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Critical Constraints on Chiral Hierarchies

arXiv:hep-ph/9210276v2 17 Nov 1992BUHEP-92-35HUTP-92/A058hep-ph/9210276Critical Constraints on Chiral HierarchiesR. Sekhar Chivukulaa,1, Mitchell Goldenb,2, and Elizabeth H. Simmonsb,3ABSTRACTWe consider the constraints that critical dynamics places on models with a topquark condensate or strong extended technicolor (ETC).

These models require that chiral-symmetry-breaking dynamics at a high energy scale plays a significant role in electroweaksymmetry breaking. In order for there to be a large hierarchy between the scale of the highenergy dynamics and the weak scale, the high energy theory must have a second order chi-ral phase transition.

If the transition is second order, then close to the transition the theorymay be described in terms of a low-energy effective Lagrangian with composite “Higgs”scalars. However, scalar theories in which there are more than one Φ4 coupling can have afirst order phase transition instead, due to the Coleman-Weinberg instability.

Therefore,top-condensate or strong ETC theories in which the composite scalars have more than oneΦ4 coupling cannot always support a large hierarchy. In particular, if the Nambu–Jona-Lasinio model solved in the large-Nc limit is a good approximation to the high-energydynamics, then these models will not produce acceptable electroweak symmetry breaking.10/92aBoston University, Department of Physics, 590 Commonwealth Avenue, Boston, MA 02215bLyman Laboratory of Physics, Harvard University, Cambridge, MA 021381sekhar@weyl.bu.edu2golden@physics.harvard.edu3simmons@physics.harvard.edu

1. IntroductionMuch recent work has focused on top quark condensate (and related) models [1]–[5]as well as models with strong extended technicolor interactions [6].

In these theories chiralsymmetry breaking driven by dynamics at a high scale (Λ ≫1 TeV) plays a significantrole in electroweak symmetry breaking. Typically, the high-energy dynamics is assumedto be a broken gauge theory – either extended technicolor (ETC) dynamics in strongETC models or the dynamics of some grand unified theory in top-condensate models.The high-energy dynamics is usually modeled by a local Nambu–Jona-Lasinio (NJL) four-fermion interaction [7] that is attractive in the chiral-symmetry-breaking channel.

Whenthe strength of the four-fermion interaction is tuned close to the critical value for chiralsymmetry breaking, it would appear possible for the high-energy dynamics to play a rolein electroweak symmetry breaking without driving the electroweak scale to be of order Λ.The argument that high energy dynamics can play a role in electroweak symmetrybreaking is independent of the NJL approximation [8]: If the coupling constants of thehigh energy theory are small, only strong low-energy dynamics (such as technicolor) cancontribute to electroweak symmetry breaking. On the other hand, if the coupling constantsof the high-energy theory are large and the interactions are attractive in the appropriatechannels, chiral symmetry will be broken by the high-energy interactions and the scaleof electroweak symmetry breaking will be of order Λ.

If the transition between these twoextremes is continuous, i.e. if the chiral symmetry breaking phase transition is second orderin the high-energy couplings, then it is possible to adjust the high energy parameters sothat the dynamics at scale Λ can contribute to electroweak symmetry breaking.

Moreover,if the transition is second order, then close to the transition the theory may be described interms of a low-energy effective Lagrangian with composite “Higgs” scalars – the Ginsburg-Landau theory of the chiral phase transition.It is crucial that the transition be second order in the high energy couplings. If thetransition is first order, then as one adjusts the high-energy couplings the scale of chiralsymmetry breaking will jump discontinuously from approximately zero at weak couplingto approximately Λ at strong coupling.

In general it will not be possible to maintain ahierarchy between the scale of electroweak symmetry breaking and scale of the high energydynamics, Λ.In this note we show that there are cases in which the transition cannot be self-consistently second order. A scalar theory in which there is more than one Φ4 coupling1

can have a first order phase transition instead, due to the Coleman-Weinberg instability[9]. Therefore, top-condensate or strong ETC theories in which the composite scalars havemore than one Φ4 coupling cannot always support a large hierarchy.2.

