Non-decoupling effects related to a large $m_t$ affecting non-oblique
radiative corrections in vertices ($Z\bar{b}b$) and boxes ($B$-$\bar{B}$ mixing
and $\epsilon_K$) are very sensitive to the particular mechanism of spontaneous
symmetry breaking. We analyze these corrections in the framework of a chiral
electroweak standard model and find that there is only one operator in the
effective lagrangian which modifies the longitudinal part of the $W^+$ boson
without touching the oblique corrections. The inclusion of this operator
affects the $Z\bar{b}b$ vertex, the $B$-$\bar{B}$ mixing and the CP-violating
parameter $\epsilon_K$, generating interesting correlations among the hard
$m_t^4 \log m_t^2$ corrections to these observables, for example, the maximum
vertex $Z b\bar{b}$ correction allowed by low energy physics is about one
percent.Comment: LaTex, 8 pages, 1 postscript figure include

A. Akundov

A. C. Longhitano

A. J. Buras

A. Pich

A. Santamaria

D. Comelli

F. Feruglio

G. Altarelli

G. Buchalla

I. Tomalin

J. Bernabéu

L. Gibbons

L. Wolfenstein

M. E. Peskin

R. D. Peccei

T. Appelquist

W. Beenakker

W. Marciano

English

arXiv

Crossref

Hard
                    
                      
                        
                          
                            m

Nondecoupling effects related to a large m(1) affecting nonoblique radiative corrections in vertices (Z (b) over bar b) and boxes (B-(B) over bar mixing and epsilon(K)) are sensitive to the mechanism of spontaneous symmetry breaking. In the framework of the effective chiral electroweak standard model there is only one O(p(4)) operator which modifies the longitudinal part of the W+ boon without touching the oblique corrections. This operator affects the Z (b) over bar b vertex, the B-(B) over bar mixing, and the CP-violating parameter epsilon(K) generating interesting correlations among the hard m(t)(4)lnm(t)(2) corrections to these observables

