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Gauge-Higgs systems with fermions
Nuclear Physics B (Proc. Suppl.) 9 (1989) 77-81 North-Holland, Amsterdam
77
GAUGE-HIGGS SYSTEMS WITH FERMIONS Robert E. SHROCK Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794 USA Recent results on lattice gauge-Higgs models with fermions are discussed. Implications for electroweak models are noted.
1.
INTRODUCTION
The (zero-temperature) chiral transition in lattice gauge theories with Higgs and fermions fields has recently been the subject of intensive study. ~-ls Following initial numerical observations for SU(2) (with quenched fermions) 1, the existence of the chiral transition was demonstrated and the order (continuous) and approximate critical point were determined analytically for both U(1) and Z(N) 2 and for SU(2) e using mean field methods applicable in the strong gauge coupling limit,/3 9 = O. These results agreed very well with/3 e = 0 simulations of U(1) with quenched 3 and dynamical 12 fermions and of SU(2), again with both quenched s and dynamicals'15 fermions. The SU(2) theory with I --- 1/2 Higgs fields and ! = 1/2 fermions is of particular physical interest since it constitutes a lattice formulation of the SU(2) sector o{ the standard SU(2) × U(1) electroweak theory. Although the analytic studies applied for ~g = 0, the behavior of the theory in a strip adjacent to the /3g = 0 edge of the phase diagram is qualitatively the same as that at this edge, owing to the finite radius of convergence of a small- ~'g expansion; hence, the chiral transition point at #g = 0 is part of a line of chiral transitions which extends into the interior of the diagram with ~g > 0. This line cannot end anywhere in the interior and consequently the chiral boundary completely separates the phase diagram into two (sets of) phases characterized by whether or not chiral symmetry is spontaneously broken, a (The same is true for the ~g < 0 region, but this is not of direct physical interest.) Since gauge-Higgs theories with Higgs fields transforming according to the fundamental representation have analytically connected confinement and Higgs phases, the chiral transition means that the addition of fermions to such theories significantly changes
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
their phase structure. Our analytic methods yield the usual mean field value for the critical exponent describing the order parameter < XX >, viz., j3 = 1/2. 2'9'13 In contrast to the mean field predictions for the critical point, which only become exact in the limit where the dimensionality d --* oo, this prediction is exact for d > d=.... where du.c. is the upper critical dimensionality. We expect that du.c. = 4 for the chiral transition, and hence the mean field prediction for ~3 should presumably be exact here. We have found that this prediction is consistent with our U(1) and SU(2) data. 9,13 The origin of the chiral transition can be explained as follows: spontaneous chiral symmetry breaking is due to the gauge-fermlon interaction. In a theory with Higgs in the fundamental representation, as the gauge-Higgs coupling/3h increases, the gauge-Higgs interaction term has the effect of increasingly restricting the gauge field configurations which make a significant contribution to the path integral. This weakens the effective gauge-fermion interaction, and, as we showed both analytically and numerically, eventually the latter is rendered too weak to break the chiral symmetry spontaneously, and this symmetry is restored via the chiral phase transition. 2.
IMPLICATIONS OF THE CHIRAL TRANSITION The chiral transition has important implications for the electroweak theory: (i) It is clearly important to have a formulation of electroweak interactions which treats the full path integral defining the quantum theory, and which subsumes, but is not limited to, perturbation theory. The lattice formulation provides a powerful means for making progress toward this goal. It is especially useful for investigating nonperturbative phenomena such as spontaneous chiral symmetry breaking. It should be noted
78
R.E. Shrock/ Gauge-ttiggs systems with fermions
that the well-known problems associated with attempts to put chiral fermions on the lattice do not prevent one from using lattice techniques to study at least the SU(2) sector of the standard electroweak theory. ~,e The reason for this is that one can write this theory in vectorlike form, which follows from the fact that the representations of SU(2) are real. This is exact in the absence of Yukawa couplings; as we showed in Ref. 10, in the presence of Yukawa interactions, it is not exact, but is a good approximation for fermions with masses mf < < 250 GeV. The point here is that the mass terms have the form 1°, e.g., for quarks, ml(~u -k Ss) -k m2(~c -k rid), so that certain pairs of fermions are given degenerate masses. This shortcoming should not be too severe, however, since we are studying spontaneous chiral symmetry breaking, and the natural scale for the dynamically generated fermion masses in the chirally broken phase is of order the electroweak scale, 250 GeV, much larger than known fermion masses. However, since the representations of U(1) are complex, the full SU(2) x U(1) theory cannot be rewritten in such a manner. (ii) The effective continuum limit should be taken in the phase without