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arXiv:h

ep-ph/0102056v1

6Feb2001

Fundamental Forces as Gauge Theories

Heui-Seol Roh

BK21 Physics Research Division, Department of Physics, Sung Kyun Kwan University, Suwon 440-746, Republic of Korea

(February 1, 2008)

This study proposes that all the known fundamental forces including gravity may be described bylocal gauge theories. Gravitational, electroweak, and strong interactions on length scales from 1033

cm to 1028 cm are systematically discussed from the unified gauge theory point of view toward a

ultimate theory for fundamental forces. New concepts such as dynamical spontaneous symmetrybreaking, gauge group hierarchy, coupling constant hierarchy, effective coupling constant hierarchy,cosmological constant, massive gauge bosons, massless gauge bosons, quantum weakdynamics, anal-ogy between quantum weakdynamics and quantum chromodynamics, a possible gauge theory forthe universe expansion, the relation between time and gauge boson mass, other quantum tests, etc.are briefly reported based on experiments.

PACS numbers: 12.10.-g, 04.60.-m, 12.60.-i, 11.15.-q

The longstanding problem in physics is the unificationof gravitation theory and the other gauge theories forelectroweak and strong interactions, which has attracteda great deal of work. Two fundamental theories, generalrelativity and quantum gauge theory, can together cover

the energy regime from the elementary particle to theuniverse in describing the behavior of particles. How-ever, the problem of unifying gravity and the other fun-damental forces in nature is that general relativity the-ory is difficult to formulate as a renormalizable gaugetheory. All the known covariant or canonical quantiza-tion methods are not yet successful in quantizing thegravitational field, even though they work well for theother physical fields. General relativity or standard bigbang theory based on general relativity has outstand-ing problems: the singularity, cosmological constant orvacuum energy, flat universe, baryon asymmetry, hori-zon problem, the large scale homogeneity and isotropyof the universe, dark matter, galaxy formation, discrep-ancy between astrophysical age and Hubble age, etc. Onthe other hand, according to the recent experiments,BUMERANG-98 and MAXIMA-1 [1], the universe is flatand there exists repulsive force represented by non-zerovacuum energy, which plays dominant role in the universeexpansion. The outstanding problem as well as the otherchallenging problems may be resolved systematically inthis paper, which uses local gauge theories to describe allthe known fundamental forces on length scales from thePlanck scale 1033 cm to the universe scale 1028 cm. Thepresent work is mainly restricted to the low, real dimen-

sions of spacetime without considering supersymmetryand is based on phenomenology below the Planck scale:Newtonian mechanics, Einsteins general relativity andcosmological constant [2], quantum gauge theories [35],and vacuum [6] are taken into account. This paperonly briefly reports common features of gauge theoriesfor fundamental forces, whose details are addressed byfollowing papers, quantum gravity (QG) as an SU(N)gauge theory for gravitational interactions [7,8], quan-tum weakdynamics (QWD) as an SU(3)I gauge theory

and Glasshow-Weinberg-Salam (GWS) model for weakinteractions [9,3], and quantum chromodynamics (QCD)and quantum nucleardynamics (QND) for strong inter-actions [4,10,11].

Toward quantum gravity as a gauge theory, its features

are investigated in quantum gravity (QG) [7,8]; Newtongravitation constant GN and cosmological constant arediscussed from the viewpoint of quantum gauge theoryand quantum tests supporting this scheme are suggestedin addition to its relation to Einsteins general relativity.QG holds underlying principles such as special relativ-ity, gauge invariance, and quantum mechanics under theconstraint of the flat universe

1 = 1061. (1)

The flat universe condition is required in quantum gaugetheories and inflation scenario [12] and is verified by

the experiments, BUMERANG-98 and MAXIMA-1 [1].1061 represents the 1030 expansion in one dimension:NR = i/( 1)1/2 1030. An SU(N) gauge theorywith the vacuum term as a trial theory is introducedand dynamical spontaneous symmetry breaking (DSSB)is adopted even though the exact group for gravity is notknown at present. The Lagrangian density for QG [7,8]is, in four vector notation, given by

LQG = 12TrGG

+

i=1

iiDi +

g2g162

TrGG

(2)

where the bare term [6] is added as a single, additionalnonperturbative term to the perturbative Lagrangiandensity with an SU(N) gauge invariance. The subscripti stands for the classes of pointlike spinor and D = iggA for the covariant derivative with the gravita-tional coupling constant gg. Particles carry local chargesand gauge fields are denoted by A =

a=0A

a

a/2 withmatrices a, a = 0,.., (N2 1). The field strength tensoris given by G = A A igg[A, A ] and G is

