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On the large-N dynamics of gauge symmetry breaking
Volume 133B, number 3,4
PHYSICS LETTERS
15 December 1983
ON THE LARGE-N DYNAMICS OF GAUGE SYMMETRY BREAKING N.I. KARCHEV 1 International Centre for Theoretical Physics, Trieste, Italy Received 9 August 1983
We consider a Gw XUTC(N) gauge theory. A method of colour singlet bilocal collective coordinates is proposed to show, large-N colour dynamics is responsible for the Gw gauge symmetry breaking if the large-N Schwinger-Dyson equation admits anomalous solutions. The dynamically generated mass matrix is computed through these solutions. The technicolour model is discussed.
For a variety of reasons [ 1 - 3 ] the assumption that the spontaneous breakdown of gauge symmetry is due to the vacuum expectation values of a set of spin-zero fields, seems to be unnatural. Some believe that the masses of gauge vector bosons arise from dynamical symmetry breaking and that the scalar bosons (in particular, Goldstone bosons) are bound states [ 4 - 7 , 2 ] . The symmetric gauge group of such theory is GW X U T c ( N ) . Here UTC(N) describes technicolour strong interaction, and GW describes weak and electromagnetic interactions. There is a set of fermion fields which transform under the fundamental representation of the colour group and under appropriate representation of Gw. The lagrangian density is
Z(W) = f d / a ( f ) dff d ~ exp {i[-¼FuvaFa tw + i~Tu(au - igfu a T a - igr Wu r pr) ~] },
£(x) = - ¼ F . v a ( x ) FaUV(x) -- ¼ W.vr(x) WrUU(x) + i~(x) 3~ [Ou -- igfua(x) 1~ _ igr wur(x) pr]~ ¢ ( x ) .
Our task is to describe the dynamics of the vector boson mass generation without any phenomenological consideration. In an approximation when only strong interaction is taken into account it is shown that the large N dynamics is responsible for the G w gauge symmetry breaking if the large N Schwinger-Dyson equation admits anomalous (symmetry breaking) solutions. The large-N mass matrix is computed through these solutions. The possibility to construct an effective action of GW gauge fields and colour singlet bilocal fields which describe fermion-antifermion bound states is discussed. Let us consider the generating functional
(1)
The strengths F u r and Wuvr are constructed from the vector gauge fields fu a and Wu r, respectively, and Ta, p r are matrices representing the corresponding group generators. 1 On leave of absence from Department of Physics, Sofia University, Sofia 1126, Bulgaria. 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
where d # ( f ) includes gauge-fixing conditions and Faddeev-Popov determinant. The G w gauge fields are considered as classical sources. Integrating overfua(x) one obtains
z(w) = f d ¢
d ~ exp {i[i~TU(3# - igr w r p r)
+~(g~v.~)]},
(3)
where S is generating functional for the connected diagrams of the pure Yang-Mills theory. To analyze 201
Volume 133B, number 3,4
PHYSICS LETTERS
the N-dependence of S it is convenient to introduce a diagram technique, indicating explicitly the flow of colour indices [8]. Because a gluon carries two colour indices fu(x)~ =fua(x)(Ta)~, the gluon propagator is represented by two lines with different direction and an analogous representation holds for the ghost field propagator as well. To see where the indices go it is enough to follow the arrows. The index lines flow continuously and they either close to make a loop, or beginning from some ~bi(x) (lower index) they end in some ~i(y) (upper index). To every such a line corresponds fii'/,. Summing over the colour indices will give N, for every index loop, and colour singlet combination of fermion fields ~i(x) ~kio') in the other case. So that the ffi and ~i dependence of S is only through singlet combinations. It is convenient to introduce a colour singlet bilocal collective coordinates [9,10] by the relations
f x,yH dUo~(x,y)f(NUc~#(x,y)-
t~ai(x)
or,[3 = const.
(4)
Then
exp [is(g f7 ~ U ~,)] =f x~,y dU,~(x,y) ~ (NUo¢ - ~ai~io) exp[iV(U)] or,f3
=const. f
H dC(x,y) H dU(x,y) x,y x,y
× exp {i[Ntr CU+ ~ict~i + V(U)]}.
