- Email: [email protected]
Dynamical gauge symmetry breaking on the lattice
Volume 249, number 1
PHYSICS LETTERS B
11 October 1990
Dynamical gauge symmetry breaking on the lattice K. F a r a k o s , G. K o u t s o u m b a s a n d G. Z o u p a n o s Phystcs Department, Nattonal Technical Untverszty of Athens, Zografou Campus, GR-15 7 73 Athens, Greece
Received 25 June 1990
We study, using lattice techniques, the dynamical symmetry breaking of a three-dimensional theory that m~mlcsthe electroweak sector of the standard model We show that m the strong couphng hmlt ofa QCD-hke theory the ferm~on condensates which are produced reduce dynamical symmetry breaking of the sector corresponding to the electroweak gauge group
1. Introduction
Nowadays it is widely believed that the original suggestion of N a m b u and Jona-Lasinio [1 ] on the chiral symmetry breaking (xSB) is realised by the dynamics of QCD. In addition, the plausible assumption of Welnberg [2 ] and Susskmd [ 3 ] that the 2SB due to Q C D could induce the spontaneous symmetry breaking o f the electroweak SU(2)LXU(1 ) model, has also been widely appreciated. This assumption, which ~s a modern version of an earlier proposal due to Schwinger [ 4 ], is at the heart of the dynamical symmetry breaking scenaria such as the technicolour [ 5 ] and high colour [ 6 ]. Unfortunately, so far, nobody was able to derive analytically the xSB and its immediate consequences such as the Goldstone nature of the n meson starting from the underlying gauge theory QCD. Therefore, any approximation that supports the c o m m o n belief that Q C D produces the xSB is very welcome. In this respect it is worth mentioning a n u m b e r o f lattice calculations as more reliable [ 7 ]. Moreover so far not even an approximate calculation exists indicating that the zSB can induce the spontaneous symmetry breaking of gauge theories. It is our purpose here to provide a lattice calculation, using a slmphfied model, which demonstrates both xSB and spontaneous symmetry breaking of a gauge theory. The framework in which we shall make our study
is an abehan three-dimensional gauge theory with two fermionic flavours [ 8,9 ]. This model will play a role which we expect will mimic the four-dimensional non-abehan QCD, or a QCD-hke strongly interactIng theory, in the feature of producing spontaneous chiral symmetry breaking. A very Important feature shared by the toy model and the four-dimensional Q C D is that they are both confining theories [ 10 ]. The model that we consider here is a very interesting field theory in its own right. Exploration o f its glueball spectrum has recently been done [ 11 ] using both quenched and dynamical fermlons. Quenched calculations on the meson spectrum are reported in ref. [ 9 ]. There has also been some controversy concerning the parity properties of the theory [ 12 ]. An important aspect of the model most interesting for our purposes is the choral symmetry breaking, which has been estabhshed using either quenched [ 13 ] or dynamical [ 11,14 ] fermxons. We shall formulate the model in the continuum and then turn to the lattice calculations, in order to demonstrate our mare point, namely that the gSB triggers the spontaneous breaklng of an appropriately coupled gauge theory.
2. The model The lagrangian density of the " Q C D " - Q E D (in this paper) with one flavour is LP= - ~ (F~,,) 2 + 1~PD u~,~,~ - m ~°~v,
Partially supported by the Greek Ministry of Research and Technology
( 1)
with D u = 0 u - leA~,. The fermion field is a four-com-
0370-2693/90/$ 03 50 © 1990 - Elsevier Soence Pubhshers B.V ( North-Holland )
101
Volume 249, number 1
PHYSICS LETTERS B
ponent spinor. The 4 × 4 y-matrices are chosen to be
~0
(;3
2 [10"2
_00.3) , ,l
(1;1
--10-10) ,
_0o)
11 October 1990
Al=}Py3 T ,
A2=t/glysT,
A3=~T ,
(11)
and
Bl,u-~- ~ft~,uY3T , (2)
There exist two 4 × 4 matrices which antlcommute with 7 o, yl, 22:
B2u=~P?u75T,
B3u= !PYuAT,
/t=0, 1 , 2 ,
(12)
transform as triplets under SU (2), while the
A4=~JAT,
B4u = ~PyuT
(13)
T ~ e x p (10y3) T ,
(4)
are SU (2) slnglets. The choice for the g a m m a matrices given in eqs. (2), (3) permits to separate the lagranglan density given in eq. ( 1 ) into two pieces by writing T = ( Tl, T2) and ~Pz= T~,0-3, l = 1,2. Then the fermion lagranglan becomes
T ~ exp (1~075) T ,
(5)
Y=tP~(10-~,D~'-m)Tl+~(iauDU+m)T2.
Therefore the massless theory is invariant under the global "chiral" transformations
which both correspond to
(14)
in four dimensions. The generators Y3, )'5 together with and d-=Iy3y5 are the generators of a global U(2)~SU(2)×U(1) symmetry that the theory possesses. In particular ~ generates a U ( 1 ) while the other three form the S U ( 2 ) part of the group. The corresponding global symmetry for N flavours [ 15 ] is U ( 2 N ) Let us make the S U ( 2 ) part of the global symmetry a little bit more explicit. The currents corresponding to the transformations
When m = 0 the two fields Tl, T2 are indistinguishable and may be considered as two flavours, whose mixture produces a global SU (2) flavour symmetry. This global SU (2) flavour symmetry is precisely the one that we referred to above. Loosely speaking, for m e 0 , the fields Tl and T2 have opposite "spin" components along the (missing) z-axis and are images of each other under parity; they also interact differently with the electromagnetic field. By expressing the bihnears given in eqs. ( 1 1 ), (12), ( 13 ) in terms of T~, T2 we can illustrate better their S U ( 2 ) structure:
T~exp(io~F)T,
(triplet)
T--* exp (moy5) T
(6)
F = y 3 , ys,
A=ly3Y5
(7)
are given by (8)
The j r are conserved in the absence of any fermionic mass term, while in the presence of non-vanishing mass rn one obtains
O~Jr = 2m!PFT.