spontaneous breaking of the chiral symmetry, since in the broken phase(s) the fermions pick up large, dynamically generated masses, naturally of order the electroweak scale. (iii) It is thus important to determine where this phase without spontaneous chiral symmetry breaking lies in the SU(2) case with I = 1/2 Higgs and fermions. In addition to the analytic work which applies at fig = 0 and, qualitatively, in the adjacent strip, the phase structure of this SU(2) theory has been mapped out by simulations using quenched 1,s,ll and dynamical8,1s fermions. (iv) The line of chiral transitions which completely separates the chirally symmetric and spontaneously broken phases obviously alters the renormalization group flows in the theory. It is clearly important to understand the renormalization group flows and their fixed points with fermions included. This is necessary for a satisfactory understanding of the continuum limit of the lattice theory. (v) Our work on the chiral transition may provide the first ab iniLio answer 4 to the longstanding question of why the known weak nonsinglet fermions have ] = 1/2. The point is that the strength of the effective gauge-fermion interaction increases with the Z of the fermion. It was thus expected theoretically, and has been demonstrated analyticallye and numerically, via both quenched 1'6'11 and dynamical8,1s simulations, that the chiral boundary occurs at larger values of/3h for I > 1 fermions than for
I = 1/2 fermions. It follows that if, as is now conventional, one takes an effective continuum limit from within the Higgs phase, in the vicinity of the 0(4) transition at/3g = oo, then this can be done while in the chirally symmetric phase for fermions with I = 1/2. However, since the chiral boundary for I :> 1 fermions lies at larger/3h than that for I = 1/2 fermions, one would presumably be in the chirally broken phase for the former. Hence, according to this picture, only I = 1/2 fermions could be light compared to 250 GeV. (vi) In the case of SU(2), the lattice can be used to study the strongly coupled standard model) ,16 This model requires that chiral symmetry not be broken spontaneously, since this would produce large, dynamically generated masses for the fermions of order 250 GeV (and would violate electric charge); at the same time, however, it also assumes a strongly coupled, confined behavior with the resultant spectrum of excited bound states. It was motivated in part by early work with bosonic gauge-Higgs models with Higgs in the fundamental representation, which showed that the confinement and Higgs phases are analytically connected, giving rise to the principle of complementarity of states and thus the interesting implication that the fermions and W and Z bosons would have excited states analogous to the baryon and meson resonances
(e.g., N, N*'s,...;p, p(1600),...). However, because of the chiral transition separating the old confinement and Higgs phases, this complementarity does not hold in the presence of fermions. One can also investigate this possibility in a toy U(1) model; studies with both quenched ~ and dynamical12 fermions again find no evidence for such a phase. The continued search for such a hypothetical phase remains an interesting pursuit (as does the experimental search for excited fermions and excited W and Z bosons). (In the full SU(2) × U(1) electroweak theory, even without fermions, there is a non-analytic boundary completely separating the interior of the phase diagram into two phases17 so again there is no complementarity between states in these two phases.) 3.
ANALYTIC TECHNIQUES In order to review how these results are obtained, we consider a general lattice gauge theory with a Higgs field q~ in the fundamental representation and a fermion field X; this theory is described, in standard notation, by the path integral
Z = f H dU'~,~,dc~,~dc~dx,~dxne-s rh , u
(3.1)
R.E. Shrock / Gauge-Higgs systems with [ermions where S = SG + SH + SF,
as = - i + 4/'b2 - 3r~ + 12/'~/~(/', -/'~/~)
s~ = Z~ ~ [~ - P]
(3.2)
plaq.
s.~ = - u ~ a ~ { ¢ ~ D , ( V . , . ) ¢ . + . A
(a.3)
1 SE = ~ E ~7,,,uX,,[Dx(U,,,u)X,,+.,, - Dx(U~-.,,,u)X,,-.,,] (3.4) Sere, Z~ = ~ / f for UO) and 4 / f for SU(~); P = R~{U~,o~.} for UO) and (1/2)T/'{U~,o~.} for SO(2), and y = 2~h. D~(U) is the representation of the gauge group element U corresponding to the Higgs field ¢, and similarly for D x ( U ). The lattice is d-dimensional hypercubic with d _> 3 and lattice spacing a -- 1. The Higgs fields satisfy ¢t¢~ = 1; as is well known, this does not imply that the continuum Higgs fields are of fixed length. With no loss of generality, we take ~h _> 0. We use staggered fermions X,~ because of their advantages for studies of chiral symmetry. The ~/,~,u = {1 for # = 1; (--1) '~+'''+''-~ for 2 < p < 4} are factors from the Dirac matrices. The analytic strong gauge-coupling results are obtained using the following technique. First, we perform the integrals over Higgs and gauge fields, obtaining the exact result Z = jN~ j. ~ n dxnd2ne-S', where Nz is the number of links in the lattice, and ,/1 = I0(2fla) for U(1)
and ~(2Z~)/Z~ for SU(2), where ~(~) is the modified Bessel function. For U(1) we find ~
-sb.> = ~.[-(1/2)~.,./'.(~.x.,
- ~.,x.)
where ,~' - ~ + ~. and/'. = ~(U)/Io(U), where q is the fermion charge. For SU(2) with I = 1/2 fermions ~, -s;uc,),
~=,/, = ~[-2-'~.,./',/,(~.x.
, - ~.,x.)
