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KEK-78-11 July 1978 T/E
KEK LECTURE NOTE
UNIFICATION OF ALL ELEMENTARY-PARTICLE FORCES INCLUDING GRAVITY
Hidezumi TERAZAWA, Keiichi AKAMA, Yuichi CHIKASHIGE
and Takayuki MATSUKI
NATJONAL LABORATORY FOR HIGH ENERGY PHYSICS
OHO-MACHI, TSUKUBA-GUN IBARAKI, JAPAN
KEK Reports are available from
Technical Information Office National Laboratory for High Energy Physics Oho-machi, Tsukuba-gun Ibarafct-ken, 300-32 JAPAN
Phone: 02986-4-1171 Telex: 3652-534 (Domestic)
(0)3652-534 (International) Cable: KEKOHO
Unification of All Elementary-Particle Forces Including Gravity
Hidezumi TERAZAWA, Keiichi AKAMA, Yuichi CHIKASHIGE and Takayuki MATSUKI
Institute for Nuclear Study University of Tokyo, Tanashi, Tokyo 188 tSaitama Medical College, Kawakado, Moroyama, Iruma-gun, Saitama 350-04
Abstract A unified model of the Nambu-Jona-Lasinio type for all
elementary-particle forces including gravity is reviewed in some detail. Starting with a nonlinear fermion Lagrangian of the Heisenberg type and imposing the massless conditions of Bjorken on vector auxiliary fields, we construct an effective Lagrangian which combines the unified SU{2)xU(l) gauge theory of Weinberg and Salam for the weak and electromagentic interactions of leptons and quarks and the Yanr;-Mills gauge theory of color SU(3) for the strong interaction of quarks. The photon y, the weak vector bosons W" and Z, and the physical Higgs scalar n appear as collective excitations of lepton-antilepton or qiark-antiquark pairs while the color-octet gluons G (a = 1,2,3r ••-,8) appear as those of quark-antiquark pairs.
The most important results of our unified model of the
* This review is based on the KEK lectures presented by H. Terazawa at National Laboratory for High Energy Physics, Japan, on March 6-7, 1978 and the invited talk presented by the same author at the 1978 Annual Meeting of Physical Society of Japan, on April 3, 1978.
Nambu-Jona-Lasinio type for strong, weak, and electromagnetic interactions are the following: the Weinberg angle 9 is determined to be
sin 29 w = (Z I2)/(I Q 2) = 3/8
where I and Q are the weak isospin and charge of leptons and quarks. The gluon coupling constant is also determined to be 8/3 times the fine-structure constant a. These results coincide with those of Georgi and Glashow in their unified SU(5) gauge model. However, our results are due not to such an assumed higher symmetry as SU(5) but to the Nambu-Jona-Lasinio dynamics in our model with only SU(3) c o l o r»< [SU(2) *D(1) ] W e i n b e r g _ S a l a m - The masses of the weak vector bosons are predicted to be
m j = (na//2 G p s i n 2 6 w ) 1 / 2 =60.9 GeV
and
m z = m H/cos9 w =77.0 GeV
where G p is the Fermi coupling constant. Entirely new and proper to our model are the following
relations between the masses of the physical Higgs scalar and weak vector boson and those of leptons and quarks:
m n = 2[(Z m4)/(I m 2 ) ] 1 / 2
and
n^ = /3 [<m 2>] 1 / 2
where E and < > denote the summation and arithmetic average over all leptons and quarks. These relations together with the previous results predict the arithraetic-like-average mass of
- 2 -
leptons and quarks to be
[<m 2>] 1 / 2 = 35.2 GeV
and the mass of the physical Higgs scalar to be bounded by
m 1 UZ/Dm^ = 70.3 GeV.
Another important result is the relation between the fine-structure constant and the sum of the charge squared of leptons and quarks:
a = 3ir/!£ Q 2) Hn(A2/m2) ,
where A is the universal cutoff momentum and m is the geometric-like-average mass of charged leptons and quarks defined by
m - HmQ 2/^ 2"' .
This relation is essentially the old result o Gell-Mann and Low in their renormalization group approach.
In our model of the Nambu-Jona-Lasinio type for gravity, the graviton is also a collective excitation of a fermion-antifarmion pair. We start again with a very simple nonlinear fermion Lagrangian and impose the massless condition of Bjorken on a tensor field, the gravitational field. The effective Lagrangiar derived, then, reproduces the familiar Newtonian gravitational potential if the gravitational constant G is related with the total number N_ of leptons and quarks:
G = 4Tr/K0NQA2 ,
where K. = 2/3 or 1/3 depending on the invariant or Pauli-Villars cutoff procedure. It is also shown that a more sofisticated model of this type defined on the curved space effectively
- 3 -
reproduces the Einstein-Weyl•s theory of general relativity. We further unify the unified model of strong, weak, and
electromagnetic interactions and the model of gravity into a unified model of the Narabu-Jona-Lasinio type for all elementary-particle forces including gravity. The mo3t exciting result of this grand unification is a simple relation (the G-a relation) between the fine-structure constant and the Newtonian gravitational constant:
a = 3n/(Z Q2)£n(4ir/c N QGm 2) .
This relation can be easily derived from combining the above two relations for a and G. Historically, a relation of this type was conjectured by Landau in 1955.
Based on the G-a relation, we predict that there exist a dozen lepton (six neutrinos and six charged leptonsi and a dozen flavors and three colors of quarks (6x3 up quarks ard 6x3 down quarks). The geometric-like-average mass of the charged leptons and quarks is also predicted to be
m = (4ir/K0N0G)1/2exp[-3Tr/2a(IQ2)] =23.7 GeV .
Furthermore, an answer is given to the question,"Why so many leptons and quarks?". It is a "spinor-subquark" model of leptons and quarks in which leptons and quarks are made of three subquarks of spin 1/2, w £ (i = 1,2), h ± (i = 1,2,---,N), and C± (i = 0,1,2,3). The left-handed • w and the right-handed w. R and w__ are a doublet and singlets of the Weinberg-Salam SD(2), respectively. The h.'s form an N-plet of the unknown H symmetry. Also, the C Q and C.'s (i = 1,2,3) are singlet and triplet under the SO(3) color symmetry. Leptons (v. and SL.) and quarks (u.
and d.) are expressed in terms of these subquarks as follows:
v. = ( w l h j C 0 ) , *. = ( W 2 h j C o ) , U j i = (w^C.), and d j i = (w^C.)
for j = 1,2,---,N and i = 1,2,3. In the unified subquark model of all elementary-particle
forces, which is an alternative to the unified lepton-quark model, the gauge bosons y, W~, and Z and the physical Higgs scalar appear as collective excitations of a w-w pair while the color-octet gluons G appear as those of a C-C pair. As a result, we
+ derive the following relations between the masses of n, W , and Z and those of w's:
m n ^ 2m w, n^ i /Imw, and raz 'k (/I/cos8w)mw .
These relations predict the masses of the physical Higgs scalar and the w subquark to be
m £ 70.3 GeV and m„ = 35.2 GeV . n w
Finally, discussions are given on some fundamental problems for future investigations in our unified model.
- 5 -
Contents
I. Introduction 7 II. Unified Model of Strong, VIeak, and Electromagnetic Interactions
2-1. Unified Lepton-Quark Model n 2-2. Spontaneous Generation of Masses 19
III. Model of Gravity 24 IV. Unifying Gravity and All Other Forces •••• 28 V. How Many Leptons and Quarks?
5-1. A Dozen Leptons and a Dozen Flavors and Three Colors of Quarks • • 31
5-2. Spinor-Subquark Model 33 VI. Recent Developments and Related Topics
6-1. Pregeometry 38 6-2. Miscellaneous Topics 48
VII. Future Problems 50
- 6 -
I. Introduction This review, which is an extended version of the second part
of KEK lectures presented by one of the present authors (H.T.), is to summarize and discuss in some detail what we have found for the last two years in our unified model of the Nambu-Jona-Lasinio type for all elementary-particle forces including gravity. Some other short reviews are recommended for impatient readers. Let us first sketch the historical background leading to our model.
In 1961, by adopting the nonlinear fenaion interaction of the 1) 2)
Heisenberg type, Nambu and Jona-Lasinio proposed a dynamical model of elementary particles based on an analogy with superconductivity. In the original model, it was made clear, among other things, that an idealized pion, the massless pseudoscalar bound state of a nucleon-antinucleon pair, appears as a Nambu-4) Goldstone boson when the nucleon mass is generated sp.. ataneously, breaking the chiral symmetry possessed by the Lagrangian. Subsequently in 1963, in the model with a nonlinear vector interaction, Bjorken ' demonstrated that the photon can be considered as a "collective excitation" of a fermion-antifermion pair. He showed that the model is equivalent to quantum electrodynamics in spite of its different appearances. In 1974, in the modified Nambu-Jona-Lasinio model, Eguchi and Sugawara found a set of equations which describes the collective motion of fermior.s and which is equivalent to the Higgs Lagrangian. They thereby clarified the nature of interactions among bound states of fermion-antifermion pairs. Furthermore, Kikkawa and, independently, Kugo ' made this approach to the collective motion very transparent by using the functional integration method, which gives the easiest way to find an effective Lagrangian for bosonic bound states.
