---

Complementarity and Chiral Fermions in SU(2) Gauge Theories

arXiv:hep-ph/9302235v1 8 Feb 1993January, 1993HUTP-92-A047Complementarity and Chiral Fermions in SU(2) Gauge TheoriesStephen D.H. Hsu∗Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138AbstractComplementarity - the absence of a phase boundary separating the Higgs and con-finement phases of a gauge theory - can be violated by the addition of chiral fermions.We utilize chiral symmetry violating fermion correlators such as ⟨¯ψψ⟩as order param-eters to investigate this issue. Using inequalities similar to those of Vafa-Witten andWeingarten, we show that SU(2) gauge theories with Higgs and fermion fields in thefundamental representation exhibit chiral symmetry breaking in the confined phaseand therefore do not lead to massless composite fermions.

We discuss the implicationsfor the Abbott-Farhi strongly interacting standard model.∗Junior Fellow, Harvard Society of Fellows. Email: Hsu@HUHEPL.bitnet, Hsu@HSUNEXT.Harvard.edu

1Complementarity and all thatCertain gauge theories with scalars in the fundamental representation can be shown to exhibita remarkable property known as complementarity [1, 2, 3]. Complementarity means that theHiggs phase (large vacuum expectation value v, small gauge coupling g) and confinementphase (small v, large g) are not separated by a phase boundary.

(Here both g = g(Λ) and vare defined in terms of some lattice spacing Λ−1.) The result, proved by Fradkin and Shenker[2] using results of Osterwalder and Seiler [1] (see also Banks and Rabinovici [4] for a similarresult for U(1) theories), consists of demonstrating that in a lattice formulation of the theoryall correlators (ie free energy, n-point Greens functions) are analytic functions of g and v ina connected region which contains both the Higgs and confinement phases.

(See figure 1 fora typical phase diagram.) Therefore, quantities such as the free energy of the theory varysmoothly without discontinuity as we interpolate between the two regions.The rigorous demonstration of complementarity coincided with observations by ’t Hooft[5] and Susskind (unpublished) that there exists a strong similarity between the spectrumof states in the standard Higgs picture and the confined picture of a gauge theory withscalars in the fundamental.

This led ’t Hooft to remark that the question of confinementin this class of models could only be answered dynamically - there being no fundamentaldifference between the confining and spontaneously broken phases. (Indeed, these remarksapply equally to QCD, despite its lack of fundamental colored scalars, because of compositefields which can be formed out of glue and fermions.

)In this letter we wish to examine complementarity in gauge-Higgs models when chiralfermions are included. We will demonstrate in the SU(2) case that the addition of chiralfermions is capable of drastically altering the phase diagram of the theory.The modelswe study exhibit a phase transition associated with chiral symmetry breaking (χSB) as wemove from the broken to confined phase.

The above result was established previously byI.-H. Lee and R. E. Shrock [6] using analytical and numerical techniques on the lattice.Our analysis will be in the continuum, which makes it less rigorous from the viewpoint ofconstructive quantum field theory but perhaps easier to understand to theorists who workin the continuum.It is straightforward to argue that addition of chiral fermions to a purely bosonic the-ory can lead to a violation of complementarity. One has merely to consider the ’t Hooftanomaly matching conditions [5], which are necessary but not sufficient conditions for theexistence of massless composite fermions.

If the anomalies resulting from the fundamentalfermion triangle graphs do not match those of the (putative) massless composite fermions,one can immediately conclude that there are no massless composites and chiral symmetriesare broken. The effects of the anomaly are then reproduced by the Goldstone modes via theWess-Zumino term [7] in the chiral lagrangian.

Here we will examine a more interesting class1

of models, where the ’t Hooft matching conditions are satisfied by the composite fermions,and therefore dynamical information is necessary to determine the status of the chiral sym-metries. There has been much interest in models of this type in which the known fermionsof the standard model are massless bound states of more fundamental preons [8].

Dimopou-los, Raby and Susskind [3] gave a physically motivated construction involving tumbling (viafermionic condensates) of chiral gauge theories which always yields solutions to the ’t Hooftconditions.More information is required to conclude that models of the above sort actually yieldmassless composites. The only cases this author is aware of where the existence of masslesscomposites can be demonstrated is in models where the large-N approximation (see Eichtenet.

al. [9]) can be used to show the absence of goldstone bosons at leading order in 1/N.