(9)
In fact the current corresponding to the generator ~ is always conserved. When rn = 0 the three conserved currents j F lead to the conserved charges Qr= fd2x 7-rtFT, which satisfy an SU (2) algebra: [ 05, Qa ] = 21Q3,
[Q,J, Q3] =21Q5 . One can also easily see that the bilinears 102
~rJ2-- ~2~/1 ,
A2 = ~Ol T 2 + ~P2 T l ,
j~r = !PT~FT.
[ Q3, Qs ] = 2IQ~,
Al =~Pl
(10)
(triplet)
A3 = ~tl TI - ~O2T2 ,
Bl~= Tl0-uT2 + ~'2a~,Tl ,
B2# = ~l 0-~,T2 - ~2 o,u Tl , B3,u = ~l 0-,u~[/l --~-120-lt~-12, (slnglets)
Aa = ~Pl T~ + ~P2Tz,
B4,~----~l 0-. T l -+-~t20-. T 2 .
(15)
A mass term which is parity invarlant and breaks the "chiral" S U ( 2 ) transformations is m T T . This becomes, writing the four-component spinor in terms o f T l , T:,
m ~ T = m ~ l Till -- mgP2 ttJ2= mA3
(16)
This mass term breaks the global symmetry U ( 2 )
Volume 249, number 1
PHYSICS LETTERS B
down to U ( 1 ) × U ( 1 ) with generators ~, A. In the case o f N flavours the U ( 2 N ) breaks to S U ( N ) × SU(N) ×U(I) ×U(1). In o r d e r to keep the analogies we recall that Q C D with two flavours has a chtral global s y m m e t r y U ( 2 ) L × U (2) R, which m the presence o f a mass term breaks down to U(2)L+R. Such a chlral s y m m e t r y breaking mass t e r m is expected to emerge d y n a m i cally m QCD. Switching on the S U ( 2 ) L × U ( 1 ) , which corresponds to gauging an SU ( 2 ) subgroup o f the global symmetry, given in eq. ( 1 0 ) , the zSB is expected to induce the spontaneous s y m m e t r y breakmg o f the gauge group S U ( 2 ) L X U ( 1 ) down to U ( 1 )EM. Keeping the analogies in our toy model, it is suggested to gauge the SU ( 2 ) subgroup o f U ( 2 ) as well. [We note that the U ( 1 ) part has already been gauged. ] T h e lagrangtan density o f our toy m o d e l is o~--
- ~ (| F ~ v )
2
' -/2 yu~V-m~P~, - ~ (1 F u v ) 2 +I~ttD
I I October 1990
similar to the one used to study spontaneous magnetization in ferromagnets: In the beginning an external magnetic field is present, which is gradually r e m o v e d and the residual magnetization is measured at the h m i t o f zero external field. In our case the need to take the massless limit m a y be rephrased by pointing out that the f e r m i o m c mass t e r m gives masses to the would-be G o l d s t o n e bosons. Thus it forbids the appearance o f the pole contribution to the massless gauge boson p r o p a g a t o r which is expected to provide mass to the gauge boson ~j We now give the lattice action employed, which is just the lattice transcription o f the action corres p o n d m g to the lagrangian density given in eq. ( 17 ):
S=½ ~ K[~n(1-~u)UnuVnu~ rl,/2
+ Lp.+~(1+~,~)uL vL~v.] - Z ~p,~v +fl, Z ( 1 - T r Up)
(17) where D u = O,-ig~Au-xg2r.B u a n d Fu,, Fu~ are the U ( 1 ) and S U ( 2 ) gauge field kinetic terms. The dynamtcal emergence o f a mass term as in eq. ( 1 6 ) [or equivalently (A 3) :# 0, due to the f o r m a t i o n o f condensates by the strong U ( 1 ) coupling ] is expected to trtgger the breaking o f the gauge group SU ( 2 ) down to U ( 1 ) . The two other m e m b e r s o f the trtplet, namely At, A2, will generate the would-be G o l d s t o n e particles, which are expected to provide masses to the two SU ( 2 ) gauge bosons. Thus the m o d e l has been p r o m o t e d to a theory based on the gauge group S U ( 2 ) × U ( 1 ) which is expected to b r e a k down to U(1)×U(1).
3. Lattice calculations Considering the m o d e l on the lattice, one should introduce a mass term, in o r d e r to regulate the functional mtegral. In addition, this mass acts like a gauge n o n - i n v a r l a n t term, that is, it orients the VEV o f the compostte triplet o f Higgs fields given in eq. ( 11 ) [which is going to emerge after integrating out the U ( 1 ) field] in the ( ~ ) direction. We will conclude that there is genuine s y m m e t r y breaking if the ( 7 6 u ) remains non-zero after the removal o f the mass (gauge breaking) term. The above p r o c e d u r e is very
n
p
+f12 •
(1-Tr
Vv).
