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Search for Dynamical Symmetry Breaking Physics by Using Top Quark

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Search for Dynamical Symmetry Breaking Physics by Using Top Quark April 1995 Search for Dynamical Symmetry Breaking Physics by Using Top Quark T.Asaka, N.Maekawa, T.Moroi, Y.Shobuda, and Y.Sumino Department of Physics, Tohoku University Sendai, 980 Japan Abstract We report first results from the investigation on. probing dynamical mech- anism of the electroweak symmetry breaking using top quark. We consider the case where the top quark mass originates from a fermion-antifermion pair condensate, which necessitates (i)strong interaction to cause condensation, and (ii)4-fermi interaction to give top mass. From the observed top mass and the unitarity constraint, we obtain, for the 4-fermi interaction (ii), lower bounds for its strength and upper bounds for its intrinsic new mass scale as we vary the type of the strong interaction (i). Bethe-Salpeter and Schwinger-Dyson equations are solved numerically to study the dynamical symmetry breaking effect semi-quantitatively. — 108 1. Introduction The SU(2) x U(l) gauge theory for describing elcctroweak interactions are very suc- cessful both theoretically and experimentally. However all the experimental tests has been done only for its gauge part. We know that weak bosons and various fermions have their own masses. Because these mass terms are forbidden by gauge symmetry, a mechanism of symmetry breaking is needed, but we have few knowledge about this mechanism so far. Two possibilities have been considered for a symmetry breaking mechanism. In the so- called standard model (SM), symmetry breaking sector consists of an SU(2) gauge doublet of elementary scalar field (Higgs field). By the non-zero vacuum expectation value of this field, gauge symmetry is broken. On the other hand, in models such as technicolor (TC) model [1], new fermions are introduced for its symmetry breaking sector and their condensate breaks symmetry dynamically. Here we consider the latter possibility and intend to search for dynamical symmetry breaking physics by using top quark. Why do we use top quark? Top quark mass is much heavier than other fermions and is of the same order value as the electroweak symmetry breaking scale [2]. When gauge symmetry is dynamically broken by a fermion-antifermion pair, two things are needed to explain fermion masses. First, we need a strong attractive force to form condensate. Second, 4-fermi interaction, which corresponds to the fermion-Higgs Yukawa interaction in the standard model, are needed. Top quark, for example, acquires its mass through a 4-fermi interaction, C = -^HmLtR + h.c. , (1) where M is the new physics scale and \& denotes a new fermion which is introduced in the symmetry breaking sector. From the fermion condensate (XP\I/), top gets mass as This condensate (ty$\ also gives mass to the gauge bosons, because the condensate has a non-zero SU(2) charge. Therefore in the TC-like model, we expect a following inequality: 1/3 < v ~ 0(250 GeV), (3) where v represents the electroweak symmetry breaking scale. The reason we have inequality rather than equality is because there may exist a condensate, which does not give mass to the top quark while giving masses to the gauge bosons, other than this /\&\I/\. From the 109 — observed value of mt a 175 GeV [2], the lower bound for the strength of 4-fermi interaction (~ 1/M2 ) in eq.(l) is obtained via eqs.(2) and (3). We also get the lower bound for the non-standard effect to e.g. Ztt vertex as shown in fig.l. Similar argument holds for other fermions, but the most stringent lower bound tr 4-fermi interaction Figure 1: The Feynman diagram for the non-standard correction from the dynamical sym- metry breaking to the Ztt vertex. for the non-standard gauge-fermion coupling is posed in the case of top quark. This is the reason why we use top quark in searching for a mechanism of dynamical symmetry breaking. To study the dynamical symmetry breaking effect, we solve numerically the Schwinger- Dyson (SD) and Bethe-Salpeter (BS) equations. For the QCD case, taking /„. as the only input parameter, AQCDX^^), Mp,Mai,Ma0,fp, and fai have been calculated by this approach, which meets the experimental value within 20 ~ 30 % accuracy; see refs.[4, 5]. Therefore we expect to calculate the oblique corrections, the Ztt vertex, etc., semi- quantitatively. In this note, we report, as our first results, the numerical estimation of 4-fermi in- teraction which gives rise to the top mass. From the experimental value of mt and the unitarity constraint, we obtain lower bounds for the strength of the 4-fermi interaction and upper bounds for its intrinsic new mass scale by considering various types of the strong interaction which is needed to cause a fermion-antifermion condensate. 