic iona y is consis en wi h SMEFT no maliza ion.
1. SMEFT dipole ope a o a e EWSB
In he Wa saw basis he elec oweak dipole ope a o s
ele an o lep ons a e
Op
eB = (¯
Lpσµν e )ϕ Bµν ,(F1)
Op
eW = (¯
Lpσµν τIe )ϕ WI
µν ,(F2)
wi h co esponding Wilson coe icien s. De ining
Op
eγ ≡
cWOp
eB −sWOp
eW , he ele an Lag angian is:
LSMEFT ⊃X
p,
Cp
eγ
Λ2
EFT Op
eγ +h.c. (F3)
A e elec oweak symme y b eaking, his gene a es:
L(ℓ)
dip =−
√2ℜ[Ceγ,ℓℓ]
Λ2
EFT
¯
ℓ σµν ℓ Fµν +. . . (F4)
2. Canonical de ini ion o aℓ
Compa ing Eq. (F4) wi h he canonical de ini ion:
L(ℓ)
e ⊃e Qℓ
4mℓ
aℓ¯
ℓ σµν ℓ Fµν ,(F5)
we ob ain he ma ching ela ion:
∆aℓ=4√2mℓ
eℜ[Ceγ,ℓℓ]
Λ2
EFT
.(F6)
3. Geome ic dic iona y and consis ency check
We pos ula ed he "Mas e Fo mula":
∆a(ϕ)
ℓ=m2
ℓ
4¯
λ2
C.(F7)
Using he dic iona y Eq. (17):
ℜ[Ceγ,ℓℓ]
Λ2
EFT
=e mℓ
8√2
¯
λ2
C.(F8)
Subs i u ing his in o Eq.
(F6)
exac ly ep oduces
Eq. (F7), p o ing ull consis ency.
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