U(Nf) × U(Nf) modelsFor simplicity, we first consider a theory of Nf left- and right-handed fermions Ψ witha chiral U(Nf) × U(Nf) symmetry. As usual, we assume that the high-energy dynamicsis attractive in the ¯ΨΨ channel.

Therefore, the order parameter Φ of chiral symmetrybreaking transforms as an (N f, Nf) under the chiral symmetry. If it is possible to arrangefor a large hierarchy, then at energies below Λ the dynamics can be described in terms ofa Ginsburg-Landau theory for the order parameter Φ coupled to the fermionsL =Ψi/DΨ +πypNfΨLΦΨR + h.c.+ tr(∂µΦ†∂µΦ) −M 2tr(Φ†Φ)−π23λ1N 2f(trΦ†Φ)2 −π23λ2Nftr(Φ†Φ)2 + OΦ†ΦΛ2 , ∂2Λ2.

(2.1)The quantities y, M 2, λ1 and λ2 are functions of the couplings of the fundamental high-energy theory.This effective Lagrangian can be considered the theory of a compositeU(Nf) × U(Nf) “Higgs” boson Φ.At tree level, if the high-energy couplings can be chosen so that M 2 ≪Λ2, thenit is possible to establish a large hierarchy. This prediction can be affected by quantumcorrections: as shown by Coleman and Weinberg [9], if M 2 is adjusted to be close tozero, then quantum corrections can destabilize the minimum at Φ = 0.

More precisely,if one computes the renormalization-group-improved effective potential and requires thatthe second derivative of the potential at Φ = 0 be small, one finds that the potential isminimized far away from the origin. Consequently, if one adjusts the couplings in the high-energy theory so that M 2 goes through zero, one finds that the location of the effectivepotential’s absolute minimum jumps discontinuously from Φ = 0 to some large nonzerovalue of Φ.

In other words, the transition which at tree level was second order is drivenfirst order by quantum fluctuations1.1 The stability of the U(Nf) × U(Nf) linear sigma-model, without fermions, was considered in[10].2

Yamagishi [11] has shown that the condition that the effective potential be minimizedaway from the origin can be stated purely in terms of the couplings λ1(µ) and λ2(µ), byfollowing their flows from µ ≈Λ as the scale µ is decreased. We apply the results of [11]to the Lagrangian (2.1).

The effective potential is minimized away from the origin if thecouplings cross the line4(λ1 + λ2) + β1 + β2 = 0(2.2)in a region where λ2 > 0, λ1 + λ2 < 0 and4(β1 + β2) +Xi,j=1,2βi∂βj∂λi> 0 . (2.3)Here β1 and β2 are the beta functions for the couplings λ1 and λ2, respectively.

We willrefer to the line (2.2) as the “stability line”.If the couplings never cross the stability line, quantum corrections do not drive thetransition first order and the high-energy theory may self-consistently have a second ordertransition. However, if the couplings do cross the stability line, the low-energy effective the-ory has a first-order transition and therefore the high-energy theory cannot self-consistentlyhave a second order transition [12].In practice, of course, one can only compute the beta functions in perturbation theory.At one-loop the beta functions are2β1 = 112 1 + 4N 2f!λ21 + 13λ1λ2 + 14λ22 + y2Nc4Nfλ1,(2.4)andβ2 =12N 2fλ1λ2 + 16λ22 + y2Nc4Nfλ2 −3Nc8Nfy4.

(2.5)Here Nc is the number of colors or technicolors of fermions Ψ. Note that, if y is constant,the one-loop β-functions for the quantities λ1/y2 and λ2/y2 are independent of y. Theβ-functions (2.4) and (2.5) have a fixed point which is the analog of the fixed point for theHiggs self-coupling noted in [2].In these theories, typically the Yukawa coupling is drawn quickly to a low-energy“fixed point” [14] [15], where its value runs very slowly due to the running of a relatively2 The contributions from λ1 and λ2 differ from those given in [13] and [10] by a factor of1/4.