Bernabeu Alberola, José

Comelli, Denis

Pich Zardoya, Antonio

Santamaria, Arcadi

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-ph/9612207v1  29 Nov 1996FTUV/96-85IFIC/96-94Hard mt Corrections as a Probe of theSymmetry Breaking SectorJ. Bernabe´u, D. Comelli, A. Pich and A. SantamariaDepartament de Fı´sica Teo`rica, IFIC,CSIC-Universitat de Vale`ncia46100 Burjassot, Vale`ncia, SpainAbstractNon-decoupling effects related to a large mt affecting non-oblique radiative cor-rections in vertices (Zb¯b) and boxes (B–B¯ mixing and ǫK) are very sensitive to theparticular mechanism of spontaneous symmetry breaking. We analyze these correc-tions in the framework of a chiral electroweak standard model and find that there isonly one operator in the effective lagrangian which modifies the longitudinal part ofthe W+ boson without touching the oblique corrections. The inclusion of this operatoraffects the Zb¯b vertex, the B–B¯ mixing and the CP-violating parameter ǫK , generatinginteresting correlations among the hard m4t logm2t corrections to these observables, forexample, the maximum vertex Z bb¯ correction allowed by low energy physics is aboutone percent.November 25, 2013One of the basic ingredients of the standard model (SM) is the spontaneous breaking of theelectroweak gauge symmetry. In the SM it is implemented through the Higgs mechanism in whichthe would-be Goldstone excitations are absorbed into the longitudinal degrees of freedom of thegauge bosons. The spontaneous symmetry breaking (SSB) is realized linearly, that means, bythe use of a scalar field which acquires a non-zero vacuum expectation value. The spectrum ofphysical particles contains then not only the massive vector bosons but also a neutral scalar Higgsfield which must be relatively light.In a more general scenario, the SSB can be parametrized in terms of a non-renormalizablelagrangian which contains the SM gauge symmetry realized non-linearly1, 2 . This non-linearlyrealized SM is also called the chiral realization of the SM (χSM ) due to of its similarity withlow energy QCD chiral lagrangians. It includes, with a particular choice of the parameters of thelagrangian, the SM, as long as the energies involved are small compared with the Higgs masswhich is not present in the effective Lagrangian. In addition it can also accommodate any modelthat reduces to the SM at low energies as happens in many technicolour scenarios. The priceto be payed for this general parametrization is the lose of renormalizability and, therefore, theappearance of many couplings which must be determined from experiment or computed in a morefundamental theory.Since the SSB is related to the bosonic sector, one would expect that any deviation from theSM SSB mechanism would affect especially the gauge boson propagation properties, the so-calledoblique corrections, which are parametrized in terms of the S,T,U parameters3 (or the ǫ1, ǫ2 and ǫ3pameters4). In fact these corrections have been extensively studied in the framework of the χSM 5 .In particular, one would think that one should look into quantities which are MH-dependent in theSM to test the SSB sector. However, it is interesting to realize that the onlyMH -dependent radiativecorrection, ∆ρ, has an agreement with the SM prediction at the per-mil level. Vertex corrections,whose MH dependence appears only at the two loop level, are not so well known∗.On the other hand, the would-be Goldstone bosons coming from SSB also couple to fermions.In fact, all non-decoupling effects of the SM related to a large top quark mass, mt, come fromthe coupling of the would-be Goldstone bosons to the top quark. Therefore, we can expect anynon-decoupling quantity related to a heavy top-quark to be sensitive to the would-be Goldstoneboson propagation properties and couplings, that is, to the specific mechanism of SSB.In the SM, large m2t effects appear, in addition to the oblique corrections, in the vertex Zbb¯, thatis in Rb = Γb/Γh, and in B–B¯ and K–K¯ mixing†. Then, we will use these quantities to explorepossible deviations of the SM