Buhep051
hepph/0501204
Accidental Goldstone Bosons
Abstract
We study vacuum alignment in theories in which the chiral symmetry of a set of massless fermions is both spontaneously and explicitly broken. We find that transitions occur between different phases of the fermions’ CP symmetry as parameters in their symmetry breaking Hamiltonian are varied. We identify a new phase that we call pseudoCPconserving. We observe first and secondorder transitions between the various phases. At a secondorder (and possibly firstorder) transition a pseudoGoldstone boson becomes massless as a consequence of a spontaneous change in the discrete, but not the continuous, symmetry of the ground state. We relate the masslessness of these “accidental Goldstone bosons” (AGBs) bosons to singularities of the order parameter for the phase transition. The relative frequency of CPphase transitions makes it commonplace for the AGBs to be light, much lighter than their underlying strong interaction scale. We investigate the AGBs’ potential for serving as light composite Higgs bosons by studying their vacuum expectation values, finding promising results: AGB vevs are also often much less than their strong scale.
I. Introduction
In this paper we describe phenomena we believe to be very general but which appear to have received almost no attention in particle physics.^{1}^{1}1The principal exception is a brief passage in Dashen’s classic paper on vacuum alignment [1]; see the discussion at the end of his section III. These are the presence of various phases of CP symmetry, of transitions among these phases, and of anomalously light bosons which become massless at these phase transitions. While, in the model calculations we present, the massless state is a pseudoGoldstone boson (PGB) of an approximate chiral symmetry, the boson’s masslessness is not due to the restoration of its associated continuous chiral symmetry. Rather, it is due to a change in the phase of the discrete CP symmetry. Following Dashen, who first observed this phenomenon in the context of QCD, we call these “accidental Goldstone bosons” (AGBs). Unlike Dashen, however, we do not believe the AGB’s mass necessarily is restored by higherorder corrections. Rather, we suspect that corrections only shift the values of parameters at which phase transitions occur.
This study grew out of earlier work on vacuum alignment in technicolor theories of dynamical electroweak symmetry breaking [2, 3]; also see Ref. [4] for a recent summary. However, although our calculations are similar to those used in technicolor, we are confident our conclusions extend beyond that setting. Some of the phenomena we observe also occur in QCD when one allows an odd number of real quark masses to become negative (so that ) [1, 5]). We see no reason they would not also occur in models different from the type we investigate here. They may even have relevance to condensed matter systems.
The model we use assumes massless Dirac fermions , , transforming according to a complex representation of a stronglycoupled gauge group. The fermions’ chiral flavor symmetry, , is spontaneously broken to an subgroup. It is convenient to work in a “standard vacuum” whose symmetry is the vectorial defined by the invariant condensates^{2}^{2}2We work in vacua with the instanton angle rotated to zero. Condensates are assumed to be CPconserving.
(1) 
The condensate is renormalized at the scale , which (for of ) is commonly assumed to be and, then, . Here, is the decay constant of the massless Goldstone bosons, , , resulting from the spontaneous chiral symmetry breaking. It is normalized by the relation , with so that .
The chiral symmetry is also broken explicitly by the invariant fourfermion interactions
(2) 
where the unexhibited LL and RR terms are irrelevant for our further discussion. The are inverse squared masses of gauge bosons (or scalars) exchanged between the “currents” (or their Fierz transforms) in . They are chosen in numerical calculations so that all symmetries are explicitly broken. Ordinarily, then, we would expect the PGBs to acquire positive masssquared. Finally, we assume that is Tinvariant, i.e.,
(3) 
Now, there may be a mismatch between the standard vacuum and the one in which of Eq. (2) gives positive masssquared to all . It is therefore necessary to “align the vacuum”, more precisely, to determine the correct ground state of the theory [1]. In this state, , varied over , is a minimum at . We follow Dashen’s procedure, which is based on lowestorder chiral perturbation theory and minimize the vacuum energy defined over the infinity of perturbative ground states :
(4)  
Here, we used (since transform as a complex representation of )
(5) 
Since is invariant under transformations, the matrix is the only physically meaningful combination of and . The fourfermion condensate is renormalized at the scale of the exchangedboson masses making up the . This scale is likely to be much greater than . In a QCDlike theory, . If is a walking gauge theory [6, 7, 8, 9], could be much larger than , partially overcoming the suppression by the .