Herewe will prove that in a certain class of SU(2) gauge models, which satisfy both the bosoniccomplementarity conditions and ’t Hooft’s matching conditions, chiral symmetry is indeedbroken and no massless composites are formed. The essence of the argument is that sincerepresentations of SU(2) are real, the model can be rewritten as a vectorlike gauge theory.Weingarten [13] and Vafa and Witten [12] have shown that in vectorlike theories rigorousinequalities apply to the correlators of certain conserved currents.

Using a similar analysis,we show that the mass of a composite fermion formed of a scalar and a fundamental fermionis nonzero in the chiral limit of the model. This is sufficient to rule out the massless compositerealization of ’t Hooft’s conditions and therefore implies the existence of Goldstone bosonsand χSB.

The above result stands in contradiction to one of the key dynamical assumptionsof the Abbott-Farhi strongly interacting standard model, which relies on the existence of(nearly) massless composites.In the following section we will present our argument forχSB in certain SU(2) theories, and discuss the implications for the phase diagrams of thosetheories. In the final section we will discuss the implications for the Abbott-Farhi model andpreonic models.2Fermions and chiral symmetriesConsider an SU(N) gauge theory with (N-1) Higgs bosons in the fundamental representation(sufficient to completely break the gauge symmetry) †.

It can be rigorously shown usinglattice methods [1, 2] that this theory exhibits complementarity and exhibits a phase diagramsimilar to the one in figure 1. Now consider adding Nf massless, chiral fermions ψi to thetheory.

At the classical level there is an exact SU(Nf) chiral symmetry associated with†For the lattice proofs [1, 2] to apply it is necessary that the gauge symmetry be completely broken by theHiggs fields. It is often incorrectly stated in the literature that having the Higgs fields in the fundamentalrepresentation is sufficient for complementarity.2

the Nf fermions. In what follows we will use chiral symmetry violating correlators such as⟨¯ψψ⟩≡⟨¯ψαi ψiα⟩as order parameters to investigate the phase diagram for this theory.

Hereα is an SU(N) index and the flavor index i is left arbitrary. A nonzero value of ⟨¯ψψ⟩for anyvalue of i will be sufficient to show a violation of complementarity.In the perturbative Higgs regime (g small, v large) these symmetries remain unbroken byquantum effects.

We can argue this result as follows: suppose the weak coupling effects aresufficient to break some of the chiral symmetries. Then by the Goldstone theorem there mustexist massless composite Goldstone bosons, formed from the massless fermions.

However thebinding energy of the composite must be sufficient to cancel the positive kinetic energy ofthe two fermions confined to a region of the size of the Goldstone boson. Since in weakcoupling one expects the binding energy to be proportional to the fermion mass mf, thisis impossible if the size of the Goldstone boson is to remain finite in the zero mass limit.

(A more rigorous, lattice argument for the absence of spontaneous symmetry breaking atarbitrarily weak coupling has been given by Lee and Shrock. See the early papers in [6].

)The above argument holds for sufficiently small coupling g. Therefore there must exist asmall patch in the upper left hand corner of figure 1, in which there is no χSB and the orderparameter ⟨¯ψψ⟩is exactly zero. However, by analyticity‡, the vanishing of this correlatorcan be extended throughout the entire region where complementarity applies.

In particular,if figure 1 truly represents the phase diagram of the theory we can conclude that χSB doesnot occur in the confined phase. If, on the other hand, we can demonstrate that chiralsymmetries are broken in the confined phase it will imply the existence of a phase boundarybetween the Higgs and confinement regions, and a violation of complementarity.We will now proceed to show that chiral symmetries are indeed broken in the confinedphase of the above theory when N = 2.

(We continue to assume as above that there areno Yukawa couplings between the scalar and fermions and no explicit χSB.) This is a veryplausible result for SU(2) gauge theories for the following reason: because representations ofSU(2) are real, it is always possible to rewrite a Nf flavor SU(2) theory as a vectorlike Nf/2flavor theory.