spontaneous symmetry breaking mechanism. To do so we will usea χSM only for the bosonic sector of the theory and leave fermion couplings as in the linear SM.It turns out that there is only one operator in the effective lagrangian that affects the Zbb¯ vertexwithout touching the oblique corrections (which as commented before agree with the SM at the per-mil level). This operator modifies the propagation properties of the charged would-be Goldstonebosons, that is, the longitudinal component of the W+ boson. Therefore, it will also affect anyobservable in which the non-decoupling effects of a large mt are important, in particular B–B¯∗And, in fact, in the last years there has been a big controversy about the Rb = Γ(Z → bb¯)/Γ(Z → hadrons)value.†Of course, non-decoupling effects appear in other observables, but only in the quantities we just mentioned presentexperiments are sensitive enough to see the effects1mixing, and ǫK .In the non-linear realization of the SM the Goldstone bosons πa associated to the SSB ofSU(2)L × SU(2)R → SU(2)L+R are collected in a matrix field U(x) = Exp{i πaτ a/v}. Theoperators in the effective chiral Lagrangian are classified according to the number of covariantderivatives acting on U(x).The lowest-order operators just fix the values of the Z and W mass at tree level and do notcarry any information on the underlying physics. Therefore, in order to extract some informationon new physics we must start studying the effects coming from higher-order operators. Departureof those coefficients from the SM predictions can be a hint for the existence of new physics.The lowest order effective chiral lagrangian can be written in the following way:L = LB + Lψ + LY , (1)whereLB = −12Tr{WˆµνWˆ µν + BˆµνBˆµν}+ v24Tr{DµU+DµU} , (2)with Wˆµν = W aµντa/2 Bˆµν = Bµντ 3/2, and DµU = ∂µU + ig2Wµa τaU − ig′2BµUτ 3. Lψ is theusual fermionic kinetic lagrangian andLY = −Q¯LUMqQR + h.c , (3)where Mq is a 2× 2 block-diagonal matrix containing the 3× 3 mass matrices of the up and downquarks and QL and QR are doublets containing the up and down quarks for the three families inthe weak basis.At the next order, that is containing at most four derivatives, the CP and SU(2)L ⊗ U(1)Yinvariant effective chiral Lagrangian with only gauge bosons and Goldstone fields, is described bythe 15 operators reported in ref. 2: L = ∑14i=0 aiOi.The usual oblique corrections are only sensitive to a0, (a8 + a13) and (a1 + a13). On the otherhand, the operators proportional to a2, a3, a9 and a14 parametrize the effective non-abelian gaugecouplings that are tested by LEP2. All the other couplings remain not tested because they onlycontribute to four-point Green functions (a4, a5, a6, a7, a10) or because, although quadratic in theGoldstone fields, they do not contribute to the one-loop oblique corrections (a11, a12). For instance,the operator proportional to a11:O11 = Tr{(DµV µ)2} , (4)with Vµ = (DµU)U+ and DµVµ = ∂µVµ+ig[Wˆµ, V µ] generates corrections to the two point Greenfunction of the W+, Z and would-be Goldstone bosons:O11 = g2W+µ ∂µ∂νW−ν +g2Z2Zµ∂µ∂νZν − 4π+∂4v2π−− 2π3∂4v2π3 +2gvW+µ ∂µ∂2π− +2gvW−µ ∂µ∂2π+ +2gZvZ+µ ∂µ∂2π3 +O(φ3) . (5)However, all these interactions involve always the longitudinal components of the gauge bosonsand so do not enter directly into the ǫi parameters. The same happens to the operators O12 and O13which affect only the longitudinal part of the neutral Z boson.2The effects of the operator O11 can be seen more easily once we use the following equation ofmotion involving the operators of the Lagrangian to lowest order ‡:DµVµ =iv2Dµ(Q¯LγµτaQLτa), (6)i 6DQL = UMqQR , i 6DQR =M+q U+QL . (7)Then the operator O11 can be rewritten asO11 =g48M4W[Q¯(τaUMqPR −M+q U+τaPL)Q]2, (8)where PL and PR are the left and right chirality projectors. By writing (8) in terms of the masseigenstates and keeping only the terms proportional to the top quark mass we obtainO11 =g48M4Wm2t(t¯γ5t)2 − 4d,s,b∑f,f ′(f¯ ′LtR)(t¯RfL)VtfV∗tf ′ . (9)Therefore, the effect to lowest order of the modification of the would-be Goldstone propagatorcan be written as a four-fermion interaction proportional to quark masses. This kind of opera-tors appears also in the analysis of new physics with an effective Lagrangian with SSB realizedlinearly6 .Four fermion interactions are much more convenient for explicit calculations and also to un-derstand the effects of the new operator. For instance, it is clear that the four-fermion interactioncan only contribute to the gauge-boson self-energies at two loops and therefore do not contributeto the ǫi parameters at one loop.We discuss now some observables affected by the new interaction.Rb .