Note that CPinvariance of implies that . The which minimizes is defined up to a factor , . If, apart from this trivial ambiguity, , then the correct chiralperturbative ground state is discretely degenerate and CP symmetry is spontaneously broken. Equivalently, and more conveniently, the Hamiltonian correctly aligned with the standard violates CP.
It is convenient to make the transformation . This amounts to computing with , . Then, dropping the subscript “0” from now on,
(6) 
After this vectorial transformation, the PGB masssquared matrix is still calculated using the axial charges, formally defined by . To lowest order in chiral perturbation theory [10],
(7) 
For the Hamiltonian in Eq. (I. Introduction).
(8)  
Finally, it is very useful to parameterize in the form
(9) 
Here, are diagonal matrices, each involving independent phases , and is an –parameter CKM matrix which may be written in the standard HarariLeurer form [11].
The remainder of this paper is organized as follows: In Sec. II we review vacuum alignment for the model we’ve described. The fourfermion form of implies a linking of the phases in which allows the possibility of three “phase phases” with different CP properties. We call these three phases CPconserving (CPC), pseudoCPconserving (PCP), and CPviolating (CPV). In the CPC phase, is times a real matrix and is real. In the PCP phase, is not simply times a real matrix, but the phases in are rational multiples of and the CKM matrix is real. The phases in are also rational, but the Hamiltonian is not merely real up to an overall phase. However, introducing the aligning matrix , we show that the LR terms in are real, i.e., CPconserving, in both the CPC and PCP phases. In the CPV phase, the phases of are not rational multiples of , is not real, and is definitely CPviolating.
We carry out vacuum alignment numerically in a threeflavor () model, varying one in . We observe each of these CP phases and note that the transitions between them are either first or second order — defined here as whether the first or second derivative of with is discontinuous. Although we are varying just one of the parameters in , it is obvious that the phase transitions occur on surfaces in the space of ’s. Our calculations are merely along a single trajectory in this space. There are two PGBs whose is much less than those of the other six. These are the accidental Goldstone bosons of this model. At all secondorder (and, apparently, firstorder) transitions one of these light PGBs becomes massless. We explain why this happens.^{3}^{3}3Most of the features of vacuum alignment described in Sec. II and this model were discussed in Ref. [2]. The treatment in the present paper is much more incisive. Our calculations indicate that light AGBs are commonplace, at least as long as the are the same order of magnitude. Then, competition among the ’s means that one is never very far from a CP phase transition and a surface in space on which an AGB mass vanishes.
Sections III and IV are devoted to understanding the phase transitions in more depth. In Sec. III we present a remarkable formula for which connects the vanishing of to singular behavior of the “diagonal phases” of , the phases , , of in its diagonal form. This formula also directly relates the AGBs to the diagonal phases. The formula is derived in Appendix A. An analytic example of how it works is given in Appendix B using Dashen’s model — three quarks with negative masses. We also illustrate it numerically for the model. In Sec. IV we focus on the aligning matrix . In the CPC and in what we call PCP1 phases, where is real. In PCP2 phases, cannot be written this way. In the CPC phase of the model we study, appears to be symmetric.^{4}^{4}4As we discuss in Sec. IV, this is very nearly true numerically, but it appears to be an artifact of how we chose the model’s . It is shown that this implies the normalized diagonal phases are rational multiples of . In PCP1 phases, is not symmetric. In this case, some but not all the are rational. We spell out the conditions for determining how many are rational. In the PCP2 phase, none of the are rational. This is startling since all the phases in are.