15 December 1983
Let us now replace exp(iS) in eq. (3) by the righthand side of eq. (5). Integrating over qJ and ~, one obtains for a generating functional
z(~o =f d C dU
exp [iSeff(U , C, I41)] ,
(7)
where
Serf(U, C, I42)= N t r CU + V(U) - iNtr ln(~ - iC - igr fcrpr).
(8)
We can draw up Feynman diagrams for any term in V(U) just as in QCD. The only difference is that we have to associate with every fermion line fit).U(x, y), instead of the free fermion propagator. The diagrams allowable in V(U) are vacuum ones and fulfill the requirements: (i) They are connected diagrams which remain connected after removing all of the fermion lines. (ii) Any index loop contains no more than one fermion line. As a result of this, summing over the colour indices of a loop with a fermion line we obtain N, which reproduces N entered by eq. (5). Now to determine the Ndependence of V(U) we have to follow the well-known considerations in QCD [8]. In particular, the coupling constant rescahng g = (1/~'--N)go must be done. The diagrams contain at least one fermion loop, so that the leading ones are of order N. This results in an expression for the functional V(U) as a power series inN V(U) = N[ V0(U) + N - 1 V1(U) + N -2 V2(U) + ...]. (9)
(5)
Here V(U) is obtained from S after replacement of all singlet cotnbinations ~b~i(x) ~i~ (Y) by N U ~ (x,y). The short designations are deffmed as follows: tr c u = f dx dy C~(x,y) U~(y,x) ,
The planar diagrams with only one fermion loop (the boundary of the graphs) which obeys the condition (i) take part in the formation of V0(U). All these diagrams s~tisfy the condition (ii). In accordance with (8) and (9), Serf can be represented in the form Serf = N(So + N-1S1 + N-2S2 + ...),
(10)
where S O= tr CU+ Vo(U)
'cv:e=fox dy fio~(X ) Coq3(x,y)
~#i(y),
(6)
where a and/3 are common symbols for spinor and GW group indices. They will be systematically dropped. 202
- i tr In (~ - iC - igr I~r F r)
Si = Vi(U) ,
i>0.
(11)
To compute the generating functional (-i) In Z(W)
Volume 133B, number 3,4
PHYSICS LETTERS
15 December 1983
in the leading order of N one must fred the stationary point of the effective action S O with respect to Uand C, when Wur = 0. The equations are
the self-energy of the gauge fields and a mixing term of the War and C fields. For the self-energy of W mesons we have
6 S o / 6 U ~ ( x , y ) = O,
irl.:~(p)=Nff'~f~d4
~So/SCa~(x,y ) = O.
We look for solutions in the form
Ua43(x,y)
"
"
~ ~ tr SL(p-q)TuprsI(q)7 v p s , (15)
where SL(q) is the Fourier transform of the solution
= Saj3L(x -- Y ) ,
SL(x - y).
Ca~(x,y) = Ea~L(x - y ) .
(12)
Then, the first equation expresses the connection between the fermion propagator and the fermion selfenergy
Sc~flL(x -
y ) = i [ ( i b + ]~L)-l]ot/3(x -- y ) .