(Nf must be even to guarantee vanishing of Witten’s global SU(2) anomaly[11]. ) The latter theory, in the absence of fundamental scalars, is merely two color QCDwhich certainly breaks its chiral symmetries.

Therefore, unless the presence of fundamentalscalars somehow dramatically alters the dynamics of the theory, we expect the same tobe true here. In particular, it is clear that if the scalar mass is taken to infinity, therebydecoupling it from the low energy dynamics, χSB must occur.We will show that χSB occurs for a large range of values of the scalar mass.

Our strategy‡Technically, to apply analyticity we must verify that the correlator still vanishes when the parametersg and v are given infinitessimal imaginary parts. We will assume this is the case.

The lattice proofs ofanalyticity of course still apply when g and v have small imaginary parts.3

is to prove that composites with interpolating fields given byQi≡φ∗αψiα(1)˜Qi≡φαǫαβψiβ(2)are not massless as long as the scalar mass is nonzero§. In the confined phase of the theorythere are only two candidates for matching the anomalies of the fundamental fermions ψi:the SU(2) singlet composites Qi, ˜Qi and the Goldstone bosons πij ∼¯ψiψj.

If Qi, ˜Qi are notmassless, the matching conditions must be satisfied by the Goldstone bosons and thereforeχSB must occur.We first rewrite the model in a vectorlike manner. Defineχi = (ψi+Nf /2)c = iγ2(ψi+Nf/2)∗.

(3)Note that the χ fields have the opposite chirality of the ψ fields, but are still doublets. Thetheory is clearly vectorlike as we can now add gauge invariant mass terms to the Lagrangian,pairing χi with ψi.Now let Ψi = ψi + χi.

With the addition to the Lagrangian of mass terms mi ¯ΨiΨi ourtheory is now simply Nf/2 flavor, two color QCD with massive dirac fermions and an extracolored scalar. The fermion masses and the “vectorization” of the model are necessary fortechnical reasons in order to apply certain rigorous results similar to those first derived byVafa and Witten [12].

At the end of the calculation we will take mi →0 to reduce it to theoriginal, with classical chiral symmetries intact.Let us temporarily redefine the interpolating fields Qi, ˜Qi so that they each contain aDirac fermion Ψi rather than a Weyl fermion as previously defined. The index i now runsfrom 1 to Nf/2.

If massless composite fermions are to exist in the limit mi →0 correspondingto the old Qi, ˜Qi, then the new Qi, ˜Qi must also be massless in that limit. We will nowdemonstrate that this is not the case.Consider the Euclidean propagator for the Q field: (From here on we selectively suppressflavor and color indices for simplicity.

Q refers to either of Qi, ˜Qi. )⟨T( ¯Q(x)Q(y))⟩= Z−1ZDA Dφ D ¯Ψ DΨ exp(−SE[A, φ, Ψ])¯Q(x)Q(y),(4)where Z−1 is the standard normalization factor and the Euclidean action isSE[A, φ, Ψ] =Zd4x12g2TrFF + |Dφ|2 + M2φ2 +Xi¯Ψ(D/ + mi)Ψ.

(5)Figure 2 gives a pictorial description of the expectation value in equation 4. The normal-ization factor Z−1 divides out all vacuum bubbles, so we are left with φ and Ψ propagators§In other words, we will prove that the N = 2 theories satisfy a “persistence of mass” condition [10].4

summed over all possible gauge backgounds, with scalar and fermion loops included. Con-servation of flavor and scalar number (for the moment we neglect scalar self-interactions)prevents either the Ψ or φ lines from terminating except on an insertion of Q.We can rewrite (4) in the following manner by integrating out the scalar and fermionfields:⟨T( ¯Q(x)Q(y))⟩=Z−1ZDµ (D2 + M2)−1A,xy (D/ + m)−1A,xy(6)Dµ≡DA exp(−Zd4x12g2TrFF) det−1/2(D2 + M2)Yidet(D/ + mi).

(7)Here the determinants and propagators are evaluated in an arbitrary gauge backgound Aµ,which is then integrated over. The key point is that the measure of integration Dµ canbe shown to be positive definite.