Finally, in Sec. V we discuss one potential application of AGBs: light composite Higgs bosons for electroweak symmetry breaking [12, 13]. A light composite Higgs boson is a bound state whose mass and vacuum expectation value (vev) are naturally much less than the energy scale at which its binding occurs. The effort to construct realistic models of light composite Higgses has been driven by the strong experimental evidence in favor of the standard model with a light Higgs boson. Recently, much of this effort has focused on the little Higgs scenario [14, 15, 16, 17]. Little Higgs bosons are PGBs that are anomalously light because interlocking continuous symmetries need to be broken by several weaklycoupled interactions, making their nonzero mass a multiloop effect. In most models so far, little Higgses acquire masses in two loops so that a compositeness scale of yields a mass and vev of and –.
Accidental Goldstone bosons can easily have , the fermion scale. The challenges are (1) a vev , (2) embedding the AGB structure into electroweak symmetry, and (3) coupling the AGBs to quarks and leptons to account for their masses and mixings (without running afoul of flavorchanging neutral current and precision electroweak constraints). In Sec. V, we study the first of these, the magnitudes of the AGB vevs, and find that they too are often much smaller than .
II. Vacuum Alignment and the Phase Phases
There are several useful forms of the alignment matrix :
(10) 
There are . The CKM matrix has angles () and phases ().
It was shown in Ref. [2] that there are three possibilities for the phases . Consider an individual term, , in . If , this term is least if ; if , it is least if . Thus, links and , and tends to align (or antialign) them. However, the constraints of unitarity may partially or wholly frustrate this alignment. This then gives the three phase phases:

All are linked to one another and unitarity allows them to be equal. Unimodularity of implies all (mod ) for fixed . Then times a real orthogonal matrix, and all the terms in are real. This is the CPC phase.

Not all are linked to one another. Still, if unitarity allows it, the are again rational multiples of , but generally not equal to one another (mod ). Rather, their values are various multiples of for one or more integers . As explained in Ref. [2], is real and this is a necessary condition for rational phases. We also showed there that, while is not real, the phases in the are rational. Thus, we call this the PCP phase. We repeat the proof: If is real and then and are linked and, in this phase, (all phase equalities are mod ). The phase of an individual term in the sum for is then . This is a rational phase which is the same for all terms in the sum over . Indeed,
(11) We see from Eqs. (I. Introduction,11) that the vectorial change of variable makes all the LR terms real in Eq. (11). Under this transformation, the aligning matrix becomes and, of course, . Although the LR terms are made real by this transformation, the LL and RR terms generally are not because there is no phaselinking argument for the . Whether they have rational phases or not is a modeldependent (and conventiondependent) question.

Whether or not the are linked, unitarity frustrates their alignment so that they are all unequal, irrational multiples of , random except for the constraints of unitarity and unimodularity. This is the CPV phase in which the phases in are irrational hash.
A demonstration of these three phases is provided by a model with three flavors.^{5}^{5}5This model was studied in Ref. [2], but only over the range –1.1. The phase transitions near and 2.8 were missed in that discussion. The chiral symmetry is broken in the vacuum to . The model’s eight Goldstone bosons get mass from a Hamiltonian with nonzero couplings
(12) 
These tend to align and . The phases and are not linked by these ’s.
Vacuum alignment was carried out numerically. For , an initial guess is made for the phases and angles in and , and these are varied to search for a minimum.^{6}^{6}6We have not systematically established that we have found global minima, but searches with widely different inputs have not produced deeper ones. When an aligning matrix is found that minimizes , it is used to calculate the rotated Hamiltonian in Eq. (I. Introduction) and the PGB matrix in Eq. (7). The eigenvalues and eigenvectors of this matrix are then determined. Then, is increased slightly, the phases and angles of the just obtained are used as new inputs, and the procedure is repeated. This works well everywhere except at the discontinuous transition occurring near . Following the mass eigenstates through that transition is a matter of some judgement — but not much import. The results are shown in Figs. 1–5. There we display the variation of the minimized vacuum energy, , the phases and magnitudes of , and (these contain phases unlinked to each other), and the masses of the two lightest PGBs alone and then compared to the model’s other six PGBs.