(13)
It is evident that W mesons acquire masses if IluvrS(O) = NguvM rs is unequal to zero, and the matrix M rs has only non-negative proper values. Then, the large-N mass term in the effective lagrangian will be
12N WurMrs ws ~ •
To obtain an effective action containing only IV and C fields, we must perform the integration over the U fields. To do this we have to rescale the bilocal fields Uand C
The second one is a Schwinger-Dyson large N equation for a fermion self-energy. For simplicity reasons we will represent it only graphically
U-~ 1 / x / ~ , (14) To every fermion line - and
(16)
corresponds Sc~#L(x-y )
C~ l/x/~,
then the quadratic form of V defines the propagator 0 0 and higher-order terms de'termine the interaction vertices. As a result the effective lagrangian is £ e f f = - ~ ,I, tIMa v
r Di! I~I/4-
,,r
-
} N W~rMrs Ws u
+ N£W + £C + £WC, stands for a sum of coefficient functions of all diagrams, with a definite number of external lines, which come from the leading order diagrams in V(U) after removing all of the fermion lines. Eq. (14) reduces to a simpler one, in the case of two dimensional U(N) gauge theory, and coincides with 't Hooft's equation [11]. l e t us represent the trace in eq. (11) in the form tr ln(~ - i C - igrfCrP r) = tr ln(1 + [i/(i~ + EL)] [ i ( C - EL)+ igrWrrr]) + tr ln[b - iE L] ." The last term does not depend on the fields Wur and C, so we will drop it. E x p a n d i n g S e f f around the solution (U = S L, C = E L, W = O) and making a fields translation ( U - S L U , C - ]~L ~ (7, W ~ I4/) w e can separate from S O the large-N quadratic form of the U and C bilocal fields,
(17)
(18)
and the coefficient functions can be computed, in principle, in the large-N limit. For example, the selfinteraction of W bosons is of order N, and the selfinteraction [C] K of C-fields is of order (l/N) K/2-1. The action (18) describes the interaction of W mesons and particles which are fermion-antifermion bound states, because the inverted large-N, C-fields quadratic form satisfies the inhomogeneous BetheSalpeter equation. To obtain this equation let us consider the quadratic part of Serf, in the large-Nlimit [see (10) and (11)] S~2)= tr CU + ½CRC + ½UR U, where
K#,la,#,(v,x;x',y' ) = -g2oTo#,UDuvL(y-x) Ta,c~V8( y ' - y ) 8 ( x ' - x ) , R#,~I cg;3'(Y , x ; x', y') = --isafL (y - y') Sc,,,~t (x ' - x) . (19) 203
Volume 133B, number 3,4
PHYSICS LETTERS
Here DlavL(x-y) is the large-N gluon propagator and SL(x'-x) is the solution ofeq. (14). Performing the gaussian integral over U, we obtain for a C-field quadratic form
- G - I = [R_ K-1] . Straightforward computations lead to the large-N Bethe-Salpeter equation for G
15 December 1983
The function B(p 2) is non-zero because the vacuum expectation value of the order parameter is non-zero. Substituting the solution (22) and the vertices I "r eq. (21) into eq. (15), we obtain the non-vanishing elements of the vector-boson mass matrix
1 2r+
M11 = M22 = M33 = ~g2~" 2,
MOO = g2{ [(1102 + ½] F f + (½102F~" + 4(½ y)2F1}
Gmel sr (X, x'; y', y)
--12
= (igO)2~m~Ou~L(x - x') "r~e~(x - y) ~(x' - y')
= gglFo, where
+ ig8 f d4z d 4 ' Tm #S##'L(x -Z) G[3'a'isr(Z, t;y'y)
1 f dpp5[A+(_p2)]2
F2 -*= 16rr 2 X S~,c~L(t
- x') 7aeVD#vL(x - x').
4r=g2~Yu½rr½(1--75)~,
r=1,2,3,
+ 75)1 ~b,
(21)
F1: ~
,f
d# p3 [B(_p2)] 2.