+ 2-3(1-/'~ )(~.X.)(~., X., )+ 2-~(/'~/~-rz )(:~.X., )(~n,X. ) + 2 - 3 ( 1 - - 1,2
, V~){(X.X. ) + (~.,X.) ~)
+2-%.,./'~/:ao(~.x.,
- £.,x.)(£.x.)(£.,x.,)
+2-Z as(~.X.)~(~.,X.,) 2]
(3.6)
where r z = I2z+,(y)/Iz(y) and
a6 = 1 - 4r~l 2 + 3rl
(3.7)
79
(3.8)
Again, this result is exact. (Eq. (3.6) was written in an equivalent form in Ref. 6, using an identity given there.) Results for other values of ] were also presented there. Next, we use a mean field technique to replace operator products higher than quadratic by a quadratic part times the mean value of the rest. This renders the action quadratic, so that we can perform the fermionic integrations exactly. We then derive a mean field consistency condition for the chiral symmetry breaking fermion condensate. In the U(1) case this equation is
< )~X > = (d/2)(1 - / ' ~ ) < :~X > / ( d P ) D-~
(3.9)
where f(dp) -= (2~) -d H~=I f:~ alp. and D = [{(d/2)(1 - / ' ~ ) < ~ x >}~ + / ' ~ ]
(3.10)
with s = ~.=~ s i o ' p . . As usual, (3.9) always has the solution < XX > = 0, but if there is a solution for nonzero < .~X >, this minimizes the mean field free energy and hence is the physical solution. We have shown analytically that (3.9) implies the existence of a (continuous) chiral transition in the theory s-7. We have also solved (3.9) numerically. A 1/d expansion of (3.9) to order 1/d ~ has been calculated and gives excellent agreement with the numerical solution? '9 Indeed, in the large d limit, we have obtained an exact solution for the chiral transition. In terms of the scaled condensate/~ = (ff/2)1/2/~, and ----/',z'2we obtain 37/2 = (1--2¢)(1--a~)-~0(1/2-~). For SU(2), the critical point can be located by an equation ~ similar to (3.9), with z = r~/~. As noted, these results agree with numerical simulations in both the U(1) and SU(2) cases at least as well as the 10 - 20% accuracy typical of mean field theory in d = 4 spin systems. The chiral boundary has also been mapped out in the interior of the phase diagram. We specialize here to the physically more important case of SU(2). In the limit ~'g -~ oo, the partition function factorizesS: Z = Zo(4)(~h)ZII, where Zo(4) refers to the 0(4) spin model, and Z H is the partition function for free (massless) fermions. Since the latter do not undergo any spontaneous chira[ symmetry breaking, it followss that for /3g --~ oo, < XX > = 0 for all fl~. There are several possibilities for the chiral boundary which are consistent with this exact result. Quenched simulations1,11 found that for ] = 1/2 fermlons, the chiral boundary appears to coincide with the confinement-Higgs boundary (for values
80
R.E. Shrock / Gauge-Higgs systems with fermions
of fig where the latter was present in the pure bosonic theory), which would imply that as/39 --* oo, the chiral boundary ends at the 0(41 critical point. In our dynamical fermion simulation s with ] = 1/2 fermions, we again found that, for the value of/99 where we measured,/39 = 1.9, the chiral boundary coincided with the confinement-Higgs boundary. Of course without dynamical simulations at arbitrarily large fig, it is not possible to determine precisely where the chiral boundary ends in the limit fig --* oo. (For this reason the chiral phase boundary presented in Ref. 8 was not drawn past our measured point, although in accordance with our theorem, it could not end in the interior.) Unfortunately, it becomes very difficult to measure the chiral boundary at values of fig significantly larger than about 2 for several reasons. First, in order to locate this boundary, one must detect where < XX > drops to zero as/3h increases at fixed /39, but in the spontaneously chirally broken phase this condensate is decreasing exponentially rapidly as a function of ~g Cat fixed fib)- Eventually, even where the condensate is nonzero, it is so small that one cannot accurately measure where it would vanish, as flh increases, across the chiral boundary. A related fact is that in the broken phase the correlation length is increasing exponentially rapidly, so that eventually finite-size effects become important. Finally, even with symmetric lattices, there can be finitetemperature effects for sufficiently large /99, and these can lead to a finite-temperature transition, which, while interesting in its own right, is spurious if interpreted as a zero-temperature chiral transition. However, the most important result of our work concerning the strongly coupled standard model is independent of the details of the location of the chiral boundary (which could be changed anyway by the addition of irrelevant operators to the action): this result is the loss of the complementarity of states between the chirally broken, confinement phase and the chirally symmetric Higgs-like phase, and depends only on the fact that the chiral boundary completely separates these two phases. Recently, we have mapped out the chiral boundary for SU(2 / with I = 1 dynamical fermions, is. We find (a / good agreement with our analytic results at/~g -~ 0 and (b) at/39 = 1.9, the chiral boundary occurs at a larger/3h than the conflnement-Higgs phase boundary, in contrast to the I = 1/2 case. Property (b) was also observed in quenched data. T M
4.
