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DOI: 10.14738/aivp.85.9116
Publication Date: 05th October, 2020
URL: http://dx.doi.org/10.14738/aivp.85.9116
Unification of Strong-Weak Interactions and Possible Unified
Scheme of Four-Interactions
Yi-Fang Chang
Department of Physics, Yunnan University, Kunming 650091, China
yifangch@sina.com
ABSTRACT
First, various known unified theories of interactions in particle physics are reviewed. Next, strong and
weak interactions are all short-range, which should more be unified. Except different action ranges
their main character is: strong interactions are attraction each other, and weak interactions are mutual
repulsion and derive decay. We propose a possible method on their unification, whose coupling
constants are negative and positive, respectively. Further, we propose a figure on the unification of
the four basic interactionsin three-dimensionalspace, and search some possible tests and predictions,
for example, strong-weak interactions transform each other, some waves may be produced. Finally,
based on the simplest unified gauge group GL(6,C) of four-interactions, a possible form of Lagrangian
is researched. Some relations and equations of different interactions are discussed.
Key words: interaction; unification; short-range force; coupling constant; Lagrangian; equation
DOI ·10.13140/RG.2.2.18002.71368
1 Introduction
In nature have four different strong, electromagnetic, weak and gravitational interactions. Their
distinction justified by relative strengths. The strength of weak interaction is very small, but it involves
all observed particles (hadrons and leptons) except photon. Only proton, electron, neutrino and
photon are stable, which are just the smallest particles with four interactions Other particles all decay.
Another important property is the range of interactions. Electromagnetic and gravitational
interactions have infinite range. Strong and weak interactions have short range, and weak interaction
possesses the shortest range of all interactions [1].
It is known that the coupling constant of strong interaction is / 4 15 2
g π ≅ , the coupling constant of
weak interaction is -6 ≅ 10 . The range of strong interaction is 1 -13 ≅ 1.4×10 − mπ cm [1], the range of weak
interaction should be 1 -16 ≈ 2×10 − mWZ cm. In this paper, we research unification of strong and weak
interactions as short-range, and some results of the simplest unified gauge group GL(6,C) of four- interactions.
2 Various Known Unifications on Interactions
The unification of various interactions is always an important question in physics. The early
unifications are mainly some theories on the gravitational and electromagnetic fields, which are all
long-range [2-4]. We proposed the gravitational field and the source-free electromagnetic field can be
unified preliminarily by the equations in the Riemannian geometry [5,6]:
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* i
klm
i Rklm = κT . (1)
Both are contractions of im and ik, respectively. If * i
klm
i Rklm = κT =constant, so it will be equivalent to
the Yang’s gravitational equations [7]:
0 Rkm;l − Rkl;m = , (2)
which include Rlm = 0 . From Rlm = 0 we can obtain the Lorentz equations of motion, the first system
and second source-free system of Maxwell field equations. This unification can be included in the
gauge theory, and the unified gauge group is SL(2,C)×U(1)=GL(2,C), which is just the same as the
gauge group of the Riemannian manifold. Another unification on the general nonsymmetric metric
field with high-dimensional space-time is analyzed mathematically, and we proposed an imaginative
representation on the ten dimensional space-time [6].
The mathematical basis in unified theories of particle physicsis the gauge groups. Weinberg and Salam
proposed a well-known electroweak theory unified the weak and electromagnetic interactions, whose
unified gauge group is SU(2)×U(1)=U(2) [8,9]. Further, various grand unified theories (GUT) of the
strong, weak and electromagnetic interactions are researched [10-16], whose pioneer is Bars-Halpern- Yoshimura model [10,11]. Pati and Salam proposed the unified lepton-hadron symmetry and a gauge
theory SU(2)×U(1)× SU(3') of the basic interaction [12]. There is the same gauge group in Itoh- Minamikawa-Miura-Watanabe model [13]. A famous theory is the Georgi-Glashow SU(5) theory [14],
which is only a non-Abel field in which proton will decay to electron. Moreover, there are Fritzsch- Minkowski SU(n)×SU(n) (n=8,12,16) and SO(n) (n=10,14) unified interactions theories of leptons and
hadrons [15], and a universal gauge theory model based on E(6) [16]. Calmet, et al., showed grand
unification and some enhanced quantum gravitational effects [17]. Blumenhagen investigated gauge
coupling unification for F-theory SU(5) GUT with gauge symmetry breaking via nontrivial hypercharge
flux [18].