0 The masses of W+-bosons are equal to ~Ng2F2 +. To ensure the photon mass vanishes and that Mz = Mw cos 0 the conditions 1
+
1
$F2 = F 1 = s F 0 have been imposed [2,7,3]. In our case they lead to the relations /72- = XF2+ '
F 1 = (I/N/r2) F2 + ,
X = (1 - }Y2)/(1 + } y 2 ) . The first one can be simply satisfied and for a model with Y= 0 [3] it is enough to put A+fp 2) = A - ( p 2) = A (p2) but the second condition leads to a complicated relation between A (p2) and B(p2). The author is grateful to Professor A.A. Slavnov for helpful correspondence and to Professor N.S. Craigie for a stimulating conversation. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References
Sc~3L(P) = 8o4~[PA+(P 2) ½(1 + 75)
204
,
where
where r r are Pauli isospin matrices. In the limit gl, g2 -+ 0 the technicolour interaction is invariant under a global SUr(2 ) × SUR(2 ) symmetry. We assume that the technicolour forces break the symmetry. It was shown in ref. [12] that, in the largeN limit the symmetry necessarily breaks down to diagonal "isospin" SU(2). The Green's function of the T product of three chiral currents can be represented, in large-N, by a triangle diagram only, in which one has to associate with the fermion lines the solution of eqs. (13), (14). Then, from the anomaly equation [12], it follows that the solution must be in the form
+ PA-(.p 2) {(1 - US) + B(p2)] •
0
(20)
The poles in the solution GmeIsr(X, x'; y', y) determine the fermion-antifermion bound states, so it is clear that the bilocal fields describe composite particles. In the case of the technicolour model Gw = SUL X Uy(1). We have a multiplet of fermions (~bl, tb2) T which form a left-handed SUL(2 ) doublet (~L 1, ~L2) T with hypercharge - ½ Y and a right-handed SUL(2) singlet qJR1, ~R 2 with hypercharges equal to ½(Y + I) and - } ( Y - 1) correspondingly. The gauge bosons Wu, Bu couple to the following fermion currents
4 0 =gl~v~[~r+x'3!~' : - =,*
M30 = M03 = glg2F1,
(22)
[1] G. 't Hooft, Lectures Carg~se Summer Institute (1979). [2] L. Susskind, Phys. Rev. D20 (1979) 2619.
Volume 133B, number 3,4
PHYSICS LETTERS
[3] E. Farhi and L. Susskind, Phys. Rep. C74 (1981) 277. [4] J. Cornwall and R. Norton, Phys. Rev. D8 (1973) 3338; R. Jackiw and K. Jonson, Phys. Rev. D8 (1973) 2386. [5] E. Eichten and P. Feinberg, Phys. Rev. D10 (1974) 3254; S. Tye, E. Tomboulis and E. Poggio, Phys. Rey. D l l (1975) 2839. [6] J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. [7] S. Weinberg, Phys. Rev. D13 (1976) 974.
15 December 1983
[8] G. 't Hooft, Nucl. Phys. B72 (1974) 461. [9] V. Pervushin and D. Ebert, Teor. Mat. Fiz. 36 (1978) 313; E. Witten, Nucl. Phys. B160 (1979) 57. [10] A. Slavnov, Phys. Lett. l12B (1982) 154; Teor. Mat. Fiz. 51 (1982) 307. [11] G. 't Hooft, Nucl. Phy~ B75 (1974)461. [12] S. Coleman and E. Witten, Phy~ Rev. Lett. 45 (1980) 100.
205
PHYSICS LETTERS
15 December 1983
ON THE LARGE-N DYNAMICS OF GAUGE SYMMETRY BREAKING N.I. KARCHEV 1 International Centre for Theoretical Physics, Trieste, Italy Received 9 August 1983
We consider a Gw XUTC(N) gauge theory. A method of colour singlet bilocal collective coordinates is proposed to show, large-N colour dynamics is responsible for the Gw gauge symmetry breaking if the large-N Schwinger-Dyson equation admits anomalous solutions. The dynamically generated mass matrix is computed through these solutions. The technicolour model is discussed.
For a variety of reasons [ 1 - 3 ] the assumption that the spontaneous breakdown of gauge symmetry is due to the vacuum expectation values of a set of spin-zero fields, seems to be unnatural. Some believe that the masses of gauge vector bosons arise from dynamical symmetry breaking and that the scalar bosons (in particular, Goldstone bosons) are bound states [ 4 - 7 , 2 ] . The symmetric gauge group of such theory is GW X U T c ( N ) . Here UTC(N) describes technicolour strong interaction, and GW describes weak and electromagnetic interactions. There is a set of fermion fields which transform under the fundamental representation of the colour group and under appropriate representation of Gw. The lagrangian density is
Z(W) = f d / a ( f ) dff d ~ exp {i[-¼FuvaFa tw + i~Tu(au - igfu a T a - igr Wu r pr) ~] },
£(x) = - ¼ F . v a ( x ) FaUV(x) -- ¼ W.vr(x) WrUU(x) + i~(x) 3~ [Ou -- igfua(x) 1~ _ igr wur(x) pr]~ ¢ ( x ) .
