Curvature-Projection as an Effective Field Operator in Quantum Chromodynamics

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Cu a u e-P ojec ion as an E ec i e Field Ope a o in Quan um Ch omodynamics
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
We show ha he mass–cha ge binding and cu a u e-p ojec ion mechanism p oposed o
explaining had onic and lep onic mass hie a chies can be exp essed as a
legi ima e e ec i e ope a o wi hin Quan um Ch omodynamics (QCD) and Quan um
Field Theo y (QFT).
By in oducing a scala cu a u e media o ha couples o local mass and absolu e-cha ge
densi ies, in eg a ing i ou yields a non-local gauge-in a ian in e ac ion
−𝑔!" ∫ 𝜇(𝑥) 𝐷(𝑥−𝑦) 𝜎(𝑦) 𝑑#𝑥 𝑑#𝑦.
This e m ep esen s an en opic co ec ion o he QCD acuum ene gy associa ed wi h
gluonic cu a u e luc ua ions and is nume ically consis en wi h la ice de e mina ions o
he opological suscep ibili y 𝜒$ . The esul ing ope a o p ese es Lo en z in a iance,
locali y, and eno malizabili y up o he con inemen scale, p o iding a igo ous QFT
basis o cu a u e-induced mass–cha ge coupling and cha ge-o ien a ion p ojec ion.
1 In oduc ion
Non-pe u ba i e QCD is domina ed by con inemen , chi al symme y b eaking, and
opological s uc u e encoded in he ield s eng h
𝐹%&
' and i s dual 𝐹
.'%& [1–3]. Ins an ons, monopole condensa ion, and dual
supe conduc i i y o he acuum p o ide geome ic insigh in o colo con inemen [4–6].
La ice calcula ions o he opological suscep ibili y 𝜒$ demons a e ha cu a u e
luc ua ions con ibu e di ec ly o had on masses [7–9].
To desc ibe hese e ec s analy ically, we in oduce a cu a u e-media ed e ec i e ield
heo y ha ep oduces he empi ically obse ed mass–cha ge co ela ions in ba yons [10–
12]. The cons uc ion espec s all axioms o QFT and embeds seamlessly in o he QCD
Lag angian.
2 E ec i e-Field De i a ion
S a ing om he QCD Lag angian
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ℒ()* =−1
4𝐹%&
'𝐹'%& +4𝜓
¯+(𝑖 /𝐷−𝑚+)𝜓+
+
we supplemen a eal scala cu a u e ield 𝜙(𝑥) ep esen ing coa se-g ained gluonic
cu a u e modes:
ℒ,=1
2(∂%𝜙)-−1
2𝑚,
-𝜙-+𝜆!𝜙 𝜇(𝑥)+𝜆"𝜙 𝜎(𝑥)
whe e 𝜇(𝑥)=> 𝑚.𝛿(𝑥−𝑥.)@@
.and 𝜎(𝑥)=4 ∣𝑞/∣ 𝛿(𝑥−𝑥/)@
/
a e gauge-in a ian
composi e ope a o s.
Func ional in eg a ion o e 𝜙 gi es he non-local e m
ℒ011 =−𝑔!" ∫ 𝜇(𝑥) 𝐷(𝑥−𝑦) 𝜎(𝑦) 𝑑#𝑦,@@@@@@@@@𝑔!" = 𝜆!𝜆"
wi h 𝐷(𝑥−𝑦) he p opaga o o 𝜙.
The esul ing Hamil onian
𝐻
E!" =−𝑔!"∑𝑚.∣ 𝑞/∣𝐷./
is he cu a u e-p ojec ion ope a o p e iously in oduced phenomenologically.
Gauge in a iance is p ese ed because bo h 𝜇 and 𝜎 a e colo single s, and
eno malizabili y ollows o ene gies 𝐸 <𝑚,.
3 Physical In e p e a ion
The e ec i e e ex ep esen s exchange o cu a u e quan a be ween egions o mass
densi y and cha ge ampli ude, gene a ing an en opic co ec ion o he bound-s a e
ene gy:
𝐸!" ≃ −𝐺!"
𝑅 𝑀2𝑄345
I s sign co esponds o cu a u e o ien a ion: ou wa d (posi i e) o inwa d (nega i e) lux
h ough he colo mani old de e mines elec ic-cha ge sign.
Minimiza ion o he o al ee-ene gy unc ional
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𝐹 =𝐸()* +𝐸!" +𝛽𝑆6789
selec s he physically ealized p ojec ion, ep oducing empi ical cha ge-mass o de ings
wi hin isomul iple s.
4 Consis ency wi h La ice QCD and QFT
Iden i ying 𝐺!"/𝑅 wi h a cu a u e-suscep ibili y scale yields
𝐺!" ∼ 𝜒$
:/#/Λ()*
gi ing e ec i e binding ene gies ≈ ens o MeV o adii 0.5–0.8 m—consis en wi h
la ice da a [7–9].
Because 𝜙 is a scala single , Lo en z and gauge in a iance emain in ac , and he heo y
unc ions as a s anda d e ec i e QFT below 𝑚,.
The β sel -consis ency loop co esponds o esumma ion o sel -ene gy diag ams
ensu ing s abili y o he composi e adius, pa alleling a ia ional ea men s in non-
pe u ba i e QCD [13–15].
5 Ou look
This cons uc ion demons a es ha he cu a u e-p ojec ion ope a o can be de i ed
di ec ly om he QCD Lag angian by in eg a ing ou a cu a u e media o .
I hus p o ides a i m ield- heo e ic ounda ion o mass–cha ge coupling and cha ge-
o ien a ion p ojec ion wi hou al e ing gauge s uc u e.
Fu u e la ice s udies could es he p edic ed mixed co ela o
𝐶%<(𝑟)=⟨𝜇(0)𝜎(𝑟)⟩
linking opological cu a u e o had onic mass spli ings and po en ially e ealing deepe
composi eness such as ishon subs uc u e [16–20].
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