- 7 -
On the other hand, as a result of the extensive theoretical works performed and experimental supports presented for the last several years, an attractive picture for interactions of elementary particles has been given by the unified gauge theory of Weinberg
9) and Salam for the weak and electromagnetic interactions of leptons and quarks and by quantum chromodynamics, the Yang-Hills gauge theory of color S0(3) for the strong interaction of quarks In addition to the familiar photon, the charged and neutral weak vector bosons and the color-octet massless vector gluons form a set of elementary gauge fields inherent to these theories.
Recently, two of us (K.A. and H.T.) have suggested the possibility that all of these gauge bosons are composite states of "spinor-subquarks" which are the building blocks of the ordinary 14) quarks. Independently, Saito and Shigemoto, starting with a Lagrangian of self-interacting leptons, have constructed an effective Lagrangian of the Weinberg-Salam type and have proposed that the photon, the weak vector bosons, and the Higgs scalars are all composites of lepton-antilepton pairs. They have fixed, among other things, the Weinberg angle to be sin28„ = g . Three of us (H.T., K.A., and Y.C.) have extended their model to a more realistic one including quarks. In our picture, the photon, the weak vector bosons, and the Higgs scalars are considered as collective excitations of lepton-antilepton or quark-antiquark pairs, while the color-octet gluons are considered as those of quark-antiquark pairs. As a result, the Weinberg angle is determined to be sin28„ = 5 , which coincides with the prediction of Georgi and Glashow ' in their unified SO(5) gauge model of all elementary-particle forces. The gluon coupling constant is also determined to be = times the fine-structure constant. In
- 8 -
Sec. II, we shall present all the details of this unified model
of the Nambu-Jona-Lasinio type for "all" elementary-particle
forces. They include an important result that the masses of the
weak vector bosons and the Higgs scalars are unambiguously related
to those of leptons and quarks.
What is excluded by the "all" elementary-particle forces is
the gravitational forces. We have proposed a model of the Nambu-17) Jona-Lasinio type for gravity in which the graviton is also
a collective excitation of a fermion-antifermion pair. A similar 18) idea was first proposed by Phillips in 1966, much influenced
2) 5) by the works of Nambu-Jona-Lasinio and Bjorken. The details
of our model will be presented in Sec. III. We have further
unified the unified model of strong, weak, and electromagnetic
interactions and the model of gravity into a unified model of the
Nambu-Jona-Lasinio type for all elementary-particle forces including 17) gravity. A main result of this grand unification is a simple
relation between the fine-structure constant a and the Newtonian 19) gravitational constant G, which has been conjectured by Landau
for a long time. We shall reproduce our derivation of the relation
in Sec. IV.
What are left as the most fundamental problems in our unified
model are the following: (1) How many leptons and quarks? (2) Why
the lepton and quark masses? (3) Why the Cabibbo angle? and 20) (4) Why CP violation? One of us (H.T.) has presented an
answer to the first question and a possible clue to the second
problem: (1) There exist a dozen leptons (six neutrinos and six
charged leptons) and a dozen flavors and three colors of quarks
(6*3 up quarks and 6x3 down quarks) , and (2) the arithmetic- and
geometric-like-average masses of the charged leptons and quarks
_ 9 _
are about 35.2 GeV and 23.7 GeV, respectively. The reason for these will be given in Sec. V. The first question will then likely be converted into "Why so many leptons and quarks?". Also in Sec. V, we shall present a possible answer by two of us (K.A. and H.T.) to this question. It is the previously mentioned "spinor-subquark" model of leptons and quarks in which leptons and quarks are made of three subquarks of spin 5 .
Since we proposed the unified model of all elementary-particle forces including gravity, our main efforts have been devoted to refinemants of the model of gravity. After a few steps of
21) improvement, we have made a generally covariant formulation of the model, showing that the model is effectively equivalent
22) to the Einstein-Weyl' s theory of general relativity. Very 23) recently, one of us (K.A.) ' has shown that this model defined
on the curved space can further be derived from a basic "determinant Lagrangian" written only in terms of fermions. This accomplishes a model of Sakharov's pregeometry. ' These recent developments and other related topics will be presented in Sec. VI.
Finally in Sec. VTI, discussions will be given on some fundamental problems for future investigations in our unified model.
- 10 -
II. Unified Model of Strong, Weak, and Electromagnetic Interactions 2-1. Unified Lepton-Quark Model
~~£ uftifjed lepton-quark model of the Nambu-Jona-Lasinio type for strong, weak, and electromagnetic interactions consists of the leptons I and I and the quarks q , u R , and d . They belong to representations of (1, 2_, 1), U, 1, 1), (3_, 2_, 1), (3_, ±, 1) , and (3_, 1 , X) , respectively, of the Sl7(3)xSU(2)xU(l) group, where SU(3) is the color symmetry of quarks and SU(2)xU(l) is the symmetry of Weinberg and Salam. The left- and right-handed fields are defined by 1)1. = (l-YcliJi and I|JR = gd+Yg) *- The lepton a. and £ can be identified with, for example,
* • & ) , and * R = e R
while the quarks q , u , and d p can be identified with, for exmaple,
"(II and d R
'L jji extention of the model to the one including an arbitrary number of leptons and quarks will be discussed at the end of Sec. 2-2. The Cabibbo rotation and possible rotations alike are ignored in this model although inclusion of such rotations is easy. A modification of the model to contain a unified model of weak and electromagnetic interactions other than the Weinberg-Salam" s can also be made trivially, if necessary.
Let us start with the nonlinear Lagrangian for massless leptons and quarks.
11
L = V * V I R w V * L i * V 5 R 1 * V V ' a R
+ f i 'V^/L^V/RVLWVRWXWR' 2
+ £ 2 ' V A V A | 2 + vW^vWv2
+ f4 ( b^ LV bu <jK + bd5 LdR>< bAV bu"! ,'R + bd aR <JL' • t2'1'
where jfl = y 3 / f's and b's are real constants, Y's are the weak hypercharges related to the charges and the weak isospins
1 a of leptons and quarks by Q = I, + =Y, x. (i=l,2,3) and X (a=l,2,»••,8) are the 2x2 Pauli matrices for S0(2) and the 3x3 Gell-Mann matrices for SU(3) , and u c and q are the charge-conjugate field of u and the G-parity-conjugate field of q (H ix_q ), respectively. Obviously, this Lagrangian is invariant under the global SU(3)xsu(2)xu(l) symmetry.
To analyze this nonlinear fermion Lagrangian, we use the 8) Kikkawa's algorism which consists of the following steps: ~*" a
1) Introducing the auxiliary vector fields V , V , and v and the scalar field K, which transform as (1 , 1 , 1) , (1_, 3_, 1 ) ,
<i' i< i.)» a n d (i' !• D o f SO(3)xSn(2)xn(l), respectively, we construct the Lagrangian
L '= I L i ^ U ( 3 u + i Y J l T V " - % ) £ L + V^'V^OV^ L R
+ 5 L i Y
P ( 3 +iY V +iT-^ + 1 1 * ^ ) q L
Xi
+ VVVVX* + b u ( 5 L K G u R + S R K G «L» + "d'^AV^
+ ^ ( V y ) 2 + c 2 ( V y ) 2 + c 3 ( v ^ ) 2 + c 4K +K , (2.2)
- 12 -
where K is the G-parity-conjugate field of K (= ix.K ) and c.
(i=l,2,3,4) are constants. Variations with respect to these
ausiliary fields give the"equations of motion"
vl = 2c7 ( V/W/ a VV P
x a V - < 2 - 3 >
K = " c T ( b H J R £ I . - b u ^ R + b d d R % » •
Substi tution of these re la t ions shows tha t the Lagrangian (2.2) i s
effect ively equivalent to the or ig ina l Lagrangian (2.1) i f
4f. for i= l ,2 ,3 and c i ~ ~ T~ (2 .4 )
2) Define the effective Lagrangian Le f f ^ o r t a e auxiliary
fields by the Feynman path integrals over the lepton and quark
fields:
exp(ijd x Le f f )
= (ilL) (dfcL) (d£ R) (dH R) (di L) (dq L) (d5 R) (du R) (dd R) (dd R)exp(iJd 4xL').