(18)
P
U,u=exp(iO,u) represents the U ( 1 ) field, while Vnu= exp (t,Bn u) is the SU (2) gauge field. The SU (2) generators are the matrices (~3, ~'5, A), as discussed above. The two last terms in eq. (18) are the kinetic terms for the U ( 1 ) and the S U ( 2 ) gauge fields respectively. We use Wilson fermions to a v o i d fermion doubling. In this way we introduce a mass term in the model, which breaks the S U ( 2 ) mvariance. As already discussed above, SU ( 2 ) will be recovered in the massless limit, that is at K = Kermcal. In order to exhibit the d y n a m i c a l SU ( 2 ) breaking we restrict ourselves to the case o f infinitely strong U ( 1 ) coupling, that is fit = 0. In this case, the U ( 1 ) gauge field can be easily mtegrated out in the functional integral [ 17,18 ]. The p a r t i t i o n functton We use the familiar mass term m ~ v There is also another type of mass term, which could have been used, namely m ~vA7-1, which is invanant under the SU(2) group This mass term has the advantage of bemg SU(2) slnglet, but it has several problems: it breaks parity and reduces Chern-Slmons gauge lnvanant mass terms for all the gauge bosons Moreover, due to the fact that it does not orient the possible VEV to a specific direction, ( ~ r t ) is expected to be always zero for this mass term, independently of the existence or not of VEV! (Ehtzur's theorem [ 16 ] ) 103
Volume 249, number 1
PHYSICS LETTERS B
f
Z= J [dUdVdetd~]exp(-S),
(19)
If we only keep the linear term in the above expansion, the logarithm becomes
In Io
after performing the U integration becomes Z= f [dVd~d~,] exp(-Sen),
(20)
11 October 1990
¼yo
= - ¼KETr [M(n) ( 1 -Yu) Vn,M("+u)( 1 +7u) V~u] • (28)
where
Seff=~2E p
( 1 - T r Vv) + Z In I o ( x / ~ u ) - Z ~, ~,. n,lt
n
(21) We have denoted by Ynu the expression Ynta =- K Z ~ n (
1 - 7 u ) Vnu ~-'ln+U ~tn+u( 1 + 7u)
V*nu~n , (22)
and Io IS the zeroth order Bessel function. With the ~-(n) extra notation M ~b = 7~,~ ~P,b where n is the site and a, b the Dirac indices, Y,u becomes y . . = - K 2 Tr [M(") ( 1 -Yu) V~uM("+u) × (l+Tu)V*,,].
(23)
Next we switch from the functional integral over fermions to a functional integral over the mesons M ("). In doing so, one also has to include in the action the effect of the "jacoblan" of this transformation [ 17 ]. The partition function becomes Z= f [dVdM] exp(-Ae~r),
(24)
where Aeff=Seff-- ~ T r l n M (") . n
(25)
The last term in eq. (25) comes from the aforementioned "jacoblan". Our strategy will be to expand the term ~,,u l n l o ( x / ~ ) and keep only the lowest orders, which are likely to yield gauge boson masses, while the next ones will produce interactions. The expansion gives In l o ( x / ~ u ) = ~ Ck(y~u) k,
(26)
k
Ck= 2 ["
(1
l>m,( , f
.~m2 (27)
104
It is fairly evident from this expression that if symmetry breaking is to occur, it will emerge out of a nonzero VEV of the matrices M (n). A method to study the problem is to expand M (n) in the basis of the gamma matrices and identify meson states with definite quantum numbers; one may then consider the VEVs of each meson separately. In order to find the vacuum expectation value o f M (n), one can make use of the saddle point calculations described in ref. [ 17 ]. However, we prefer to use the results of refs. [9,11,13,14], since Monte Carlo simulations are more reliable than simple saddle-point estimates. We restrict ourselves to weak S U ( 2 ) coupling, which is justified by the fact that SU (2) plays the role of the electroweak gauge group. Thus the results derived for the model with global SU (2) can be carried over to our case without any qualitative changes. What has been found in ref. [ 9 ] is that for •1 small enough and K = K c r l t , C a l , the states generated by A~ and A 2 a r e massless, that is they are Goldstone bosons, while A3 acquires a non-zero mass. This is a first indication for the breaking of the SU (2) symmetry, since the members of an SU (2) triplet acquire different masses. It would be desirable to verify that the chlral condensate develops non-zero VEV. The results of ref. [9] are inconclusive in this respect, because they refer to rather large values offl~. However, the results for small fll have been obtained by one of us [13], performing calculations with quenched fermlons; a non-zero chlral condensate has been found forfl~ < 1.5. What is more important, these results remain valid when dynamical fermions are considered [ 11,14]. Similar results have been obtained by Kawamoto [ 19 ] using the strong coupling expansion. In particular, for fl~ = 0 , the A3 mass and the condensate are infinite m this approximation. In a d d m o n , no other meson state develops non-zero VEV Isolating the VEV (7nP) - u the expansion o f M (n) reads
Volume 249, number 1
PHYSICS LETTERS B
M ( ' ° = u ' ~ + l [ A 3 ( n ) ~ W A l ( n ) y 3 +A2(n)~5 + A4( n )AW B4u( n )yU + Blu( n )TUy3 + B2~,( n )yUy5 + B3u( n )yUA ] ,
(29)
where ¢t = 0, 1, 2. If we substitute M (") by its VEV (namely u~) in eq. (20), lnlo(x/~,u) is reminiscent of the scalargauge interaction part of the SU(2) adjoint gaugeHiggs model [ 20 ]. The experience with such models shows that the symmetry is broken down to U(1 ) whenever the constant multiplying the above-mentinned interaction part is large enough; then the model ~s m the so-called Higgs phase. In our case this constant ~s proportional to u 2, which may grow infinitely large, as/71 goes to zero. Thus for small values of ill we expect a broken SU (2) gauge symmetry. Let us now give a derivation of the breaking of the symmetry and an estimate of the masses acquired by the gauge bosons. By substituting eq. (29) into eq. (28) and taking into account that
Vnu = exp (leB, u) =cos(IB.ul)+ilrB,,usln(IB,,ul)/IB,,ul
,
(30)
we find In Io ( x / / ~ ) ~ ½K2crU2(sin21B,u I / I B , u 12)
11 October 1990
the functional integrations. In the strong coupling hmit of the U ( 1 ) gauge theory we have shown that the gSB of the fermlons, signaled by fermion-antlfermion condensates leads to a dynamical breaking of the SU (2) gauge group. The SU (2) gauge group breaks down to U ( 1 ), a fact which is established by determining the mass spectrum of the gauge bosons. It would be very interesting to study the symmetry breaking by performing a Monte Carlo simulation of the model. The relative simphclty of our toy model as compared to the reahstic four-dimensional electroweak gauge group coupled to a QCD-hke theory has been found to be very useful techmcally. A study of the realistm theory we feel will face an appremable technical complication as compared with the present one, but no fundamentally new difficultms.