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the dual of the field strength tensor. A current anomalyis taken into account to show DSSB in analogy with theaxial current anomaly, which is linked to the vacuumin QCD as a gauge theory [10]. Since the GG term isa total derivative, it does not affect the perturbative as-pects of the theory. The term plays important roleson the DSSB of the gauge group and on the quantiza-tion of the matter space and vacuum space. The gauge

boson identified as the graviton is proposed as massivevector boson. Newton gravitation constant is defined asthe effective coupling constant

GN/

2 = gfg2

g/8M2

G 1038 GeV2 (3)

with the gravitational coupling constant gg, and thegauge boson mass MG MPl 1019 GeV in QG [7,8]

just as Fermi weak constant is defined as GF/

2 =g2w/8M

2

W 105 GeV2 with the weak coupling con-stant gw and the intermediate vector boson mass MWin weak interactions [3,9]. The gravitational factor gfin gravitational interactions is a factor depending on a

gauge group just as the color factor cf in strong inter-actions is a factor depending on color dynamics: thesefactors are closely related to charge mixing angles [9,10].The effective cosmological constant e representing thevacuum energy density is related to the gauge boson massMG by

e = 8GNM4

G (4)

with e M2Pl 1038 GeV2 with MG 1019 GeV atthe Planck epoch and e 1084 GeV2 with MG 1012 GeV at the present epoch. The gauge boson massMG decreases through the condensation of the singlet

graviton:

M2G = M2

Pl gfg2g2 = gfg2g[A20 2] (5)

where A0 is the singlet gauge boson with even parity and is the condensation of singlet gauge boson with oddparity. DSSB consists of two simultaneous mechanisms;the first mechanism is the explicit symmetry breakingof gauge fields, which is represented by the gravitationalcoupling constant gfg and the gravitational fine struc-ture constant g, and the second mechanism is the spon-taneous symmetry breaking of gauge fields, which is rep-resented by the condensation of singlet gauge fields. The

effective potential is related to the gauge boson mass byVe = M4G = M4

Pl 2gfg2gM2Pl2 + g2fg4g4. It reducesto nearly zero, MG 1012 GeV, at the present universe;the present universe expansion [13] and cosmic microwavebackground radiation (CMBR) [14] are suggested as con-clusive evidences for nearly massless gauge bosons withthe mass MG 1012 GeV and the number NG 1091and for massless gauge bosons (photons) as Nambu-Goldstone bosons [15] with the energy E 1013 GeVand the number Nt 1088. For the gauge boson mass

MG 1012 GeV at the present epoch, the correspond-ing cosmological constant 0 1084 GeV2 or Hubbleconstant H0 1042 GeV suggests new types of fun-damental forces responsible for the universe expansionand CMBR: the possible gauge groups for this process isSU(3)R SU(2)BU(1)A U(1)g with the extremelystrong effective coupling constant GS 1024 GeV2 1061GN. This is supported by the recent experiments

BUMERANG-98 and MAXIMA-1 [1]. This is consistentwith the theoretical, large cosmological constant at thePlanck epoch and the empirical, small cosmological con-stant at present. This is also compatible with no de-tection of gravitons at present since they are extremelyheavy particles with the Planck mass MPl or extremelyhigh energy particles with EMPl.

The extremely flat universe 1 = 1061 is definedby the relation = (m m)/G where m is thezero point energy density, m is the matter energy den-sity, G = M4G is the vacuum energy density, and isthe parameter of the vacuum term representing thesurface term [6]. This scheme suggests quantum tests to-ward resolutions to the inflation [12] with the universesize R = NR/MG and the maximum wavevector mode ofthe vacuum NR = i/( 1)1/2 = (G/m)1/2 1030,dark matter as strongly interacting massive particles(SIMPs) with the mass around 1012 GeV in additionto weakly interacting massive particles (WIMPs) [16],structure formation due to massive gauge bosons anddark matter, primordial nucleosynthesis, etc. of the uni-verse: they constitute conclusive evidences of massivegauge bosons at different energy scales. The condition 1 = 1061 of the flat universe leads to the value = 1061 G/m. If the conserved matter en-

ergy density in the universe is close to the critical densitym c 1047 GeV4, the constant depends on thegauge boson mass MG:

= 1061 M4G/c. (6)

values become Pl 1061 at the Planck energy,EW 104 at the weak energy, QCD 1012 atthe strong energy, and 0 1061 at the present en-ergy of the universe. This is consistent with the ob-served results, < 109 in the electric dipole moment ofthe neutron in strong interactions [17] and 103 inthe neutral kaon decay in weak interactions [18]. If thebaryon mass density is B