(2.5)
3) Performing the path integration formally, we then obtain
J d 4 x Leff = j d ^ [ = i ( V M ) 2 + c 2 ( V y ) 2+ c 3 ( v a ) 2
+ c 4 K + K ]
-iTrln 1 " hrvl\\*'V Wi"
1 - T S f l T * ^
- 13 -
-iTrJln
^^V^'W *V° I ? b d K
l - ^ Y ^ f Y , , V„+X av a) " R "
^^"•V^1
(2.6)
where Tr denotes the trace operation with respect to the space-time
points, the y matrices, and the matrices for the internal symmetry
SU(3)xSU(2)*U(l). The second and third terms in (2.6) correspond
to a series of fermion-loop diagrams if they are expanded into
a Taylor series in the auxiliary fields. Among them, one-fermion-
loop diagrams, to which two, three, and four auxiliary fields are
attached as external fields, involve quadratically and logarithmical
ly divergent integrals. We, however, suppose that there exists
the natural and universal cutoff momentum A beyond which
the nonlinear interactions described by the Lagrangian (2.1) are
suppressed. The magnitude of A is related to and, therefore,
determined by the Newtonian gravitational constant G, which will -1/2 be shown in Sec. III. Here, just imagine it is of order G ,
the Planck mass ( 10
which we call
19 GeV) . Then, the "divergent" part of !• eff L . , can be calculated to be
Ldiv = " S I 2 [ ( N l + 2 ) n 2 t ' a v > 2 + H 2 ( 1 + n l ) ^ V > 2 + ? 2 ( V ) 2 ]
- |lxI(Nj+2)n2(va)2+N2(1+i^) (V^) 2+Y 2(V p) 2 ]
.2.2 +12& | D Kl " + 2 1 ^ | K| d-X2B9 ( | K| ') (2.7)
- 14 -
w i t h
v
a = 3 v a - 3 v a - 2 f a b c v b v c , yv y \> v y v y v '
V = 3 V - 3 V - 2 V X V . yv y V V y y V '
and
where
V = 3 V - 3 V , [2 s i yv y v v y ' i ^ - o j
D K = (3 + iV + i x - V )K , y y u v
N, = t r I 2 = 2 and N _ 6 . . = t r T "1 2 « i j = t r T ^ = 2 5 i ; j f o r SD(2) ,
n x = t r i 2 = 3 , n 2 6 a b = t r X a X b = 2 6 a b , and [ X a , X b ) = 2 i f a b c f o r SU(3) ,
* 2 - N I Y L + Y L + N i n i Y L + n i Y l + n i ¥ L < = t a Y 2 ) • ( 2 - 9 )
-E l d R
b " b I + n l b u + nl bd •
B" = b l + n l b u + nl bd '
and the constants I, _ are the guadratically and logarithmically "divergent" integrals defined by
/•A „4_ ,2 11 = (2^)4i _iO_ =
(2™)' J p2+iE (4ir)2
and (2.10)
* (2TT) IT) 4 J (pN-iEr <4irr
4) Let us now construct the new Lagrangian
L" = L- + L_. , (2.11) div '
- 15 -
where the auxiliary fields, V , v , v a , and K, have been promoted to become "genuine" Bose fields. Then the relation
L ' = L ° - Ldiv ' < 2- 1 2>
which looks trivial, indicates that the original Lagrangian L is effectively equivalent to the now one 1" if the divergent parts of one-fennion-loop diagrams are subtracted.
5) Furthermore, the Lagrangian L" becomes invariant under the local SO(3)xsu(2)xu(l) gauge symmetry if we require the massless conditions for the vector fields, v , v , and v :
1 2 1 \ Y \ + 4f7 = ° ' \ Njd+n^Ij + jf- = 0 , (2.13)
and
| ( N 1 + 2 ) n 2 I 1 + 5 i - = 0 .
It is only by imposing these massless conditions of Bjorken on vector fields that our results will follow from the original Lagrangian of the Nambu-Jona-Lasinio type (2.1). The condition (2.13) fix the original coupling constants f. for i=l,2;3 in terms of the cutoff momentum A, hence also fixing the Weinberg angle, etc., as we shall see in what follows. It should also be emphasized here that we must impose the massless conditions of this type in order by order when more than one fermion loops are considered.
6) Rescaling the Bose fields by
V" " * (**J 1/2
2'
- 16 -
Vy ~ 2 [N2(l+ni)I2J \ • J V 2 * lV2 v a = - i T 6, 1 G a (2-")
VU 2 IjN^n^J Gy ' and
K = -(fe2I,)V2 + , 2'
we finally obtain the desired form of the Lagrangian
L-- V* W ( V i g , Kv i 9 § ? -*u ) l L + V^ ( V i g , 3VV*R + 5 L i T P ' 3 M
+ i 9 , K L B 1 J - i g i T . A l i - i f l x a G a ) q L
+ u ^ Y ^ O ^ i g - i y u B - i f i x^ 3 ) ^ R
- ! (v> 2-i<v> 2-Kv> 2 + ' V 2 - V 2 I * I 2 - x ( i * i 2 ) 2 • (2.15)
where
B = 3 B - 3 B , yv y v v p '
V - 3/v - \ \ + *V*v • G a = 3 G a - 3 G a + f f a b C G b G C , uv y v v y y v '
(2.16)
V = ( 3y + i g'5 By " " V * ' G * and 4> = IT24>
This Lagrangian is exactly what is expected from the combination of the unified gauge theory of Weinberg and Salam for the weak and electromagnetic interactions of leptons and quarks and the
- 17 -
Yang-Mills gauge theory of color SU(3) for the strong interaction of quarks. The difference here lies in the fact that the coupling constants f, g, g*, X, G^ , G , and G d and the mass parameter u are arbitrary in the combined theory, whereas in our model they are completely fixed by the quantum numbers of leptons and quarks, the cutoff momentum, and the coupling constants in the original Lagrangian L . in fact, they are given by
f = [6/(N 1+2)n 2I 2] 1 / 2 .
g = [6/^(1+^) I 2 ] 1 / 2 ,
g- = (B/Y 2!,) 1' 2 ,
-"2 = < 2 Ii - wr)n2 -b f 4
X = B 4/(S 2) 2I 2 . and
G ± = b i / ( £ 2 I 2 ) 1 / 2 for i = 1, u, and d.
From these relations, we reach the important results in our model. The Weinberg angle defined by tan8 = g'/g and the ratio of the coupling constants f and g are determined as follows:
2„ - 2
(2.17)
sin 'V g 2+ g - 2
N 2 (1+nj ) [•$$) (2.18)
and N 2(l+n 1)+?! 2
f2. _ M2 ( 1"*" nl > [ _ the number of isodoublets \ 2 n_(H,+2) \ the number of color triplets /
9 2 1 = 1 . (2.19)
- 18 -
Since Y = -1, Y = -2, Y = \ , Y = \ , and Y, = - \ . t *R qL 3 UR 3 "hi 3
for fractionally charged quarks <*2 = ^ - ) , the relation (2.18) . 2 3
gxves sin 6 W = g . Also, the relation (2.19) together with the 9) 2 2 2 familiar one g = e /sin 6 W fixes the gluon coupling constant
2 2 8 to be f /4ir = a/sin 8 = =a , where a is the fine structure
constant. These results coincide with the predictions of Georgi
and Glashow in their unified SU(5) gauge theory. Here, however,
it cannot be stressed too strongly that the relatione (2.18) and
(2.19) are due not to such an assumed higher symmetry but to the
dynamics in our model with only SO(3)xSU(2)xtJ(l). In any case,
all the gauge coupling constants are related to a single coupling
constant, the fine-structure constant a. It should also be noted
here that the fine-structure constant is related to the sum of
the charge squared of leptons and quarks and the cutoff momentum
by
a = 3Tr/(i:Q 2Hn(A 2/m 2) (2.20)
where m is the geometric-like-average mass of charged leptons and
quarks defined by
m = H m Q 2 / ( J : Q 2 > . (2.21)
This relation, which can be easily derived from (2.17) and the 9) 2 2 1/2
familiar relation ' e = gg'/(g +g" ) , is essentially the 27)
old result of Gell-Mann and Low.
2-2. Spontaneous Generation of Masses
Although we have started with the massless leptons and quarks
and have required the massless conditions (2.13) for the vector
fields, the Higgs potential ' in the Lagrangian L" (2.15) products 9) the familiar breakdown of SU(2)xU(l) gauge symmetry of the vacuum
- 19 -
if -y 2 > 0, i.e., 2I X - -£— > 0 . (2.22)
b f 4
Notice that this condition is satisfied either with an enough large cutoff momentum A or with an enough large coupling constant f. of the fundamental interaction, which is quite natural. The Higgs scalar <|p will then acquire the nonvanishing vacuum expectation value
<*'<> • -% U ) """ v " m ' ' • l 2 - ! " *C) — ' - f t This vacuum expectation value generates, through the Yukawa interactions in l>" , the masses of leptons and quarks given by
— G.v = b. 1 v ° I for i = I, u, and d. (2.24) 3. v = b t [Z 1 V2 X X \ 2B 4J Equation (2.24) is essentially the same as the Nambu-Jona-Lasinio self-consistency condition for the fermion mass. In fact, it can be transformed into
/m.\2 B 4I = 1 - ^ 1 ^i-2- , (2-25) m 2f 4ft 2I 1 V bii S 2 ^ which corresponds to Eq. (3.9) in Ref. 2.
Define as usual the four real components g. and n of the 28) Higgs scalar 41 by
* = exp(i?-f/2v)^- ( v°J (2.26) The three massless components % are "fictituous" Nambu-Goldstone bosons4' which will be absorbed by the Higgs mechanism, whereas the physical Higgs scalar has the mass given by
2,1/2 = p ™ r y V M . ( 2 . 2 7 ) m n = (-2u') ( 4, 4 . 4 \1 /2
K t + n l m u + n l m d '
- 20 -
This result-that the mass of the physical scalar boson is roughly twice the fermion mass-also agrees with the result of Nambu and
2) Jona-La s inio. The nonvanishing vacuum expectation value of the Higgs scalar
also geenrates the masses of the charged weak vector bosons B* [= 1— 2 3
(1/V2) (A +iA ) ] and the neutral one Z (= cosB^+sine^yB ), leaving the photon A (= -sine^ -fcosBjjB ) massless in a familiar way. The acquired masses of the weak vector bosons are given by
i „ r -3u 2 ( fe 2 ) 2 1 1 / 2
r -3uz(fev i L2N 2(l+n 1)B 4J
L H 2 < 1 + n l > J 2, -.1/2
(2.28)
and
"fc±/««8w
J" 3 [ N 2 ( l + n 1 ) + ? 2 ] ( m ^ + n ^ + n ^ l ) "> 1 / 2
S - ( l + n , ) Y 2 ;j (2.29)
These relations between the masses of the weak vector bosons and those of the leptons and quarks are entirely new and proper to our model. More practically, however, the weak-vector-boson
9) 2 masses can be predicted by the well-known relations g /8m„+ = 2 -5
Gp//2~ [where G_ i s the Fermi coupl ing constant (Gpm = 1.026x10 ) ] , 2 2 2 2 3
g = e / s i n 8 , and the present r e s u l t s i n 8„ = g t o be
( /Sirg V 2G„sin2BMJ
1/2 = 6 0 . 9 GeV
F V a a A (2.30)
m z = n^+ZcosB^j = 7 7 . 0 GeV .