Acknowledgement We would hke to thank J. Ihopoulos for useful discussions at the early stages of this work. K.F. and G.K. would also like to thank G. Tlktopoulos for enlightening discussions. Financial support by the Greek Ministry of Research and Technology ~s gratefully acknowledged.
× [ (B~.u):+ (B2u): ] + interaction terms,
(31 )
out of which we identify the mass term: 1/4"2 .it2[ 1 2 ~..¢n,t(B.u) + ( B .2u ) 2 ]
(32)
for the two gauge bosons, while the third one, B 3, only develops interactmn terms, remaining massless. Therefore the theory breaks from SU (2) X U ( l ) down to U ( 1 ) × U ( 1 ) .
4. Conclusions In the present paper we have used a three-dimensmnal toy model in order to mimic the electroweak sector of the standard model coupled to a strongly interacting QCD-like theory. The latter is represented by a U ( 1 ) model whde the electroweak gauge group is represented by an SU (2). We have formulated the theory on the lattice in order to be able to perform
References [ 1 ] Y. Nambu and G Jona-Lasmto, Phys. Rev 122 ( 1961 ) 345 [2]S Welnberg, Phys. Rev. D 13 (1976) 974, D 19 (1979) 1277 [3] L Susskmd, Phys. Rev D 20 (1979) 2619 [4] J Schwmger, Phys. Rev 128 (1962) 2425 [ 5 ] For revmws see E Farhl and L Susskind, Phys. Rep. 74 (1981) 277, R.K Kaul, Rev Mod Phys. 55 (1983) 449. [6] W.J Marclano, Phys. Rev. D 21 (1980) 2425, G Zoupanos, Phys Lett B 129 (1983) 315, D Ltist, E Papantopoulos, K H Streng and G Zoupanos, Nucl Phys B 268 (1986)49 [7] See for example J Kogut et al, Nucl. Phys B 265 [FSI5] (1986) 293. [ 8 ] S Deser et al., Ann Phys 140 (1982) 372, D. Boyanovsky et al, Nucl Phys B 270 [FSI6] (1986) 483, T. Appelqmst et al., Phys. Rev Lett 55 ( 1985 ) 1715, C Burden and N Burkltt, Europhys Len 3 (1987) 545. [9] K Farakos and G Koutsoombas, Phys Lett. B 178 (1986) 260.
105
Volume 249, number 1
PHYSICS LETTERS B
[10]J Ambjorn et al., Nucl Phys. B210 [FS6] (1982) 347, G Bhanot and M Creutz, Phys. Rev D 21 (1980) 2892, C Peterson and L Skold, Nucl Phys. B 255 (1985) 365 [ I I ] A N B u r k l t t a n d A C Irvlng, Nucl Phys B 295 [FS21] (1988) 525 [ 12 ] A Coste and M Luscher, Panty anomaly and fermlon boson transmutation m 3-&menslonal lattice QED, DESY prepnnt DESY 89-017, and references thereto [ 13 ] K Farakos, Ph.D Thesis, National Technical University of Athens (1987) [14] E Dagotto et al, Chlral symmetry breaking m threedimensional QED with Nf flavours, preprmt ILL-(TH)89#10
106
11 October 1990
[ 15 ] C. Vafa and E Wltten, Commun. Math Phys 95 (1984) 257 [16] S Ehtzur, Phys Rev D 12 (1975) 3978 [ 17 ] N Kawamoto and J Smlt, Nucl. Phys B 192 ( 1981 ) 100 [ 18 ] H Kluberg-Stern et al., Nucl Phys B 190 [FS3 ] ( 198 ! ) 504 [ 19 ] N. Kawamoto, Nucl Phys B 190 [FS3 ] ( 1981 ) 617 [20] C B L a n g e t a l , Phys Lett B 104 (1981) 294, V.K MltrIushkm and A.M Zadorozhny, Phys Lett B 181 (1986) 111, K. Farakos and G Koutsoumbas, Z Phys. C 43 (1989) 301.