Our task is to describe the dynamics of the vector boson mass generation without any phenomenological consideration. In an approximation when only strong interaction is taken into account it is shown that the large N dynamics is responsible for the G w gauge symmetry breaking if the large N Schwinger-Dyson equation admits anomalous (symmetry breaking) solutions. The large-N mass matrix is computed through these solutions. The possibility to construct an effective action of GW gauge fields and colour singlet bilocal fields which describe fermion-antifermion bound states is discussed. Let us consider the generating functional
(1)
The strengths F u r and Wuvr are constructed from the vector gauge fields fu a and Wu r, respectively, and Ta, p r are matrices representing the corresponding group generators. 1 On leave of absence from Department of Physics, Sofia University, Sofia 1126, Bulgaria. 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
where d # ( f ) includes gauge-fixing conditions and Faddeev-Popov determinant. The G w gauge fields are considered as classical sources. Integrating overfua(x) one obtains
z(w) = f d ¢
d ~ exp {i[i~TU(3# - igr w r p r)
+~(g~v.~)]},
(3)
where S is generating functional for the connected diagrams of the pure Yang-Mills theory. To analyze 201
Volume 133B, number 3,4
PHYSICS LETTERS
the N-dependence of S it is convenient to introduce a diagram technique, indicating explicitly the flow of colour indices [8]. Because a gluon carries two colour indices fu(x)~ =fua(x)(Ta)~, the gluon propagator is represented by two lines with different direction and an analogous representation holds for the ghost field propagator as well. To see where the indices go it is enough to follow the arrows. The index lines flow continuously and they either close to make a loop, or beginning from some ~bi(x) (lower index) they end in some ~i(y) (upper index). To every such a line corresponds fii'/,. Summing over the colour indices will give N, for every index loop, and colour singlet combination of fermion fields ~i(x) ~kio') in the other case. So that the ffi and ~i dependence of S is only through singlet combinations. It is convenient to introduce a colour singlet bilocal collective coordinates [9,10] by the relations
f x,yH dUo~(x,y)f(NUc~#(x,y)-
t~ai(x)
or,[3 = const.
(4)
Then
exp [is(g f7 ~ U ~,)] =f x~,y dU,~(x,y) ~ (NUo¢ - ~ai~io) exp[iV(U)] or,f3
=const. f
H dC(x,y) H dU(x,y) x,y x,y
× exp {i[Ntr CU+ ~ict~i + V(U)]}.
15 December 1983
Let us now replace exp(iS) in eq. (3) by the righthand side of eq. (5). Integrating over qJ and ~, one obtains for a generating functional
z(~o =f d C dU
exp [iSeff(U , C, I41)] ,
(7)
where
Serf(U, C, I42)= N t r CU + V(U) - iNtr ln(~ - iC - igr fcrpr).
(8)
We can draw up Feynman diagrams for any term in V(U) just as in QCD. The only difference is that we have to associate with every fermion line fit).U(x, y), instead of the free fermion propagator. The diagrams allowable in V(U) are vacuum ones and fulfill the requirements: (i) They are connected diagrams which remain connected after removing all of the fermion lines. (ii) Any index loop contains no more than one fermion line. As a result of this, summing over the colour indices of a loop with a fermion line we obtain N, which reproduces N entered by eq. (5). Now to determine the Ndependence of V(U) we have to follow the well-known considerations in QCD [8]. In particular, the coupling constant rescahng g = (1/~'--N)go must be done. The diagrams contain at least one fermion loop, so that the leading ones are of order N. This results in an expression for the functional V(U) as a power series inN V(U) = N[ V0(U) + N - 1 V1(U) + N -2 V2(U) + ...]. (9)
(5)
Here V(U) is obtained from S after replacement of all singlet cotnbinations ~b~i(x) ~i~ (Y) by N U ~ (x,y). The short designations are deffmed as follows: tr c u = f dx dy C~(x,y) U~(y,x) ,
The planar diagrams with only one fermion loop (the boundary of the graphs) which obeys the condition (i) take part in the formation of V0(U). All these diagrams s~tisfy the condition (ii). In accordance with (8) and (9), Serf can be represented in the form Serf = N(So + N-1S1 + N-2S2 + ...),
(10)
where S O= tr CU+ Vo(U)
'cv:e=fox dy fio~(X ) Coq3(x,y)
~#i(y),
(6)
where a and/3 are common symbols for spinor and GW group indices. They will be systematically dropped. 202
- i tr In (~ - iC - igr I~r F r)
Si = Vi(U) ,
i>0.