Notice t h a t the SU(3) color-gauge symmetry i s unbroken so that
- 21 -
the color-octet gluons G" remain massless. In concluding this section, let us briefly discuss a possible
extension of the present model to an arbitrary number of leptons and quarks which form a series of the Weinberg-Salam multiplets
( S ,L' £R' qL' ™H' "Vn ' n = 1, 2, •••, N . (2.31)
A necessary modification of all the results so far presented is easy and almost trivial. The predictions for the Weinberg angle and the ratio of the gauge-coupling constants are unchanged. The mass sum rules (2.27)-(2.29) should be modified as
E(m* +n,mf, +n.m^ ) - 1 / 2 ? l m * i+ n i r a u i
+ n i m a i) ]
(m2 +n m 2 +n m| )J i l l
13? (m2 +n m 2 + n im 2 )Y'* i "i x uj L ai \ _ yrr.,,,2.,1/2 ( 1 , 1 - /3[<m >]
= 2[(Zm 4)/(Zm 2)] 1 / 2 , (2.32)
and
m z = n^t/cose^j . (2.34)
where Z and < > denote the summation and arithmetic average -over all leptons and quarks. These extended relations indicate that the Higgs-scalar and weak-vector-boson masses are related to certain arithmetic-like averages of the lepton and quark masses. This situation strongly suggests that there exist much heavier leptons and/or quarks whose masses are of order of the weak-vector-boson masses. In fact, the relation (2.33) together with the previous result (2.30) predicts the arithmetic-like-average mass of leptons and quarks to be
[<nT>] 12 > ] 1 / 2 = 35.2 GeV . (2.35)
- 22 -
Also, these relations (2.32) and (2.33) together with the same result (2.30) predict the mass of the physical Higgs scalar to be bounded by
m n 2 (2//3)mw± = 70.3 GeV . (2.36)
In any case, the mass sum rules (2.32)-(2.35) and the mass bound (2.36) provide the best and most important tests of our unified model of strong, weak, and electromagentic interactions.
In short, what we have discussed in this section is a modernization of the unified Nambu-Jona-Lasinio model by rewriting in terms of leptons and quarks in stead of nucleons.
- 23 -
III. Model of Gravity In our model of the Nambu-Jona-Lasinio type for gravity,
the graviton is also a collective excitation of a fennion-18) antifermion pair. Let U3 consider the following nonlinear
Lagrangian for N Q fundamental Dirac fermions i|/, (i=l,2, • • • ,N„) as a model of gravity:
(3.1) L Q = Jifft + f Q ( T y v ) 2
with T p u = •ij(Yp?v+Tv'yil)* where 3 = 3 - 3 and f n is the coupling constant. Variations or refinements of this simplest model will be discussed
g \ in Sec. VI. Ks use again the KikJcawa's algorism in analyzing this nonlinear Lagrangian.
25) 1) Introducing the auxiliary tensor field H , we construct the Lagrangian
L- = frXfr + T y y V + c 0 ( H ) j u ) 2 (3.2)
where c Q is a constant. Variation with respect to H gives the "equation of motion" for H :
V = - ( 2 c C ) _ \ v • <3-3)
which shows that H is symmetric and, furthermore, traceless and divergenceless up to the order of f- . Substitution of this relation indicates that the Lagrangian LJ is effectively equivalent to the original one LQ if
c„ = -(4f 0)- 1 . (3.4)
eff 2) Define the effective Lagrangian I-0 for the auxiliary field by the Feynman path integrals over the fermion fields:
- 24 -
exp(ijd 4x I.Q f ) = (dif))di/))exp(i\d4x L^) . (3.5)
3) Performing the path integration formally, we then obtain
j d 4 x I ^ f f = jd 4X[c 0(H i J U) 2]-iTrS.n[l+(i^)- 1i|(Y 1 J
,r+Y u'?)E , J V] (3.6)
where Tr denotes the trace operation with respect to the space-
time points as well as the y matrices and the internal symmetry
indices i of ijj. The second term in (3.6) corresponds to a series
of fermion-loop diagrams if it is expanded into Taylor series in
the auxiliary field. All actual calculations of such diagrams
involve quartically divergent integrals. As in our unified model
of strong, weak, and electromagentic interactions, however, we
assume the universal momentum cutoff A. Then, the "divergent" ef f div
part of L Q , which we call L n , can be calculated to be
d i v 1„ T ,„ ,2 1 „,.,„„ ,2 3.7)
L0 =tNoVHuv> ^ M ' W +
.4 with I - * ' " H -- -i f ^
J (2it) 0 I ,,_)4 2(4TT ) 2
where K„ = 5/9 or 2/3 depending on the invariant cutoff with
and without introducing the Feynman integratir •_, respectively.
All the remaining terms are irrelevant for the purpose of this
section. We define L^ 1 v by the first two terms in L0
1 V •
4) Let us now construct the new Lagrangian
L 5 = LQ + ^ 0 1 V ( 3- 8 )
where the auxiliary field H y has been promoted to become a
"genuine" tensor field, the gravitational field. Then, the relation
_ „ ~div L 0 _ 0 " ^0 (3.9)
- 25 -
indicates that the original Lagrangian is effectively equivalent
to the new one LjJ if L Q is subtracted as a counter term in
renormalization.
5) Furthermore, in order to make the graviton massless we
require the massless condition of Bjorken :
hoh - \f~ol - ° • < 3 - i o > 6) Rescaling the gravi ta t ional field by
V - - « K 0 H 0 I l ) " 1 / \ v ' ( 3 - 1 1 > we obtain the following form of L" :
L5 = JipV + | 0 x h y v ) 2 - g o T l j y v
with gQ = K^fa)'1'2 . (3.12)
To the lowest order in the coupling constant g« , this form
of the Lagrangian reproduces the familiar Newtonian gravitational
potential if
G = gg/4n = 4 T T / K Q N 0 A 2 (3.13) —8 3 —1 where G is the Newtonian gravitational constant (= 6.67x10 cm g
sec ) . This result shows that the universal cutoff momentum is
determined by the Newtonian gravitational constant:
A = ( 4 ¥ / K 0 N 0 G ) 1 / 2 = 4 . 3 3 ( K 0 N 0 ) " 1 / 2 X 1 0 : L S GeV . (3.14)
The "genuine cutoff" '' at such high energies around the Planck
mass G - 1'' 2 (= 1.221xl0 1 9 GeV) or, equivalently, at such short
distances as
r Q E A - 1 = 0. 4 5 5 ( K n N 0 ) 1 / 2 x l O - 3 3 cm (3.15)
- 26 -
would not harm the present success of quantum field theories in high energy physics, but would eliminate all existing infinities together with the "superficial cutoff" at the weak-vector-boson
2 mass nu (VLO GeV). It is further expected that at distances shorter than this fundamental length r„ some presently reliable basic principles in high energy theories such as special relativity, the microscopic causality, and the flatness of space-time would break down.
The momentum cutoff around the Planck mass was first introduced 19) by Landau, based on the idea that the effects of gravitational
interaction may exceed the electromagnetic effects at such high 29) energy. Later, Isham, Salam, and Strathdee demonstrated in
a model that gravity realistically regularized all infinities including its own. Our assumption is, however, somewhat different from theirs at the point that, in our picture, all the basic nonlinear fermion interactions including the gravitational one are cutoff universally at a certain short length, not due to gravitation.
An important question as to whether our model of gravity is equivalent to Einstein's general relativity will be answered in Sec. VI.
- 27 -
IV. Unifying Gravity and All Other Forces To unify gravity with all other elementary-particle forces,
let us suppose that there exist N Weinberg-Salam SU(2)*U(1) multiplets of leptons and quarks:
U n L ' W qniL' qniR' dniR J
with £ T nL (y„ - w ( y „ «•« for i = 1,2,3 and n = 1,2,---,N.
where i stands for the color SU(3) indices. The fields i|i now consists of 15N fermion components (or 16N fermion components if the neutrinos v are not massless). By combining (2.1) and (3.1), our fundamental Lagrangian for leptons and quarks is then written in the form
Ltot " L0 + fl<*V* ) 2 + f 2 ( * V * ) 2 + f 3(*Y uX a*) 2
+ f4 l ^ n R ^ u *n>nL + bd %L dnR| 2 • <4"2>
n n n Of course, the analysis of this Lagrangian for our unified model of the Nambu-Jona-Lasinio type for all elementary-particle forces including gravity produces both the results of our unified model for strong, weak, and electromagnetic interactions discussed in Sec. II and those of our model for graivty discussed in Sec. III. In addition to both of these results more will come out from combining them, which we shall discuss in what follows.