PHYSICS LETTERS B
11 October 1990
Dynamical gauge symmetry breaking on the lattice K. F a r a k o s , G. K o u t s o u m b a s a n d G. Z o u p a n o s Phystcs Department, Nattonal Technical Untverszty of Athens, Zografou Campus, GR-15 7 73 Athens, Greece
Received 25 June 1990
We study, using lattice techniques, the dynamical symmetry breaking of a three-dimensional theory that m~mlcsthe electroweak sector of the standard model We show that m the strong couphng hmlt ofa QCD-hke theory the ferm~on condensates which are produced reduce dynamical symmetry breaking of the sector corresponding to the electroweak gauge group
1. Introduction
Nowadays it is widely believed that the original suggestion of N a m b u and Jona-Lasinio [1 ] on the chiral symmetry breaking (xSB) is realised by the dynamics of QCD. In addition, the plausible assumption of Welnberg [2 ] and Susskmd [ 3 ] that the 2SB due to Q C D could induce the spontaneous symmetry breaking o f the electroweak SU(2)LXU(1 ) model, has also been widely appreciated. This assumption, which ~s a modern version of an earlier proposal due to Schwinger [ 4 ], is at the heart of the dynamical symmetry breaking scenaria such as the technicolour [ 5 ] and high colour [ 6 ]. Unfortunately, so far, nobody was able to derive analytically the xSB and its immediate consequences such as the Goldstone nature of the n meson starting from the underlying gauge theory QCD. Therefore, any approximation that supports the c o m m o n belief that Q C D produces the xSB is very welcome. In this respect it is worth mentioning a n u m b e r o f lattice calculations as more reliable [ 7 ]. Moreover so far not even an approximate calculation exists indicating that the zSB can induce the spontaneous symmetry breaking of gauge theories. It is our purpose here to provide a lattice calculation, using a slmphfied model, which demonstrates both xSB and spontaneous symmetry breaking of a gauge theory. The framework in which we shall make our study
is an abehan three-dimensional gauge theory with two fermionic flavours [ 8,9 ]. This model will play a role which we expect will mimic the four-dimensional non-abehan QCD, or a QCD-hke strongly interactIng theory, in the feature of producing spontaneous chiral symmetry breaking. A very Important feature shared by the toy model and the four-dimensional Q C D is that they are both confining theories [ 10 ]. The model that we consider here is a very interesting field theory in its own right. Exploration o f its glueball spectrum has recently been done [ 11 ] using both quenched and dynamical fermlons. Quenched calculations on the meson spectrum are reported in ref. [ 9 ]. There has also been some controversy concerning the parity properties of the theory [ 12 ]. An important aspect of the model most interesting for our purposes is the choral symmetry breaking, which has been estabhshed using either quenched [ 13 ] or dynamical [ 11,14 ] fermxons. We shall formulate the model in the continuum and then turn to the lattice calculations, in order to demonstrate our mare point, namely that the gSB triggers the spontaneous breaklng of an appropriately coupled gauge theory.
2. The model The lagrangian density of the " Q C D " - Q E D (in this paper) with one flavour is LP= - ~ (F~,,) 2 + 1~PD u~,~,~ - m ~°~v,
Partially supported by the Greek Ministry of Research and Technology
( 1)
with D u = 0 u - leA~,. The fermion field is a four-com-
0370-2693/90/$ 03 50 © 1990 - Elsevier Soence Pubhshers B.V ( North-Holland )
101
Volume 249, number 1
PHYSICS LETTERS B
ponent spinor. The 4 × 4 y-matrices are chosen to be
~0
(;3
2 [10"2
_00.3) , ,l
(1;1
--10-10) ,
_0o)
11 October 1990
Al=}Py3 T ,
A2=t/glysT,
A3=~T ,
(11)
and
Bl,u-~- ~ft~,uY3T , (2)
There exist two 4 × 4 matrices which antlcommute with 7 o, yl, 22:
B2u=~P?u75T,
B3u= !PYuAT,
/t=0, 1 , 2 ,
(12)
transform as triplets under SU (2), while the
A4=~JAT,
B4u = ~PyuT
(13)
T ~ e x p (10y3) T ,
(4)
are SU (2) slnglets. The choice for the g a m m a matrices given in eqs. (2), (3) permits to separate the lagranglan density given in eq. ( 1 ) into two pieces by writing T = ( Tl, T2) and ~Pz= T~,0-3, l = 1,2. Then the fermion lagranglan becomes
T ~ exp (1~075) T ,
(5)
Y=tP~(10-~,D~'-m)Tl+~(iauDU+m)T2.
Therefore the massless theory is invariant under the global "chiral" transformations
which both correspond to
(14)
in four dimensions. The generators Y3, )'5 together with and d-=Iy3y5 are the generators of a global U(2)~SU(2)×U(1) symmetry that the theory possesses. In particular ~ generates a U ( 1 ) while the other three form the S U ( 2 ) part of the group. The corresponding global symmetry for N flavours [ 15 ] is U ( 2 N ) Let us make the S U ( 2 ) part of the global symmetry a little bit more explicit. The currents corresponding to the transformations
When m = 0 the two fields Tl, T2 are indistinguishable and may be considered as two flavours, whose mixture produces a global SU (2) flavour symmetry. This global SU (2) flavour symmetry is precisely the one that we referred to above. Loosely speaking, for m e 0 , the fields Tl and T2 have opposite "spin" components along the (missing) z-axis and are images of each other under parity; they also interact differently with the electromagnetic field. By expressing the bihnears given in eqs. ( 1 1 ), (12), ( 13 ) in terms of T~, T2 we can illustrate better their S U ( 2 ) structure:
T~exp(io~F)T,
(triplet)
T--* exp (moy5) T
(6)
F = y 3 , ys,
A=ly3Y5
(7)
are given by (8)
The j r are conserved in the absence of any fermionic mass term, while in the presence of non-vanishing mass rn one obtains
O~Jr = 2m!PFT.