This is because both the scalar and gauge field Euclideanactions are real and the fermion determinant is always real and positive in a vectorlike theory[12]. (It is also necessary to choose the topological θ term to be zero.

)Since the measure is positive definite, any Aµ independent bound that can be placed onthe integrand will yield a bound on the Q propagator. The above integrand consists of theproduct of the scalar and fermion propagators in arbitrary gauge background.

A great dealis known about the behavior of such propagators at large separations |x −y|. For example,Kato’s inequality [15] asserts that |(D2 + M2)−1A,xy| ≤|(D2 + M2)−1A=0,xy|.

That is, the scalarpropagator in an arbitrary gauge background falls offfaster than its free counterpart (ie inzero gauge field background). A similar result, involving for technical reasons a smearedfermion propagator also applies.

Here we will sketch the arguments from [12] which apply toa smeared propagator of either scalar or fermionic type. Consider the smeared propagator(D/ + m)−1A,αβ ≡⟨α|(D/ + m)−1A |β⟩,(8)where |α⟩, |β⟩are localized wave packet states, rather than position eigenstates |x⟩, |y⟩.

By awave packet state we mean that Ψ(x)|α⟩= φ(x)|α⟩= 0 outside a compact region α centeredat x with size much smaller than |x −y|.We can bound the smeared propagator by the following trick:⟨α|(D/ + m)−1A |β⟩=Z ∞0dt ⟨α|exp[−(D/ + m)At]|β⟩(9)=Z ∞0dt e−mt ⟨α|exp[−i(−iD/)t]|β⟩. (10)The last expression has the form of a quantum mechanical transition amplitude in a (4+1)dimensional theory with Dirac Hamiltonian H = −iD/.

By causality, we have ⟨α|e−iHt|β⟩= 0for 0 ≤t < |x −y|. Therefore⟨α|(D/ + m)−1A |β⟩=Z ∞t=|x−y| dt e−mt ⟨α|e−iHt|β⟩,(11)|⟨α|(D/ + m)−1A |β⟩|≤1/m e−m|x−y||α||β|,(12)5

where we have used (4+1) unitarity and the Schwarz inequality to obtain the last expression,and |α|, |β| are the norms of the states |α⟩, |β⟩. A similar result applies in the scalar case.Note that for position eigenstates the corresponding norms are infinite and hence do notyield a useful bound.Putting the above results together, we have:|⟨T( ¯Q(x)Q(y))s⟩| ≤Cmexp(−(M + m)|x −y|),(13)where the subscript “s” means smeared and C is a numerical constant.

This bound precludesthe existence of a massless bound state Q in the limit of zero fermion mass and massivescalar (m →0, M fixed). Note that this type of bound does not preclude the existence ofa massless bound state consisting of two massless fermions, as in that case as m →0 thebound disappears.

This is crucial, as we expect to find massless composite Goldstone bosonsπij ∼¯ψiψj in the chiral limit.An important technical point is that the above arguments require a cutoff, Λ, for thetheory because the masses M, m appearing in the various inequalities are actually baremasses, M0, m0. We can either imagine that this entire analysis has been conducted onthe lattice (see [13]), or that a suitable, gauge invariant regularization such as Pauli-Villarshas been carried out.

Because of this technical requirement, there is a problem with theinequality (13) which stems from the unnaturalness of scalar models. The problem is that asthe cutoffΛ is taken to ∞, the bare scalar mass required to yield a fixed physical scalar massbecomes negative.

This is easy to see in perturbation theory, as the one loop correction tothe bare mass has the formM2physical = M20 +λ32π2Λ2 +3g232π2Λ2,(14)where λ, g are respectively the φ4 and gauge couplings and we have computed the latterin Landau gauge.We have suppressed subleading logarithmic corrections.Note that ifone wishes to take the cutoffarbitrarily large with respect to the physical scalar mass, anarbitrarily large and negative bare mass is required. Therefore, as Λ →∞our bound (13)becomes useless.It is possible to choose bare parameters such that (13) holds with positive M20 if thecutoffis not chosen too large.

