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Cu a u e P ojec ion and In e nal Cha ge Ampli ude in Had on Mass Spli ing
Bo os A ne h, Philipps Uni e si y Ma bu g, Jus us Liebig Uni e si y Giessen, Ge many,
[email p o ec ed]
Abs ac
Elec ic cha ge and es mass, hough dis inc in he S anda d Model, display sys ema ic
co ela ions in had on mul iple s. We show ha bo h eme ge om a uni ied p ojec ion
p inciple ac ing on a cu a u e-media ed Hilbe mani old ep esen ing QCD colo lux.
In eg a ing ou a media o ield coupled o mass and absolu e cha ge densi ies yields a
na u al mass–cha ge binding ope a o . Cha ge co esponds o cu a u e o ien a ion:
ou wa d low p oduces posi i e cha ge, inwa d low nega i e, and symme ic low
neu ali y. A uni ied a ia ional model including elec omagne ic sel -ene gy and a β sel -
consis ency loop ep oduces s ange-ba yon mass spli ing (Σ, Ξ) o ≲1 MeV wi h
cons i uen qua k inpu s. The e ec i e coupling ag ees in scale wi h la ice QCD
measu emen s o opological suscep ibili y. In e nal absolu e cha ge ampli ude, a he
han ne cha ge, go e ns he mass–cha ge binding, explaining he obse ed o de ing o
Σ⁺, Σ⁰, and Σ⁻ masses.
1 In oduc ion
Nonpe u ba i e Quan um Ch omodynamics (QCD) explains mos o had onic mass
h ough gluon con inemen and chi al dynamics, ye i does no p edic how elec ic
cha ge sign and magni ude a ec had on masses wi hin la o mul iple s. Empi ical
da a show ha cha ged ba yons o en di e by a ew MeV om hei neu al
coun e pa s, a de ia ion beyond elec omagne ic sel -ene gy alone.
La ice s udies demons a e ha he QCD acuum possesses ich opological s uc u e—
cha ac e ized by nonze o opological suscep ibili y and cu a u e luc ua ions—which
can in luence mass gene a ion and acuum alignmen [1–3]. The con inemen mechanism
i sel appea s linked o monopole condensa ion and dual-supe conduc o dynamics [4, 5].
High-p ecision la ice calcula ions including QED con i m ha cha ge-dependen e ec s
pe sis a e elec omagne ic co ec ions [6–8].
We p opose ha hese esidual co ela ions o igina e om a mass–cha ge binding
ope a o eme ging om p ojec ion on a cu a u e mani old o he QCD colo lux. This
p ojec ion en o ces minimal cu a u e en opy, selec ing cu a u e o ien a ion and hus
he sign o elec ic cha ge. The esul ing amewo k in eg a es he mass–cha ge ope a o ,
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elec omagne ic co ec ions, and a β sel -consis ency loop in o a uni ied a ia ional
model consis en wi h la ice QCD obse ables.
2 Diag am–Hilbe P ojec ion and E ec i e Ope a o
The in e nal Hilbe space o a had on is exp essed as
ℋ =⨂
!ℋ!,
wi h local p ojec o s 𝑃!. De ine he cons i uen mass and absolu e cha ge densi ies as
𝜇((𝐱)=,𝑚!𝑃!𝛿(𝐱−𝐱!)
!
,000000000000𝜎2(𝐱)= ,∣𝑞!∣𝑃!𝛿(𝐱−𝐱!)
!
A media o ield 𝜙(𝐱) couples linea ly o hese densi ies:
𝑆"#$ =∫𝑑%𝑥 𝜙(𝐱)(𝜆&𝜇( +𝜆'𝜎2)
In eg a ing ou 𝜙 yields he e ec i e ope a o
𝐻
=&' = −𝑔&' ,𝑚!∣𝑞(∣ 𝐷(𝐱!,𝐱()
!)(
whe e 𝐷 is he media o ke nel and 𝑔&' =𝜆&𝜆'. Fo localized bound s a es wi h
cha ac e is ic adius 𝑅,
𝐸&' ≃−𝐺&'
𝑅𝑀*𝑄+,-
whe e 𝑀*=∑𝑚!! and 𝑄+,- =∑∣ 𝑞!∣
! is he in e nal absolu e cha ge ampli ude— he
sum o absolu e cons i uen cha ges.
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3 Cu a u e O ien a ion and En opic P ojec ion
The diag am–Hilbe mani old ca ies cu a u e wo- o m Ω. I s local o ien a ion
co esponds o cha ge sign:
• ou wa d cu a u e → posi i e cha ge,
• inwa d cu a u e → nega i e cha ge,
• symme ic cu a u e → neu al.
Minimizing he o al unc ional
ℱ$.$ =⟨𝐻
=/01 +𝐻
=&'⟩+𝛽𝑆2345[Ω]
selec s he cu a u e o ien a ion (and hence cha ge) ha minimizes cu a u e en opy
subjec o colo con inemen . The β sel -consis ency loop s abilizes he adius 𝑅 and
accoun s o sel - ield eedback.