The most exciting result of our grand unification of gravity with other forces is a simple relation between the fine-structure constant a and the Newtonian gravitational constant G:
o = 3Tr/(IQ2Hn(4Tr/K0N0Gm2) (4.3)
- 28 -
where Q, N Q and m are the charge, the total number (where massless neutrinos be counted as = for each), and the geometric-like-average mass [defined in (2.21)] of leptons and quarks, respectively. This relation (let us call it the G-o relation) can be easily derived from combining the two relations (2.20) and (3.13). Thus, we have succeeded in calculating the fine-structure constant in the model. Historically, a relation of this type was first conjectured in an implicit from by Landau in 1955, as mentioned earlier.
In order to derive some interesting results from the C-ot relation (4.3), we first notice that the right-hand side of the relation does not strongly depend on the ambiguous quantities K. or m. Next, notice that Q = |n a n d N. = =^- for N
2 Weinberg-Salam multiplets. Take, for example, K„ = = and im*l GeV, then the G-a relation together with the experimental data G~ 1 / 2 = 1.221xl019 GeV leads to a^l/25N. Since there are at least three charged leptons, the electron, the muon, and the heavy lepton x discovered by Perl and others, the integer N is equal to or larger than 3. lie can, therefore, understand in this model why the fine-structure constant is less than 1/75. Furthermore, the relation a ^ 1/25N with the familiar experimental value of a i 137 leads to an exciting expectation, N = 5 or 6. In the next section, it will be shown that the magic number N is not 5 but 6.
In concluding this section, it should be emphasized that in our unified model, given a set of fundamental fermions, the leptons and quarks, a single parameter, the Newtonian gravitational constant, is enough to determine not only all the other coupling strengths including the fine-structure constant [see (4.3)] and
- 29 -
the s t rong , semi-weak, and Fermi coupl ing cons tant s , but a l s o
the Weinberg angle and the weak-boson and Higgs s c a l a r masses
[see ( 2 . 3 2 ) - ( 2 . 3 4 ) ] . In f a c t , the strong coupling cons tant f
of the g luon , the semi-weak coupl ing constant g o f the weak-
vector bosons W , and the Weinberg angle B- are a l s o determined
f = [72Tr 2/2:in(4Tr/K 0N 0Gm 2)] 1 /' ? , (4.4)
g = [96 ir 2 / ££n(4 ir /K 0 N Q Gm 2 ) ] 1 / 2 , (4.5)
and
- Z[(I , ) 2 to(4Tt/K n N.Gm 2 )] s i n 2 6 „ = ^ 5-5- , (4.6)
W UQ tn(4Tr/ic0N0Gnr)]
where m's denote the lepton and quark masses. The summation £
runs over a l l l eptons and quarks except in (4.4) where i t does
over quarks on ly . By combining ( 2 . 1 7 ) , ( 2 . 3 3 ) , and ( 3 . 1 3 ) , the
Fermi coupl ing constant G_ i s a l s o determined as
G F / / 2 = g 2 / 8 m 2
7 ± = 4ir2/[<m2>Z*n(4Tr/K0N0Gm2)] . (4.7)
- 30 -
V. How Many Leptons and Quarks? 5-1. A Dozen Leptons and a Dozen Flavors and Three Colors of Quarks
In this section, we shall present an answer to a question, "How many leptons and quarks?", and a possible clue to another, "Why the lepton and quark masses?". ' To do that, let us transfrora the G-a relation (4.3) into the form of
m = (47T/K0N0G)1/2exp[-3Tr/2a(ZQ2)] . (5.1)
Next, notice again that EQ = ?N and N. = M for N Weinberg-Salam multiplets of leptons and quarks. Furthermore, take
2 21) K0 = 3 ' w * l l c n * s m o r e favorable to reproduce Einstein's gravity. Then, the transformed G-a relation becomes
m = (4ir/5NG)1/'2exp(-9TT/16aN) . (5.2)
This shows that the geometric-like-average mass of the charged leptons and quarks is given as a known function of the integer N when both the fine-structure constant and the Newtonian gravitational constant are fixed. Given the experimental value of a and G(a = 137.03G and G~1/'2 = 1.221* 1 0 1 9 GeV), the G-a relation finally becomes
m = (1.936xl019//N)exp(-242.163/N)GeV _ - (5_3)
The numerical results are, for example, m = 8.11 HeV for N = 5, m = 23.7 GeV for N = 6, and m = 7.02 TeV for N = 7.
Since 8.11 MeV is even smaller than the muon mass, the case of N = 5 must be excluded. Also, as 7.02 TeV seems too large to be the average mass of leptons and quarks, the case of N = 7 should be physically excluded. What is left is the case where H = 6 and m = 23.7 GeV. The geometric-like average of 23.7 GeV
- 31 -
seems to be quite natural for the six charged leptons and a dozen flavors and three colors of quarks since the (kinetic) mass of the fourth lightest quark, the charmed quark, seems to lie around 1.5 GeV and since the mass of the third lightest charged lepton, the heavy lepton T, has been reported to be around 1.8 GeV. ' Notice
32) also that the recent discovery of r (9.5 GeV) suggests that the mass of the fifth lightest quark is around 5 GeV, supporting our argument more strongly. Furthermore, the previous result (2.35) that the arithmetic-like-average mass of leptons and quarks is predicted to be 35.2 GeV, which is of the same order of magnitude as 23.7 GeV, would give an additional circumstantial evidence for the case of m = 23.7 GeV.
For the above reasons, we reach the following conclusion: The case of N = 6 is the most plausible and all the other cases are very unlikely. We, therefore, predict with confidence that there exist a dozen leptons (six neutrinos and six charged leptons) and a dozen flavors and three colors of quarks (6*3 up quarks and 6x3 down quarks) and also that the arithmetic- and geometric-like-average masses of the charged leptons and quarks are about 35.2 GeV and 23.7 GeV, respectively. Let us call these leptons and quarks as follows:
CI. CI. CI. Cj emeu:) for n=4,5,6 (5.4)
In concluding this sub-section, l e t us show the results for 12) 2
integrally charged quarks of the Han-Nambu type. Since ZQ = 4N (N even) in this case, the G-a relation becomes m =
- 32 -
1/2 (4H/5NG) exp(-3u/8aN). The numerical results are, for example, m = 1.22x10" eV for N = 2, m = 29.1 GeV for N = 4, and m = 823 TeV for N = 6. This strongly indicates that the case of
N = 4 is the right answer. Namely, there exist eight leptons (four neutrinos and four charged leptons) and eight flavors and three colors of quarks (4x3 up quarks and 4x3 down quarks) and the geometric-like-average mass of the charged leptons and quarks is about 29.1 GeV if quarks are integrally charged.
5-2. Spinor-Subquark Model Some physicists may now ask themselves why there exist so
many leptons and quarks. In this sub-section, we shall present a possible answer by two of us (K.A. and H. T.) to this question. It is a "spinor-subquark" model of leptons and quarks in which leptons and quarks are made of three "subquarks" of spin | , w. (i=l,2), h. (i=l,2,- -• ,N) , and C.(i=0,1,2,3). The left-handed w and the right-handed w. and w_ are a doublet and singlets of the Weinberg-Salam SU(2), respectively. The h.'s form a N-plet of the unknown "H-symmetry". Also, the C_ and C.'s (i=l,2,3) are singlet and triplet under the SU(3) color-symmetry. One of us (H.T.) has proposed to call them "wakem", "hakam", and "chrcm", which stand for spinor-subquarks concerning the weak and electromagnetic interactions, the heaviness
35) (according to Harari ) or the horizontal degree of freedom, and the color symmetry, respectively. A dozen leptons and a dozen flavors and three colors of quarks are expressed in terms of these wakems, hakams, and chroms as follows:
- 33 -
Ue = '"ihlCo1 \ = ( wl h2 C0> V T " < wl h3«V v£ " t wl hn Co' n
e = (w 2h l C o) y = (w 2h 2C 0) T = (w 2h 3C 0) * n = tw 2h nC 0) (5.5)
Pi = (*1»iCi> Ci = ' W i 1 fci " ( wl h3 Ci» uni= ( wl hn Ci'
ni " ( w2 hl Ci' Xi " l*2h2Ci> b i - ( w2 h3 Ci» dni= ( w2 hn Ci'
for i = 1,2,3 and n = 4,5,6.
