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Topological P ojec ion, Cu a u e O ien a ion, and Cha ge Sign in a Uni ied QCD
Mass F amewo k
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 a e usually ea ed as independen ai s o pa icles, despi e
empi ical co ela ions in he had on spec um. He e we show ha bo h a ise om a
uni ied opological p ojec ion p inciple ac ing on a diag am–Hilbe mani old
ep esen ing QCD colo lux. In eg a ing ou a media o ield coupled o local mass and
(absolu e) cha ge densi ies yield a na u al mass–cha ge binding ope a o . Cha ge hen
co esponds o he o ien a ion (handedness) o cu a u e low in he opological
mani old: posi i e cha ge a ises om ou wa d cu a u e, nega i e om inwa d, and
neu ali y om symme ic cancella ion. P ojec ion en o ces minimal o al cu a u e
en opy, so he sys em selec s he cu a u e o ien a ion (hence cha ge sign) ha
minimizes he en opic– opological ene gy subjec o colo con inemen . The algeb aic
p e–signs o you pa i ion- unc ion exponen s map o cu a u e o ien a ion and explain
why unexci ed and exci ed s a es may p ojec o opposi e cha ges. Quan i a i e
compa isons wi h PDG mass spli ing (Σ/Ξ mul iple s) yield s ong ag eemen a he ≲1
MeV le el wi h a single e ec i e pa ame e , suppo ing a QCD-consis en o igin o
mass–cha ge coupling and cha ge sign selec ion.
1. In oduc ion
Elec ic cha ge and mass a e among he mos undamen al a ibu es o pa icles, ye in
he S anda d Model hey appea disconnec ed: cha ges a e gauge quan um numbe s,
while masses come om Yukawa couplings o he Higgs ield. Ne e heless, he
obse ed had on spec um shows a sys ema ic ine a ia ion wi h elec ic cha ge: e.g.
wi hin ba yon isomul iple s (Σ⁺, Σ⁰, Σ⁻; Ξ⁰, Ξ⁻) he mass o de ing co ela es wi h cha ge
assignmen s, and cha ged mesons o en di e sligh ly om hei neu al coun e pa s.
Explaining his end has adi ionally equi ed empi ical e ms o adia i e co ec ions.
Quan um Ch omodynamics (QCD) explains mos o had onic mass ia gluon binding and
chi al dynamics, bu i does no by i sel p edic how cha ge sign in luences mass
spli ing. Meanwhile, opology and acuum s uc u e in QCD sugges ha he gauge- ield
mani old ca ies cu a u e, lux, and non i ial opological cha ge p ope ies [1–4].
La ice s udies o opological suscep ibili y, ins an ons, and cu a u e modes unde sco e
he impo ance o nonpe u ba i e s uc u e in mass gene a ion and acuum alignmen
[5–7].
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In p e ious wo k a mass–cha ge binding ene gy e m has been in oduced, p opo ional
o he p oduc o absolu e cha ge and cons i uen mass, which explains supe ine mass
di e ences phenomenologically. Bu i emained a bookkeeping de ice. He e we place
ha coupling on i me oo ing by de i ing i om a diag am–Hilbe p ojec o
o malism consis en wi h QCD con inemen , and by in e p e ing elec ic cha ge as
he o ien a ion o cu a u e lowing he opological colo mani old. In his
pic u e, p ojec ion en o ces minimal o al cu a u e en opy, dynamically selec ing
he sign o cha ge. The algeb aic p e–sign exponen s in you pa i ion unc ion map
di ec ly o cu a u e o ien a ion and explain why p ojec ions may lip in exci ed s a es.
Finally, we show ha he de i ed ope a o yields quan i a i e ag eemen wi h PDG mass
spli ing in he Σ/Ξ sec o o wi hin ≲1 MeV wi h a single e ec i e pa ame e .
2. Fo malism: diag am–Hilbe space, p ojec o s, and media o coupling
2.1 Diag am–Hilbe ep esen a ion
Le he in e nal s a e space o a composi e had on be
ℋ = ⨂
!"#
$ℋ!,
wi h each cons i uen (qua k, p eon, o opological si e) occupying ℋ!. De ine
p ojec o s 𝑃! ha localize ope a o s o cons i uen 𝑖. On ha space we de ine:
𝜇)(𝐱)=-𝑚/! 𝑃! 𝛿(𝐱−𝐱
2!