Einstein gravitational Lagrangian possesses two invariances: the GL(4,R) invariance of Einstein under
coordinate transformations, and the SL(2,C) gauge invariant of Weyl. The strong interaction of quarks
possesses internal SU(3) symmetry. From these symmetries Isham, Salam and Strathdee proposed the
unified scheme on gravitational and strong interactions, whose gauge group is SU(3)×SL(2,C)=SL(6,C)
[19-21].
The supersymmetry theory describes a basic symmetry between bosons and fermions, and arouses
the supergravity in the gravitational theory, and derives a superstring combining a string model. This
is related with the unified theory of interactions. Barr and Raby proved minimal SO(10) unification in
the supersymmetric grand unified theory [22]. Kakushadze and Tye researched the classification of
three-family grand unification of SO(10), E(6), SU(5) and SU(6) models in string theory [23]. Das and
Jain discussed dynamical gauge symmetry breaking in an SU(3)×U(1) extension of the standard model
[24]. Albright and Barr discussed explicit SO(10) and U(1)×Z(2)×Z(2) supersymmetric grand unified
model for the Higgs and Yukawa sectors [25]. Moreover, the unified forms of supersymmetry are also
connected with the statistics unifying BE and FD statistics, and with the possible violation of Pauli
exclusion principle [26,5].
A possible development is the higher dimensional complex space [27]. Triantaphyllou and Zoupanos
searched strongly interacting fermions from higher (4-12) dimensional E(8) × E(8)’ unified gauge
theory [28].
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European Journal of Applied Sciences, Volume 8 No 5, October 2020; pp: 29-45
URL: http://dx.doi.org/10.14738/aivp.85.9116 30
3 Unification of Strong-Weak Interactions with Short-Range
3.1 Some Possible Unified Methods
It is known that the strong and weak interactions are all the short-range, which should more be
unified. But, so far their unification is almost neglected.
Recently, some confusion exists on weak interactions. How do weak interactions may overcome strong
interactions leading to particle decay? This should be for the shorter distance, and smaller distance
corresponds to higher energy. The weak interactions include two types: one determines particle
decay, and another exists for interaction of all particles except photon, especially leptons, which are
mainly electron-neutrino (e-ν e ) and ν -ν interactions, and which exchange large mass W and Z,
respectively. But, neutrino ν is introduced to ensure energy conservation, and it can run through the
Earth without hindrance, so ν -ν is very small interaction. Strong interactions between nucleons (p,
n) pass through 0 π ,π + and gluon. Weak interaction among all baryons and leptons pass through
0 W , Z + (m=80, 91GeV).
A base of electroweak unification is that -γ 0 Z are neutral particles and neutral currents. Base of the
unification of weak and strong interactions is that 0 W - Z ± and 0 π -π ± (for u and d, m=140,135MeV)
are symmetry, and the difference of both interaction scales is 90/0.14=64.286. This is strong
interaction as 13 10− cm, so weak interaction scale is 15 16 10 -10 − − cm. Further, 0 K - K ± (add s,
m=494,497MeV), 0 D - D ± (add c, m=1870,1865MeV), 0 B - B ± (add b, m=5279MeV) all are
completely the systems.
We propose the possible theoretical approach on unification of strong and weak interactions can be:
(1) The removal of electromagnetic fields from the grand unified theory (GUT) seems to be
the easiest way.