(We also require λ(Λ) = 0 in order to perform the scalarfunctional integration in Eq. (4) ¶.) One would like to keep the cutofflarge compared tothe scale Λ2 at which SU(2) becomes strongly interacting, while keeping Mphysical ≃Λ2.¶This choice of λ(Λ) does not imply the theory is unbounded from below.

If one computes the renor-malization group improved effective potential, it is easy to see that a positive renormalized mass squaredcompensates for λ(µ) being driven (logarithmically) negative in the infrared.There is also no vacuumexpectation induced for the scalar unless the ratio Λ/Mphysical is taken very large.6

Whether this is possible depends on the evolution of the gauge coupling constant betweenΛ and Λ2. This in turn depends on the number of fermion flavors in the model.

If we fixMphysical = Λ2, for Nf = 2 we get Λ/Λ2 ≃3.5 while for Nf = 12 (the Abbott-Farhi case) weget Λ/Λ2 ≃2. (Note that for Nf = 2 the axial chiral symmetry is anomalous, and thereforealready explicitly violated by quantum effects.) Larger ratios of Λ/Λ2 are possible if weallow Mphysical > Λ2.

This verifies our intuition that very heavy scalars should decouplefrom the strong SU(2) dynamics, leaving behind a theory with broken chiral symmetries.However, the Abbott-Farhi model assumes a phase transition between χSB and no χSB asthe mass of the scalar is lowered, and hence we are more interested in what happens whenMphysical ≃Λ2.Should it concern us that the cutoffmust be taken so low? If we think of the renormal-ization group in Wilson’s language we know that we can start with a continuum theory (orone with arbitrarily large cutoff, momentarily ignoring triviality problems of scalar theories)and relate it to an effective theory at scale µ by systematically integrating out degrees offreedom.

The information from the high momentum modes will be contained in the runningcoupling constants gi(µ) and the higher dimension operators Oi(µ) induced by this proce-dure.The theory defined with cutoffΛ, “bare” couplings equal to gi(Λ) and additionalnon-renormalizable interactions given by Oi(Λ) is then completely equivalent to the originalcontinuum theory.In deriving our inequalities we assumed that there were no non-renormalizable operatorsin the bare theory. Perhaps this is justified because we are attempting to determine whetheran exactly massless state exists in a certain channel.

We are therefore interested in physics atvery low momentum scales, and at arbitrarily long distances, where higher-dimension oper-ators should be irrelevant. One might expect that the long distance behavior of two theoriesdiffering by some set of such irrelevant operators Oi should be the same.

(By assumptionthe operators Oi respect the fermion chiral symmetries, otherwise complimentarity is al-ready violated.) However, a loophole in this line of reasoning is that the higher-dimensionoperators may actually shift the ground state of the theory from chiral symmetry preservingto breaking.

This is possible in principle if spontaneous χSB is due only to gauge dynam-ics of momentum scale k ≃Λ2 rather than k << Λ2. In that case the operators Oi areonly supressed by powers of Λ2/Λ to some power, and may be important if this ratio is notlarge.

Because of this possibility, our results can probably only be rigorously applied forMphysical > (few) Λ2. This leaves a small window for the Abbott-Farhi model with a verylight Higgs scalar.

However, the lattice work of Lee and Shrock [6] is valid for all values ofthe scalar mass, and hence is sufficient to close this window.7

3Further implications: the Abbott-Farhi modelThe existence of a chiral phase boundary in the above class of models has strong implicationsfor the Abbott-Farhi model. Abbott and Farhi [16] formulated a strongly coupled versionof the electroweak theory, in which the the Higgs vacuum expectation value is small andthe SU(2) coupling large.This model is successful in roughly reproducing the observedspectrum of the electroweak theory, generating W and Z bosons as bound states of thescalar doublet field, although its current phenomenological viability is subject to certainunproven dynamical assumptions (see last reference in [16]).One of the key dynamical assumptions is that chiral symmetries remain unbroken inthe confined phase and that in the limit of zero fermion-Higgs Yukawa couplings there aremassless, left handed fermionic bound states QiL.

