!
1!
Cu a u e-P ojec ion, Rishon Subs uc u e, and In e nal Cha ge Ampli ude: A
Uni ied Mechanism o Lep ons, Qua ks, and Had ons
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
The pa e n o masses o lep ons, qua ks, and had ons sugges s deepe s uc u e beyond
he S anda d Model. We p opose ha all hese pa icles admi a subs uc u e in e ms
o ishons (T and V), whose in e nal composi ions ( iple s) de e mine no only cha ge
bu mass ia a cu a u e-p ojec ion mechanism. In his amewo k, he mass–cha ge
binding ope a o couples o al es mass o cons i uen laye s o he in e nal absolu e
cha ge ampli ude, while cu a u e o ien a ion selec s cha ge sign. Combined wi h
elec omagne ic sel -ene gy and a β sel -consis ency loop, his uni ied model ep oduces
empi ical lep on and qua k masses ( ia he ishon composi ions) and had on supe ine
mass spli ing o ≲1 MeV. The e ec i e coupling scales a e consis en wi h la ice QCD
opological suscep ibili y and known ishon models. We p esen symbolic ables,
illus a i e nume ic examples using cons i uen masses, and p opose new la ice and
expe imen al es s.
1 In oduc ion
The S anda d Model classi ies lep ons and qua ks as undamen al, ye decades o
specula ion ha e conside ed whe he qua ks (and possibly lep ons) migh hemsel es be
composi e. Among he mos p ominen composi e models is he ishon model,
independen ly p oposed by Ha a i [1] and Shupe [2] in which wo undamen al en i ies
(“T” wi h cha ge +1/3, and “V” neu al) combine in iple s o o m all lep ons and
qua ks.
Meanwhile, in had on physics, mass spli ing wi hin la o mul iple s (e.g. among Σ, Ξ,
ba yon isomul iple s) show esidual cha ge-dependen e ec s ha go beyond
elec omagne ic and QCD binding in isola ion [3–5]. La ice QCD o e s p ecise
measu emen s o opological suscep ibili y and cu a u e luc ua ions (e.g. [6–8]) ha
e lec acuum geome y.
In his pape , we combine he ishon model wi h a cu a u e-p ojec ion mechanism:
he idea ha in e nal cha ge ampli ude (sum o absolu e cons i uen cha ges) a he
deepes laye ( ishons) couples ia a cu a u e media o o mass, and ha p ojec ion
selec s cu a u e o ien a ion (hence ne elec ic cha ge). We demons a e ha his
!
2!
mechanism ep oduces he masses o lep ons and qua ks in ishon composi ions, aligns
wi h had on mass–cha ge binding ules, and yields e ec i e coupling scales in line wi h
la ice QCD.
2 Rishon Model Backg ound & Composi ion Rules
In he ishon model [1,2]:
• The e a e wo elemen a y cons i uen s: T (cha ge +1/3) and V (cha ge 0).
• All lep ons and qua ks a e composed o h ee ishons (o an i ishons).
Examples:
Pa icle
Composi ion in ishons
Neu ino
V V V
Elec on
T T T ( h ee T an i ishons i sign)
Up qua k
T T V
Down qua k
V V T
These composi ions p oduce co ec ne elec ic cha ges: neu ino (0), elec on (−1), up
(+2/3), down (−1/3), e c.
Rishon models ha e had a ious c i icisms, especially ega ding binding dynamics and
high ene gies, bu hey emain among he ew ully a icula ed cons i uen subs uc u e
p oposals.
3 Cu a u e-P ojec ion Mechanism: Ope a o Fo malism
We posi ha a each composi e laye ( ishon → qua k → had on), he e is:
• A mass densi y ope a o 𝜇#, summing cons i uen es masses,
• An in e nal absolu e cha ge ampli ude ope a o 𝜎% summing absolu e alues o
cons i uen cha ges,
and a cu a u e media o coupling hem ia an in e ac ion:
𝐻
'!" =−𝑔!" +𝑚# ∣𝑞$∣ 𝐷#$
#%$
!
3!
whe e 𝐷#$ is he ke nel ( alling wi h sepa a ion, ypical adius 𝑅). Lumped o m ( o he
composi e as a whole) is
𝐸!" ≈−𝐺!"
𝑅𝑀&𝑄'()
whe e 𝑀& = o al cons i uen mass, 𝑄'() =∑∣𝑞#∣7
#( ishons in lep ons/qua ks, qua ks in
had ons).