–We start with the evaluation of the corrections to the Zb¯b vertex. We parametrize the effectiveZb¯b vertex as:gcWZµ(gbLb¯LγµbL + gbRb¯RγµbR)] , (10)with the values of the tree level couplings, gbL = −1/2 + s2W/3 and gbR = s2W/3.At one loop we parametrize the effect of new physics as a shift in the couplings:gbL,R → gbL,R + δgbL,R . (11)We calculate the one-loop contribution of the operator O11 keeping only the divergent logarithmicpiece. This means we neglect any possible local contribution from the chiral lagrangian at order‡This is allowed in the effective Lagrangian, even at the one loop level, as long as we keep only the dominantpieces. The use of the equations of motion is equivalent to a redefinition of the fields which affects only higher orderoperators in the effective Lagrangian3p6. The relevant diagram§ is depicted in fig. (1.a) and the result is:δgL = − α4πs2wa11g24m4tM4WlogΛ2m2t. (12)A shift in the Zbb¯ couplings gives a shift in Rb given byRb = RSMb1 + δNPbV1 +RSMb δNPbV, (13)withδNPbV =δΓbΓSMb≈ 2 gbL(gbL)2 + (gbR)2δgbL = −4.58 δgbL . (14)The ALEPH collaboration has presented a new analysis of Rb data which leads to results whichare compatible with the standard model predictions at the one sigma level7 . In fact the new worldaverage is8 Rb = 0.2178 ± 0.0011 to be compared with the SM expectation for mt = 175 GeVRSMb = 0.2157 ± 0.0002. Clearly the new value of Rb is within two standard deviations of thestandard model predictions9 .Using these data on Rb we getδNPbV = 0.012± 0.007 . (15)K–K¯ and B–B¯ Mixing .–In the SM, the mixing between the B0 meson and its antiparticle is completely dominated bythe top contribution. The explicit mt dependence of the corresponding box diagram is given by theloop function10S(xt)SM =xt4[1 +91− xt −6(1− xt)2 −6x2t ln xt(1− xt)3], xt ≡ m2tM2W, (16)which contains the hard m2t term, S(xt) ∼ xt/4, induced by the longitudinal W exchanges. Thesame function regulates the top–quark contribution to the K–K¯ mixing parameter εK . The mea-sured top–mass,mt = 175±6 GeV [mt ≡ mt(mt) = 167±6 GeV], impliesS(xt)SM = 2.40±0.13.The correction induced by the new operator, O11, can be parametrized as a shift on the functionS(xt). The calculation of the diagrams§ in fig. (1.b) leads to the result:S(xt) = S(xt)SM + δS(xt) , δS(xt) = −a11 g2m4t2M4WlnΛ2m2t. (17)§The same result is of course obtained using the original form (4) for O11, where the effect of this operator appearsas a modification of the longitudinal W propagator. However, one needs to consider a larger number of Feynmandiagrams in this case.4Thus, the hard m4t lnm2t contributions to δNPbV and δS(xt) are correlated:δS(xt) =32π2|Vtb|2g2 δgbL = −163 δNPbV . (18)We can use the measured B0d–B¯0d mixing11 , ∆MB0d= (0.464 ± 0.018) × 1012 s−1, to inferthe experimental value of S(xt) and, therefore, to set a limit on the δgbL contribution. The explicitdependence on the quark–mixing parameters can be resolved putting together the constraints from∆MB0d, εK and Γ(b→ u)/Γ(b→ c). Using the Wolfenstein parametrization12 of the quark–mixingmatrix, one has:∣∣∣∣ VtdλVcb∣∣∣∣ =√(1− ρ)2 + η2 = (1.21± 0.09)√S(xt)185 MeV√ηB (√2fB√BB)=(1.21+0.50−0.30)√S(xt), (19)η[(1− ρ)A2η2 S(xt) + P0]A2BK = 0.226 (20)∣∣∣∣ VubλVcb∣∣∣∣ =√ρ2 + η2 = 0.36± 0.09 . (21)We have taken λ ≡ |Vus| = 0.2205 ± 0.0018, |Vcb| ≡ Aλ2 = 0.040 ± 0.003 and |Vub|/|Vcb| =0.08± 0.02. The numerical factor on the rhs of eq. (19) should be understood as an allowed range,because the error is dominated by the large