Nonzero Yukawa couplings λi provide massterms which marry the QiL to SU(2) singlet right handed fermions ψjR yielding Dirac fermionsof mass ∼λiΛSU(2), where ΛSU(2) ≃256 GeV is approximately the weak scale.Since the electroweak theory is intrinsically chiral, one might wonder how the resultsof the previous section can be applicable.The answer is that, perhaps suprisingly, theelectroweak theory is actually vectorlike in the limit where we ignore hypercharge and color.In the absence of hypercharge, color and Yukawa couplings the electroweak theory belongsto the class of models studied in section 2. (It is easy to show that the electroweak theoryhas no physical θ angle.) Therefore, to the extent to which those couplings can be treatedas perturbations the results of the previous section should apply to the Abbott-Farhi model.Indeed, all of the relevant coupling constants are small at scale Λ2, with the possible exceptionof the top Yukawa coupling.

This suggests that the masses of the fermions in that theorydo not resemble those of their perturbative electroweak counterparts. We expect the righthanded fermions to remain nearly massless, with masses of order (λv)2/ML induced by theirinteractions with the QiL.

We also expect a plethora of relatively light pseudo-Goldstonebosons which are bound states of left handed leptons and quarks. Finally, the condensateswhich form will spontaneously break color and hypercharge.Because our results are specific to gauge groups which have (pseudo)real representations,it is not always possible to apply them to more complicated composite models [8].

However,it seems possible to this author that many models of the sort first constructed by Dimopouloset. al.

[3] may indeed undergo χSB rather than produce massless composite fermions in theconfined phase. Additional dynamical information is required to decide the issue.8

4AcknowledgmentsThe author would like to thank E. Farhi, H. Georgi, T. Gould, V. Khoze, S. Nussinovand R. Singleton for useful discussions.The author is also grateful to R.E. Shrock fordiscussions of his lattice work in this context.

SDH acknowledges support from the NationalScience Foundation under grant NSF-PHY-87-14654, the state of Texas under grant TNRLC-RGFY106, the Milton Fund of the Harvard Medical School and from the Harvard Society ofFellows.9

References[1] K. Osterwalder and E. Seiler, Ann. Phys.

(NY) 110, 440 (1978)[2] E. Fradkin and S. Shenker, Phys. Rev.

D19, 3697 (1979)[3] S. Dimopoulos, S. Raby and L. Susskind, Nucl. Phys.

B173, 368 (1980)[4] T. Banks and E. Rabinovici, Nucl. Phys.

B160, 349 (1979)[5] G. ’t Hooft, in Recent Developments in Field Theory, ed. G. ‘t Hooft et.

al., (Plenum,New York, 1980)[6] I-H. Lee and R.E. Shrock, Phys.

Rev. Lett.

59, 14 (1987), Phys. Lett.

B201, 497 (1988);S. Aoki, I-H. Lee and R.E. Shrock, Phys.

Lett. B207, 471 (1988);for a recent reviewsee R.E.

Shrock in Quantum fields on the Computer, ed. M. Creutz (World Scientific,Singapore, 1992)[7] J. Wess and B. Zumino, Phys.

Lett. B37, 95 (1971)[8] Various Preonic papers[9] E. Eichten, J. Preskill, R. Peccei and D. Zeppenfeld, Nucl.

Phys. B268, 161 (1986)[10] J. Preskill and S. Weinberg, Phys.

Rev. D24, 1059 (1981)[11] E. Witten, Phys.

Lett. B117, 324 (1982)[12] C. Vafa and E. Witten, Nucl.

Phys. B234, 173 (1984)[13] D. Weingarten, Phys.

Rev. Lett.

51, 1830 (1983)[14] S. Nussinov, Phys. Lett.

B136, 93 (1984)[15] T. Kato, Israel J. Math 13 135 (1972)[16] L. Abbott and E. Farhi, Phys.

Lett. B101, 69 (1981); Nucl.

Phys. B189, 547 (1981);M.Claudson, E. Farhi and R.L.

Jaffe, Phys. Rev.

D34, 873 (1986)figure 1: Typical phase diagram for theories exhibiting complementarity.All physicalquantities are analytic functions of v and g in the shaded region.figure 2: Q field propagator in arbitrary gauge field background.10

---

출처: arXiv:9302.235 • 원문 보기