Bc, the baryon asymme-

try B = NB/Nt 1078/1088 = 1010 with the baryonnumber NB = B at present [19] is consistent with Avo-gadros number NA = 6.02 1023 mol1 and nuclearmatter density nn 1.95 1038 cm3. Moreover, lep-ton asymmetries e = Ne/Nt 1081/1088 = 107 forelectrons, = N/Nt 1079/1088 = 109 for muons,and = N/Nt 1078/1088 = 1010 for taus are ex-pected if lepton matter has the same order with the criti-cal density e ec c [8]. The neutrino asymmetry

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is = N/Nt 1088/1088 = 1 if the neutrino massm 1 eV. Since the baryon number and the leptonnumber are good quantum numbers at low energies, ex-cess lepton matter, consisted of electrons and neutrinos,over baryon matter is suggested as dark matter: equalparticle numbers in quarks and leptons are crucial tothe renormalization in the GWS model [3], which per-turbatively requires no anomaly. Based on this scheme,

the quantization units of energy, temperature, frequency,time, and distance in the universe are respectively 1042

GeV, 1030 K, 1019 Hz, 1043 s, and 1033 cm if grav-itational and present interactions are concerned as inter-actions at the highest and lowest limits. Furthermore, theproposed relation between time and gauge boson mass [8]is given by

t = 1/He = (3/8GNM4

G)1/2, (7)

which provides scales in quantum cosmology as shown inTable I.

At the Planck energy, a phase transition takes place

from a certain group G to a group H and at thegrand unification energy, the group H breaks down toan SU(3)I SU(3)C group, in which the SU(3)I de-notes the isospin (isotope) SU(3) symmetry [9] and theSU(3)C denotes the color SU(3) symmetry [4]: the groupchain is G H SU(3)I SU(3)C. The Lagrangiandensity for the SU(3)C or SU(3)I gauge theory hasthe exactly same form with the Lagrangian density forgravitation. As the energy scale decreases the SU(3)Igroup for isospin interactions, QWD, breaks down tothe SU(2)L U(1)Y U(1)e group for electroweakinteractions, known as the GWS model [3], and theSU(3)C group for strong interactions, QCD [4], analo-

gously breaks down to the SU(2)N U(1)Z U(1)fgroup for nuclear interactions, known as QND [11]. TheU(1)e gauge theory is photon dynamics known as quan-tum electrodynamics (QED) and the U(1)f gauge the-ory is also photon dynamics. The DSSB of local gaugesymmetry and global chiral symmetry triggers the cur-rent anomaly: for example, the (V+A) current anomalyfor weak interactions and the axial current anomaly forstrong interactions. Coupling constants for weak andstrong interactions are unified around 102 GeV or slightlyhigher energy rather than the order of 1015 GeV of grandunified theory (GUT) [5]: i = s 0.12 accordingto the renormalization group analysis [9]. The unifica-tion at the order of a TeV energy is consistent with therecent GUT [20]. Important motivations of QWD arecoupling constant unification and fermion mass genera-tion and important concepts of QND are massive gluonand colorspin. In terms of the Weinberg weak mix-ing angle sin2 W = 1/4 and the strong mixing anglesin2 R = 1/4, coupling constant hierarchies for weakand strong interactions are respectively obtained. Elec-troweak coupling constants are z = izfi = i/3 0.04,

w = iwfi = i/4 0.03, y = iyfi = i/12 0.01,

and e = iefi = i/16 1/133 as symmetric isospin in-

teractions at the weak scale and 2i/3, i/2, i/6,and i/8 as asymmetric isospin interactions: iwf =sin2 W and i

ef = sin

4 W. Strong coupling constants

for baryons are b = cbfs = s/3, n = c

nfs = s/4,

z = czfs = s/12, and f = c

ffs = s/16 as symmet-

ric color interactions and 2s/3, s/2, s/6, ands/8 as asymmetric color interactions: cnf = sin2 Rand cff = sin

4 R. The isospin factors isf = (i

zf, i

wf , i

yf, i

ef)

or color factors introduced are

isf = csf = (c

bf, c

nf, c

zf, c

ff) = (1/3, 1/4, 1/12, 1/16) (8)

for symmetric interactions and

iaf = caf = (2/3,1/2,1/6,1/8) (9)

for asymmetric interactions. The symmetric charge fac-tors reflect intrinsic even parity with repulsive force whilethe asymmetric charge factors reflect intrinsic odd par-ity with attractive force; this suggests electric-magneticduality before DSSB. The coupling constant chain isg h i s for gravitation, grand unifica-tion, weak, and strong interactions respectively. Mas-sive gravitons at the Planck scale produce massive inter-mediate vector bosons and massive gluons at weak andstrong interaction energies, respectively. The effectivecoupling constant chain is GN GH GF GR forgravitation, grand unification, weak, and strong inter-actions respectively: effective strong coupling constantsare GF/