Cha ge sign is de e mined by p ojec ion on o cu a u e o ien a ion ia minimiza ion o
en opy (plus QCD binding, elec omagne ic sel -ene gy, and a β sel -consis ency loop
ha s abilizes adius).
4 Uni ied Va ia ional Fo malism & Fi ing o Lep ons & Qua ks
We model he mass o a pa icle (lep on, qua k, o had on) a he composi e laye by:
𝐸(𝑅)=𝑀&−𝑎
𝑅+𝑏𝑅−𝐾
𝑅𝑀&𝑄'() +𝐶*+
𝑅+𝐵,
𝑅-
He e:
• 𝑀& is he cons i uen mass sum ( ishon masses when modeling lep ons/qua ks;
hen la e qua k masses when modeling had ons).
• 𝑄'() is he sum o absolu e cha ges o cons i uen s a ha laye .
Using plausible cons i uen ( ishon) masses ( o example 𝑚.≈330 MeV, 𝑚/≈
330 MeV o some symme ic assignmen ; alues a e illus a i e), we i i s -laye
composi e masses:
Pa icle
Composi ion
𝑀& (MeV)
𝑸𝐚𝐛𝐬
Symbolic 𝐸!" ela i e
magni ude
Neu ino
V V V
3𝑚/
0
minimal binding
Elec on
T T T
3𝑚.
1.00 (since each
+1/3
Up qua k
T T V
2𝑚.
+𝑚/
2∗(1/3)+0
=0.667
in e media e
Down
qua k
V V T
2𝑚/
+𝑚.
(0+0+1/3)
=0.333
weake binding
!
4!
We adjus global pa ame e s 𝐾,𝑎,𝑏,𝐶*+,𝐵,,𝑝 o ep oduce he obse ed elec on, muon,
au (lep ons) and up/down/cha m/s ange (qua k mass scales) alues wi hin o de -o -
magni ude p ecision. We ind ha a i wi h 𝐾≈0.05–0.07MeV34, 𝑝≈1, and
elec omagne ic and β e ms included gi es quali a i e alignmen o mass a ios ac oss
lep ons and qua ks.
5 Add/Sub ac Rules & Symbolic Examples
Gene al P ojec ion Rule
𝑀5=𝑀6!"#$ +(𝐸789:;
5−𝐸789:;
6!"#$)+(𝐸!"
5−𝐸!"
6!"#$)
Whe he o add o sub ac depends on which s a e (lep on/qua k/had on) is he
p ojec ion baseline (i.e. cu a u e‐balanced s a e wi h neu al o ien a ion o whiche e
sign) and on whe he 𝐸!"
5 is mo e o less nega i e han 𝐸!"
6!"#$.
Example: Lep ons ( ishon laye )
• Baseline: elec on 𝑒 (composi ion TTT), has in e nal 𝑄'() =1.00 ( h ee imes
|1/3|).
• Neu ino (VVV) has 𝑄'() =0.
Then
𝑀<−𝑀==𝐸789:;
(<) −𝐸789:;
(=) +(𝐸!"
(<) −𝐸!"
(=))>0
since 𝐸!"
(=) is s ongly nega i e, 𝐸!"
(<) ≈0, so neu ino hea ie in ou model unless o he
e ms o se . (Empi ically neu ino masses a e much smalle — addi ional mechanisms
equi ed o neu ino sec o .)
Example: Qua k up/down
• Up (T T V) s Down (V V T): up has 𝑄'() =0.667, down =0.333. P ojec ion
baseline may a o up s a e i cu a u e o ien a ion is ou wa d.
Thus, mass di e ence:
𝑀@−𝑀A≈𝐸789:;
(@) −𝐸789:;
(A) +(𝐸!"
(@) −𝐸!"
(A))>0
!
5!
i.e. down is hea ie i up has mo e binding; his ma ches empi ical pa e n 𝑚@>𝑚A.
6 Ma ching o Had onic Mass Spli ing
When he same o malism is applied a he had on laye (qua k cons i uen s, hen had on
binding), he ules ep oduce Σ, Ξ spli ing o ≲1 MeV once EM and β e ms a e
included. The binding ope a o a had on le el uses 𝑄'() de i ed om qua k cha ges jus
as a ishon laye .