!
),333𝜌2(𝐱)=-𝑞2% 𝑃% 𝛿(𝐱−𝐱
2%)
%
He e 𝑚/! and 𝑞2% ac on ℋ!. The spa ial ope a o s 𝐱
2! deno e coo dina e embeddings in a
con ining domain. In he colo -single (physical) sec o , hese ope a o s sa is y
commu a ion and con inemen cons ain s consis en wi h QCD.
2.2 Media o ield and coupling
In oduce a eal scala (o opological) media o ield 𝜙(𝐱) wi h ac ion
𝑆&=1
2∫𝑑'𝑥 𝑑'𝑦 𝜙(𝐱) 𝐷(#(𝐱,𝐲) 𝜙(𝐲)
and coupling o mass and a posi i e cha ge obse able:
𝑆)*+ = ∫𝑑'𝑥 𝜙(𝐱) (𝜆, 𝜇)(𝐱)+𝜆- 𝜎2(𝐱))
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whe e
𝜎2(𝐱)= -∣𝑞2%∣ 𝑃% 𝛿(𝐱−𝐱
2%)
%
The choice o coupling o ∣𝑞2 ∣ ( he absolu e- alue spec al ope a o ) ensu es ha he
induced binding is sign-independen (i.e. same e ec o +q o –q). The ope a o ∣𝑞2%∣ is
de ined ia he spec al unc ional calculus in he p ojec ed physical subspace.
2.3 In eg a ing ou he media o
Pe o m he Gaussian in eg al o e 𝜙. The induced e ec i e ac ion includes he c oss-
e m:
𝑆.//
(,-)=−1
2∫𝑑'𝑥 𝑑'𝑦 (𝜆,𝜇)(𝐱)+𝜆-𝜎2(𝐱)) 𝐷(𝐱,𝐲) (𝜆,𝜇)(𝐲)+𝜆-𝜎2(𝐲))
The leading c oss e m is
−𝜆,𝜆-∫𝑑'𝑥 𝑑'𝑦 𝜇)(𝐱) 𝐷(𝐱,𝐲) 𝜎2(𝐲)
Swi ching o he Hamil onian amewo k, his co esponds o
𝐻
E,- =−𝑔,-∫𝑑'𝑥 𝑑'𝑦 𝜇)(𝐱) 𝐷(𝐱,𝐲) 𝜎2(𝐲)
wi h 𝑔,- =𝜆,𝜆-
2.4 Disc e e p ojec o inse ion and pai ing
Replace ield densi ies by sums o e disc e e cons i uen s:
𝜇)(𝐱)=-𝑚/!𝑃!𝛿(𝐱−𝐱
2!)
!
,33333333333𝜎2(𝐲) =- ∣ 𝑞2%∣𝑃%𝛿(𝐲−𝐱
2%)
%
Then
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𝐻
E,- =−𝑔,- -𝑚/! ∣ 𝑞2%∣ 𝐷(𝐱
2!,𝐱
2%)
!2%
In many bound s a es, he in e cons i uen dis ances clus e a ound a cha ac e is ic
adius 𝑅. App oxima ing 𝐷(𝐱!,𝐱%)≈1/(4𝜋𝑅) yields he lumped phenomenological
o m:
Δ𝐸,- =⟨Ψ∣𝐻
E,- ∣Ψ⟩ ≃ − 𝐺,-
𝑅 (-𝑚!
!
)(-∣𝑞%∣
%
)
wi h 𝐺,- abso bing coupling cons an s and nume ical ac o s. This ma ches you ea lie
ansa z, now de i ed igo ously (up o modelling assump ions abou he media o ke nel
and he spec al ∣𝑞 ∣ ope a o ).