CASE OF TWO DIFFERENT FERMIONS We have also used our analytic methods to determine the phase structure of a theory with two different types of interacting ferrnions simultaneously present. 9 For simplicity, we used a U(1) theory and worked again at strong gauge coupling, but again, similar results should hold near/39 = 0 and for SU(2). Let us denote the different fermions as X and ~ with charges qx :> q~" We have shown analytically that this theory exhibits two distinct (continuous) chiral transitions, and thus three different phases. The locations of the critical points for the two chiral transitions were also given. The phases are characterized as follows:
Pl:
Zh # O ;
<~>#0
P2: t3h,c,~<_flh # O ; < ( ~ > = 0 P3:
~h>_~h,~,x:
<~>=0;
< ) ~ x > = O (4.1)
S. ARBITRARY HIGGS AND FERMION CHARGES Although in the standard electroweak theory, the Higgs field transforms according to the fundamental representation of SU(2), theories with higher unification require Higgs in higher representations. It is thus of interest to investigate the nature of chiral symmetry breaking in such a theory. The U(1) model provides a useful system with which to investigate higher Higgs representations. Extending Ref. 2, we proved a general classification theorem which completely specifies under what conditions this theory has a chiral transition at ~'g = 0. 9 Consider the U(1 / theory with a fermion f of charge q], and a Higgs field of charge qh. Then 9 if and only if qf is a (nonzero) integral multiple of qh, the theory has a chiral phase transition. If qf is not a multiple of %, then the path integral factorizes into separate, decoupled bosonic and fermionic factors, Z(qf :/: k%) = ZhZ.~, where k is a nonzero integer, Zh = Io(2flh) N~ , and Z s (given in Ref. 9/ is independent of/3h; hence, < XX > is also independent of/3h, so that in this case chiral symmetry is spontaneously broken (at fl~ = 0) for all fib. 6. YUKAWA INTERACTIONS We have studied the chiral transition in U(11 and SU(2) gauge-Higgs theories, s,1° In both theories, the fermions consisted of a field X in the fundamental representation interacting with gauge-singlet fermion(s) (and ( in the SU(2) case/ via a Higgs in the fundamental
R.E. Shrock / Gauge-Higgs systems with fermions representation. This is motivated by the Yukawa interactions in the SU(2) part of the standard electroweak theory. The analyses were carried out in the strong gauge coupling limit and used mean field techniques similar to those already discussed. In the U(1) case, the Yukawa interaction has the form Sy = h ~ ( ~ ¢ n ~ n + b.c.). In this model it was found that for a range of sufficiently weak Yukawa couplings h, the chiral transition continues to exist (and to be of second order), but is removed as the strength of the Yukawa coupling increases through a critical value, he = (d/2) 1/~. Interestingly, the neutral fermion, through its Yukawa interaction with the charged fermion, picks up a nonzero condensate < ~ > in the broken phase. Both I~h.,d and < XX > decrease as h increases. For the SU(2) case1°, there are two types of Vukawa interactions:
81
REFERENCES 1. I-H. Lee and J. Shigemitsu, Phys. Lett. 178B (1986) 93. 2. I-H. Lee and R. E. Shrock, Phys. Rev. Lett. 59 (1987) 14. 3. I-H. Lee and R. E. Shrock, Nucl. Phys. B290 (1987) 275.
4. I-H. Lee and R. E. Shrock, Phys. Lett. 196B (1987) 82. 5. I-H. Lee and R. E. Shrock, Phys. Lett. 199B (1987) 541. 6. {-H. Lee and R. E. Shrock, Phys. Lett. 201B (1988) 497. I-H. Lee, R. E. Shrock, in Proceedings of the International Symposium on Field Theory on the Lattice, Seillac, Nucl. Phys. (eroc. Suppl. ) B4 (1988), 373, 438. 8. S. Aoki, I-H. Lee, and R. E. Shrock, Phys. Lett. 207B (1988) 471.
n
where e~j = -ej~. We find that if and only if h,~4 + h,~-4 > 8d -2, then the theory has a spontaneous chiral symmetry breaking phase transition (at ~g -- 0). This condition is always satisfied if either h,1 or h2 is smaller than 2-3/4dl/~; in this case, the transition remains even for arbitrarily large values of the other Yukawa coupling. These mean field results are, of course, not exact, but should give some insight into the true behavior of the theory. As in the other cases, the transition was found to be continuous and approximate values for the critical point were given. The research reported here was done in collaboration with I-H. Lee and, in Refs. 8 and 15, S. Aoki and was partially supported by the grant NSF-PHY-8507627.
g. I-H. Lee and R. E. Shrock, Nuc{. Phys. B305 (1988) 286. 10, I-H. Lee and R. E, Shrock, Nucl. Phys, B305 (1988) 305. 11. A. De and J. Shigemitsu, Nucl. Phys, 307]3 (1988) 376. 12. E. Dagotto and J. Kogut, Phys. Lett. 208B (1988) 475. 13. I-H. Lee, R. E. Shrock, in the Proceedings of the FSU-SCRI Lattice Higgs Workshop (May, 1988), in press. 14. I-H. Lee, R. E. Shrock, in the Proceedings of the APS Division of Particles and Fields Meeting, Storrs, Conn. (Aug. 1988), in press. 15. S. Aokl, I-H. Lee and R. E. Shrock, preprint BNLITP-SB-S8-67. 16. M. Claudson, E. Farhi, and R. L. Jaffe, Phys. Rev. D34 (1986) 873; L. Abbott and E. Farhi, Phys. Lett. 101B (1981) 69. 17. R. E. Shrock, Phys. Lett. B162 (1985); Nucl. Phys. B267 (1986) 301; Phys. Rev. Lett. 56 (1986) 2124; Nucl. Phys. B278 (1986) 380.