The energy is constant and from to 0.7215; this is a CPC phase.^{7}^{7}7Recall that is defined only up to a power of . At this point, there is a transition to a PCP phase in which becomes nondiagonal; still equals but and . The phases in are 0, and . The lightest PGB’s goes to zero, and starts to increase surpassing that of the second lightest PGB near . That PGB’s vanishes at , then rises and quickly falls back to zero at . This small region is a CPV phase with irrational phases. The region from to 1.854 is a CPC phase with all phases equal (mod ). Up to this point, the energy, , and all have varied continuously, although there are obvious discontinuities in the slopes of all but .^{8}^{8}8The phases and are not defined below and above 2.85, so their behavior there is not discontinuous. Here, there is a jump in these quantities and, as can be seen in Fig. 1, in the slope of . To see it better, we plot in Fig. 6. This transition is from the CPC phase to a PCP one. The lightest PGB appears to become massless, but it is difficult to tell numerically because of the discontinuous change from one set of vacua to the another. Finally, there is another transition back to a CPC phase near . There, is so large that becomes blockdiagonal with the mixing elements and vanishing.
We classify the transitions between different CP phases as being of first order (1OPT) or second order (2OPT) depending on whether or is discontinuous at the transition. The second derivative is plotted in Fig. 7; we will discuss it in the next section. Firstorder transitions involve discontinuous changes in matrix elements. They occur only at CPC–PCP transitions. The elements of are continuous at secondorder transitions. They occur at the boundaries between CPC or PCP regions and CPV ones, or at CPC–PCP boundaries such as and 2.85 where elements of continuously become nonzero or vanish.
We stress that the vanishing of an eigenvalue at a phase transition is not a consequence of increased chiral symmetry; the current corresponding to the massless boson is still not conserved at the transition. Rather, the boson’s masslessness is associated with a change in the discrete CP symmetry. We refer to the two chronically light PGBs of this model as accidental Goldstone bosons. They remain light because — in this model and others we have looked at — one is never very far from a phase transition. We explain in Sec. III why there are two AGBs in this model.
It is easy to understand why one PGB’s at a 2OPT, . As is increased, the true vacuum corresponding to one CP phase is becoming less stable, while the false vacuum corresponding to a different phase is becoming more stable. In this false vacuum, one PGB has .^{9}^{9}9There cannot be more than one. In a true vacuum, all , and it seems most unlikely that two PGB masses will vanish at the same on their way from negative to positive values. In the true vacuum this PGB’s positive is decreasing while it is increasing in the false one. Since the 2OPT is continuous, the two trajectories must cross at . For a 1OPT, there is a discontinuous jump in the lightest as there is for all the others. Hence, there seems to be no argument for . Nevertheless, in our calculations for this and other models, the lightest AGB mass appears to approach zero on one side of the 1OPT as well. It is obvious that there are surfaces in the space of the that separate the different CP phases and, at least for 2OPT surfaces, an AGB mass vanishes there.^{10}^{10}10We suspect that the order of the phase transition does not change as long as new ’s are not introduced. We also note that adding new ’s can change the character of a phase, e.g., from PCP to CPV if too many phases are linked to be consistent with unitarity.
There is a clear levelcrossing phenomenon in Fig. 4, in the CPC region near . There we see the two lightest PGBs’ masses approach other and repel.^{11}^{11}11The two levels cross, but without interaction, in PCP regions, near and 2.15. The effect of this will be seen on the vevs of these states, discussed in Sec. V.
A comment on the units used for in Figs. 4 and 5 is in order: The quantity being plotted in these figures is actually . In our numerical calculations, we set so that . But, up to an anomalous dimension factor for the fourfermion condensate, where . If, for example, , then the vertical scale in Figs. 4,5 is in units of . The AGB masses are then .