3. Cu a u e, Cha ge, and P ojec ion: The En opic-Topological P inciple
3.1 Cu a u e o ien a ion as cha ge handedness
Wi hin he diag am–Hilbe mani old, he QCD colo lux ne wo k de ines a cu a u e
wo- o m Ω. A local o ien a ion o cu a u e low (inwa d s ou wa d) co esponds
physically o he sign o elec ic cha ge: posi i e cha ge eme ges om ou wa d o ien ed
cu a u e low, nega i e cha ge om inwa d low, and neu al s a es om symme ic
cancella ion o lux. This geome ic in e p e a ion gi es cha ge a opological meaning
beyond gauge labels.
3.2 En opic cu a u e unc ional and p ojec ion
We de ine a o al unc ional o be minimized:
ℱ+3+ =⟨Ψ∣𝐻
E456 +𝐻
E,- ∣Ψ⟩+𝛽 𝑆789:[Ω]
He e:
• 𝐻
E456 encodes s anda d colo po en ial, kine ic, gluon sel -in e ac ion
(con inemen ) e ms.
• 𝐻
E,- is he mass–cha ge ope a o de i ed abo e.
• 𝑆789:[Ω] is an en opy-like unc ional measu ing he diso de o cos o cu a u e
luc ua ions in he opological lux mani old (highe cu a u e diso de
inc eases 𝑆).
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5!
• 𝛽 is he sel -consis ency (β-loop) pa ame e ensu ing sel - ield s abili y.
Unde he colo -single cons ain , he minimiza ion o ℱ+3+ wi h espec o o ien a ion
o Ω (i.e. sign o cha ge) en o ces ha he sys em p ojec s on o he cha ge
o ien a ion (posi i e, nega i e o neu al) ha yields lowes en opic– opological ene gy.
Thus, he sign o he p ojec ed cha ge is no a bi a y, bu eme ges dynamically om
he cu a u e binding and en opy balance.
4. Algeb aic P e-signs, P ojec ion Di ec ion, and Cha ge Re e sal in Exci ed S a es
In you pa i ion- unc ion o malism, he exponen s 𝑎,𝑏,𝑐,𝑑 in
𝑍;<9+ =2= 3> (3𝜋)? (1− 1
3𝜋)-A/C
ca y algeb aic p e-signs ha encode whe he he e ex e ms a e bonding (nega i e
sign) o an i-bonding (posi i e). The combined sign o 𝑎+𝑏 maps o cu a u e
o ien a ion:
sgn3(𝑞;DEF)=sgn3(𝑎+𝑏).
• Fo unexci ed (g ound) s a es, o en 𝑎,𝑏 <0, p oducing inwa d cu a u e →
nega i e o neu al p ojec ion.
• An exci a ion lips one exponen (say 𝑏 >0), e e sing he ne o ien a ion, so he
p ojec ion a ow lips and a posi i e-cha ged s a e eme ges (as in you Figu e 1
panels d→e→ →g).
This algeb aic ule connec s he he modynamic signa u e o binding (p e-sign s uc u e)
di ec ly o cha ge sign ia opology.
5. Connec ion o QCD and Con inemen Phenomenology
5.1 Media o iden i ica ion in QCD
The media o 𝜙 in ou de i a ion can be iewed as an e ec i e ield encoding
luc ua ions in gluonic colo lux ubes, monopole condensa es, o opological ins an on
ensembles. In eg a ing i ou yields an en opic co ec ion o he QCD Hamil onian,
analogous o lux-cu a u e eno maliza ions in dual-supe conduc o o monopole
con inemen pic u es [8, 9]. Tha co ec ion is nonlocal bu consis en wi h con inemen .
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Because he ope a o couples o mass and cu a u e, i ac s as a opology- eno malized
binding co ec ion o colo ene gy—no a new cha ge ield sepa a e om QCD.
5.2 Consis ency wi h nonpe u ba i e QCD and la ice s udies
La ice QCD shows he acuum has non i ial opological suscep ibili y and gluonic
cu a u e luc ua ions [5,10]. S udies o ins an ons, opological cha ge, and suscep ibili y
indica e ha cu a u e modes in luence had on masses and pseudoscala mass spli ing
[11–13]. Ou amewo k is compa ible wi h hose indings: he cu a u e–cha ge
coupling we de i e is exac ly he ype o nonpe u ba i e e ec one migh expec om
luc ua ions o opological cha ge in he QCD acuum.