theoretical uncertainties in the hadronic matrix elementof the ∆B = 2 operator; it corresponds to13, 14 √ηB (√2fB√BB) = (185± 45) MeV. In eq. (20),η2 = 0.57 ± 0.01 is the short–distance QCD correction15 , while P0 = 0.31 ± 0.02 takes intoaccount the charm contributions13 . For the ∆S = 2 hadronic matrix element we have chosen therange14 BK = 0.6± 0.2.Both the circle (19) and the hyperbola (20) depend on the the value of S(xt). The intersectionof the two circles (19) and (21) restricts S(xt) to be in the range 0.39 < |S(xt)| < 9.7. The requestof simultaneous intersection with the hyperbola ǫK imposes a further constraint. Since a positivevalue of BK is obtained by all present calculations and S(xt)SM > 0, the SM implies a positivevalue for η. In our case, the constraint that the total S(xt) = S(xt)SM + δS(xt) is positive does notexist and this opens the possibility of solutions also with η < 0; however, this would imply a hugecorrection δS(xt). Taking η > 0, the three curves (bands) intersect if S(xt) > S(xt)min = 1.0.The minimum value of S(xt) is reached for V maxcd , BmaxK and |Vub/Vcb|max. Taking a moreconservative ±0.14 error in eq. (21) (corresponding to |Vub/Vcb| = 0.08 ± 0.03) would resultin S(xt)min = 0.8.The shift in gbL required by Rb [eq. (15)] and the relation (18) implyδS = −2.0± 1.1 , (22)i.e., −0.7 < S(xt) < 1.5. Thus, the present experimental measurements of Rb and the low-energyconstraints from the usual unitarity triangle fits are compatible with the introduction of the operatorO11. From eq. (18) and eq. (15) and the constraint S ≥ Smin = 1 we can see that the maximum(positive) value of δNPbV allowed by low-energy physics isδNPbV < 0.01 ,5which is even stronger than the values obtained by present direct measurements of Rb (eq. (15).We acknowledge interesting discussions with Misha Bilenky, Domenec Espriu and JoaquimMatias. One of us (D.C.) is indebted to the Spanish Ministry of Education and Science for apostdoctoral fellowship. This work has been supported by the Grant AEN-96/1718 of CICYT,Spain.References[1] T. Appelquist, “Gauge Theories and Experiments at High Energies”, ed. K.C. Brower andD.G. Sutherland (Scottish University Summer School in Physics, St. Andrews); T. Ap-pelquist and C. Bernard, Phys. Rev. D22, 200 (1980)[2] A.C. Longhitano, Phys. Rev. D22, 1166 (1980); Nucl. Phys. B188, 188 (1981)[3] M.E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D46, 381 (1992);W. Marciano and J. Rosner, Phys. Rev. Lett 65, 2963 (1990)[4] G. Altarelli and R. Barbieri, Phys. Lett. B253, 161 (1990); G. Altarelli, R. Barbieri andJ. Jadach, Nucl. Phys. B369, 3 (1992)[5] See for instance F. Feruglio, Int. J. Mod. Phys. A8, 4937 (1993)[6] J. Bernabeu, D. Comelli, A. Pich and A. Santamaria in preparation.[7] I. Tomalin, “Rb with Multivariate Analysis”, talk at the ICHEP 96 Conference, Warsaw, July1996.[8] The LEP Electroweak Working Group and the SLD Heavy Flavor Group, “A Combinationof Preliminary LEP and SLD Electroweak Measurements and Constraints on the StandardModel”. Report LEPEWWG/96-02, 1996.[9] A. Akundov, D. Bardin and T. Riemann, Nucl. Phys. B276, 1 (1986); J. Bernabe´u,A. Pich, and A. Santamaria, Phys. Lett. B200, 569 (1988); Nucl. Phys. B363, 326 (1991);W. Beenakker and W. Hollik, Z. Phys. C40, 141 (1988)[10] G. Buchalla, A.J. Buras and M.E. Lautenbacher, “Weak Decays Beyond Leading Loga-rithms” Submitted to Rev. Mod. Phys. [hep-ph/9512380].[11] L. Gibbons, Quark Masses and Mixing from Weak Processes, Plenary talk at the 28th Interna-tional Conference on High Energy Physics (Warsaw, 1996)[12] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983)6[13] A.J. Buras, Flavour Changing Neutral Current Processes, Plenary talk at the 28th Interna-tional Conference on High Energy Physics (Warsaw, 1996), [hep-ph/9610461].[14] A. Pich and J. Prades, Phys. Lett. B346, 342 (1995);[15] A.J. Buras, M. Jamin and P.H. Weisz, Nucl. Phys. B347, 491 (1990)7Zbbt(a)bd bd(b)ttpiFigure 1: a) Contribution of the effective operator O11 to Z → bb¯. b) Contribution of the effectiveoperator O11 to B–B¯ mixing8