(2) In the electroweak unified theory the electromagnetic field is transformed into a strong
interaction. If the gluon is simplified to one type, it is similar to a photon, and can be
obtained as a similar electroweak unified theory, in which positive and negative charges
correspond to quarks u and d. It has neutral charged (pn, nn) and (pp) strong interaction,
but p-p has weaker charge interactions. In experiments pp, pn (elastic and total), and
p p + − π ,π and K p K n K p K n + + − − , , , are all similar, but they are not the same, so
there have symmetry violations.
(3) The juxtaposed strong and weak interactions and their interactions each other, whose
Lagrangian is:
L = Ls + Lw + Lin . (3)
We may establish a simplified theory of strong and weak unity: the first generation quark- lepton has basic symmetry, and are all SU(2). The second and third generations are some
excited states, and are heavy electrons and heavy quarks. SU(3) is originally three kinds
of quark u, d, s, and now become three kinds of color, and total gauge group should be
SU q SU c (6) × (3) . Leptons with weak interaction are three generations SU(2).
(4) The best way seems to transform each other with distance, energy-momentum, and
action strength, etc., so the interaction direction and coupling constants are opposite.
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Except different action ranges their main character is: strong interactions are attraction
each other, and weak interactions are mutual repulsion and derive decay. Now we
propose a new possible method on their unification, whose coupling constants are
negative and positive, respectively. Any attract forces cannot obtain decay of particles.
In electromagnetic field attraction or repulsion is determined by opposite charges. In
particle physics strong and weak interactions should be determined by opposite coupling
constants.
3.2 Change and Unification of Coupling Constants
It is known that the strong interaction is a big attraction, which corresponds to the quarks
confinement. When the distance decreases, it is asymptotic freedom, i.e., no interaction. They
correspond to that strong interaction, SU(3) and QCD first become zero along with distance decrease.
Then, the distance becomes smaller and the corresponding energy becomes larger, it becomes weak
interaction, SU(2) and QWD (quantum weak dynamics), and derive decay. It has a critical point of
transformation between strong and weak interactions. Further, probably it may include
electromagnetic interactions.
The asymptotic freedom is the biggest characteristic of the QCD. The formula is:
ln( / )
4
4 ( ) 2 2
0
2
2
S Λ = = Q
g Q S
β
π
π
α . (4)
When energy
2 Q ∞, the coupling constant of strong interaction ( )
2 αS Q 0. Further, reduction of
distance should be weak interaction with repulsive force. It can be inaccessibility and exclusion, are
repulsion. This can be the unity of strong and weak interactions with short-range.
Defining a new kind of renormalized coupling constant g(μ) depends on a sliding energy scale μ
[29]. The ‘t Hooft-Weinberg renormalization group equations are [30-32]:
[ ] ( ,... , , , ) 0 2 2
2 1
2
2
2 − Γ = ∂
∂
+
∂
∂
+
∂
∂
n p p M m
m
m
M
M γ γ n λ
λ β θ , (5)
where 2
2 ( ) dM
d M λ β λ = , etc.
The coupling satisfies the renormalization group equations [33-36]:
( ) ( ...) 4
2
3
1
2
2 0
2 = s = − s + s + s +
R
s
R b b b
d
d β α α α α
μ
α
μ , (6)
where (33 2 )/(12 ) b0 = − nf π is referred to as the l-loop beta function coefficient, etc.
Some possible forms of the function ( )l β g depend on the running coupling gl . ( ) β gl may be
positive or negative or 0. Fig.1 is Fig.18.5b [37] and Fig.18.4c (but, ( ) ( ) β g → −β gl and g → q * )
[29] and Fig.3.3.2b [38], ( ) β gl <0 (for 0< l g <q), ( ) β gl =0 (when l g =q), and ( ) β gl >0 (for l g >q).
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European Journal of Applied Sciences, Volume 8 No 5, October 2020; pp: 29-45
URL: http://dx.doi.org/10.14738/aivp.85.9116 32
Fig.1 l g - ( ) β gl
In Fig.1 we propose that ( )l β g >0 is strong interaction with big gl and attraction, ( ) β gl <0 is weak
interaction with small l g and opposite repulsion, and ( ) β gl =0 with l g =q is the asymptotic freedom.