2 = ifg2i /8M

2

G 105 GeV2 and GR/

2 =cfg

2

s/8M2

G 10 GeV2. Just as the gauge boson massbelow the Planck energy becomes M

2

G = gfg2

g[A2

02

],the gauge boson mass below the grand unification energybecomes

M2G = M2

H cfg2s2 = cfg2s [A20 2] (10)

with the grand unification mass MH 102 GeV: MG 300 MeV at the QCD cutoff scale. In this scheme, all theelementary fermions such as quarks and leptons possessthree types of charges, (color, isospin, spin). Leptons arecolor singlet fermions, which are governed by the isospindynamics of QWD as well as spin dynamics but quarksare color triplet fermions, which are governed by the color

dynamics of QCD as well as isospin dynamics and spindynamics. Combined charges are asymmetric configura-tions for both quarks and leptons to satisfy the Pauliexclusion principle for fermions. The essential point forDSSB is that the gauge boson decreases its mass as thegauge boson condensation increases or the charge factor(gravitational factor gf, isospin factor if, or color factorcf) decreases as the energy scale of the system decreases.The factors described above are pure color (or isospin)factors due to color (or isospin) charges but the effective

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color factors used in nuclear dynamics must be multi-plied by the isospin factor iwf = 1/4 since the proton andneutron are an isospin doublet as well as a color doublet:

cefff = iwfcf = i

wf(c

bf, c

nf, c

zf, c

ff) = (1/12, 1/16, 1/48, 1/64)

(11)

for symmetric configurations. For example, the electro-

magnetic color factor for the U(1)f gauge theory becomesefff = /64 1/137 when s = 0.48 at the strongscale. Note that the color mixing is the origin of theCabbibo angle for quark flavor mixing in weak interac-tions. Massless gauge bosons with different energies areaccompanied with massive gauge bosons during DSSB atdifferent energy scales. Table II summarizes the typicalfeatures of fundamental forces. In gravitational interac-tions, it is predicted that the typical cross section at atemperature T = 1 GeV is G2NT2 1070 m2 andthe typical lifetime for a particle with the mass 1 GeV is = 1/ 1/G2Nm5 1050 years, which is much longerthan the proton decay time predicted by GUT [5].

All the known fundamental forces in nature, strong,electroweak, and gravitational forces on length scalesfrom 1033 cm to 1028 cm, may be described by localgauge theories toward a ultimate theory for fundamen-tal forces. Comparison between QG and general rel-ativity is summarized by Table III. QG is quantumtheory holding principles of gauge invariance and spe-cial relativity while Einsteins general theory of relativ-ity is classical theory holding principles of equivalenceand general relativity. Newly introduced concepts in thispaper are the flat universe rather than the curved uni-verse, DSSB rather than spontaneous symmetry break-

ing, gauge group hierarchy G H SU(3)I SU(3)C,coupling constant hierarchy (,/3, /4, /12, /16) forboth weak and strong interactions, effective coupling con-stant hierarchy GN GF GR, effective cosmologi-cal constant e = 8GNM4G, massive gauge boson masssquare M2G = M

2

Pl gfg2g2 = gfg2g[A20 2], mass-less gauge bosons as Nambu-Goldstone bosons includingCMBR, the dynamical spontaneous breaking of discretesymmetries (C, P, T, and CP), QWD as an SU(3)I gaugetheory beyond the SU(2)LU(1)Y weak dynamics, anal-ogy between QWD and QCD, Newtonian mechanics asquadrupole (tensor) interactions, the possible modifica-tion of Einsteins general relativity, the other quantum

tests such as SIMPs in addition to WIMPs as the candi-date of dark matter, the universe inflation with the orderof magnitude 1030, the baryon asymmetry B 1010,the relation between time and gauge boson mass, theinternal and external quantization of spacetime, protondecay time much longer than one predicted by grand uni-fied theory, mass generation mechanism with the surfaceeffect, etc. The gauge boson mass MG 1012 GeV asso-ciated with the cosmological constant 0 1084 GeV2and the Hubble constant H0 1042 GeV especially

indicates gauge theories for new fundamental forces re-sponsible for the universe expansion and CMBR. Thesepredictions are compatible with the recent experiments,BUMERANG-98 and MAXIMA-1. The potential QG asa gauge theory may resolve serious problems of GUTsand the standard model: different gauge groups, Higgsparticles, the inclusion of gravity, the proton lifetime, thebaryon asymmetry, the family symmetry of elementary

particles, inflation, fermion mass generation, etc. Thisscheme may also provide possible resolutions to the prob-lems of Einsteins general relativity or the standard hotbig bang theory: the spacetime singularity, cosmologicalconstant, quantization, baryon asymmetry, structure for-mation, dark matter, flatness of the universe, and renor-malizability, etc. This work may thus significantly con-tribute to the unification of fundamental forces since allthe forces in nature might be formulated in terms of localgauge theories toward the theory of everything.

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