77
GAUGE-HIGGS SYSTEMS WITH FERMIONS Robert E. SHROCK Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794 USA Recent results on lattice gauge-Higgs models with fermions are discussed. Implications for electroweak models are noted.
1.
INTRODUCTION
The (zero-temperature) chiral transition in lattice gauge theories with Higgs and fermions fields has recently been the subject of intensive study. ~-ls Following initial numerical observations for SU(2) (with quenched fermions) 1, the existence of the chiral transition was demonstrated and the order (continuous) and approximate critical point were determined analytically for both U(1) and Z(N) 2 and for SU(2) e using mean field methods applicable in the strong gauge coupling limit,/3 9 = O. These results agreed very well with/3 e = 0 simulations of U(1) with quenched 3 and dynamical 12 fermions and of SU(2), again with both quenched s and dynamicals'15 fermions. The SU(2) theory with I --- 1/2 Higgs fields and ! = 1/2 fermions is of particular physical interest since it constitutes a lattice formulation of the SU(2) sector o{ the standard SU(2) × U(1) electroweak theory. Although the analytic studies applied for ~g = 0, the behavior of the theory in a strip adjacent to the /3g = 0 edge of the phase diagram is qualitatively the same as that at this edge, owing to the finite radius of convergence of a small- ~'g expansion; hence, the chiral transition point at #g = 0 is part of a line of chiral transitions which extends into the interior of the diagram with ~g > 0. This line cannot end anywhere in the interior and consequently the chiral boundary completely separates the phase diagram into two (sets of) phases characterized by whether or not chiral symmetry is spontaneously broken, a (The same is true for the ~g < 0 region, but this is not of direct physical interest.) Since gauge-Higgs theories with Higgs fields transforming according to the fundamental representation have analytically connected confinement and Higgs phases, the chiral transition means that the addition of fermions to such theories significantly changes
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
their phase structure. Our analytic methods yield the usual mean field value for the critical exponent describing the order parameter < XX >, viz., j3 = 1/2. 2'9'13 In contrast to the mean field predictions for the critical point, which only become exact in the limit where the dimensionality d --* oo, this prediction is exact for d > d=.... where du.c. is the upper critical dimensionality. We expect that du.c. = 4 for the chiral transition, and hence the mean field prediction for ~3 should presumably be exact here. We have found that this prediction is consistent with our U(1) and SU(2) data. 9,13 The origin of the chiral transition can be explained as follows: spontaneous chiral symmetry breaking is due to the gauge-fermlon interaction. In a theory with Higgs in the fundamental representation, as the gauge-Higgs coupling/3h increases, the gauge-Higgs interaction term has the effect of increasingly restricting the gauge field configurations which make a significant contribution to the path integral. This weakens the effective gauge-fermion interaction, and, as we showed both analytically and numerically, eventually the latter is rendered too weak to break the chiral symmetry spontaneously, and this symmetry is restored via the chiral phase transition. 2.
IMPLICATIONS OF THE CHIRAL TRANSITION The chiral transition has important implications for the electroweak theory: (i) It is clearly important to have a formulation of electroweak interactions which treats the full path integral defining the quantum theory, and which subsumes, but is not limited to, perturbation theory. The lattice formulation provides a powerful means for making progress toward this goal. It is especially useful for investigating nonperturbative phenomena such as spontaneous chiral symmetry breaking. It should be noted
78
R.E. Shrock/ Gauge-ttiggs systems with fermions
that the well-known problems associated with attempts to put chiral fermions on the lattice do not prevent one from using lattice techniques to study at least the SU(2) sector of the standard electroweak theory. ~,e The reason for this is that one can write this theory in vectorlike form, which follows from the fact that the representations of SU(2) are real. This is exact in the absence of Yukawa couplings; as we showed in Ref. 10, in the presence of Yukawa interactions, it is not exact, but is a good approximation for fermions with masses mf < < 250 GeV. The point here is that the mass terms have the form 1°, e.g., for quarks, ml(~u -k Ss) -k m2(~c -k rid), so that certain pairs of fermions are given degenerate masses. This shortcoming should not be too severe, however, since we are studying spontaneous chiral symmetry breaking, and the natural scale for the dynamically generated fermion masses in the chirally broken phase is of order the electroweak scale, 250 GeV, much larger than known fermion masses. However, since the representations of U(1) are complex, the full SU(2) x U(1) theory cannot be rewritten in such a manner. (ii) The effective continuum limit should be taken in the phase without spontaneous breaking of the chiral symmetry, since in the broken phase(s) the fermions