Hard m(t) corrections as a probe of the symmetry breaking sector

4 páginas, 1 figura.-- PACS numbers: 12.15.Lk, 11.30.Qc, 12.60.NzNondecoupling effects related to a large mt affecting nonoblique radiative corrections in vertices (Zb̅ b) and boxes ( B- B̅ mixing and εK) are sensitive to the mechanism of spontaneous symmetry breaking. In the framework of the effective chiral electroweak standard model there is only one O(p4) operator which modifies the longitudinal part of the W+ boson without touching the oblique corrections. This operator affects the Zb̅ b vertex, the B- B̅ mixing, and the CP-violating parameter εK, generating interesting correlations among the hard mt4Inmt2 corrections to these observables.One
of us (D. C.) is indebted to the Spanish Ministry of
Education and Science for a postdoctoral fellowship. This
work has been supported by Grant No. AEN-96/1718 of
CICYT, Spain.Peer reviewe

Bernabéu, José

Comelli, D.

Pich, Antonio

Santamaría, Arcadi

Digital.CSIC

VOLUME 78, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 14 APRIL 19972902Hard mt Corrections as a Probe of the Symmetry Breaking SectorJ. Bernabéu, D. Comelli, A. Pich, and A. SantamariaDepartament de Fı´sica Teòrica, IFIC, CSIC-Universitat de València, 46100 Burjassot, València, Spain(Received 3 December 1996)Nondecoupling effects related to a large mt affecting nonoblique radiative corrections in verticessZb¯bd and boxes (B-B¯ mixing and eK ) are sensitive to the mechanism of spontaneous symmetrybreaking. In the framework of the effective chiral electroweak standard model there is only oneOsp4d operator which modifies the longitudinal part of the W1 boson without touching the obliquecorrections. This operator affects the Zb¯b vertex, the B-B¯ mixing, and the CP-violating parametereK , generating interesting correlations among the hard m4t lnm2t corrections to these observables.[S0031-9007(97)02873-1]PACS numbers: 12.15.Lk, 11.30.Qc, 12.60.NzOne of the basic ingredients of the standard model (SM)is the spontaneous breaking of the electroweak gaugesymmetry. In the SM it is implemented through the Higgsmechanism in which the would-be Goldstone excitationsare absorbed into the longitudinal degrees of freedom ofthe gauge bosons. The spontaneous symmetry breaking(SSB) is realized linearly, that means, by the use of ascalar field which acquires a nonzero vacuum expectationvalue. The spectrum of physical particles contains thennot only the massive vector bosons but also a neutralscalar Higgs field which must be relatively light.In a more general scenario, the SSB can be parame-trized in terms of a nonrenormalizable Lagrangian whichcontains the SM gauge symmetry realized nonlinearly[1,2]. This nonlinearly realized SM is also called thechiral realization of the SM sxSMd due to its similaritywith low-energy quantum chromodynamics (QCD) chiralLagrangians. It includes, with a particular choice of theparameters of the Lagrangian, the SM, as long as the en-ergies involved are small compared with the Higgs masswhich is not present in the effective Lagrangian. In ad-dition, it can also accommodate any model that reducesto the SM at low energies as happens in many techni-color scenarios. The price to be paid for this generalparametrization is the loss of renormalizability and, there-fore, the appearance of many couplings which must bedetermined from experiment or computed in a more fun-damental theory.Since the SSB is related to the bosonic sector, onewould expect that any deviation from the SM SSB mecha-nism would affect especially the gauge-boson propagationproperties, the so-called oblique corrections, which are pa-rametrized in terms of the S,T ,U parameters [3] (or thee1, e2 and e3 parameters [4]). In fact, these correctionshave been studied extensively in the framework of thexSM [5]. In particular, one would think that one shouldlook into quantities which are MH dependent in the SM totest the SSB sector. However, it is interesting to realizethat the only MH -dependent radiative correction, Dr, hasan agreement with the SM prediction at the per-mil level.0031-9007y97y78(15)y2902(4)$10.00Vertex corrections, whose MH dependence appears onlyat the two-loop level, are not so well known. [And, infact, in the past there has been a big controversy about theRb ­ GsZ ! bb¯ dyGsZ ! hadronsd value.]On the other hand, the would-be Goldstone