Finally, we do not believe that these phase transitions and the associated vanishing of a PGB mass are mere artifacts of our using lowestorder chiral perturbation theory. Higherorder corrections may shift the surfaces in space separating the phases (not to mention expanding the dimensions of the space), and they may even eliminate existing transitions or add new ones. But we see no reason that phase linking, the transitions between various rational and irrational phase solutions, and the associated massless states would not occur for with higher dimensional than fourfermion operators and vacuum energies involving higher powers of and .
III. Understanding the Phase Transitions I:
The Formula for
Considerable insight into the AGBs — their number and the connection between their vanishing masses and the behavior of the phases — can be gained from studying . For definiteness, we continue to consider a theory in which chiral flavor symmetry is spontaneously broken in the vacuum to .^{12}^{12}12This discussion and Eq. (14) apply to any symmetry groups and . The chiral symmetry is also explicitly broken by an interaction as in Eq. (2), for example. Suppose that depends linearly on a parameter . Write the vacuum energy of the properly aligned Hamiltonian as^{13}^{13}13The reason is that, for our model’s symmetry groups, with .
(13) 
where , , is a phase at the minimum. Then (sum on repeated indices)
(14) 
Equation (14) is derived in Appendix A. Here, is the PGB squaredmass matrix and is the matrix
(15) 
and is the adjoint representation of . At a minimum, is a positivesemidefinite matrix, so that , as seen in Fig. 7.
To go further with Eq. (14), it is convenient to replace by its diagonalized form:
(16) 
Here, is the matrix which diagonalizes to and to . There are diagonal phases , with . They depend in complicated ways on the phases and the parameters in . For , define the real orthogonal matrix by . Then, implies
(17) 
Next, define by
(18) 
For of the form in Eq. (2), is given by
(19)  
(20) 
The relation and Eq. (17) imply , where is a diagonal generator in the adjoint representation. Thus, Eq. (14) can be cast in the form
(21) 
This is our key equation.
In Sec. II we saw that all phases are rational multiples of in the CPC and PCP phases. In Fig. 8 we plot the normalized diagonal phases for the model (). We see that in CPC phases, both are rational multiples of ; in PCP phases, only is rational; in the CPV phase, both are irrational. This will be explained in Sec. IV. It is remarkable that, even though is irrational in the PCP phases, all are rational there.^{14}^{14}14The definition of the is conventiondependent. The scheme we use for calculating the is this: Starting at the initial , here zero, the matrix is diagonalized and the phases of its eigenvalues — its eigenphases — are determined. A multiple of is subtracted from them so that . The eigenvalues are then ordered so that . Then, , , etc. As is increased, the procedure is repeated, requiring the changes in the and the to be continuous, except at a 1OPT. If necessary, the multiple of subtracted from the is changed to keep their evolution continuous. These subtraction changes typically occur at 2OPTs. The discontinuous changes at a 1OPT are also kept as small as possible. In the CPC and PCP phases, nonzero are actually rational multiples of to about a part in , whereas the are rational to computer accuracy. As we discuss in Sec. IV, the near rationality of the appears to be an unintentional artifact of the way we chose the .
One sees in Fig. 8 that the slope of one or both of the is singular at every 2OPT ( is merely discontinuous at ) while both are discontinuous at the 1OPT at . Looking back at Fig. 2, this behavior is clearly reflected in all the ; it is especially dramatic at the 2OPTs near . The slopes are plotted in Fig. 9. Away from the phase transitions, they are not large except in the narrow CPV phase where the are rapidly varying.^{15}^{15}15We have numerically studied an model and found very similar features to the ones described here. One difference is that the CPV phase in that model is wider. This is not important; in fact, it is surprising that the CPV phase in the model is so narrow. The singular behavior of the in Fig. 8 is just what we expect of order parameters at first and secondorder phase transitions. Therefore, we interpret the diagonal phases as the order parameters for the phase transitions we’ve been observing. Here, however, the transitions are between different phases of a discrete symmetry.