We note that in [10], the complex scalar field is incorrectly normalized and this explains thediscrepancy.3

weak gauge coupling (color or technicolor). For the purposes of illustration, therefore, weignore the running of the Yukawa coupling y.In fig.

1 we plot the renormalization group trajectories of λ1/y2 and λ2/y2 for themodel with Nf = 2 and Nc = 3.These figures show that the couplings λ1 and λ2run toward the fixed point of (2.4) and (2.5) discussed above. If λ2 > λ1 and if bothare sufficiently strong at µ = Λ, the couplings run in such a way as to intersect thestability line.

In fact, these trajectories intersect the line twice. One can check that, as onescales to the infrared (toward the fixed point), condition (2.3) is satisfied only at the firstintersection and this intersection corresponds to the minimum of the effective potential.We have numerically checked that the picture does not qualitatively change with a runningYukawa coupling or for different values of Nf and Nc.In the cases where the two λ’s start at reasonably large values, they run quickly andintersect the stability line after a small change in µ.

At one loop, the value of Φ at theminimum of the potential is equal to the value of µ at which the stability line is crossed.Therefore, if the couplings cross the stability line quickly, then ⟨Φ⟩is of order Λ and therecan be no large hierarchy.Of crucial importance, then, is what values the couplings λ1(µ) and λ2(µ) take whenµ = Λ. This is a non-perturbative problem.

In the NJL model one may show [2] that toleading order in 1/Nc, λ1(µ) →0 and λ2(µ) →∞as µ →Λ. This boundary conditionputs the U(Nf) × U(Nf) model in the region which flows rapidly toward the stability lineand therefore suggests that it is not possible to obtain a large hierarchy3.One may be concerned that we are investigating the Coleman-Weinberg phenomenonin perturbation theory, but have been forced to consider potentially large values of thecouplings λ.

However, since the phenomenon depends only on the qualitative features ofthe renormalization group flows, we do not expect that higher order effects will qualita-tively change the conclusions. This issue may be tested by simulating the model (2.1)nonperturbatively using lattice techniques.

While this has not been done in four dimen-sions, numerical simulations in three dimensions without fermions (where the lowest-orderrenormalization group analysis also predicts a first order transition [17]) confirm that thetransition is first order [18].3 These predictions will be modified in a generalized NJL model [16]. However, even in gen-eralized models λ1(µ) →0 as µ →Λ to leading order in 1/Nc.

Therefore, we expect that thetransition will still be first order if λ2(Λ) is not small.4

The point is that it is not sufficient to adjust the couplings of the high-energy theoryso that the second derivatives of the scalar potential at the origin are small. One will alsohave to adjust the theory so that, at µ ≈Λ, one is in a region of coupling constant spacewhich does not quickly flow toward the stability line.

In a spontaneously broken gaugetheory with a simple gauge group, however, having fixed the scale of symmetry breakingone can only adjust one parameter: the value of the gauge coupling at the symmetrybreaking scale.One cannot, therefore, simply assume that a large hierarchy of scalesis possible. One must check that the effective low-energy theory does not suffer from aColeman-Weinberg instability.

As we have seen the large-Nc limit of the high-energy NJLmodel places the U(Nf) × U(Nf) low-energy model in a region which has this instability.3. Other modelsWe now consider some other examples.Consider first a generic theory withoutfermions.As before, we can introduce a field Φ to represent the order parameter ofchiral symmetry breaking.

If the symmetry of the high-energy theory is such that theGinsburg-Landau theory for Φ has more than one coupling of dimension four, then, atleast in the ǫ-expansion, the only fixed point is the infrared-unstable Gaussian fixed point.One therefore expects that the couplings generally flow toward the unstable region, i.e.most trajectories are pushed away from the origin and flow toward large negative valuesof the couplings.Now consider the theory with fermions. As we have seen, there will in general beinfrared-stable fixed-points.