bosonscoming from SSB also couple to fermions. In fact, allnondecoupling effects of the SM related to a large top-quark mass, mt , come from the coupling of the would-be Goldstone bosons to the top quark. Therefore, wecan expect any nondecoupling quantity related to a heavytop quark to be sensitive to the would-be Goldstoneboson propagation properties and couplings, that is, to thespecific mechanism of SSB.In the SM, large m2t effects appear, in addition tothe oblique corrections, in the vertex Zbb¯, that is inRb ­ GbyGh, and in B-B¯ and K-K¯ mixing. [Of course,nondecoupling effects appear in other observables, butpresent experiments are sensitive enough to see the effectsonly in the quantities we just mentioned.] Then, we willuse these quantities to explore possible deviations of theSM spontaneous symmetry breaking mechanism. To doso, we will use a xSM only for the bosonic sector ofthe theory and leave fermion couplings as in the linearSM. [Possible modifications of the fermionic couplingsof the gauge bosons have been investigated in Ref. [6].However, these couplings affect the oblique correctionsas well.]It turns out that there is only one operator in the effec-tive Lagrangian that affects the Zbb¯ vertex without touch-ing the oblique corrections (which, as mentioned before,agrees with the SM at the per-mil level). This operatormodifies the propagation properties of the charged would-be Goldstone bosons, that is, the longitudinal componentof the W1 boson. Therefore, it will also affect any ob-servable in which the nondecoupling effects of a large mtare important, in particular, B-B¯ mixing and eK .In the nonlinear realization of the SM the Gold-stone bosons pa associated with the SSB of SUs2dL 3SUs2dR ! SUs2dL1R are collected in a matrix fieldUsxd ­ expsipatayyd. The operators in the effective© 1997 The American Physical SocietyVOLUME 78, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 14 APRIL 1997chiral Lagrangian are classified according to the numberof covariant derivatives acting on Usxd.The lowest-order operators just fix the values of theZ and W mass at tree level and do not carry anyinformation on the underlying physics. Therefore, inorder to extract some information on new physics, wemust start studying the effects coming from higher-orderoperators. Departure of those coefficients from the SMpredictions can be a hint for the existence of new physics.The lowest-order effective chiral Lagrangian can bewritten in the following way:L ­ LB 1 Lc 1 LY , (1)whereLB ­ 212TrsWˆmnWˆmn 1 BˆmnBˆmnd1y24TrsDmU1DmUd , (2)with Wˆmn ­ Wamntay2, Bˆmn ­ Bmnt3y2, and DmU ­›mU 1 ig2 Wma taU 2 ig02 BmUt3. Lc is the usualfermionic kinetic Lagrangian andLY ­ 2Q¯LUMqQR 1 H.c. , (3)where Mq is a 2 3 2 block-diagonal matrix containing the3 3 3 mass matrices of the up and down quarks, and QLand QR are doublets containing the up and down quarksfor the three families in the weak basis.At the next order that contains, at most, four deriva-tives, the CP and SUs2dL › Us1dY invariant effectivechiral Lagrangian with only gauge bosons and Goldstonefields is described by the 15 operators reported in Ref. [2]:L ­P14i­0 aiOi .The usual oblique corrections are sensitive to a0sa8 1 a13d and sa1 1 a13d; the present data bound thesecouplings below the 1% level. On the other hand, theoperators proportional to a2, a3, a9, and a14 parametrizethe effective non-Abelian gauge couplings that are testedby LEP2. The operators contributing to three- and four-point Green functions sa2, a3, a4, a5, a6, a7, a9, a10, a14dmodify the oblique corrections only at the one-loop level;thus, the present bounds on those couplings are ratherweak s,10%d. The other couplings sa11, a12d remainuntested because, although quadratic in the Goldstonefields, they do not contribute to the oblique correctionseven at one loop. For instance, the operator proportionalto a11,O11 ­ TrfsDmV md2g , (4)with Vm ­ sDmUdU1 and DmVm ­ ›mVm 1 igfWˆm, V mggenerates corrections to the two-point Green function ofthe W1, Z, and would-be Goldstone bosons:O11 ­ g2W 1m ›m›nW2n 1g2Z2Zm›m›nZn 2 4p1 ›4y2p2 2 2p3›4y2p3 12gyW1m ›m›2p2 12gyW2m ›m›2p112gZyZ1m ›m›2p3 1 Osp3d . (5)However, all these interactions always involve the longi-tudinal components of the gauge bosons and so do notenter directly into the ei parameters. The same happensto the operators O12 and O13, which