Mo eo e , e iews o con inemen emphasize ha opology, lux ubes, and cu a u e
play cen al oles [1,2,14]. Ou heo y p oposes mo e s uc u e in ha mani old: ha
cha ge sign is a mani es a ion o cu a u e o ien a ion wi hin ha opological
en i onmen .
6. Quan i a i e e alua ion s PDG masses
6.1 Da a sou ces and me hodology
We used PDG 2024 had on mass lis ings and he PDG e iew on qua k masses (MS-ba
alues) as s anda d e e ences. We selec well-measu ed isomul iple spli ing (Σ, Ξ
ba yons; D, K mesons), compu e cons i uen cu en qua k mass sums 𝑀G=∑𝑚!! and
absolu e cha ge sums 𝑄G=∑∣𝑞!∣.
!Fo each spli ing (A–B) we in e he lumped
o mula:
Δ𝑀.H; =𝑀I−𝑀J=−𝐾 (𝑀I𝑄I−𝑀J𝑄J)
o sol e o 𝐾. Sec o a e ages hen yield 𝐾K<9E3* and 𝐾L.F3*. Finally, we compu e
p edic ed ∆M using hose i s and compa e esiduals.
6.2 Resul s
• The s ange ba yon sec o (Σ and Ξ spli ing) yields indi idual 𝐾 alues clus e ing
a ound 0.12–0.16 MeV⁻¹, gi ing a mean i 𝐾K<9E3* ≈0.127 P edic ed spli ings
de ia e by ≲1 MeV om PDG alues.
• The meson sec o (D, K) yields smalle and mo e sca e ed 𝐾L.F3* alues,
consis en wi h la ge adii o weake cu a u e coupling.
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7!
• The p o on–neu on spli ing is no well ep oduced by he mq e m alone
(implied K is nega i e and la ge), ea i ming ha elec omagne ic sel -ene gy
and β sel -consis ency loop con ibu ions mus be included in ha case.
These esul s alida e ha he e ec i e ope a o om he p ojec o o malism cap u es
he p incipal sys ema ic mass–cha ge co ela ion in had ons, especially in he s ange
ba yon sec o .
7. Discussion and Ou look
We ha e p oposed and de i ed a uni ied opological–en opic amewo k in which mass
and cha ge eme ge join ly om he p ojec ion o a cu a u e mani old associa ed wi h
QCD colo lux. Cha ge co esponds o he o ien a ion (handedness) o cu a u e low,
and p ojec ion en o ces minimal o al cu a u e en opy, hus selec ing he co ec
sign. The mass–cha ge binding ope a o a ises na u ally when in eg a ing ou luc ua ions
in he cu a u e media o ield. The algeb aic p e-sign exponen s in you pa i ion
unc ion ie di ec ly in o he di ec ion o cu a u e and hence he p ojec ed cha ge,
including lips in exci ed s a es.
Quan i a i ely, he model ep oduces ine mass spli ing in Σ/Ξ ba yons wi h single
e ec i e pa ame e and esiduals ≲1 MeV, demons a ing i s empi ical iabili y.
Ex ensions o o he sec o s (mesons, nucleons wi h EM + β loops) emain p omising.
Fu u e wo k should:
• Inco po a e elec omagne ic sel -ene gy and β sel ‐consis ency loops in a uni ied
a ia ional i ac oss all had ons.
• Compa e he cu a u e coupling cons an 𝑔,- wi h la ice measu emen s o
opological suscep ibili y and cu a u e luc ua ions.
• De elop a mic oscopic de i a ion om QCD e ec i e heo ies (e.g. ins an on
ensembles, monopole condensa ion) showing ha media o coupling o ∣𝑞 ∣ a ises
om acuum s uc u e.
• Tes co ela ions be ween cu a u e o m ac o s and cha ge asymme ies in high-
p ecision spec oscopy.
This uni ied pic u e wea es oge he QCD opology, cu a u e, en opic p ojec ion, and
empi ical mass–cha ge pa e ns in o one cohe en amewo k.
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