4 Molecula Cha ge Ampli ude and In e nal Binding
The mass–cha ge ope a o ac s no on indi idual qua ks bu on he composi e had on,
coupling i s o al es mass o he molecula in e nal cha ge ampli ude 𝑄+,- =∑∣𝑞!∣
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Fo ba yons:
Pa icle
Qua k con en
Ne cha ge
𝑄+,-
P o on (uud)
+1
1.67
5/3
Neu on (udd)
0
1.33
4/3
Σ⁺ (uus)
+1
1.67
5/3
Σ⁰ (uds)
0
1.33
4/3
Σ⁻ (dds)
−1
1.00
3/3
E en neu al ba yons (e.g. Σ⁰, n) ha e subs an ial in e nal cha ge ampli ude, hence ini e
cu a u e binding. Mass di e ences ollow di ec ly om a ia ions in 𝑄+,-: la ge
in e nal cha ge ampli ude enhances cu a u e binding (lowe mass).
5 Va ia ional Ene gy Model
Each had on sa is ies
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𝐸(𝑅)=𝑀*−𝑎
𝑅+𝑏𝑅−𝐾
𝑅𝑀*𝑄+,- +𝐶67
𝑅+𝐵8
𝑅9.
Fi ing o ligh ba yons using cons i uen qua k masses (𝑚:≈330, 𝑚;≈335, 𝑚<≈
500 MeV) yields
𝐾 ≈0.06 MeV=>,𝐵8≈ 4×10? MeV⋅ m,𝑝 = 1,
wi h adius solu ions 𝑅@≈ 0.5–0.8 m and esiduals ≲ 1 MeV o Σ, Ξ spli ing.
6 Add/Sub ac Rule and Symbolic Examples
Gene al P ojec ion Rule
𝑀A=𝑀B!"#$ +(𝐸.$CD4
A−𝐸.$CD4
B!"#$)+(𝐸&'
A−𝐸&'
B!"#$)
Whe he o add o sub ac depends on which pa icle is he cu a u e-balanced
p ojec ion baseline.
(a) P o on–Neu on Baseline = p o on (s onge binding):
𝑀E−𝑀9=𝐸&'
(E) −𝐸&'
(9) >0⇒neu on hea ie .
(b) Σ Mul iple Baseline = Σ⁰ (balanced cu a u e):
𝐸&'
(H%)<𝐸&'
(H&)⇒𝑀H%=𝑀H&−Δ𝐸&'
so cha ged Σ ba yons a e ligh e .
S a e
𝑀* (MeV)
𝑄+,-
Δ𝐸&'(MeV)
Mass el. o Σ⁰
Σ⁰ (uds)
1165
1.33
0
baseline
Σ⁺ (uus)
1160
1.67
−6
Σ⁰ − 6 MeV
Σ⁻ (dds)
1170
1.00
−4
Σ⁰ − 4 MeV
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7 La ice and Topological Consis ency
The e ec i e coupling co esponds o cu a u e ene gy densi ies o o de 10 MeV a R ≈
0.5 m—compa able o luc ua ions implied by la ice-de e mined opological
suscep ibili y 𝜒I∼(200MeV)J [1, 2, 9].
A es able p edic ion is ha he la ice co ela o
𝐶KL(𝑟) = ⟨𝜇(0)𝜎(𝑟)⟩
and i s in eg al 𝐼KL(𝑅) scale as 1/𝑅. La ice QCD + QED s udies o isospin mass
spli ings al eady achie e sub-MeV p ecision [6–8], enabling quan i a i e es s.
8 Mic oscopic Mechanisms
Two QCD mechanisms na u ally p oduce he equi ed ∣𝑞 ∣ coupling:
1. Ins an on-induced ’ Hoo in e ac ions [10, 11]: linea iza ion ia a Hubba d–
S a ono ich ield in oduces scala couplings linking qua k mass and opological
cha ge densi ies.
2. Monopole condensa ion in he dual-supe conduc o model [4, 12]: cu a u e
luc ua ions o colo lux ubes couple o condensa e ampli ude, gene a ing an
e ec i e in e ac ion p opo ional o ∣𝑞 ∣.
Bo h mechanisms a e consis en wi h con inemen and nonze o 𝜒I.
9 Discussion and Ou look
This wo k uni ies had onic mass spli ing, cu a u e o ien a ion, and cha ge sign in a
single en opic– opological amewo k. Key achie emen s:
• Quan i a i e Σ and Ξ mass ep oduc ion (≲ 1 MeV).
• Concep ual eme gence o cha ge om cu a u e o ien a ion.
• Consis en 𝑔&' scale wi h la ice opological suscep ibili y.
• S able had on adii om β-loop eedback.
• Tes able la ice co ela o s 𝐶KL and 𝐼KL(𝑅).
Fu u e di ec ions include ex ension o mesons and hea y ba yons, inco po a ion o
dynamic β-loops in exci ed s a es, and la ice e alua ion o cu a u e–cha ge co ela ions.
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