The theory of strong interactions based on the gauge group SU(3) is asymptotically free, which first is
discovered by Gross and Wilczek [39] and Politzer [40] in 1973.
The solution of Eq.(6) may obtain [29]:
1 2 2 1/( 1)
2
[1 ( 1) ln( / )] (Q ) − − + − Λ = s n n b n α Q
α
α . (7)
The effective ‘running’ coupling constant with SU(N) is [36-38]:
[11 ( ) 4 ( )]ln( / ) 12
1 4 (Q )
2 2
2
2
2
+ − Λ
= =
C G n T R Q
g
f
s
π
α
α
π
α , (8)
where nf is the number of quark flavors participating in the interaction at this Q. It is determined by
momentum and energy. When → ∞ 2 Q (higher energy and shorter range), α s =0, i.e., the
asymptotically freedom. The experiments shown α s =0.1184 decrease, but it is not α s 0.
For SU(N) ( ) C2 G =N. For SU(3) and QCD, ( ) C2 G =3 and T(R)=1/2,
(11 2 / 3)ln( / )
4 (Q ) 2 2
2
− Λ = nf Q s
π
α . (9)
General nf <33/2, α s >0. Strong-weak interactions and QED may be unified by δ
2
g0 , in which δ >0
for QCD, δ <0 for SU(2), and δ =0 for QED [37]. Let nf =33/2, α s =∞. Let nf =6 and β =0, α f =12.726
is an inflection point, from infrared attraction to ultra-violet repulsion [38].
In each energy range there have a different scale Λ, for example, Λ=253MeV [29] or Λ=500MeV
[41]. For unified α s in Eq.(8), if nf <33/2, α s >0 for Q> Λ ; when Q∞, α s 0 is the asymptotic
freedom; when Q=Λ, α s ∞ is an impossible state; α s <0 for Q<Λ.
For SU(2) ( ) C2 G =2 and T(R)=1/2,
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(11 )ln( / )
4 (Q ) 2 2
2
− Λ = nf Q w
π
α . (10)
For weak interaction and QWD, the coupling constant should be the same change. The known Fermi
coupling constant 3 5 2 /( ) 1.16637 10− − GF c = × GeV <0. For Eq.(10) if nf >11, α w <0; or α w <0 for Q<
Λ. Or Eq.(8) becomes:
(11 )ln( / ) 6
1 4 (Q )
2 2
2
2
+ − Λ
= =
n Q
g
f
w
π
α
α
π
α . (11)
For α w in Eq.(11), if nf <11, α w decrease for Q>Λ; when Q∞, α w0 is the asymptotic freedom;
when Q=Λ, α w is an inflection point; α w increase for Q<Λ. Probably, it corresponds to the “strong
decay”. Assume that nf =2, if ln(Q / Λ) < −(π / 3α) , it will be α w <0. If α =1, −1.05 Q < Λe . If α =15
is strong coupling constant, −0.07 Q < Λe .
For QED and U(1), ( ) C2 G =0 and T(R)=1,
1 ( / 3 ) ln( / ) (Q ) 2 2
2
− Λ = nf Q em α π
α
α . (12)
For αem in Eq.(12), if nf =2, αem increase for Q>Λ; when Q∞, αem 0 is the asymptotic freedom;
when Q=Λ, αem is an inflection point; α decrease for Q<Λ. If ln(Q / Λ) > (3π / 4α), it will be αem
<0. If α =1, 2.355 Q > Λe . If α =15 is strong coupling constant, 0.157 Q > Λe .
For QED, β function is positive. But, for Yang-Mills (YM) field and strong interaction, β function is
negative. For weak interaction, β function should be positive. So β function is related with QCD and
QWD, and unified strong and weak interactions.