pick up large, dynamically generated masses, naturally of order the electroweak scale. (iii) It is thus important to determine where this phase without spontaneous chiral symmetry breaking lies in the SU(2) case with I = 1/2 Higgs and fermions. In addition to the analytic work which applies at fig = 0 and, qualitatively, in the adjacent strip, the phase structure of this SU(2) theory has been mapped out by simulations using quenched 1,s,ll and dynamical8,1s fermions. (iv) The line of chiral transitions which completely separates the chirally symmetric and spontaneously broken phases obviously alters the renormalization group flows in the theory. It is clearly important to understand the renormalization group flows and their fixed points with fermions included. This is necessary for a satisfactory understanding of the continuum limit of the lattice theory. (v) Our work on the chiral transition may provide the first ab iniLio answer 4 to the longstanding question of why the known weak nonsinglet fermions have ] = 1/2. The point is that the strength of the effective gauge-fermion interaction increases with the Z of the fermion. It was thus expected theoretically, and has been demonstrated analyticallye and numerically, via both quenched 1'6'11 and dynamical8,1s simulations, that the chiral boundary occurs at larger values of/3h for I > 1 fermions than for
I = 1/2 fermions. It follows that if, as is now conventional, one takes an effective continuum limit from within the Higgs phase, in the vicinity of the 0(4) transition at/3g = oo, then this can be done while in the chirally symmetric phase for fermions with I = 1/2. However, since the chiral boundary for I :> 1 fermions lies at larger/3h than that for I = 1/2 fermions, one would presumably be in the chirally broken phase for the former. Hence, according to this picture, only I = 1/2 fermions could be light compared to 250 GeV. (vi) In the case of SU(2), the lattice can be used to study the strongly coupled standard model) ,16 This model requires that chiral symmetry not be broken spontaneously, since this would produce large, dynamically generated masses for the fermions of order 250 GeV (and would violate electric charge); at the same time, however, it also assumes a strongly coupled, confined behavior with the resultant spectrum of excited bound states. It was motivated in part by early work with bosonic gauge-Higgs models with Higgs in the fundamental representation, which showed that the confinement and Higgs phases are analytically connected, giving rise to the principle of complementarity of states and thus the interesting implication that the fermions and W and Z bosons would have excited states analogous to the baryon and meson resonances
(e.g., N, N*'s,...;p, p(1600),...). However, because of the chiral transition separating the old confinement and Higgs phases, this complementarity does not hold in the presence of fermions. One can also investigate this possibility in a toy U(1) model; studies with both quenched ~ and dynamical12 fermions again find no evidence for such a phase. The continued search for such a hypothetical phase remains an interesting pursuit (as does the experimental search for excited fermions and excited W and Z bosons). (In the full SU(2) × U(1) electroweak theory, even without fermions, there is a non-analytic boundary completely separating the interior of the phase diagram into two phases17 so again there is no complementarity between states in these two phases.) 3.
ANALYTIC TECHNIQUES In order to review how these results are obtained, we consider a general lattice gauge theory with a Higgs field q~ in the fundamental representation and a fermion field X; this theory is described, in standard notation, by the path integral
Z = f H dU'~,~,dc~,~dc~dx,~dxne-s rh , u
(3.1)
R.E. Shrock / Gauge-Higgs systems with [ermions where S = SG + SH + SF,
as = - i + 4/'b2 - 3r~ + 12/'~/~(/', -/'~/~)
s~ = Z~ ~ [~ - P]
(3.2)
plaq.
s.~ = - u ~ a ~ { ¢ ~ D , ( V . , . ) ¢ . + . A
(a.3)
1 SE = ~ E ~7,,,uX,,[Dx(U,,,u)X,,+.,, - Dx(U~-.,,,u)X,,-.,,] (3.4) Sere, Z~ = ~ / f for UO) and 4 / f for SU(~); P = R~{U~,o~.} for UO) and (1/2)T/'{U~,o~.} for SO(2), and y = 2~h. D~(U) is the representation of the gauge group element U corresponding to the Higgs field ¢, and similarly for D x ( U ). The lattice is d-dimensional hypercubic with d _> 3 and lattice spacing a -- 1. The Higgs fields satisfy ¢t¢~ = 1; as is well known, this does not imply that the continuum Higgs fields are of fixed length. With no loss of generality, we take ~h _> 0. We use staggered fermions X,~ because of their advantages for studies of chiral symmetry. The ~/,~,u = {1 for # = 1; (--1) '~+'''+''-~ for 2 < p < 4} are factors from the Dirac matrices. The analytic strong gauge-coupling results are obtained using the following technique. First, we perform the integrals over Higgs and gauge fields, obtaining the exact result Z = jN~ j. ~ n dxnd2ne-S', where Nz is the number of links in the lattice, and ,/1 = I0(2fla) for U(1)
and ~(2Z~)/Z~ for SU(2), where ~(~) is the modified Bessel function. For U(1) we find ~
-sb.> = ~.[-(1/2)~.,./'.(~.x.,
- ~.,x.)
where ,~' - ~ + ~. and/'. = ~(U)/Io(U), where q is the fermion charge. For SU(2) with I = 1/2 fermions ~, -s;uc,),
~=,/, = ~[-2-'~.,./',/,(~.x.
, - ~.,x.)