However, if the scalar self-couplings are large compared tothe Yukawa couplings, the coupling constant flows will (at least initially) look the same asthey did without fermions and should, therefore, still cross the stability line.Accordingly, in a model of composite scalars in which there is more than one Φ4coupling and in which the scalar self-interactions become strong at the compositenessscale Λ, the chiral phase transition may not be second order. Such a model will not alwayssustain a large hierarchy between the compositeness-scale Λ and the weak scale.In top-condensate-inspired models with two composite “Higgs” bosons [19] [20], forexample, one has five Φ4 couplings and three mass terms.

It can be argued that one hasenough freedom to adjust the three mass terms to be close to zero, but for the reasons5

discussed above the theory can still have a fluctuation-induced first-order phase transition.Again, if large-Nc arguments apply, the model will not sustain a large hierarchy4.By contrast, the standard O(4) model [1] has only one quartic coupling.In thiscase, the “stability line” is a point, and it is at a lower value of λ than the fixed point.Therefore, if, as in [2], the value of λ(Λ) is large, then the trajectory hits the fixed pointwithout crossing the stability point and it may be possible to sustain a large hierarchy.Note that our results apply only in cases in which the scalar self-interactions becomestrong at the compositeness scale. In composite-Higgs models in which all of the scalarsare Goldstone Bosons of some chiral symmetry breaking transition at a higher energy scale[21], the nonderivative self-couplings of the scalars are related to small symmetry breakingeffects and can naturally be small at µ = Λ.

Although the transition may in principle befirst order, it may take a very large change of scale before the couplings cross the stabilityline since the couplings are weak. In this case the hierarchy can be large.4.

ConclusionsIn conclusion, theories of composite “Higgs” scalars may have a first order chiralsymmetry breaking phase transition if there is more than one Φ4 coupling and if thescalar self-interactions become strong at the compositeness scale. One must check thatthe theory does not suffer from the Coleman-Weinberg instability.

In particular, in strongETC models or generalized top-condensate models with more than one Φ4 coupling in thelow-energy theory, one may not be able to adjust the high-energy theory to obtain a largehierarchy between the scale of the high-energy dynamics and the weak scale. If the NJLmodel solved in the large-Nc limit is a good approximation to the high-energy dynamics,then these models will not produce acceptable electroweak symmetry breaking.5.

AcknowledgementsWe would like to thank Andrew Cohen, Mike Dugan, Marty Einhorn, Howard Georgi,and Chris Hill for useful conversations and comments. R.S.C.

and E.H.S. thank RobertJaffe and Emil Mottola for organizing the Sante Fe Workshop on Hadrons and PhysicsBeyond the Standard Model where some of this work was completed.

R.S.C. acknowledgesthe support of an Alfred P. Sloan Foundation Fellowship, an NSF Presidential Young4 The instability was noted in [19], but its implications were not discussed.6

Investigator Award, and a DOE Outstanding Junior Investigator Award. R.S.C.

and M.G.acknowledge support from the Texas National Research Laboratory Commission under aSuperconducting Super Collider National Fellowship. This work was supported in partunder NSF contracts PHY-90-57173 and PHY-87-14654 and DOE contracts DE-AC02-89ER40509 and DE-FG02-91ER40676, and by funds from the Texas National ResearchLaboratory Commission under grants RGFY92B6 and RGFY9206.7

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Figure CaptionsFig. 1.The trajectories for the quantities λ1/y2 and λ2/y2 in the U(Nf)×U(Nf) model.The arrows indicate the behavior as one scales toward the infrared.

Here we havetaken Nf = 2 and Nc = 3. Because of the form of equations (2.4) and (2.5), thisplot is independent of y.

The “stability line” is shown in dashes. Note that thecurves that start at large λ2 and small λ1 cross the stability line twice, and thushave a Coleman-Weinberg instability.9

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