In the unified spinor-subquatk model of all elementary-particle forces proposed by three of us (H.T., Y.C., and K.A.), which is an alternative to the unified lepton-quark model discussed in Sec. II, the gauge bosons y, W~, and Z appear as collective excitations of a wakem-antiwakem pair which behave as (w Q w), (w T w), and (w R w) (where R is orthogonal to the charge Q) , respectively, while the color-octet gluons G appear as those of a chrora-antichrom pair tC A aC). This picture can be realized by assuming the Lagrar->ian for wakems,
L w - w Li^w L + 5 l g i * r u + w 2 Ri/Jw 2 R
+ F l ( Y w A V L + V _ » 1 R T V » 1 R + Y „ - R " 2 R ^ W 2 R ) 2
L 1R 2R ( 5 6 )
+ F 2 ( ™ L V V 2
+ F 4 ( _ a l " R W l L + a 2"L W 2R» ( - a l " l L W R ^ ^ R V '
where F's and a's are real constants, Y's are the weak hypercharges of wakems, and the superscripts c and G denote the charge-conjugate and G-parity-conjugate states, respectively. Without repeating the same procedure as in Sec. II, we simply present the following
- 34 -
results. This model can be effectively equivalent to the unified gauge theory of Weinberg and Salam for the w subquarks. The gauge bosons A and B appear as the collective excitations of wakem-antiwakem pairs which behave as w TY TW, and Y w, v w T+Y w, „y w, „
L'y L w L'y L w. R lR'y 1R +Y W
W 2 R ^ UW 2 R ' r e sP ectively. This is exactly what we have suggested
in Ref. 13. The Weinberg angle is determined to be
2 N 2 I s i n 6 = 2-^2 2— = -5—5 ? , (5.7) N 2 + N 1 Y „ « „ 1 R
+ < 2 R 02+lO.-l)2
2 where Q is the charge of w 1 . For Q = 0, =•, and 1, for example,
2 1 9 1 sin 8 = j, 20', and j, respectively. Although Q is rather arbitrary, 2 < 1 the Weinberg angle is bounded by sin 8 = =•. The relations between
the masses of the Higgs scalar and weak vector bosons and those of the wakems are
m n = 2|-^ ^ | = 2m„ (5.8)
m W ± =
and nZ = ^ i / c o s e w = ^ mi/'
(5.9)
= /3 m /cos8„. (5.10) W"
These relations together with the previous results (2.30) predict the masses of the physical Higgs scalar and the wakem to be
m = 70.3 GeV and m = 35.2 GeV. (5.11) n w
- 35 -
Also, the model Lagrangian for chroms is assumed to be
L c = Ci?C + F 3(CY pX aC) 2, (5.12)
where F, is the real coupling constant. This model can be effectively equivalent to the StJ(3) color-gauge theory for the C subquarks. The color-octet gluons appear as the collective excitations of chrom-antichrom pairs which behave as Cy X aC. This also coincides with our conjecture in Ref. 13. The C subquarks are massless in this model since the S0(3) gauge symmetry is unbroken.
We shall not discuss interactions of the h subquarks, which are ambiguous, though some speculations are made in Ref. 13, nor shall we discuss the mechanism of binding subquarks into a lepton or a quark, which is more ambiguous at this stage of high energy physics.
Although some contents in this sub-section seem to be still premature, an interesting picture of the gauge fields as composite states of subquark-antisubquark pairs has emerged in a model of the Nambu-Jona-Lasinio type. The result (5.11) especially suggests that the masses of the physical Higgs scalar and weak-vector bosons may be very close to the threshold of wakem-pair production, if any. This possible situation is very much similar to the one in which the masses of J/I|I and <i' are very close to the threshold of the reported charmed-meson-pair production in the e e colliding-beam experi nients. <?e, therefore, strongly urge experimentalists to be still alert for producing possible wakem pairs even after the anticipated exciting discovery of the weak vector bosons in
- 36 -
80's. In any case, "subauark diagrams" or "wakem-hakam-chrom diagrams" would become relevant and useful in discussing strong, weak, and electromagnetic interactions of leptons and quarks as
37) quark diagrams in discussing those of hadrons. In concluding this sub-section, let us introduce one more
38) suggestion made by one of us (H.T.). In the wakem model of weak interaction, the Cabibbo angle 9 appears as the ratio of the transition matrix element of the charged current between the p-quark and X-quark to that between the p-quark and n-quark:
tan9 = _ 1 L u 2 L . (5.13) <p| W 1 L Y ( J W 2 L |n>
Although this picture does not provide an easy solution of the Cabibbo angle because of ambiguous subquark dynamics, it strongly suggests that the Cabibbo angle would vary at higher momentum transfers where the subquark structure of quarks may become relevant. This effect would be observed in future high energy neutrino and lepton reactions.
- 37 -
VI. Recent Developments and Related Topics 6-1. Pregeometry
There are two important questions left unanswered in the model of gravity discussed in Sec. III. The first one is whether our •jiodel is equivalent to Einstein's gravity and the second is ' whether it is consistent with the presently existing data on gravity. In this sub-section, we shall answer these questions both affirmatively. To this end, we shall rewrite the model in a generally covariant way and calculate by adopting the Pauli-Villars regulator method which is invariant under the general coordinate transformation as welll as the local Lorentz and gauge transformations. Before introducing the generally covariant model, let us briefly sketch various steps of improvement of the original model leading to the final version.
The first step of improvement has been made by extending the 21) original model into
L Q - •!*& + f n ( T p u ) Z + f u(T ( J V) 2
(6.1) with T = tyxk-ly 3 +Y a ) if yv r 4 'p v v y
where fA is another coupling constant. This Lagrangian is shown to be effectively equivalent to
where (lx. £ 2, E 3) are (-|, |, -—) and (-|, 1, -—) depending on the invariant cutoffs with or without the Feynman integration.
- 38 -
The effective Lagrangian (6.2) gives the following coupled equations of motion:
tf* - 9 o 4 ( V v + V p ' * h , I V = ° and
" - 2 g0 Tpv >'6-4'
where 100=-n11=-ri22=-ri33=l and n =0 for u^v. On the other hand, making the weak-field approximation in the Einstein's theory gives the following form of the equation of motion for the gravitational field:
n ^ v - v \ v - v V + V v h V v 3 K 8 X h K X - v D h X x = - 2 9 o V - ( 6 - 5 > Therefore, in the case of the invariant cutoff without the Feynman integration, our gravity coincides with Einstein's gravity in the weak-field approximation if we take a special coordinate condition
3 V = 3 v h V ( 6- 6 )
However, it has been shown by Nishino and Fujii that this choice of gauge is exceptional, allowing no solution for the graviton propagator.
The second step of improvement has been made by three of us 2i\ (K.A., Y.C., and T.M.). The Lagrangian assumed for gravity
and electromagnetism is
L = *(i^-m)^+fn(Tijv+aniJvTXx+bniJV)2+f1(?T1Ii|')2, (6-7)
where T is the symmetric energy momentum tensor of a free fermion.
- 39 -
and a and b a r e c o n s t a n t s . By a d o p t i n g t h e P a u l i - V i l l a r s
r e g u l a t o r method, t h e Lagrangian (6 .7) i s shown t o be e q u i v a l e n t
t o
L" = iMlJ -mte* )* + g 0 h y U T v v - f ( F ^ ) 2
+ i [ ( 3 > .V> 2 - 2 ! 8 \ v ) 2 + 2 3 \ v 3 V h x- ( 3 / x> 2 i
- V U V < F u \ x - k v F K X ^ X )
+ e g ^ ^ t A ^ y ^ - n ^ W ) , (6.8)
where F is the field strength of A . This Lagrangian is exactly the same as the weak field approximation of the Lagrangian of general relativity with the photon field A and the charged fermion field i|). Consequently, this model can explain the deflection of light by the sun. We cannot, however, yet answer whether it is consistent with the precession of perihelia.
Now let us discuss the covariant formulation of our model 22) of gravity. We start with the Lagrangian for a fundamental
fermion field <j/ with the mass m moving on a curved space,
L = e ^ i t T ^ - D ^ ) * - mW+C , (6.9)
where e v is the vierbein and D is the covariant differentiation defined by
D = 3 - |Y S^ 1 , U u 2'mny
Y = i ( c - c - c ) , (6.10) 'ranu 2 V mnji nmp limn'
c = 3 e - 3 e m\i\) u mv v my
and s mn = i [ T * f Y n ]
- 40 -
The constant C is a counter term which will later partially cancel the quartically divergent cosmological terra. The Lagrangian L in (6.9) is invariant not only under the general coordinate transformation (GCT),
6^ = e and 6e k l J = B*\SV , (6.11)
but under the local Lorentz transformation (LLT),
6I|I = - ;i]i S 1)1 and <5e = m „e (6.12)
with in = -u mn nm
ku Apparently, L has no kinetic term for the vierbein field e . As will be seen immediately, the divergent parts of the fermion loop integrals for quantum corrections play a role of the kinetic
ku term, so that the field e becomes the "genuine" gravitational field as a collective excitation of fermion-antifermion pairs. The invariant action on the curved space is given by
Cd 4x/=gL (6.13) k where g = det g and g = e u e f e u . To analyze the Lagrangian
(6.9), we shall again follow the Kikkawa's algorithm. The Lagrangian (6.9) is already at the step 1).