2. Numerical Estimation of the Four-Fermi Interaction Now we consider that top quark mass comes dynamically from a fermion-antifermion pair condensate. In this case, a strong interaction which causes the condensate and a 4-fermi interaction which gives a top mass are required. Here we show the theoretical — 110 — estimation of this 4-fermi interaction using the observed top quark mass and the unitarity constraint. Model First, we explain the model we consider here for the symmetry breaking sector. We consider the one-doublet TC-like model in which the SU(NTC) gauge interaction cause fermion condensation. New fermions which are fundamental representation of SU(NTC) are introduced as QL = I „ I , UR, DR, (4) with the weak hypercharge, Y(QL) = 0, Y(UR) = \, Y(DR) = -\ , (5) where these quantum numbers are chosen to cancel the anomaly. JVd sets of above new fermions are introduced in the breaking sector. We also introduce the kernel K which characterizes the strong interaction between \& and *, where ty denotes the above fermion U or D. Now in the case where the SU(NTc) gauge interaction cause the condensation, we can approximately write down the explicit form of the kernel K. In the improved ladder approximation in Landau gauge [3], the kernel K in the momentum space is written as [4] (the momentum configuration is defined as fig.2 ) <P~k) 2 where C2 is the second Casimir invariant of the SU(N) fundamental representation and the running coupling constant g(p, k) is used. Figure 2: The Feynman diagram for the kernel K(p, k). The helix denotes the techni-gluon propagator. Ill — Once the kernel is fixed as in eq.(6), we can write down the SD eq. and BS eq. which are the self-consistent equations for the full 2-point and 4-point Green functions of $, respectively. In fig.3, these equations are shown diagrammatically. By solving these equa- tions numerically following ref.[4], we obtain the mass function of \&, S(p), and the decay constant, F^. SDeq. BSeq. Figure 3: The graphical representations of the SD eq. and the BS eq. The line with a blob denotes the full propagetor of ty. Top Quark Mass In order to give top quark mass, we also introduce a 4-fermi interaction j^ h.c., (7) where qi denotes the ordinary quark weak doublet and G is the dimension-less coupling. In this 4-fermi interaction, M denotes a new physics scale. Because 4-fermi interaction cannot be a fundamental interaction, there should be a scale M above which this interaction will resolve. Although above this scale there will exist some fundamental theory (such as extended technicolor model[6]) which induce the 4-fermi interaction, we will leave the whole theory and consider only this 4-fermi interaction below the scale M. Now top quark acquires mass through the 4-fermi interaction (7) as shown in fig.4. Using the full propagator of "[/" which is a solution to the SD equation W, top mass can be calculated as [*] In fact, the SD eq. shown in fig.3 is incomplete. Because of the 4-fermi interaction (7), we should consider the mixing of SD eqs. of top quark and "[/", but here we neglect this effect. — 112 — Figure 4: The diagram for the top quark mass. where with x — p% = — p2. Here the new physics scale M is defined as the upper bound of the integral. In order to set the mass scale of the theory, we use the decay constant F* which is obtained by solving the BS eq. of "£/" and "Z?". From the gauge bosons masses, the Fv is normalized as, Mw > ^, (11) where g is the SU(2)L gauge couping constant. Here we have inequality rather than equality, because there may be fermion condensates other than \UU\ and \DD\ , which do not contribute to the top mass while giving masses to the gauge bosons. By substituting the observed top mass mt — 175 GeV [2], we obtain the coupling G for a given M. We consider the case where the strong interaction is given by the SU(2) and SU(3) gauge interaction, and show the G~M contours in figs.5 and 6, respectively. In each figure, the upper region of the lines are allowed because of the inequality in eq.(ll). The behavior of the coupling G can be understood by the momentum dependence of the mass function Y,(x). The E(x) is almost flat for x < A^c, while decreasing rapidly for x 3> A^c as S(z) ~ -(lnx)^"1 , (12) x where B represents the coupling flow and, using the lowest order coefficient of the /? function (/?o), B = /?o/12C2. Now Arc is defined as the scale where the leading-logarithmic running — 113 — coupling constant diverges. Prom this momentum dependence, G behaves as, 2 2 G ~ ~ (for M «A TC), (13) M2 G "onus (fMM2»A->- (14» 2 For M «C Aj-C, the coupling G becomes larger as M —> 0, because the upper bound of the integral is getting smaller.
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