(11)
To compute the generating functional (-i) In Z(W)
Volume 133B, number 3,4
PHYSICS LETTERS
15 December 1983
in the leading order of N one must fred the stationary point of the effective action S O with respect to Uand C, when Wur = 0. The equations are
the self-energy of the gauge fields and a mixing term of the War and C fields. For the self-energy of W mesons we have
6 S o / 6 U ~ ( x , y ) = O,
irl.:~(p)=Nff'~f~d4
~So/SCa~(x,y ) = O.
We look for solutions in the form
Ua43(x,y)
"
"
~ ~ tr SL(p-q)TuprsI(q)7 v p s , (15)
where SL(q) is the Fourier transform of the solution
= Saj3L(x -- Y ) ,
SL(x - y).
Ca~(x,y) = Ea~L(x - y ) .
(12)
Then, the first equation expresses the connection between the fermion propagator and the fermion selfenergy
Sc~flL(x -
y ) = i [ ( i b + ]~L)-l]ot/3(x -- y ) .
(13)
It is evident that W mesons acquire masses if IluvrS(O) = NguvM rs is unequal to zero, and the matrix M rs has only non-negative proper values. Then, the large-N mass term in the effective lagrangian will be
12N WurMrs ws ~ •
To obtain an effective action containing only IV and C fields, we must perform the integration over the U fields. To do this we have to rescale the bilocal fields Uand C
The second one is a Schwinger-Dyson large N equation for a fermion self-energy. For simplicity reasons we will represent it only graphically
U-~ 1 / x / ~ , (14) To every fermion line - and
(16)
corresponds Sc~#L(x-y )
C~ l/x/~,
then the quadratic form of V defines the propagator 0 0 and higher-order terms de'termine the interaction vertices. As a result the effective lagrangian is £ e f f = - ~ ,I, tIMa v
r Di! I~I/4-
,,r
-
} N W~rMrs Ws u
+ N£W + £C + £WC, stands for a sum of coefficient functions of all diagrams, with a definite number of external lines, which come from the leading order diagrams in V(U) after removing all of the fermion lines. Eq. (14) reduces to a simpler one, in the case of two dimensional U(N) gauge theory, and coincides with 't Hooft's equation [11]. l e t us represent the trace in eq. (11) in the form tr ln(~ - i C - igrfCrP r) = tr ln(1 + [i/(i~ + EL)] [ i ( C - EL)+ igrWrrr]) + tr ln[b - iE L] ." The last term does not depend on the fields Wur and C, so we will drop it. E x p a n d i n g S e f f around the solution (U = S L, C = E L, W = O) and making a fields translation ( U - S L U , C - ]~L ~ (7, W ~ I4/) w e can separate from S O the large-N quadratic form of the U and C bilocal fields,
(17)
(18)
and the coefficient functions can be computed, in principle, in the large-N limit. For example, the selfinteraction of W bosons is of order N, and the selfinteraction [C] K of C-fields is of order (l/N) K/2-1. The action (18) describes the interaction of W mesons and particles which are fermion-antifermion bound states, because the inverted large-N, C-fields quadratic form satisfies the inhomogeneous BetheSalpeter equation. To obtain this equation let us consider the quadratic part of Serf, in the large-Nlimit [see (10) and (11)] S~2)= tr CU + ½CRC + ½UR U, where
K#,la,#,(v,x;x',y' ) = -g2oTo#,UDuvL(y-x) Ta,c~V8( y ' - y ) 8 ( x ' - x ) , R#,~I cg;3'(Y , x ; x', y') = --isafL (y - y') Sc,,,~t (x ' - x) . (19) 203
Volume 133B, number 3,4
PHYSICS LETTERS
Here DlavL(x-y) is the large-N gluon propagator and SL(x'-x) is the solution ofeq. (14). Performing the gaussian integral over U, we obtain for a C-field quadratic form
- G - I = [R_ K-1] . Straightforward computations lead to the large-N Bethe-Salpeter equation for G
15 December 1983
The function B(p 2) is non-zero because the vacuum expectation value of the order parameter is non-zero. Substituting the solution (22) and the vertices I "r eq. (21) into eq. (15), we obtain the non-vanishing elements of the vector-boson mass matrix
1 2r+
M11 = M22 = M33 = ~g2~" 2,
MOO = g2{ [(1102 + ½] F f + (½102F~" + 4(½ y)2F1}
Gmel sr (X, x'; y', y)
--12
= (igO)2~m~Ou~L(x - x') "r~e~(x - y) ~(x' - y')
= gglFo, where
+ ig8 f d4z d 4 ' Tm #S##'L(x -Z) G[3'a'isr(Z, t;y'y)
1 f dpp5[A+(_p2)]2
F2 -*= 16rr 2 X S~,c~L(t
- x') 7aeVD#vL(x - x').