+ 2-3(1-/'~ )(~.X.)(~., X., )+ 2-~(/'~/~-rz )(:~.X., )(~n,X. ) + 2 - 3 ( 1 - - 1,2
, V~){(X.X. ) + (~.,X.) ~)
+2-%.,./'~/:ao(~.x.,
- £.,x.)(£.x.)(£.,x.,)
+2-Z as(~.X.)~(~.,X.,) 2]
(3.6)
where r z = I2z+,(y)/Iz(y) and
a6 = 1 - 4r~l 2 + 3rl
(3.7)
79
(3.8)
Again, this result is exact. (Eq. (3.6) was written in an equivalent form in Ref. 6, using an identity given there.) Results for other values of ] were also presented there. Next, we use a mean field technique to replace operator products higher than quadratic by a quadratic part times the mean value of the rest. This renders the action quadratic, so that we can perform the fermionic integrations exactly. We then derive a mean field consistency condition for the chiral symmetry breaking fermion condensate. In the U(1) case this equation is
< )~X > = (d/2)(1 - / ' ~ ) < :~X > / ( d P ) D-~
(3.9)
where f(dp) -= (2~) -d H~=I f:~ alp. and D = [{(d/2)(1 - / ' ~ ) < ~ x >}~ + / ' ~ ]
(3.10)
with s = ~.=~ s i o ' p . . As usual, (3.9) always has the solution < XX > = 0, but if there is a solution for nonzero < .~X >, this minimizes the mean field free energy and hence is the physical solution. We have shown analytically that (3.9) implies the existence of a (continuous) chiral transition in the theory s-7. We have also solved (3.9) numerically. A 1/d expansion of (3.9) to order 1/d ~ has been calculated and gives excellent agreement with the numerical solution? '9 Indeed, in the large d limit, we have obtained an exact solution for the chiral transition. In terms of the scaled condensate/~ = (ff/2)1/2/~, and ----/',z'2we obtain 37/2 = (1--2¢)(1--a~)-~0(1/2-~). For SU(2), the critical point can be located by an equation ~ similar to (3.9), with z = r~/~. As noted, these results agree with numerical simulations in both the U(1) and SU(2) cases at least as well as the 10 - 20% accuracy typical of mean field theory in d = 4 spin systems. The chiral boundary has also been mapped out in the interior of the phase diagram. We specialize here to the physically more important case of SU(2). In the limit ~'g -~ oo, the partition function factorizesS: Z = Zo(4)(~h)ZII, where Zo(4) refers to the 0(4) spin model, and Z H is the partition function for free (massless) fermions. Since the latter do not undergo any spontaneous chira[ symmetry breaking, it followss that for /3g --~ oo, < XX > = 0 for all fl~. There are several possibilities for the chiral boundary which are consistent with this exact result. Quenched simulations1,11 found that for ] = 1/2 fermlons, the chiral boundary appears to coincide with the confinement-Higgs boundary (for values
80
R.E. Shrock / Gauge-Higgs systems with fermions
of fig where the latter was present in the pure bosonic theory), which would imply that as/39 --* oo, the chiral boundary ends at the 0(41 critical point. In our dynamical fermion simulation s with ] = 1/2 fermions, we again found that, for the value of/99 where we measured,/39 = 1.9, the chiral boundary coincided with the confinement-Higgs boundary. Of course without dynamical simulations at arbitrarily large fig, it is not possible to determine precisely where the chiral boundary ends in the limit fig --* oo. (For this reason the chiral phase boundary presented in Ref. 8 was not drawn past our measured point, although in accordance with our theorem, it could not end in the interior.) Unfortunately, it becomes very difficult to measure the chiral boundary at values of fig significantly larger than about 2 for several reasons. First, in order to locate this boundary, one must detect where < XX > drops to zero as/3h increases at fixed /39, but in the spontaneously chirally broken phase this condensate is decreasing exponentially rapidly as a function of ~g Cat fixed fib)- Eventually, even where the condensate is nonzero, it is so small that one cannot accurately measure where it would vanish, as flh increases, across the chiral boundary. A related fact is that in the broken phase the correlation length is increasing exponentially rapidly, so that eventually finite-size effects become important. Finally, even with symmetric lattices, there can be finitetemperature effects for sufficiently large /99, and these can lead to a finite-temperature transition, which, while interesting in its own right, is spurious if interpreted as a zero-temperature chiral transition. However, the most important result of our work concerning the strongly coupled standard model is independent of the details of the location of the chiral boundary (which could be changed anyway by the addition of irrelevant operators to the action): this result is the loss of the complementarity of states between the chirally broken, confinement phase and the chirally symmetric Higgs-like phase, and depends only on the fact that the chiral boundary completely separates these two phases. Recently, we have mapped out the chiral boundary for SU(2 / with I = 1 dynamical fermions, is. We find (a / good agreement with our analytic results at/~g -~ 0 and (b) at/39 = 1.9, the chiral boundary occurs at a larger/3h than the conflnement-Higgs phase boundary, in contrast to the I = 1/2 case. Property (b) was also observed in quenched data. T M
4.
