We argue that extended technicolor augmented with topcolor requires that all
mixing between the third and the first two quark generations resides in the
mixing matrix of left-handed down quarks. Then, the anti-B_d--B_d mixing that
occurs in topcolor models constrains the coloron and Z' boson masses to be
greater than about 5 TeV. This implies fine tuning of the topcolor couplings to
better than 1%.Comment: 9 pages, LaTeX, with one postscript figure. Version to appear in
  Physics Letters B. Minor typos corrected and note added on top-pion
  contribution to Z to b b-bar deca

Alavi-Harati

Bertram

Bosch

Buchalla

Buras

Burdman

Bönisch

Caravaglios

Chivukula

Dashen

Dimopoulos

Eichten

Fanti

Gustavo Burdman

Hill

Kenneth Lane

Kominis

Lane

Lindner

Martin

Nuyts

Popovic

Susskind

Tonguç Rador

Weinberg

Physics Letters B

English

arXiv

We argue that extended technicolor augmented with topcolor requires that all mixing between the third and the first two quark generations resides in the mixing matrix of left-handed down quarks. Then, the anti-B_d--B_d mixing that occurs in topcolor models constrains the coloron and Z' boson masses to be greater than about 5 TeV. This implies fine tuning of the topcolor couplings to better than 1percent