(9)
In fact the current corresponding to the generator ~ is always conserved. When rn = 0 the three conserved currents j F lead to the conserved charges Qr= fd2x 7-rtFT, which satisfy an SU (2) algebra: [ 05, Qa ] = 21Q3,
[Q,J, Q3] =21Q5 . One can also easily see that the bilinears 102
~rJ2-- ~2~/1 ,
A2 = ~Ol T 2 + ~P2 T l ,
j~r = !PT~FT.
[ Q3, Qs ] = 2IQ~,
Al =~Pl
(10)
(triplet)
A3 = ~tl TI - ~O2T2 ,
Bl~= Tl0-uT2 + ~'2a~,Tl ,
B2# = ~l 0-~,T2 - ~2 o,u Tl , B3,u = ~l 0-,u~[/l --~-120-lt~-12, (slnglets)
Aa = ~Pl T~ + ~P2Tz,
B4,~----~l 0-. T l -+-~t20-. T 2 .
(15)
A mass term which is parity invarlant and breaks the "chiral" S U ( 2 ) transformations is m T T . This becomes, writing the four-component spinor in terms o f T l , T:,
m ~ T = m ~ l Till -- mgP2 ttJ2= mA3
(16)
This mass term breaks the global symmetry U ( 2 )
Volume 249, number 1
PHYSICS LETTERS B
down to U ( 1 ) × U ( 1 ) with generators ~, A. In the case o f N flavours the U ( 2 N ) breaks to S U ( N ) × SU(N) ×U(I) ×U(1). In o r d e r to keep the analogies we recall that Q C D with two flavours has a chtral global s y m m e t r y U ( 2 ) L × U (2) R, which m the presence o f a mass term breaks down to U(2)L+R. Such a chlral s y m m e t r y breaking mass t e r m is expected to emerge d y n a m i cally m QCD. Switching on the S U ( 2 ) L × U ( 1 ) , which corresponds to gauging an SU ( 2 ) subgroup o f the global symmetry, given in eq. ( 1 0 ) , the zSB is expected to induce the spontaneous s y m m e t r y breakmg o f the gauge group S U ( 2 ) L X U ( 1 ) down to U ( 1 )EM. Keeping the analogies in our toy model, it is suggested to gauge the SU ( 2 ) subgroup o f U ( 2 ) as well. [We note that the U ( 1 ) part has already been gauged. ] T h e lagrangtan density o f our toy m o d e l is o~--
- ~ (| F ~ v )
2
' -/2 yu~V-m~P~, - ~ (1 F u v ) 2 +I~ttD
I I October 1990
similar to the one used to study spontaneous magnetization in ferromagnets: In the beginning an external magnetic field is present, which is gradually r e m o v e d and the residual magnetization is measured at the h m i t o f zero external field. In our case the need to take the massless limit m a y be rephrased by pointing out that the f e r m i o m c mass t e r m gives masses to the would-be G o l d s t o n e bosons. Thus it forbids the appearance o f the pole contribution to the massless gauge boson p r o p a g a t o r which is expected to provide mass to the gauge boson ~j We now give the lattice action employed, which is just the lattice transcription o f the action corres p o n d m g to the lagrangian density given in eq. ( 17 ):
S=½ ~ K[~n(1-~u)UnuVnu~ rl,/2
+ Lp.+~(1+~,~)uL vL~v.] - Z ~p,~v +fl, Z ( 1 - T r Up)
(17) where D u = O,-ig~Au-xg2r.B u a n d Fu,, Fu~ are the U ( 1 ) and S U ( 2 ) gauge field kinetic terms. The dynamtcal emergence o f a mass term as in eq. ( 1 6 ) [or equivalently (A 3) :# 0, due to the f o r m a t i o n o f condensates by the strong U ( 1 ) coupling ] is expected to trtgger the breaking o f the gauge group SU ( 2 ) down to U ( 1 ) . The two other m e m b e r s o f the trtplet, namely At, A2, will generate the would-be G o l d s t o n e particles, which are expected to provide masses to the two SU ( 2 ) gauge bosons. Thus the m o d e l has been p r o m o t e d to a theory based on the gauge group S U ( 2 ) × U ( 1 ) which is expected to b r e a k down to U(1)×U(1).
3. Lattice calculations Considering the m o d e l on the lattice, one should introduce a mass term, in o r d e r to regulate the functional mtegral. In addition, this mass acts like a gauge n o n - i n v a r l a n t term, that is, it orients the VEV o f the compostte triplet o f Higgs fields given in eq. ( 11 ) [which is going to emerge after integrating out the U ( 1 ) field] in the ( ~ ) direction. We will conclude that there is genuine s y m m e t r y breaking if the ( 7 6 u ) remains non-zero after the removal o f the mass (gauge breaking) term. The above p r o c e d u r e is very
n
p
+f12 •
(1-Tr
Vv).