CASE OF TWO DIFFERENT FERMIONS We have also used our analytic methods to determine the phase structure of a theory with two different types of interacting ferrnions simultaneously present. 9 For simplicity, we used a U(1) theory and worked again at strong gauge coupling, but again, similar results should hold near/39 = 0 and for SU(2). Let us denote the different fermions as X and ~ with charges qx :> q~" We have shown analytically that this theory exhibits two distinct (continuous) chiral transitions, and thus three different phases. The locations of the critical points for the two chiral transitions were also given. The phases are characterized as follows:
Pl:
Zh
<~>#0
P2: t3h,c,~<_flh
~h>_~h,~,x:
<~>=0;
< ) ~ x > = O (4.1)
S. ARBITRARY HIGGS AND FERMION CHARGES Although in the standard electroweak theory, the Higgs field transforms according to the fundamental representation of SU(2), theories with higher unification require Higgs in higher representations. It is thus of interest to investigate the nature of chiral symmetry breaking in such a theory. The U(1) model provides a useful system with which to investigate higher Higgs representations. Extending Ref. 2, we proved a general classification theorem which completely specifies under what conditions this theory has a chiral transition at ~'g = 0. 9 Consider the U(1 / theory with a fermion f of charge q], and a Higgs field of charge qh. Then 9 if and only if qf is a (nonzero) integral multiple of qh, the theory has a chiral phase transition. If qf is not a multiple of %, then the path integral factorizes into separate, decoupled bosonic and fermionic factors, Z(qf :/: k%) = ZhZ.~, where k is a nonzero integer, Zh = Io(2flh) N~ , and Z s (given in Ref. 9/ is independent of/3h; hence, < XX > is also independent of/3h, so that in this case chiral symmetry is spontaneously broken (at fl~ = 0) for all fib. 6. YUKAWA INTERACTIONS We have studied the chiral transition in U(11 and SU(2) gauge-Higgs theories, s,1° In both theories, the fermions consisted of a field X in the fundamental representation interacting with gauge-singlet fermion(s) (and ( in the SU(2) case/ via a Higgs in the fundamental
R.E. Shrock / Gauge-Higgs systems with fermions representation. This is motivated by the Yukawa interactions in the SU(2) part of the standard electroweak theory. The analyses were carried out in the strong gauge coupling limit and used mean field techniques similar to those already discussed. In the U(1) case, the Yukawa interaction has the form Sy = h ~ ( ~ ¢ n ~ n + b.c.). In this model it was found that for a range of sufficiently weak Yukawa couplings h, the chiral transition continues to exist (and to be of second order), but is removed as the strength of the Yukawa coupling increases through a critical value, he = (d/2) 1/~. Interestingly, the neutral fermion, through its Yukawa interaction with the charged fermion, picks up a nonzero condensate < ~ > in the broken phase. Both I~h.,d and < XX > decrease as h increases. For the SU(2) case1°, there are two types of Vukawa interactions:
81
REFERENCES 1. I-H. Lee and J. Shigemitsu, Phys. Lett. 178B (1986) 93. 2. I-H. Lee and R. E. Shrock, Phys. Rev. Lett. 59 (1987) 14. 3. I-H. Lee and R. E. Shrock, Nucl. Phys. B290 (1987) 275.
4. I-H. Lee and R. E. Shrock, Phys. Lett. 196B (1987) 82. 5. I-H. Lee and R. E. Shrock, Phys. Lett. 199B (1987) 541. 6. {-H. Lee and R. E. Shrock, Phys. Lett. 201B (1988) 497. I-H. Lee, R. E. Shrock, in Proceedings of the International Symposium on Field Theory on the Lattice, Seillac, Nucl. Phys. (eroc. Suppl. ) B4 (1988), 373, 438. 8. S. Aoki, I-H. Lee, and R. E. Shrock, Phys. Lett. 207B (1988) 471.
n
where e~j = -ej~. We find that if and only if h,~4 + h,~-4 > 8d -2, then the theory has a spontaneous chiral symmetry breaking phase transition (at ~g -- 0). This condition is always satisfied if either h,1 or h2 is smaller than 2-3/4dl/~; in this case, the transition remains even for arbitrarily large values of the other Yukawa coupling. These mean field results are, of course, not exact, but should give some insight into the true behavior of the theory. As in the other cases, the transition was found to be continuous and approximate values for the critical point were given. The research reported here was done in collaboration with I-H. Lee and, in Refs. 8 and 15, S. Aoki and was partially supported by the grant NSF-PHY-8507627.
g. I-H. Lee and R. E. Shrock, Nuc{. Phys. B305 (1988) 286. 10, I-H. Lee and R. E, Shrock, Nucl. Phys, B305 (1988) 305. 11. A. De and J. Shigemitsu, Nucl. Phys, 307]3 (1988) 376. 12. E. Dagotto and J. Kogut, Phys. Lett. 208B (1988) 475. 13. I-H. Lee, R. E. Shrock, in the Proceedings of the FSU-SCRI Lattice Higgs Workshop (May, 1988), in press. 14. I-H. Lee, R. E. Shrock, in the Proceedings of the APS Division of Particles and Fields Meeting, Storrs, Conn. (Aug. 1988), in press. 15. S. Aokl, I-H. Lee and R. E. Shrock, preprint BNLITP-SB-S8-67. 16. M. Claudson, E. Farhi, and R. L. Jaffe, Phys. Rev. D34 (1986) 873; L. Abbott and E. Farhi, Phys. Lett. 101B (1981) 69. 17. R. E. Shrock, Phys. Lett. B162 (1985); Nucl. Phys. B267 (1986) 301; Phys. Rev. Lett. 56 (1986) 2124; Nucl. Phys. B278 (1986) 380.