Burdman, Gustavo

Lane, Kenneth

Rador, Tonguc

UNT (University of North Texas) Digital Library

    LBNL-49152         Anti-B–B Mixing Constrains Topcolor–Assisted Technicolor  Gustavo Burdman, Kenneth Lane, and Tonguç Rador  Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215                        This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.  DISCLAIMER  This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California. arXiv:hep-ph/0012073v3  26 Jun 2001BUHEP-00-25hep-ph/0012073B̄–B Mixing Constrains Topcolor–AssistedTechnicolorGustavo Burdman∗, Kenneth Lane†, and Tonguç Rador‡Department of Physics, Boston University,590 Commonwealth Avenue, Boston, MA 02215February 1, 2008AbstractWe argue that extended technicolor augmented with topcolor requires that allmixing between the third and the first two quark generations resides in the mixingmatrix of left–handed down quarks. Then, the B̄d–Bd mixing that occurs in topcolormodels constrains the coloron and Z ′ boson masses to be greater than about 5 TeV.This implies fine tuning of the topcolor couplings to better than 1%.∗burdman@bu.edu†lane@physics.bu.edu‡rador@buphy.bu.edu1The impressive agreement of the standard model’s predictions with experimental datadoes not lessen the need for new physics to explain the dynamics underlying electroweakand flavor symmetry breaking. This physics may manifest itself not only in high energycollider experiments but also in precision low energy measurements of meson decays andmixing. In turn, low energy measurements powerfully constrain flavor physics scenarios.A prime example is technicolor [1] with extended technicolor (ETC) [2,3], a natural,dynamical scheme for electroweak and flavor symmetry breaking. There, |∆S| = 2 effectsin the neutral kaon system require ETC gauge boson masses of 102–104 TeV and walkingtechnicolor to produce the correct first and second generation quark masses. With suchlarge masses, ETC by itself cannot account for the top quark’s mass. Therefore, we takeit to be augmented by topcolor, a system referred to as topcolor–assisted technicolor(TC2) [4,5,6].All TC2 models assume that color SU(3)C and weak hypercharge U(1)Y arise from thebreakdown of the topcolor groups SU(3)1 ⊗ SU(3)2 and U(1)1 ⊗ U(1)2 to their diagonalsubgroups. Here SU(3)1 and U(1)1 are strongly–coupled, SU(3)2 and U(1)2 are weakly–coupled, with the color and weak hypercharge couplings given by gC = g1g2/√g21 + g22 ≡g1g2/gV8 ≃ g2 and gY = g′1g′2/√g′ 21 + g′ 22 ≡ g′1g′2/gZ′ ≃ g′2. Top and bottom quarksare SU(3)1 triplets. The broken topcolor interactions are mediated by a color octet ofcolorons, V8, and a color singlet Z′ boson, respectively. By virtue of the different U(1)1couplings of tR and bR, V8 and Z′ exchange between third generation quarks generates alarge contribution m̂t(1 TeV) ≃ 160 GeV to the top mass, but none to the bottom mass.If topcolor is to provide a natural explanation of m̂t, the V8 and Z′ masses ought tobe O(1 TeV). In the Nambu–Jona-Lasinio (NJL) approximation—which we rely heavilyupon here—the degree to which this naturalness criterion is met is quantified by theratio [7]α(V8) + α(Z′) − (α∗(V8) + α∗(Z ′))α∗(V8) + α∗(Z ′)=α(V8) rV8 + α(Z′) rZ′α(V8)(1 − rV8) + α(Z′)(1 − rZ′). (1)Here,α(V8) =4αV8 cos4 θC3π, α(Z ′) =αZ′YtLYtR cos4 θYπ;tan θC =g2g1, tan θY =g′2g′1, ri =m̂2tM2iln(M2im̂2t), (i = V8, Z′) ; (2)and YtL,R are the U(1)1 charges of tL,R. The NJL condition on the critical couplings fortop condensation is α∗(V8) + α∗(Z ′) = 1. In this letter, we show that, for such largecouplings, TC2 is tightly constrained by the magnitude of B̄d–Bd mixing: it requiresMV8 ≃ M′Z>∼ 5 TeV [8]. This implies that the topcolor coupling α(V8) + α(Z′) must bewithin less than 1% of