In general, the in Eq. (21) are small. Thus, is well approximated by keeping only the terms in Eq. (21). Just how good this approximation is can be seen by looking at the region to 1.85 in Fig. 7. There , while is negative, but very small. If we drop the terms, Eq. (21) simplifies greatly because and when index or . Then
(22) 
In Fig. 10 we compare with the righthand side of Eq. (22). The agreement is excellent except in the narrow CPV region with rapidly varying phases. There, the discrepancy is due both to the neglect of the terms and the difficulty of computing the derivatives as they become divergent.
Equation (22) makes a clear connection between the lightest PGBs, the ones we call AGBs, and the diagonal phases . We believe the association is onetoone, and that is why the model has two AGBs.^{16}^{16}16We have examined larger models and never found more than especially light PGBs. Of course, this onetoone connection is applicable only so long as all symmetries are explicitly broken so that there are no true Goldstone bosons. At 2OPTs, the are continuous, but at least some are divergent. Meanwhile, is finite, though discontinuous. This is possible only if a zero eigenvalue of the PGB matrix appears exactly at the transition to cancel singularities in the .^{17}^{17}17An analytic example is given for Dashen’s model in Appendix B. This is another reason we believe that the vanishing of AGB masses at phase transitions is not an artifact of lowestorder chiral perturbation theory. At a 1OPT at , is discontinuous and . On the other hand, all the PGB masses are discontinuous there, so we expect , i.e., a discontinuous slope in , as well.
IV. Understanding the Phase Transitions II:
The Character of
In Sec. II we showed that, in a basis in which the aligning matrix is , the LR terms in are real in the PCP and CPC phases. The matrix has the same eigenvalues as , therefore the same diagonal phases . However, it is easier to analyze the possibilities for the by considering .
Consider first the CPC phase. In that case, where is an matrix and . Denote ’s eigenvalues by , where the eigenphases satisfy (mod ). If is even, the eigenphases form conjugate pairs, for . If is odd, one eigenvalue, say , is . The ordering of the is arbitrary. Given an ordering, we can calculate the diagonal phases from
(23) 
Because of the ambiguity in , we can set if we wish.
Now, if is also symmetric, then all its eigenvalues are real, therefore equal , with an even number of ’s. All its eigenphases of would be rational multiples of and, then, so would their linear combinations forming the . In the models we studied numerically, is symmetric to about a part in in all nontrivial CPC phases, i.e., when is not merely proportional to the identity. Hence, the are rational to about the same accuracy in these calculations. The difference from exactly rational phases is not visible in the CPC regions of Fig. 8. This closeness to rational phases is tantalizing, but we believe it is an unintended artifact of the way we chose the couplings in the model. Those couplings seem to favor minimizing with a symmetric ; we have modified them to make nonsymmetric in a CPC phase.
Turning to the PCP case, in which the phases of are different rational multiples of , we have identified two subphases: PCP1 in which with a real matrix, and PCP2 in which cannot be written in this way. In PCP1, which is what we observed in our model, if , while if . If is odd and , we can change the sign of and take . For odd , then, the eigenvalues of form pairs, plus one real eigenvalue, and, so, has truly rational eigenphases, . As in Eq. (IV. Understanding the Phase Transitions II: The Character of ), we can define . If is even and , has rational phases. In this case, there may be no rational even though all phases are rational. If , there must be a real pair of eigenphases, , so there are will be at least two rational . We can choose them to be and .
Finally, in a PCP2 phase, there is no argument that any of the are rational. The same is of course true in a CPV phase, and we find only irrational phases in both.
V. VEVs of the AGBs
In this section we investigate whether AGBs can serve as light composite Higgs bosons. We have seen that they are usually much lighter than the scale of their strong binding interaction. Having associated the AGBs with the diagonal phases and, in turn, identified these as the order parameters of the various CP phases, it is natural to connect the vacuum expectation values of the AGBs with these phases. The question studied here is whether these vevs can also be much less than .
In a nonlinear sigmamodel formulation of the model, we would replace by , where .^{18}^{18}18This normalization of guarantees that the axial current it generates creates from the vacuum with strength . Under a transformation, . Minimizing the energy in this formulation amounts to determining the vacuum expectation values