Assume that α all hold in general cases. For the extensive air shower (EAS) nf >33/2, so α <0, and
corresponds to repulsion. Further, when strong interaction becomes weak interaction and
electromagnetic interaction, the scaling parameter Λ as the earliest cutoff factors linked to the
renormalization process should be changeable, for example, Λ = me for QED [38].
Assume that the coupling constant g<0 corresponds to repulsion, ε = −g >0 in QCD. Fig.1 may
describe unification of strong and weak interactions.
It is known that the short-range potential V r g e r mr ( ) / 0
− = , in which the coupling constant g0 <0 is
attraction and strong interaction, and g0 >0 is repulsion and weak interaction, and g0 =0 corresponds
to the asymptotic freedom. = g0 ε in QCD. When the combination of positive and negative quark
forms the color singlet state, whose interaction ε = −8 / 3 < 0 is attraction.
Quarks confinement corresponds to a large scale of strong interactions. Strong interactions exist
outside the hadron, and within the hadron QCD tends to asymptotically freedom and to mutually
repulsive weak interactions. Thus two short-range interactions are unified. Gluon (meson) and W-Z
boson are unified. This no interaction must go through inevitable transition from strong interaction to
weak interaction with the smaller scale of mutual repulsion.
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Moreover, we proposed that hadrons are similar to the limit cycles, whose exterior is an attractive
strong interaction, and interior is a repulsive weak interaction which derives decays, and both
combination obtains hadron [42]. Strong interaction with longer-range force; and interior of attractor
all is repulsive each other, which is weak interaction with shorter-range force. A zero dimensional
strange attractor becomes a point charge, and corresponds to electron. Electron e in electromagnetic
interaction corresponds to stable focal point, and strong interaction is the limit cycle. At certain degree
GUT is namely unification of the limit cycles and singular attractor. Limit of strong and weak
interactions is namely electromagnetic interaction of positive and negative charges, whose mass m=0
corresponds to zero dimension. For different interactions the equations of quantum mechanics are
different. Electromagnetic interaction is Abel group U(1), and strong and weak interactions are non- Abel groups SU(2) and SU(3), and are Yang-Mills (YM) field. Further, the cycle may extend to high
dimensions. The non-Abel group equations of interactions with SU(N) symmetry obtain the period
solution, which corresponds to the limit cycle.
The decay modes and fractions in particle physics are some quantitative and very complex questions.
We discussed various decays of particles and some known decay formulas. Many important decays of
particles ( −+ 0 mesons, + (1/ 2) baryons and ± μ ) and some known decays of resonances can be
generally described by the square of some types of the associated Legendre functions l Pl and l Pl +1 .
Its universal formula is:
| | 2
2
| ( ) | 4
m P x
N
G m
i l
α
π
Γ = , (15)
in which x is the ratio of the mass-sum of the final state and the mass of the initial state of particle. It
is combined with the ∆I=1/2 rule, then the agreements are very good. The decay formulas of the
similar decay modes are the same. For the same types of particles, the coupling constants are kept to
be invariant, and only six constants are independent. The simplest dynamical basis is probably that
the decay formulas are the square of the solution of free field equation of the decay particles in the
momentum coordinate system [5,44]. Further, we discuss some general decays and their rules, and
apply the decay formulas to some massive hadrons [44]. Various baryons pass through weak
interaction to become proton. Various mesons pass through weak interaction to become electron or
photon.
3.4 Unified Figures on Interactions and Possible Tests
Based on Dirac’s negative energy state, we developed to the negative matter as the simplest candidate
of dark matter and dark energy [45-51], and proposed some tests and a judgment test [52]. Further,
we proposed the most perfect symmetrical world on the four types of positive, opposite, and negative,
negative-opposite matters (Fig.2) [51,53].
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Yi-Fang Chang; Unification of Strong-Weak Interactions and Possible Unified Scheme of Four-Interactions.