(18)
P
U,u=exp(iO,u) represents the U ( 1 ) field, while Vnu= exp (t,Bn u) is the SU (2) gauge field. The SU (2) generators are the matrices (~3, ~'5, A), as discussed above. The two last terms in eq. (18) are the kinetic terms for the U ( 1 ) and the S U ( 2 ) gauge fields respectively. We use Wilson fermions to a v o i d fermion doubling. In this way we introduce a mass term in the model, which breaks the S U ( 2 ) mvariance. As already discussed above, SU ( 2 ) will be recovered in the massless limit, that is at K = Kermcal. In order to exhibit the d y n a m i c a l SU ( 2 ) breaking we restrict ourselves to the case o f infinitely strong U ( 1 ) coupling, that is fit = 0. In this case, the U ( 1 ) gauge field can be easily mtegrated out in the functional integral [ 17,18 ]. The p a r t i t i o n functton We use the familiar mass term m ~ v There is also another type of mass term, which could have been used, namely m ~vA7-1, which is invanant under the SU(2) group This mass term has the advantage of bemg SU(2) slnglet, but it has several problems: it breaks parity and reduces Chern-Slmons gauge lnvanant mass terms for all the gauge bosons Moreover, due to the fact that it does not orient the possible VEV to a specific direction, ( ~ r t ) is expected to be always zero for this mass term, independently of the existence or not of VEV! (Ehtzur's theorem [ 16 ] ) 103
Volume 249, number 1
PHYSICS LETTERS B
f
Z= J [dUdVdetd~]exp(-S),
(19)
If we only keep the linear term in the above expansion, the logarithm becomes
In Io
after performing the U integration becomes Z= f [dVd~d~,] exp(-Sen),
(20)
11 October 1990
¼yo
= - ¼KETr [M(n) ( 1 -Yu) Vn,M("+u)( 1 +7u) V~u] • (28)
where
Seff=~2E p
( 1 - T r Vv) + Z In I o ( x / ~ u ) - Z ~, ~,. n,lt
n
(21) We have denoted by Ynu the expression Ynta =- K Z ~ n (
1 - 7 u ) Vnu ~-'ln+U ~tn+u( 1 + 7u)
V*nu~n , (22)
and Io IS the zeroth order Bessel function. With the ~-(n) extra notation M ~b = 7~,~ ~P,b where n is the site and a, b the Dirac indices, Y,u becomes y . . = - K 2 Tr [M(") ( 1 -Yu) V~uM("+u) × (l+Tu)V*,,].
(23)
Next we switch from the functional integral over fermions to a functional integral over the mesons M ("). In doing so, one also has to include in the action the effect of the "jacoblan" of this transformation [ 17 ]. The partition function becomes Z= f [dVdM] exp(-Ae~r),
(24)
where Aeff=Seff-- ~ T r l n M (") . n
(25)
The last term in eq. (25) comes from the aforementioned "jacoblan". Our strategy will be to expand the term ~,,u l n l o ( x / ~ ) and keep only the lowest orders, which are likely to yield gauge boson masses, while the next ones will produce interactions. The expansion gives In l o ( x / ~ u ) = ~ Ck(y~u) k,
(26)
k
Ck= 2 ["
(1
l>m,( , f
.~m2 (27)
104
It is fairly evident from this expression that if symmetry breaking is to occur, it will emerge out of a nonzero VEV of the matrices M (n). A method to study the problem is to expand M (n) in the basis of the gamma matrices and identify meson states with definite quantum numbers; one may then consider the VEVs of each meson separately. In order to find the vacuum expectation value o f M (n), one can make use of the saddle point calculations described in ref. [ 17 ]. However, we prefer to use the results of refs. [9,11,13,14], since Monte Carlo simulations are more reliable than simple saddle-point estimates. We restrict ourselves to weak S U ( 2 ) coupling, which is justified by the fact that SU (2) plays the role of the electroweak gauge group. Thus the results derived for the model with global SU (2) can be carried over to our case without any qualitative changes. What has been found in ref. [ 9 ] is that for •1 small enough and K = K c r l t , C a l , the states generated by A~ and A 2 a r e massless, that is they are Goldstone bosons, while A3 acquires a non-zero mass. This is a first indication for the breaking of the SU (2) symmetry, since the members of an SU (2) triplet acquire different masses. It would be desirable to verify that the chlral condensate develops non-zero VEV. The results of ref. [9] are inconclusive in this respect, because they refer to rather large values offl~. However, the results for small fll have been obtained by one of us [13], performing calculations with quenched fermlons; a non-zero chlral condensate has been found forfl~ < 1.5. What is more important, these results remain valid when dynamical fermions are considered [ 11,14]. Similar results have been obtained by Kawamoto [ 19 ] using the strong coupling expansion. In particular, for fl~ = 0 , the A3 mass and the condensate are infinite m this approximation. In a d d m o n , no other meson state develops non-zero VEV Isolating the VEV (7nP) - u the expansion o f M (n) reads
Volume 249, number 1
PHYSICS LETTERS B
M ( ' ° = u ' ~ + l [ A 3 ( n ) ~ W A l ( n ) y 3 +A2(n)~5 + A4( n )AW B4u( n )yU + Blu( n )TUy3 + B2~,( n )yUy5 + B3u( n )yUA ] ,
(29)
where ¢t = 0, 1, 2. If we substitute M (") by its VEV (namely u~) in eq. (20), lnlo(x/~,u) is reminiscent of the scalargauge interaction part of the SU(2) adjoint gaugeHiggs model [ 20 ]. The experience with such models shows that the symmetry is broken down to U(1 ) whenever the constant multiplying the above-mentinned interaction part is large enough; then the model ~s m the so-called Higgs phase. In our case this constant ~s proportional to u 2, which may grow infinitely large, as/71 goes to zero. Thus for small values of ill we expect a broken SU (2) gauge symmetry. Let us now give a derivation of the breaking of the symmetry and an estimate of the masses acquired by the gauge bosons. By substituting eq. (29) into eq. (28) and taking into account that
Vnu = exp (leB, u) =cos(IB.ul)+ilrB,,usln(IB,,ul)/IB,,ul
,
(30)
we find In Io ( x / / ~ ) ~ ½K2crU2(sin21B,u I / I B , u 12)
11 October 1990
the functional integrations. In the strong coupling hmit of the U ( 1 ) gauge theory we have shown that the gSB of the fermlons, signaled by fermion-antlfermion condensates leads to a dynamical breaking of the SU (2) gauge group. The SU (2) gauge group breaks down to U ( 1 ), a fact which is established by determining the mass spectrum of the gauge bosons. It would be very interesting to study the symmetry breaking by performing a Monte Carlo simulation of the model. The relative simphclty of our toy model as compared to the reahstic four-dimensional electroweak gauge group coupled to a QCD-hke theory has been found to be very useful techmcally. A study of the realistm theory we feel will face an appremable technical complication as compared with the present one, but no fundamentally new difficultms.