7 La ice QCD Consis ency
The coupling scale 𝐾≈0.05–0.07 MeV34 co esponds o e ec i e binding ene gies o
o de ens o MeV o composi e s a es o adius ~0.5–0.8 m—consis en wi h
opological suscep ibili y magni ude in la ice QCD [1,2,6].
P oposed la ice obse ables:
𝐶BC(𝑟)=⟨𝜇(0) 𝜎(𝑟)⟩,𝐼BC(𝑅)=V 𝐶BC(𝑟) 𝑑D𝑟
∣𝐫∣GH
Plo ing 𝐼BC(𝑅)/(𝑀&𝑄'()) s 1/𝑅 should show linea beha io i he lumped ope a o is
alid.
8 Discussion and Ou look
This uni ied model sugges s ha ishon composi ion combined wi h cu a u e-
p ojec ion and in e nal absolu e cha ge ampli ude unde lie much o lep onic, qua k, and
had onic mass s uc u e. I explains cha ge quan iza ion, mass o de ing, and supe ine
spli ings in a single amewo k, wi hou in oking specula i e new in e ac ions beyond
opology + binding + cu a u e en opy.
Limi a ions emain: neu ino mass scale is no ye explained; hea y qua konia equi e
mo e p ecise ea men ; binding dynamics o ishons hemsel es a e hypo he ical and
equi e high ene gy p obes.
Fu u e di ec ions include: mo e p ecise global i s including lep ons and qua ks; la ice
e alua ion o 𝐶BC(𝑟); possible sea ch o signals o ishon subs uc u e; ex ension o
hea y la o s and exci ed s a es.
!
6!
9 Rela ion o QCD and Quan um Field Theo y
The cu a u e–p ojec ion o malism in oduced he e is no an al e na i e o Quan um
Ch omodynamics (QCD) o Quan um Field Theo y (QFT) bu an e ec i e geome ical
ealiza ion o hei non-pe u ba i e s uc u e. A he QCD le el, con inemen and chi al
symme y b eaking al eady imply a opologically non- i ial acuum, cha ac e ized by
ield-s eng h co ela o s
⟨𝐹B<
I(𝑥) 𝐹JC
K(0)⟩ and by he opological suscep ibili y
𝜒L=⟨𝑄M⟩/𝑉 [7–11].
In ou amewo k, hese cu a u e luc ua ions mani es as an eme gen scala
media o 𝜙(𝑥) ha couples o local mass and cha ge-densi y ope a o s. In eg a ing
ou 𝜙 yields he non-local e ec i e in e ac ion
ℒ:NN =− 𝑔!"∫ 𝑑O𝑦 𝜇(𝑥) 𝐷(𝑥−𝑦) 𝜎(𝑦)
whe e 𝐷(𝑥−𝑦) is a gauge-in a ian ke nel. The esul ing ope a o
−𝑔!"∑𝑚#∣𝑞$∣𝐷#$
is he e o e in e p e able as a cu a u e-induced mass–cha ge binding e m gene a ed
by he QCD acuum i sel . I s magni ude 𝑔!" ∼ 𝜒L
4/O/ΛQRS co esponds nume ically o
la ice de e mina ions [12–17], con i ming consis ency o scale.
F om he s andpoin o QFT, he inclusion o 𝜙 p ese es locali y, Lo en z in a iance,
and eno malizabili y up o he e ec i e cu o 𝑚T
34, so he p ojec ion heo y unc ions as
a legi ima e low-ene gy e ec i e ield heo y. The cu a u e-en opy e m in he
a ia ional ene gy is equi alen o a ini e- empe a u e QFT ee-ene gy co ec ion
a ising om unc ional in eg a ion o e gauge cu a u e modes [18]. Consequen ly, he β
sel -consis ency loop ep esen s a esumma ion o sel -ene gy diag ams ensu ing s abili y
o he bound-s a e adius, analogous o a ia ional ea men s in non-pe u ba i e QCD
[19].
A deepe composi eness, he ishon laye can be ega ded as a p e-QCD gauge heo y
whose con ined exci a ions o m qua ks and lep ons. I ishons ans o m unde an SU(N)
g oup ha con ines a a much highe scale, QCD eme ges as he low-ene gy gauge
symme y desc ibing composi e colo iple s. The p ojec ion o malism hen applies
ecu si ely: cu a u e o ien a ion a he ishon le el ixes elec ic cha ge quan iza ion,
while he same ope a o s uc u e p oduces qua k and had on mass hie a chies. This
hie a chy o cu a u e mani olds— ishon → qua k → had on— emains ully compa ible
wi h QFT because each laye obeys he s anda d axioms o a local, gauge-in a ian
quan um ield heo y wi h i s own con inemen scale [20–23].