ku 2) Define the effective Lagrangian L f f for the field e M by
exp(i(d4x/=g L e f f ) = (a*) (dip)exp(i(a4x/::g L) . (6.14)
Let us write /^q L in the form
/=q~ L = iMijf -m+D* + C^q (6.15)
ku where 2 = n Yv3„ and
r = •=? f t Y ^ - D ^ ) - §7- (/=g - Dm . (6.16)
- 41 -
3) Performing the path integration formally, we then obtain
Jd4x/=g" L e f f = Jd 4x C/=g" - iTrtaU + j ^ T) , (6.17)
where Tr denotes the trace operation with respect to the space-time points and the y matrices. The second term in (6.17) corresponds to a series of one-fermion-loop diagrams if it is expanded into a Taylor series in T. Since r contains one differentiation, all the loop integrals are quartically divergent. As in the previous sections, however, we believe that there exists a realistic momentum cutoff at around the Planck mass. We further assume in this model that this cutoff is invariant under GCT and LLT. For this reason, we adopt the cutoff by Pauli-Villars regulators, which is invariant under GCT and LLT and which may offer a good approximation to the real one. We introduce three regulators with the masses M. (i=l,2,3) and the weight coefficient c. which must satisfy
3 v v I c.M. + m K = 0 for k=0,2,4 . (6.18) i=l x 1
4) Let us first proceed in the weak field approximation, writing the vierbein as
e k u = n k V + H1"1 . (6.19)
ku For the purpose of this sub-section, H can be assumed to be
symmetric. Then, r in (6.16) is given by
r = H ^ l i y ^ - nku(§?-m>] + B ^ V I ^ fcrft + !<n k vn, v +n k vV<!?-m)] + Iw^w^v,+ ° < H 3 J (6-20)
- 42 -
where the d i f fe ren t ia t ion 3 in the f i r s t two terms does not ku operate on H but on the fermions outs ide . Up to the second
ku order in H H , only three fermion-loop diagrams contribute to
L „ . Notice t h a t the spinor-connection terra [the th i rd term
in (6.20)] does not contribute to L _. to th i s order. After
somewhat lengthy calculat ions, we find the following expression
for the divergent part of L . . , which we cal l L_. :
• / ^ L d i v = J 4 t - H \ + l ( H l J V ) 2 + l ( H i > ) 2 l
+ | j 2 [ ( 3 x H 1 J V ) 2 - 2 ( 3 p H | J V ) 2 + 2 3 1 J H M V 3 U H \ - O ^ ) 2 ]
+ < T 5 V ( D H \ ' 2 " 2 a H \ 3 p 3 y V - 2 < V v H P U ) 2
- 3 ( D H y V ) 2 + 6(3 U3 VH U X) 2] + 0(H3) (6.21)
where Q = n 1 J V3 3 and the suffices of Br are raised or lowered by multiplying n v v . The Jk's are "divergent" factors defined by
J. i- - I c.M.k£n(M.2/m2) for k=0,2,4 . (6.22) K (4TT) 2 i=l x 1 x
The form in the square bracket of the first term in (6.21) coincides with (/-g - 1) up to the second order in H^ . Also, the form in the square bracket of the second term in (6.21) is nothing but the weak field approximation of /^g R , where.R is the scalar curvature defined by
R = g"vR
V\> °v ua ~ °a'uv T 'Bv'ua 'Su'liv R = a r™ - 8 r°. + r" r* - r" T* (6.23)
Furthermore; the form in the square bracket of the third term in
- 43 -
(6.21) can be identified with R2-3R R w v in the weak field approximation. Disregarding the trivial constant term, we, therefore, obtain up to the second order in H
^ Ldiv " ^ l J i + l J 2 R + ^ J O ( H 2 - 3 H J , V R , , , , ) 1 " ( 6- 2 4 )
The third term in the square bracket in (2.24) is not only practically negligible (smaller by the order of G than the second term) but taken as the divergent part of radiative corrections in the usual quantum gravity. We, therefore, ignore it hereafter (which means that we do not take it as a large amplitude).
Let us now consider what. L „ looks like to all orders in ku H . Since the original Lagrangian L and the method of momentum cutoff are both invariant under GCT and LLT, L f f is also invariant. Furthermore, since the cutoff momenta M. are arbitrary, L « should be invariant seperately in each order of divergence. From dimensional analysis, the quartically (quadratically) divergent terms in L.. involve no (two) differentiations. It is known that div the only GCT and LLT invariant scalar made of the vierbeins with no (two) differentiations for each term is a constant (the scalar curvature R) . Therefore, Eq. (2.24) is proved to all orders in ku . H without any approximation.
5) We constract the new Lagrangian
L™ = L + L,. (6.25) div
where the vierbein e^ 1 aquires the kinetic term. The orginal
Lagrangian i s then written as
L = L » - L d i v , (6-26)
where L.. becomes the counter term which subtracts divergent parts div
- 44 -
arising from the fermion loop integration due to L". Of course, perturbation theory of L" involves infinite series of "divergent" loops with internal graviton propagators. This difficulty is proper not to our model but to any theory of quantum gravity. How to avoid these divergences by renormalization is beyond the scope of this sub-section.
6) Introducing G and X by
|J2 = iks < 6" 2 7> and
C + J 4 = X , (6.28)
we finally obtain
L" = e%§(YkV2pYk>* " m « + ike R + X - ( 6 - 2 9 )
This is precisely the Lagrangian for the theory of general relativity for a fermion field, with the Newtonian gravitational constant G and the cosmological constant \ . In the Lagrangian L", the vierbein has become the genuine gravitational field. The relation (2.27) shows that the cutoff momentum is indeed determined to be around the Planck mass.
To include both gravity and electromagnetism, we further consider the Lagrangian
L = L + f1nla(*Yk>l>)(*Yj,'J>) - (6-30)
Proceeding as before, we obtain the effective Lagrangian
J- = L- + eiekVu* " h " V \ ^ v l (6-31) w h e r e Fyv = V v ' 3vAy a n d
- 45 -
2 3 e = =4J^ • (6.32)
This is precisely the Lagrandian for the general relativity of a photon A and a fermion (i with the electric charge e.
What should be noted in this model is that the effective Lagrangian £" becomes invariant under the local gauge transformation (GT)
6A = 3 4 and St|i = i*.fli (6.33)
only if we require the massless condition which implies f. = •» . Therefore, GT-invariance of L" holds only in the strong coupling limit in the original nonlinear Lagrangian L. This situation is somewhat different from that in our previous model discussed in Sec. II, where the coupling constants f. (i=l,2,3) are rather small as 1/Jn or 1/In • This is because in the present model we have adopted the gauge invariant cutoff so that the loop diagrams give no quadratically "divegent" photon-mass term.
If we take the limit of the equal regulator masses (M. •* M) for simplicity, we obtain from (6.27) and (6.32)
G = i ^ l and a E ^ p-^- . (6.34) M' 4 l r fcn(MZ/mZ)
In t h e more r e a l i s t i c case where t h e r e e x i s t a number ( t h e t o t a l
number N n ) of fundamental f enn ions ( t h e charge Q. and t h e mass m^)
as i n ou r u n i f i e d model d i s c u s s e d i n t h e p rev iuos s e c t i o n s , t h e s e
r e l a t i o n s shou ld be r e p l a c e d by
G = J2* and a = 3 * . (6.35) N 0 tT XQf£n(M z/mp
E l i m i n a t i n g VT from t h e s e r e l a t i o n s , we aga in o b t a i n t h e G-a
r e l a t i o n between t h e f i n e s t r u c t u r e and Newtonian g r a v i t a t i o n a l
- 46 -
constants:
a = 3TT/[IQ J!,n(12iT/N0Gm?)] . (6.36)
Comparing this with the previous result (4.3), we find that the G-a relation is rather insensitive to cutoff procedures. Different cutoffs may only change the argument of logarithm in the relation.
To sum up the results so far presented in this sub-section, since our model is effectively equivalent to the Einstein-Weyl's theory of general relativity, it is consistent with the presently existing data on gravity including the precession of perihelia.
In concluding this sub-section, let us briefly introduce the latest progress made by one of us (K.A.), ' an attempt to achieve
24) the "pregeometry". The point is the following: the Lagrangian (6.9) is not yet written in terms of fundamental-matter fields only. However, it can be shown that a Lagrangian of the type (6.9) is equivalent to a determinant Lagrangian of the Nambu-Goto
43) type written in terms of matter fields only. In fact, in the model of "spinor pregeometry", the more fundamental Lagrangian
L = (det ^lY^ii/lFdJ.,*) (6.37)
is assumed and shown to be equivalent to the Lagrangian (6.9) if the Lorentz invariant function F(iji,ijj) is appropriately chosen. In this type of model, the metric and, therefore, the graviton appear as composites of the fundamental matters. Following the textbook of Misner, Thorne and Wheeler, we may call such a scheme to derive Einstein's gravity or geometrodynamics, as "pregeometry". According to Misner et al ., the revolutionary
24) notion of "pregeometry" was first proposed by Sakharov in 1967. Therefore, what we have discussed in this sub-section is a realization of Sakharov's pregeometry.
- 47 -
6-2. Miscellaneous Topics
In this sub-section, we shall make a list of miscellaneous
works which are related to what we have discussed in the previous
sections.
1) Photon pairing as a micriscopic origin for gravitation? 44) Adler et a3U have extensively studied the possibility
that the gravitational fields are composite "photon pairing
amplitudes".
2) Axial-vector gluons also? 451 Saito and Shigemoto have further extended the unif ied
model discussed in Sec. I I to the one including the massive
axial-vector color-gluons which are assumed in the vortex model
of Nielsen and Olesen for dual s t r i n g s .
3) Arbi t rar iness in extract ing co l l ec t ive modes? 46) One of us (T.M.) has presented a cr i te r ion for removing
a rb i t ra r iness due to Fierz-transformation in extracting co l lec t ive
modes from models of the Nambu-Jona—Lasinio type.