European Journal of Applied Sciences, Volume 8 No 5, October 2020; pp: 29-45
URL: http://dx.doi.org/10.14738/aivp.85.9116 36
Fig. 2. A new most perfect symmetrical world
Fig. 2 as a two-dimensional plane also corresponds to gravitational and electromagnetic fields
determined by mass and charge, respectively. We may combine the four known fundamental
interactions to develop into the four-dimensional space. But in some respect, this can be simplified to
three-dimensional space, where the third dimension is assumed to be the strong and weak
interactions of the short-range. It is known that the strong interactions are attraction each other, and
the weak interactions are mutual repulsion and derive decay, both correspond to the coupling
constant G = −g0 >0 of the upper sides, and G<0 of the lower side, respectively. The two aspects are
QCD with SU (3) and QWD with SU (2), and may be unified by the YM gauge field, and between them
is asymptotically free G=0, so that Fig. 3 can uniformly describe the four basic interactions.
Fig. 3 The unification on the four basic interactions in three-dimensional space
We research some predictions of these unifications, and known experiments and possible tests. It
shows clearly that weak interaction appears for higher energy and shorter range. It is consistent
completely with those experiments show the scattering in the quark-parton model, they indicate a
strong repulsion at a smaller distance. It also seems to correspond to various resonances that are
produced by strong interactions, and then are strong decay.
At high energy scattering the cross sections, which connect the structure functions, usually agree with
Pomeranchuk theorem that all quark-quark scattering amplitudes are asymptotically equal, and the
differences between some scattering particles tend toward zero. This is related to the scaling. In some
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aspects it seems to correspond to the asymptotic freedom. But, at higher energy the cross sections
gradual increase.
The cross section for the annihilation of quark pair through gluon is [54,55]:
s
qq q q d
d s
9 ( ' ')
2 σ α
→ ≈ Ω . (16)
Here s is energy squared at c.m. For hadroproduction of heavy quarks Q=c, b, t, the formula is:
3
2
9 ( )
s
qq QQ d
dσ α s → ≈ Ω . (17)
Eq.(4) replaces into Eq.(16), we may obtain a minimum solution: When 2 2 2 2
0
− Λ
s = Q = Λ e , the
minimum value is:
2
2
0
2 2
2
0
2
9
64 | Λ
= = Λ
Ω
e
d
d
s Q β
σ π . (18)
From ( ) β gl squared in Fig.1 we can also derive the similar result, a minimum value exists at a point
l g =q. It seems to be a universal rule that the cross sections of scattering tend toward equal from big,
and then gradual increase.
From the minimum value 2 Q0 , we may derive ± Λ = Λ 2
0
i Q e . It shows the periodicity. In the standard
model three generations of quark-lepton possess some periodicities, and they are also very clear for
two systems cc ηc = and Υ = bb .
The electromagnetic fields produce each other and derive electromagnetic wave, whose equation is:
0 2 2
2
2
2
= ∂
∂ − ∂
∂
c t
A
x
A
α
. (19)
Usual gravitational fields change to produce gravitational wave, in which m=0 derives v=c.
When general strong and weak interactions are changed, strong and weak waves should also be
produced, in particular, those particle waves produced by strong interactions and weak interactions
decay. Different interactions produce different particles, which have wave property. It possesses the
uncertainty relation [5,56]. According to the de Broglie relation 2
vv = c , the particle velocity v is
related to the phase velocity v > c that obeys the generalized Lorentz transformation (GLT) [5,57],
and corresponds to the quantum entangled state [57].
Klein-Gordon equation becomes to a wave form:
0 2 2
2
2
2
2
2 2
2 2
2
2
2
= ∂
∂ − ∂
∂ − = ∂
∂ − ∂
∂
x v t
m c
x c t
φ φ φ φ φ
α α
. (20)
Here the wave velocity v<c. Assume that
amt
c
v
+ =
1
or mc t v ce−( / ) = , so the bigger m, v the smaller.
When t∞, v0.
The strong and weak interactions produce each other, which should obtain the particle waves whose
m is not 0, and v<c. It is related to duality. They are probably resonances and resonance waves. It can