Is there a Scalar Sector? Roberto Peccei

Cornwall Symposium UCLA November 2009

Is there a Scalar Sector?

•  The Cornwall – Norton Paper •  Technicolor and its Troubles •  Difficulties with CP •  Concluding Remarks

The Cornwall - Norton Paper •  In 1973 Mike Cornwall and Dick Norton

published a seminal paper entitled: “ Spontaneous Symmetry Breaking Without

Scalar Mesons” [Phys. Rev. D8 (1973) 3338] which showed that one could construct a gauge theory with spontaneous symmetry breaking without introducing elementary scalar fields

•  Similar ideas were put forth in parallel by Roman Jackiw and Ken Johnson [Phys. Rev. D8 (1973) 2386]

•  These papers provided the theoretical underpinning for dynamical models of electroweak symmetry breaking, known generically as Technicolor models, introduced by Weinberg and Susskind in the late 1970s

•  Although realistic Technicolor models are difficult to construct, experimentally the possibility remains open that the electroweak theory breakdown is generated dynamically rather than as the result of the non-vanishing vacuum expectation value of a scalar field- the Higgs mechanism

•  Nevertheless, avoiding a scalar sector is hard to do!

•  Before discussing the reasons why it is difficult to avoid a scalar sector, I want to briefly review the Cornwall-Norton argument.

•  They considered a theory with 2 degenerate fermions and two gauge fields. One of the gauge fields couples diagonally to the fermions, but the other off-diagonally. Taking ψ≡

one has: Lint = g ψγµψAµ + g’ ψγµτ2ψBµ

which is invariant under two U(1) symmetries ψ→ e iα ψ and ψ→ e iτ2β ψ

•  If the dynamics of the theory were to break the off-diagonal U(1) symmetry spontaneously then one would expect:

i.  The fermion mass degeneracy to lift m1≠m2 ii.  The gauge field Bµ to acquire a mass MB≠0 •  Cornwall and Norton addressed this

possibility by looking at the Schwinger-Dyson equations for the model.

•  The fermion self energy has two pieces, one which does not and one which does distinguish among ψ1,2:

Σ = Σs + τ3 Σv

Clear Σv obeys a homogeneous SD equation

•  To O(g2, g’2) one finds

•  There is symmetry breakdown if this equation admits a non-vanishing solution for Σv..

•  Cornwall and Norton found a self consistent solution, in which Σv. asymptotically behaves as:

Σv (p) ~ δm (p2/m2) -ε with ε>0 p2→∞ where δm is arbitrary and ε obeys the constraint ε(1- ε) = [3/16π2] (g2- g’2)

•  Cornwall and Norton were also then able to compute the dynamically induced mass for the Bµ gauge field.

•  They did this computation in the weak coupling limit g2 → 0, g’2→ 0 [ with ε = [3/16π2] (g2- g’2) → 0 simultaneously, to respect the dynamical constraint]

•  Their result, as expected, relates the dynamically generated mass MB to the mass difference between the fermions δm, as this is the only mass parameter generated in the theory:

MB 2

= [8 g’2 / 3(g2- g’2)] δm2

Technicolor and its Troubles •  In the late 70s, Weinberg and Susskind

recognized that the dynamical symmetry breaking mechanism identified by Cornwall, Norton, Jackiw and Johnson was naturally provided by the non-vanishing QCD quark condensates <q¯q>~ ΛQCD

3 for the SU(2)XU(1) electroweak theory, except that their scale was too small since ΛQCD ~ GeV

•  Susskind then proposed the existence of a QCD-like theory - Technicolor- with dynamical fermionic condensates <T¯T>~ (TeV)3 which cause the electroweak theory to break down dynamically, giving MW~ 100 GeV

•  In Technicolor, just as in QCD, a custodial SU(2) symmetry, the analog of isospin, exists and serves to guarantee that in the electroweak theory the dynamically generated gauge boson masses obey the, well satisfied, relation:

MW2=MZ

2cos2θW

where θW is the electroweak mixing angle •  Although Technicolor provides an attractive

alternative to introducing a scalar Higgs sector to break down the SU(2)XU(1) electroweak theory, present day data does not seem to favor this alternative

•  All electroweak data is consistent with the breakdown of the standard model being effected by a relatively light Higgs boson

[Figure] •  Since the Higgs self coupling is given by λ=MH

2/2v2 with v= (√2GF) -1/2 ≈ 250 GeV, it is clear that

electroweak data is consistent with a weakly coupled symmetry breaking sector

•  This result suggests that, at high energy E>> MW , the scattering of longitudinally polarized gauge bosons itself will be weak:

AWLWL << 1

•  This result follows from a famous theorem established by Cornwall, Levin and Tiktopolous [Phys. Rev. D10 (1974) 1145] which relates, for E>> MW , AWLWLto the corresponding Goldstone boson scattering amplitude Aww:

AWLWL= Aww + O( MW/E) ~ λ [Higgs case] •  In Technicolor, which is a strong coupling

theory like QCD, one would expect AWLWLto be large rather than small.