4) Collective exci ta t ions in local-gauge invariant superconductivity
models 47) Konishi, Miyata, Saito and Shigemoto have considered
a Lagrangian of the Nambu-Jona-Lasinio type which has local gauge
invariance and shown that collective excitations of fermion-
antifermion pair could occur in scalar and pseudoscalar channels
but not in vector or axial-vector channels.
5) Renormalization effects to the Weinberg angle
Saito and Shigemoto ' have calculated the renormalization
•affects to the Weinberg angle in the unified model of the Nambu-
Jona-Lasinio type for all elementary-particle forces.
- 48 -
6) Effects of the Higgs scalar on gravity 49) One of us (T.M.) has constructed an effective Lagrangian
for the gravitational field and the Higgs scalar in a model of
the Nambu-Jona-Lasinio type and shown that the gravitational field
couples to the Higgs scalar through the improved energy momentum
tensor of Callan, Coleman, and Jackiw.
7) Strongly influenced by the results discussed in Sec. II, we
have shown that weak vector bosons and Higgs scalars can form
a supermultiplet of U(2) symmetry connecting left- and right-handed
quarks (-.- leptons) .
8) Renormalizability of four-fermion theories and their
equivalence to gauge theoreis?
Many people have investigated whether four-fermion theories
of the Nambu-Jona-Lasinio type are renormalizable and whether they
are equivalent to gauge theories. Eguchi and Cooper, Guralnik,
and Snyderman have pointed out that they can be made renorma
lizable if they possess an ultraviolet-stable fixed point.
Shizuya has shown that they are renormalizable in three
space-time dimensions. On the other hand, Kerler has shown
that an equivalence between four-fermion theories and renormalizable
theories as claimed by others does not exist in general, further
investigations on these problems are most desirable in the future.
- 49 -
VII. Future Problems
What are left as the most fundamental problems for future
investigations in our unified model of the Nambu-Jona-Lasinio type
for all elementary-particle forces including gravity are the
following r
1) Why the Cabibbo angle?
In our model discussed in Sec. II, the Cabibbo rotation and
rotations alike are ignored. Although inclusion of such rotations
can be trivially made, the Cabibbo angle and angles alike are left
as free parameters.
2) Why CP violation?
Proposed models for CP violation can be classified into 52)
a) models with V+A currents, b) models with more than «3fo left-handed quark doublets, and c) models with more than one Higgs
54) scalar multiplets. If b) is the case, inclusion of CP violation
in our unified model can be most trivially made. Modifications of
our model seem to be possible also in other cases. In any case, the
CP violating parameters are left undetermined.
3) Why the lepton and quark masses?
As shown in Sees. II and V, in our unified model the arithmetic-
and geometric-like-average masses of the charged leptons and quarks
are determined to be 35.2 GeV and 23.7 GeV, respectively. The masses
of individual leptons and quarks can, however, not be predicted.
The above three problems can be all reduced to the one, how to
fix the lepton and quark mass matrix. To solve this problem in our
unified model probably needs an revolutionally new idea as an
additional input.
4) Why the genuine cutoff around the Planck mass?
In our unified model, we have assumed that there exists the
- 50 -
universal cutoff momentum A (or length r.) and shown that it is determined by the Newtonian gravitational constant G to be around the Planck mass (or length). This model seems still incomplete at that the cutoff is put in it by hand. A true theory must contain the cutoff in a more natural way. As stated in Sec. Ill, we expect that at distances shorter than this fundamental length r_ some presently reliable basic principles in high energy physics such as special relativity, the microscopic causality, and the flatness of space-time would break down. What is the origin of the genuine
19) 29) cutoff? Landau, and Isham, Salam, and Strathdee suppose that gravitational interaction gives such a cutoff. However, we suppose differently that an entirely new theory must be created with the finite Planck mass G~ or the nonvanishing Planck length G as special relativity and quantum mechanics have been created with the finite speed of light c and the nonvanishing Planck constant fi. Future investigations should be made toward such a truly unified theory.555
- 51 -
References and Footnotes
0) H. Terazawa, KEK lectures presented at National Laboratory
for High Energy Physics, Tsukuba, Ibaraki, Japan, Dec. 8-11,
1976, KEK-76-17 (National Lab. for High Energy Physics), Jan.,
1977; a talk presented at the Workshop on Fundamental Problems
in Gauge and Nonlinear Field Theories, The City College of
the City University of New York, New York, Jan. 22, 1977,
INS-Report-282 (INS, Univ. of Tokyo), Feb., 1977; an essey
selected by Gravity Research Foundation for Honorable Mention
for 1977, INS-Report-292 (INS, Univ. of Tokyo), May, 1977;
a talk presented at INS Int'l Symposium on New Particles and
the Structure of Hadrons, Tokyo, July 12-14, 1977, in the
Proceedings, edited by K. Fujikawa et al. (INS, Univ. of
Tokyo, 1978), p. 231; a talk presented at France-Japan Joint-
Seminar on New Particles and Neutral Currents, Tokyo and Kyoto,
July 14-16 and 18, 1977, in the Proceedings, edited by K.
Fujikawa et al. (INS, Univ. of Tokyo, 1978), p. 579.
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- 53 -
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18) P. R. Phillips, Phys. Rev. 146, 966 (1966). See also A. D. Sakharov, Dokl. Akad. Nauk SSSR 177, 70 (1967); H. C. Ohanian, Phys. Rev. 184, 1305 (1969) ; B. P. Diirr, Gen. Relativ. Gravit. 4_, 29 (1973); S. L. Adler, J. Lieberman, Y. J. Ng, and H. S. Tsao, Phys. Rev. D14, 359 (1976); S. L. Adler, ibid. 14, 379 (1976); D. Atkatz, ITP-SB-77-59 (SDNY, Stony Brook) , Sept. 1977.
19) L. Landau, in Niels Bohr and the Development of Physics, edited by W. Pauli (McGraw-Hill, New York, 1955), p. 52. See also Ya. B. Zel'dovich, ZhETF Pis'ma 6_, 922 (1967) [JETP Lett. 6_, 345 (1967)].
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23) K. Akama, Saitama Medical College preprint, 1978.
- 54 -
24) A. D. Sakharov, Ref. 18. The term "pregeometry" appears in C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. Freeman and Company, San Francisco, 1973), p. 1203.
25) S. Coleman, R. Jackiw, and H. D. Politzer, Phys. Rev. DIP, 2491 (1974); D. J. Gross and A. Neveu, ibid. 10, 3235 (1974).
26) This point was suggested by M. Gell-Mann. 27) M. Gell-Mann and F. E. Low, Phys. Rev. 95_, 1300 (1954) . See
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28) See for example E. S. Abers and B. W. Lee, Phys. Rep. 9C, 1 (1973).
29) C. Isham, A. Salam, and J. Strathdee, Phys. Rev. D3y 1805 (1971); ibid. 5, 2548 (1972). See also B. S. De Witt, Phys. Rev. Lett. 13_, 114 (1964); I. B. Khriplovich, J. Nucl. Phys. (U.S.S.R.) 3, 575 (1966) [Sov. J. Nucl. Phys. 3, 415 (1966)].
30) H. Terazawa, Phys. Rev. Lett. 22_, 254 (1969); ibid. 22. 442(E) (1969); Phys. Rev. D^, 2950 (1970). See also S. Weinberg, Phys. Rev. Lett. 29_, 388 (1972); Phys. Rev. 07, 2887 (1973).
31) M. L. Perl et al., Phys. Rev. Lett. 35 , 1489 (1975); Phys. Lett. 63B, 466 (1976).
32) S. W. Herb et al., Phys. Rev. Lett. 39 , 252 (1977); ibid. 3±.
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M. Nakagawa, Y. Ohnuki, and S. Sakata, Prog. Theor. Phys. 23_, 1174 (1960); C.-K. Chang, Phys. Rev. D5, 950 (1972); K. Matumoto, Prog. Theor. Phys. 52, 1973 (1974); J. C. Pati and A. Salam,
- 55 -
Phys. Rev. DIP, 275 (1974); O. W. Greenberg, Phys. Rev. Lett. 35, 1120 (1975); K. Koike, Prog. Theor. Phys. 56, 998 (1976) ,-H. Miyazawa, private communication. For a similar model suggested after our proposal, see K. Fujikawa, Prog. Theor. Phys. 58, 978 (1977).
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- 56 -
44) S. L. Adler et al., Ref. 18. See also S. L. Adler, COO-2220-120 (Inst, for Advanced Study), Aug., 1977, to be published in the Proc. of the 8th Int'l Conf. on General Relativity and Gravitation, Aug. 7-12, 1977, Waterloo, Ontario, Canada.
45) T. Saito and K. Shigemoto, Ref. 15. 46) T. Matsuki, TIT/HEP-35 (Tokyo Inst, of Tech.), Jan., 1977. 47) G. Konishi, H. Miyata, T. Saito, and K. Shigemoto, Prog. Theor.
Phys. 52, 2116 (1977). 48) T. Saito and K. Shigemoto, Prog. Theor. Phys. 59_, 964 (1978). 49) T. Matsuki, Prog. Theor. Phys. 59_, 238 (1978). 50) H. Terazawa, K. Akama, Y. Chikashige, and T. Matsuki, Prog.
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