its critical value, a tuning we regard as unnaturally fine.2There are two variants of TC2: The “standard” version [5], in which only the thirdgeneration quarks are SU(3)1 triplets, and the “flavor–universal” version [9] in which allquarks are SU(3)1 triplets. In standard TC2, V8 and Z′ exchange gives rise to flavor–changing neutral currents (FCNC) that mediate |∆B| = 2. In flavor–universal TC2, onlyZ ′ exchange can generate such FCNC. Our results constrain both types of TC2 theory.The coloron interaction at energies well below MV8 isHV8 =g2V82M2V88∑A=1JAµJAµ , (3)where the coloron current written in terms of electroweak eigenstate (primed) fields isgiven byJAµ = cos2 θC∑i=t,bq̄′iγµλA2q′i − sin2 θC∑i=u,d,c,sq̄′iγµλA2q′i . (4)The dominant coloron interactions for |∆B| = 2 come from b̄′b′b̄′b′ terms in Eq. (3). Whenwritten in terms of mass eigenstate fields, they are (neglecting smaller terms proportionalto αC alone):HV8 =2παC cot2 θCM2V8∑λ1,λ2=L,R(D∗λ1bbDλ1bdi D∗λ2bbDλ2bdi b̄λ1γµ λA2diλ1 b̄λ2γµλA2diλ2 + h.c.)(5)Here, di = d or s, and the unitary matrices DL,R arise from vacuum alignment in thedown–quark sector. We discuss them shortly.In order that Z ′ exchange not induce large |∆S| = 2 transitions, quarks of the twolight (electroweak basis) generations must have the same U(1)1 charge. Then, the Z′interaction for |∆B| = 2 isHZ′ =2παY cot2 θYM2Z′∑λ1,λ2=L,R(D∗λ1bbDλ1bdi D∗λ2bbDλ2bdi ∆Yλ1∆Yλ2 b̄λ1γµ diλ1 b̄λ2γµ diλ2+h.c.)(6)Here, ∆Yλ = Ybλ − Ydλ is the difference of U(1)1 charges.Vacuum alignment in the technifermion sector leads to unitary matrices W = (W U , W D)which represent the mismatch between the directions of spontaneous and explicit breakingof technifermion chiral symmetries. A common feature of these matrices is that all theirphases are rational multiples of π. That is, for N technifermion doublets, these phasesmay be integral multiples of π/N ′ for various N ′ from 1 to N [10]. Extended technicolorcouples quarks to technifermions and generates the “primordial” quark mass matricesMqij = ΛqTiIJj WTIJ ∆T (except for the m̂t part of Mu tt). Here (q, T ) = (u, U) or (d, D);i, j = u, c, t or d, s, b; I, J label the technifermion flavors. The ΛqTiIJj are real ETC couplingsof order (100− 1000 TeV)−2 and ∆T is the real technifermion condensate renormalized at3the ETC breaking scale. If the ΛqTiIJj properly image WT ’s rational phases onto Mq, therewill be no strong CP violation, i.e., θ̄q = arg det(Mu) + arg det(Md) <∼ O(10−10). In thisimaging, all elements of Mu and Md have (generally different) rational phases that addto zero in the determinant. We assume this can happen in ETC models.Vacuum alignment in the quark sector [11] is achieved by minimizing the quark energyEq(U, D) = −Tr(MuU† + MdD† + h.c.). The up and down–quark alignment matricesU = ULU†R and D = DLD†R are 3× 3 diagonal blocks in the (SU(6)L ⊗ SU(6)R)/SU(6)Vmatrices.∗ The ETC and TC2 interactions restrict the texture of the Mu and Md. This,in turn, determines the form of UL,R and DL,R and, ultimately, of the TC2 amplitude forB̄d–Bd mixing.Flavor–changing neutral current limits imply that ETC contributions to Mu,d are atmost a few GeV, just enough to produce mb(METC). The TC2 interactions generate m̂t,but off-diagonal elements in the third row and column of Mu come from ETC. Theyare expected to be no larger than the 0.01–1.0 GeV associated with mu and mc. Thus,Mu is very nearly block–diagonal and, so, |UL,R tui |∼= |UL,R uit|∼= δtui . Limits on B̄d–Bdmixing induced by exchange of “bottom pions” [13,14], together with the need to generateappropriate intergenerational mixing in the Cabibbo–Kobayashi–Maskawa (CKM) matrixV = U †LDL, require that dR, sR ↔ bL elements of Md are much smaller than the dL, sL ↔bR