4r=g2~Yu½rr½(1--75)~,
r=1,2,3,
+ 75)1 ~b,
(21)
F1: ~
,f
d# p3 [B(_p2)] 2.
0 The masses of W+-bosons are equal to ~Ng2F2 +. To ensure the photon mass vanishes and that Mz = Mw cos 0 the conditions 1
+
1
$F2 = F 1 = s F 0 have been imposed [2,7,3]. In our case they lead to the relations /72- = XF2+ '
F 1 = (I/N/r2) F2 + ,
X = (1 - }Y2)/(1 + } y 2 ) . The first one can be simply satisfied and for a model with Y= 0 [3] it is enough to put A+fp 2) = A - ( p 2) = A (p2) but the second condition leads to a complicated relation between A (p2) and B(p2). The author is grateful to Professor A.A. Slavnov for helpful correspondence and to Professor N.S. Craigie for a stimulating conversation. The author would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References
Sc~3L(P) = 8o4~[PA+(P 2) ½(1 + 75)
204
,
where
where r r are Pauli isospin matrices. In the limit gl, g2 -+ 0 the technicolour interaction is invariant under a global SUr(2 ) × SUR(2 ) symmetry. We assume that the technicolour forces break the symmetry. It was shown in ref. [12] that, in the largeN limit the symmetry necessarily breaks down to diagonal "isospin" SU(2). The Green's function of the T product of three chiral currents can be represented, in large-N, by a triangle diagram only, in which one has to associate with the fermion lines the solution of eqs. (13), (14). Then, from the anomaly equation [12], it follows that the solution must be in the form
+ PA-(.p 2) {(1 - US) + B(p2)] •
0
(20)
The poles in the solution GmeIsr(X, x'; y', y) determine the fermion-antifermion bound states, so it is clear that the bilocal fields describe composite particles. In the case of the technicolour model Gw = SUL X Uy(1). We have a multiplet of fermions (~bl, tb2) T which form a left-handed SUL(2 ) doublet (~L 1, ~L2) T with hypercharge - ½ Y and a right-handed SUL(2) singlet qJR1, ~R 2 with hypercharges equal to ½(Y + I) and - } ( Y - 1) correspondingly. The gauge bosons Wu, Bu couple to the following fermion currents
4 0 =gl~v~[~r+x'3!~' : - =,*
M30 = M03 = glg2F1,
(22)
[1] G. 't Hooft, Lectures Carg~se Summer Institute (1979). [2] L. Susskind, Phys. Rev. D20 (1979) 2619.
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PHYSICS LETTERS
[3] E. Farhi and L. Susskind, Phys. Rep. C74 (1981) 277. [4] J. Cornwall and R. Norton, Phys. Rev. D8 (1973) 3338; R. Jackiw and K. Jonson, Phys. Rev. D8 (1973) 2386. [5] E. Eichten and P. Feinberg, Phys. Rev. D10 (1974) 3254; S. Tye, E. Tomboulis and E. Poggio, Phys. Rey. D l l (1975) 2839. [6] J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. [7] S. Weinberg, Phys. Rev. D13 (1976) 974.
15 December 1983
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