•  However, this is just an indication against Technicolor not a proof that the electroweak theory is not broken down dynamically

•  Stronger indications against Technicolor can be garnered by looking at the modifications that the symmetry breaking sector induces to the gauge propagators [oblique corrections]

•  Expanding the vector boson self energies as: Π (q2)=Π(0) + q2Π`(0) +… it turns out that the combination S = -4π[Π`VV(0) - Π`AA(0)] = ∫o∞ ds/s [v(s)-a(s)] is particularly sensitive to the symmetry

breaking sector. •  An analysis of all electroweak data by Erler and

Langacker gives S= -0.1 ± 0.1

•  In QCD, with just u and d quarks, one can estimate the spectral function integral in S by saturating it with ρ and A1 poles, obtaining SQCD ≈ 0.3

•  If Technicolor is QCD-like one expects a similar result, modulo techniflavor and technicolor factors :

STechni ≈ SQCD n N/3 ≈ 1 •  Electroweak data disfavors a QCD-like

Technicolor theory. But, more complicated theories [Walking Technicolor ] where αWT evolves slowly can give SWT << STechni

•  However, there are other problems!

Difficulties with CP •  In a theory without scalar fields, it is difficult

to generate CP violating phenomena •  In 4 dimensions, a theory involving only

fermions and gauge fields is CP-conserving at the Lagrangian level, up to θ-terms, since there are no natural complex structures in the theory:

gi real; Aaµ Adjoint

•  Furthermore, θ-terms do not give the kind of CP violation that is observed in nature

Although the topological nature of the non-Abelian gauge theory vacuum allows the presence of CP-violating θ-terms, these terms are flavor neutral and troublesome does not contribute because θW 0

θS< 10-10 since [edm]n< 6.3 x 10-26ecm

Cannot see how any θ-term from a Technicolor theory will contribute to the observed flavor-violating CP phenomena in K and B decays

•  Nevertheless, it is possible to generate flavor changing CP violation in a Technicolor theory, as first discussed by Eichten, Lane and Peskin, because the Technicolor condensates <T¯T> can be complex

•  The possibility of having complex condensates and, therefore, spontaneous breaking of CP was suggested originally by Dashen in 1971 in theories where global symmetries are both spontaneously G→H and explicitly Ħ broken

•  The correct ground state follows by minimizing an effective potential, and the minimum can occur for complex values of the condensates

•  In Technicolor theories to generate any flavor violating interactions one must introduce a further gauge theory (ETC) which becomes strong at higher scales than Technicolor and connects fermions and technifermions

•  ETC interactions at low energy produce effective residual terms of the form

which, when SU(2)xU(1) breaks down, generate mass matrices for the quarks, and which serve to break remaining global symm.

•  The correct ground state of this theory, following Dashen, arises from minimizing the effective potential [G → H; g ⊂ G]

V(g)= - H<Ω|U(g)-1 Lresidual U(g) |Ω>H and generally gives complex condensates

and, hence, complex quark mass matrices •  Unfortunately, as Eichten, Lane and Preskill

showed, besides the ordinary CKM CP violation coming from the complex mass matrices, the aligned residual interactions can also give rise to flavor diagonal CP violation, leading to a too large neutron edm

•  Because the flavor aspects of Technicolor theories are complicated and quite model dependent, perhaps one can ignore these potential troubles

•  However, in my view, one really cannot ignore a more general problem arising from having CP being spontaneously violated by complex condensates

•  As Zeldovich, Kobzarev and Okun pointed out in 1974, spontaneous CP violation gives rise to different CP domains in the Universe and these in general spell trouble

•  The different CP domains in the Universe are separated by walls that have substantial energy density

•  Furthermore, being 2-dimensional, the walls dissipate slowly as the Universe cools

•  Roughly speaking, the surface energy density in the walls is of order

σ ~ <T¯T>~ (TeV)3 thus, if they existed, the walls would contribute

an enormous amount to the energy density of the universe now:

ρwall ~ σT ~ 10-7 GeV4 >> ρclosure ~ 10-46 GeV4

•  Unless one can find a mechanism to rapidly destroying the domain walls, one cannot countenance a theory with spontaneous CP violation below T~Tinflation

•  Theories where CP is violated at high scales exist [Nelson, Barr], but they are recondite and difficult to reconcile with experiment (which agrees perfectly with the CKM theory [Figure])

•  Typically, the CP violating phases generated at high scales induce small phases at low energy:

γeff ~ [Mw/MHS]n γ while γCKM ~ O(1)

Concluding Remarks

•  I have discussed some theoretical and phenomenological constraints which argue against the beautiful idea of Cornwall, Norton, Jackiw and Johnson that the electroweak theory is broken dynamically

- electroweak data is consistent with having a light weakly coupled Higgs boson

- CP violation is consistent with originating from explicit phases in the mass matrices coming from scalar Yukawa couplings

•  However, ultimately, we will know the truth only from experiment and not from theoretical arguments

•  Fortunately, we are at the threshold of the turn-on of the LHC, a machine which should really elucidate the nature of electroweak symmetry breaking

•  So, the only thing left to do is to wait… and wish Mike a

Happy Birthday!