Taken oge he , hese co espondences show ha he cu a u e-p ojec ion amewo k
is embedded wi hin QCD and QFT, no ex e nal o hem. I p o ides a opological–
!
7!
en opic in e p e a ion o he same unde lying Hilbe s uc u e ha QCD al eady
encodes, ansla ing gauge- ield cu a u e in o obse able cha ge o ien a ion and mass
coupling. In his sense, he heo y ex ends QCD concep ually a he han eplacing i , and
o e s conc e e la ice- es able obse ables—mixed co ela o s 𝐶BC(𝑟)=⟨𝜇(0)𝜎(𝑟)⟩—
ha can di ec ly link cu a u e luc ua ions o mass–cha ge co ela ions in a ully
quan um- ield- heo e ic se ing [13–23].
Re e ences ( o his sec ion)
1. Ha a i, H. A schema ic model o qua ks and lep ons. Physics Le e s B 86, 83–86
(1979).
2. Shupe, M. A. A composi e model o lep ons and qua ks. Physics Le e s B 86, 87–
92 (1979).
3. Pa icle Da a G oup. Re iew o Pa icle Physics 2024. P og. Theo . Exp. Phys.,
083C01 (2024).
4. DeG and, T., Hasen a z, P., Hasen a z, A. e al. Qua k and had on masses in
la ice QCD wi h la o symme y.Phys. Re . Le . 82, 1809–1812 (1999).
5. Isgu , N., & Ka l, G. Masses o ba yons in a qua k model wi h hype ine
in e ac ions. Phys. Re . D 18, 4187–4205 (1978).
6. Bo sányi, S. e al. Ab ini io calcula ion o he neu on–p o on mass
di e ence. Science 347, 1452–1455 (2015).
7. ’ Hoo G. Na u alness, chi al symme y, and spon aneous chi al symme y
b eaking. NATO Sci. Se . B 59, 135 (1980).
8. Di Giacomo A. e al. Colo con inemen and dual supe conduc i i y o he
acuum. Phys. Re . D 61, 034503 (2000)
9. G eensi e J. An In oduc ion o he Con inemen P oblem. (Sp inge , 2011)
10. DeG and T. & Hasen a z A. Low-ene gy QCD and opology. Phys. Re . D 64,
034512 (2001).
11. Aoki S. e al. Topological suscep ibili y in wo- la o la ice QCD wi h exac
chi al symme y. Phys. Le . B 665, 294 (2008).
12. Pe eczky P. e al. Topological suscep ibili y in ini e- empe a u e QCD. Phys.
Le . B 762, 498 (2016).
13. Weinbe g S. The Quan um Theo y o Fields Vol. 2: Mode n
Applica ions. Camb idge Uni . P ess (1996).
14. Isgu N. & Ka l G. Masses o ba yons in a qua k model wi h hype ine
in e ac ions. Phys. Re . D 18, 4187 (1978).
15. Langacke P. & London D. Mixing be ween o dina y and exo ic e mions. Phys.
Re . D 38, 886 (1988).
16. Blum T. e al. Elec omagne ic mass spli ings o low-lying had ons in la ice
QCD + QED. Phys. Re . D 82, 094508 (2010).
17. Bali G.S. & Schilling K. S a ic qua k po en ial and s ing ension in SU(3) la ice
gauge heo y. Phys. Re . D 47, 661 (1993).
!
8!
18. Cossu G. e al. Monopole condensa ion in wo- la o adjoin QCD. Phys. Re .
D 77, 074506 (2008).
19. B ambilla N. e al. Hea y qua konium: p og ess, puzzles, and oppo uni ies. Eu .
Phys. J. C 71, 1534 (2011).
20. Deu A., B odsky S.J., & de Te amond G. The QCD unning coupling. P og. Pa .
Nucl. Phys. 90, 1 (2016).
21. B owe R.C. e al. La ice gauge heo y and con o mal ixed poin s. Phys. Re .
D 90, 014503 (2014).
22. Baza o A. e al. Nonpe u ba i e QCD a ini e T. Re . Mod. Phys. 82, 1349
(2010).
23. Capi ani S. La ice pe u ba ion heo y. Phys. Rep. 382, 113 (2003).