elements [6]. This makes DR nearly 2 × 2 times 1 × 1 block–diagonal.Since DL contains almost all the mixing between b and d, s and, hence, between thethird generation and the first two, the mixing pattern of V = U †LDL is essentially thesame as that of DL. To an excellent approximation,|Vtdi | = |U∗LttDLbdi | = |DLbdi| ;V ∗tbVtdi = ULttD∗LbbU∗LttDLbdi = D∗LbbDLbdi . (7)Our strongest limit on the V8 and Z′ masses will arise from Eq. (7).The dominant TC2 contribution to |∆Bd| = 2 is given by the LL terms in HV8 +HZ′. The relevant observable is the B0L–B0S mass difference. Since Γ12 ≪ M12, it isgiven by ∆MBd = 2|M12| [16]. Fierzing HV8 into a product of color–singlet currents andcalculating the B̄d–Bd matrix element in the usual vacuum–insertion approximation, theTC2 contribution to M12 is2(M12)TC2 =4π3[αC cot2 θC3M2V8+αY cot2 θY (∆YL)2M2Z′]ηBMBdf2BdBBd(D∗LbbDLbd)2 . (8)∗Actually, quark vacuum alignment is based on first–order chiral perturbation theory, so it is in-applicable to the heavy quarks c, b, t. When θ̄q = 0, Dashen’s procedure is equivalent to making themass matrices diagonal, real, and positive [12]. Thus, it correctly determines the quark unitary matricesUL,R, DL,R and the magnitude of strong and weak CP violation.4Here, ηB = 0.55 ± 0.01 is a QCD radiative correction factor for the LL product of color–singlet currents. We take fBd√BBd = (200 ± 40)MeV [16], where fBd and BBd are,respectively, the Bd–meson decay constant and bag parameter. This TC2 contribution isto be added to the standard model one,2(M12)SM =G2F6π2ηBMBdf2BdM2W S0(xt)(V∗tbVtd)2 , (9)where the top–quark loop function S0(xt) ∼= 2.3 for xt = m2t (mt)/M2W and mt(mt) =167 GeV.To determine the ∆MBd–restriction on TC2, we adopt the ETC–based analysis ofquark mixing matrices made above. Then, D∗LbbDLbd = V∗tbVtd∼= Vtd, so that the standardmodel and TC2 contributions add coherently. Next, to simplify our discussion, we ignorefor now the Z ′ contribution to ∆MBd and m̂t. The NJL approximation then impliescot2 θC = 3π/4αC(1 TeV) >∼ 25 for α(V8) ≥ α∗(V8) = 1 and αC(1 TeV) ≃ 0.093. Using∆MBd = (3.11±0.11)×10−13 GeV [15], we plot in Fig. 1 the constraint in the (MV8 , |Vtd|)plane. The width of the allowed band comes mostly from the error in fBd√BBd .Figure 1: The allowed region in the (MV8 , |Vtd|) plane. The upper horizontal line comesfrom the lower limit on |Vtd| derived by making use of ǫ plus the CKM unitarity triangle.The lower line is the lowest value of |Vtd| for which Im(V∗tsVtd) and, therefore, ǫ′/ǫ are nottoo small.To set a bound on MV8 , we must find the lowest possible value of |Vtd|. If ETCcontributions to the kaon CP–violating parameter ǫ are negligible, this lowest |Vtd| can beread off the (ρ̄, η̄) plane fits using ǫ and |Vub/Vcb|. (Data on ∆MBd,s are not used as these5quantities are affected by TC2.) The most recent fit [17] gives ρ̄ < 0.40 and η̄ > 0.20 atthe 95% C.L. This yields|Vtd| = λ|Vcb|√(1 − ρ̄)2 + η̄2 > 5 × 10−3 , (10)where we used the 2σ lower limit value |Vcb| = 0.036. This bound is displayed as theupper horizontal line in Fig. 1. Its intersection with the left edge of the band gives theconservative lower limit for MV8 ,MV8 ≥ 4.8 TeV ⇒α(V8) − α∗(V8)α∗(V8)≤ 0.0075 , (11)The tuning of α(V8) is unnaturally fine.If ETC contributes significantly to ǫ, we must remove its constraint from the standardmodel (ρ̄, η̄) plane. The remaining limitation on |Vtd|, other than unitarity of V , comesfrom ǫ′/ǫ. We expect that ǫ′ is unaffected by ETC [18]. In the standard–model Wolfensteinparameterization [15], it is given byǫ′ǫ≡ Im(V ∗tsVtd)S = A2λ5η̄S . (12)Here, A = |Vcb|/λ2 = 0.83 is obtained using |Vcb| = 0.04. Also, S = P(1/2) − P (3/2) whereP (∆I) contains the hadronic matrix elements in the |∆I| = 