Acknowledgement We would hke to thank J. Ihopoulos for useful discussions at the early stages of this work. K.F. and G.K. would also like to thank G. Tlktopoulos for enlightening discussions. Financial support by the Greek Ministry of Research and Technology ~s gratefully acknowledged.
× [ (B~.u):+ (B2u): ] + interaction terms,
(31 )
out of which we identify the mass term: 1/4"2 .it2[ 1 2 ~..¢n,t(B.u) + ( B .2u ) 2 ]
(32)
for the two gauge bosons, while the third one, B 3, only develops interactmn terms, remaining massless. Therefore the theory breaks from SU (2) X U ( l ) down to U ( 1 ) × U ( 1 ) .
4. Conclusions In the present paper we have used a three-dimensmnal toy model in order to mimic the electroweak sector of the standard model coupled to a strongly interacting QCD-like theory. The latter is represented by a U ( 1 ) model whde the electroweak gauge group is represented by an SU (2). We have formulated the theory on the lattice in order to be able to perform
References [ 1 ] Y. Nambu and G Jona-Lasmto, Phys. Rev 122 ( 1961 ) 345 [2]S Welnberg, Phys. Rev. D 13 (1976) 974, D 19 (1979) 1277 [3] L Susskmd, Phys. Rev D 20 (1979) 2619 [4] J Schwmger, Phys. Rev 128 (1962) 2425 [ 5 ] For revmws see E Farhl and L Susskind, Phys. Rep. 74 (1981) 277, R.K Kaul, Rev Mod Phys. 55 (1983) 449. [6] W.J Marclano, Phys. Rev. D 21 (1980) 2425, G Zoupanos, Phys Lett B 129 (1983) 315, D Ltist, E Papantopoulos, K H Streng and G Zoupanos, Nucl Phys B 268 (1986)49 [7] See for example J Kogut et al, Nucl. Phys B 265 [FSI5] (1986) 293. [ 8 ] S Deser et al., Ann Phys 140 (1982) 372, D. Boyanovsky et al, Nucl Phys B 270 [FSI6] (1986) 483, T. Appelqmst et al., Phys. Rev Lett 55 ( 1985 ) 1715, C Burden and N Burkltt, Europhys Len 3 (1987) 545. [9] K Farakos and G Koutsoombas, Phys Lett. B 178 (1986) 260.
105
Volume 249, number 1
PHYSICS LETTERS B
[10]J Ambjorn et al., Nucl Phys. B210 [FS6] (1982) 347, G Bhanot and M Creutz, Phys. Rev D 21 (1980) 2892, C Peterson and L Skold, Nucl Phys. B 255 (1985) 365 [ I I ] A N B u r k l t t a n d A C Irvlng, Nucl Phys B 295 [FS21] (1988) 525 [ 12 ] A Coste and M Luscher, Panty anomaly and fermlon boson transmutation m 3-&menslonal lattice QED, DESY prepnnt DESY 89-017, and references thereto [ 13 ] K Farakos, Ph.D Thesis, National Technical University of Athens (1987) [14] E Dagotto et al, Chlral symmetry breaking m threedimensional QED with Nf flavours, preprmt ILL-(TH)89#10
106
11 October 1990
[ 15 ] C. Vafa and E Wltten, Commun. Math Phys 95 (1984) 257 [16] S Ehtzur, Phys Rev D 12 (1975) 3978 [ 17 ] N Kawamoto and J Smlt, Nucl. Phys B 192 ( 1981 ) 100 [ 18 ] H Kluberg-Stern et al., Nucl Phys B 190 [FS3 ] ( 198 ! ) 504 [ 19 ] N. Kawamoto, Nucl Phys B 190 [FS3 ] ( 1981 ) 617 [20] C B L a n g e t a l , Phys Lett B 104 (1981) 294, V.K MltrIushkm and A.M Zadorozhny, Phys Lett B 181 (1986) 111, K. Farakos and G Koutsoumbas, Z Phys. C 43 (1989) 301.