1/2 and 3/2 amplitudes [19].In Eq. (12), ǫ is taken from experiment, so that potential ETC contributions to it are notan issue. From Ref. [19], hadronic matrix element calculations imply S < 21.6. The worldaverage is ǫ′/ǫ = (19.3 ± 2.4) × 10−4 [20]. The experimental situation is still somewhatunsettled; we conservatively assume the lower limit ǫ′/ǫ > 12×10−4. This gives η̄ > 0.156.Using this, and the 95% C.L. upper limit |Vub/Vcb| < 0.14, we find ρ̄ < 0.60 and|Vtd| > 3.4 × 10−3 . (13)Intersecting this value with the allowed band in Fig. 1 yieldsMV8 ≥ 3.1 TeV ⇒α(V8) − α∗(V8)α∗(V8)≤ 0.016 (14)Although somewhat a matter of taste, we regard this level of fine tuning at the edge ofacceptability for a dynamical theory with naturalness as a design goal.To estimate the effect of the Z ′ on this naturalness criterion, we suppose that V8and Z ′ exchange contribute equally to generating m̂t, i.e., that α∗(V8) = α∗(Z ′) = 12inthe NJL approximation. We also assume (∆YL)2 ≃ YtLYtR. Then, cot2 θC ≥ cot2 θ∗C =3π/8αC(1 TeV) = 12.5 andαC cot2 θC3M2V8+αY cot2 θY (∆YL)2M2Z′>∼π8(1M2V8+4M2Z′). (15)6Equating the right–hand side to its maximum value π/(4(4.8 TeV)2) obtained when weneglected Z ′ givesMZ′ = 2[2(4.8 TeV)2−1M2V8]−12> 6.8 TeV ⇒α(Z ′) − α∗(Z ′)α∗(Z ′)< 0.0042 (16)when MV8 → ∞. For MV8 = 4.8 TeV, we have MZ′ = 9.6 TeV andα(V8) rV8 + α(Z′) rZ′α(V8)(1 − rV8) + α(Z′)(1 − rZ′)≃ 0.0049 . (17)In short, adding the Z ′ contribution to ∆MBd does not make TC2 more natural.It is possible to avoid this naturalness problem if, contrary to ETC expectations, allmixing between the third generation and the two light ones is contained in the up–sectormatrices UL,R. Even this possibility is constrained by V8 and Z′–exchange contributionsto the neutron electric dipole moment, dn. They affect only the up–quark moment. TheV8 contribution suffices; it is(du)TC2 =2e3αC cot2 θC4π43mt(mt)M2V8Im(U∗LtuULttU∗RttURtu) . (18)The limit [15] dn < 0.63 × 10−25 e–cm yields †MV8 >∼ 1 TeV√|Im(U∗LtuULttU∗RttURtu)|10−7. (19)Since ULtu =∑di DLbdiV∗udi∼= DLbbV∗ub implies |ULtu| >∼ 2 × 10−3, a natural V8 mass of a1–2 TeV suggests |URtu| <∼ 5 × 10−5. Thus, the analog of the Kominis constraint [13,14]applies to UR.In summary, generation of light quark masses via extended technicolor strongly sug-gests that all quark mixing occurs in the left–handed down sector. Then, the TC2 mecha-nism for mt requires, in the NJL approximation, V8 and Z′ masses exceeding 5 TeV. This,in turn, needs fine–tuning of the topcolor couplings to within less than 1% of their criticalvalues. We do not know how this difficulty will be resolved.Note added in proof: The contribution to Z0 → bb̄ from top–pions, the pseudoGold-stone bosons arising from top condensation, also significantly constrains TC2 [21]. Thiscontribution can be made consistent with experiment by increasing the top–pion decayconstant and/or its mass. Increasing either requires raising the TC2 scale embodied inMV8 and MZ′ . In the NJL approximation, we estimate that this implies a fine–tuning ofabout 1%, comparable to what we found above from ∆MBd .†Im(U∗LtuULttU∗RttURtu) = 0 in ETC, but we cannot appeal to that here.7We have benefitted from conversations with S. Chivukula, L. Giusti, C. Hill, andE. Simmons. G.B. thanks Lawrence Berkeley National Laboratory for its hospitality dur-ing the latter stages of this work. This research was supported in part by the U. S. De-partment of Energy under Grant No. DE–FG02–91ER40676.References[1] S. Weinberg, Phys. Rev. D19, 1277 (1979);L. Susskind, Phys. Rev. D20, 2619 (1979).[2] E. Eichten and K. Lane, Phys. Lett. B90, 125 (1980;S. Dimopoulos and L. Susskind, Nucl. Phys. B155, 237 (1979)[3] For a review of modern technicolor, see K. Lane, Technicolor 2000, Lectures atthe LNF Spring School in Nuclear, Subnuclear and Astroparticle Physics, Frascati(Rome), Italy, May 15–20, 2000; hep-ph/0007304.[4] C. T. Hill, Phys. Lett. B266, 419 (1991);S. P. Martin, Phys. Rev. D45, 4283 (1992);ibid D46, 2197 (1992); Nucl. Phys. B398, 359 (1993);M. Lindner and D. Ross, Nucl. Phys. B370, 30 (1992);R. Bönisch, Phys. Lett. B268, 394 (1991);C. T. Hill, D. Kennedy, T. Onogi, H. L. Yu, Phys. Rev. D47, 2940 (1993).[5] C. T. Hill, Phys. Lett. B345, 483 (1995).[6] K. Lane and E. Eichten, Phys. Lett. B352, 382 (1995);K. Lane, Phys. Rev. D54, 2204 (1996), hep-ph/9602221;K. Lane, Proceedings of the 1996 Workshop on Strongly Coupled GaugeTheories, p72, Nagoya, Japan, (November 1996), hep-ph/9703233;K. Lane, Phys. Lett. B433, 96 (1998), hep-ph/9805254.[7] R. S. Chivukula, B. A. Dobrescu and J. Terning, Phys. Lett. B353, 289 (1995),hep-ph/9503203.[8] Lower limits than this on MV8 and MZ′ were obtained in Ref. [7], in R. S. Chivukulaand J. Terning, Phys. Lett. B385, 209 (1996), hep-ph/9606233, and in I. Bertram andE. H. Simmons, Phys. Lett. B443, 347 (1998), hep-ph/9809472. The first contains amodel–independent bound from the t–b loop contributions to the ρ–parameter. Thelatter two are based on fits to precision electroweak data and Tevatron jet data,respectively.8[9] R. S. Chivukula, A. G. Cohen and E. H. Simmons, Phys. Lett. B380, 92 (1996),hep-ph/9603311;M. Popovic and E. H. Simmons, Phys. Rev. D58, 095007 (1998), hep-ph/9806287.[10] K. Lane, T. Rador, and E. Eichten, Phys. Rev. D62, 015005 (2000), hep-ph/0001056.[11] R. F. Dashen, Phys. Rev. D3, 1879 (1971)[12] J. Nuyts, Phys. Rev. Lett. 26, 1604 (1971);K. Lane, Physica Scripta 23, 1005 (1981); Proceedings of Conference on TopicalQuestions in QCD, Copenhagen, June 9–13, 1980, edited by K. Hansan, et al.[13] D. Kominis, Phys. Lett. B358, 312 (1995), hep-ph/9506305.[14] G. Buchalla, G. Burdman, C. T. Hill, and D. Kominis, Phys. Rev. D53, 5185 (1996),hep-ph/9510376.[15] Review of Particle Properties, D. E. Groom, et al., Eur. Phys. J. C15, 1 (2000).[16] A. J. Buras and R.Fleischer, Heavy Flavors II, World Scientific, A. J. Buras andM. Lindner, eds. (1997), hep-ph/9704376.[17] F. Caravaglios, et al., in the Proceedings of the Third International Conference on BPhysics and CP Violation, Taipei, Taiwan, 3-7 Dec 1999, hep-ph/0002171.[18] The details of our calculation of ETC contributions to ∆MK0 , ǫ, ǫ′, and B decays arecontained in G. Burdman, K. Lane, and T. Rador, Quark Mixing and Rare Processesin Technicolor Theories, in preparation.[19] S. Bosch et al., Nucl. Phys. B565, 3 (2000), hep-ph/9904408.[20] A. Alavi-Harati et al., the KTeV collaboration, Phys. Rev. Lett. 83, 22 (1999); V.Fanti et al., the NA48 collaboration, Phys. Lett. B465, 335 (1999).[21] G. Burdman and D. Kominis, Phys. Lett. B403, 101 (1997); hep-ph/9702265.9

Anti-B-B Mixing Constrains Topcolor-Assisted Technicolor

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B̄–B mixing constrains topcolor-assisted technicolor

We argue that extended technicolor augmented with topcolor requires that all mixing between the third and the first two quark generations resides in the mixing matrix of left-handed down quarks. Then, the B̄d – Bd mixing that occurs in topcolor models constrains the coloron and Z′ boson masses to be greater than about 5 TeV. This implies fine tuning of the topcolor couplings to better than 1%.First author draf

Lane, Kenneth D.

Rador, Tonguç

Boston University Institutional Repository (OpenBU)

(B)over-bar-B mixing constrains topcolor-assisted technicolor

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Anti-B-B Mixing Constrains Topcolor-Assisted Technicolor

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