affect only the longi-tudinal part of the neutral Z boson.The effects of the operator O11 can be seen more easilyonce we use the following equation of motion involvingthe operators of the Lagrangian to lowest order: [Thisis allowed in the effective Lagrangian, even at the one-loop level, as long as we keep only the dominant pieces.The use of the equations of motion is equivalent to aredefinition of the fields which affects only higher-orderoperators in the effective Lagrangian.]DmVm ­iy2DmsQ¯LgmtaQLtad , (6)iDyQL ­ UMqQR , iDyQR ­ M1q U1QL . (7)Then the operator O11 can be rewritten asO11 ­g48M4WfQ¯staUMqPR 2 M1q U1taPLdQg2, (8)where PL and PR are the left and right chirality projectors.By writing (8) in terms of the mass eigenstates, andkeeping only the terms proportional to the top-quark mass,we obtainO11 ­g48M4Wm2tˆst¯g5td2 2 4d,s,bXf,f 0s f¯ 0LtRd st¯RfLdVtfVptf 0!.(9)Therefore, the effect to the lowest order of the modifica-tion of the would-be Goldstone propagator can be writ-ten as a four-fermion interaction proportional to quarkmasses. This kind of operator also appears in the analysisof new physics with an effective Lagrangian with SSB re-alized linearly [7]. However, the explicit m2t yM4W factorin Eq. (9) has its origin in the bosonic operator of Eq. (4).Four-fermion interactions are much more convenientfor explicit calculations and also to understand the effectsof the new operator. For instance, it is clear that thefour-fermion interaction can only contribute to the gauge-boson self-energies at two loops and therefore do notcontribute to the ei parameters at one loop.We now discuss some observables affected by the newinteraction.2903VOLUME 78, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 14 APRIL 1997Rb .—We start with the evaluation of the corrections tothe Zb¯b vertex. We parametrize the effective Zb¯b vertexasgcWZmsgbLb¯LgmbL 1 gbRb¯RgmbRd , (10)with the values of the tree level couplings, gbL ­ 21y2 1s2W y3 and gbR ­ s2W y3.At one loop we parametrize the effect of new physicsas a shift in the couplings:gbL,R ! gbL,R 1 dgbL,R . (11)We calculate the one-loop contribution of the operatorO11, keeping only the divergent logarithmic piece. Thismeans we neglect any possible local contribution from thechiral Lagrangian at order p6. The relevant diagram isdepicted in Fig. 1(a) and the result isdgL ­ 2a4ps2wa11g24m4tM4Wln L2m2t. (12)[The same result is, of course, obtained using the originalform (4) for O11, where the effect of this operator appearsas a modification of the longitudinal W propagator.However, one needs to consider a larger number ofFeynman diagrams in this case.] A shift in the Zbb¯couplings gives a shift in Rb given byFIG. 1. (a) Contribution of the effective operator O11 toZ ! bb¯. (b) Contribution of the effective operator O11 to B-B¯mixing2904Rb ­ RSMb1 1 dNPbV1 1 RSMb dNPbV, (13)withdNPbV ­dGbGSMbø 2gbLsgbLd2 1 sgbRd2dgbL ­ 24.58 dgbL.(14)The ALEPH collaboration has presented a new analysisof Rb data which leads to results which are compatiblewith the standard model predictions at the one-sigmalevel [8]. In fact, the new world average is [9] Rb ­0.2178 6 0.0011 to be compared with the SM expectationfor mt ­ 175 GeV , RSMb ­ 0.2157 6 0.0002. Clearly,the new value of Rb is within two standard deviations ofthe standard model predictions [10].Using these data on Rb , we getdNPbV ­ 0.012 6 0.007 . (15)K-K¯ and B-B¯ mixing.—In the SM, the mixing betweenthe B0 meson and its antiparticle is completely dominatedby the top contribution. The explicit mt dependenceof the corresponding box diagram is given by the loopfunction [11]SsxtdSM ­xt4•1 191 2 xt26s1 2 xtd226x2t ln xts1 2 xtd3‚,xt ;m¯2tM2W, (16)which contains the hard m2t term, Ssxtd , xty4, in-duced by the longitudinal W exchanges. The same func-tion regulates the top-quark contribution to the K-K¯mixing parameter «K . The measured top-mass, mt ­175 6 6 GeV fmt ; mtsmtd ­ 167 6 6 GeVg, impliesSsxtdSM ­ 2.40 6 0.13.The correction induced by the new operator, O11, canbe parametrized as a shift on the function Ssxtd. Thecalculation of the diagrams in Fig. 1(b) leads to thefollowing result:Ssxtd ­ SsxtdSM 1 dSsxtd ,dSsxtd ­ 2a11g2m4t2M4Wln L2m2t. (17)Thus, the hard m4t lnm2t contributions to dNPbV and dSsxtdare correlated:dSsxtd ­32p2jVtbj2g2 dgbL ­ 2163dNPbV . (18)We can use the measured B0d-B¯0d mixing [12], DMB0d ­s0.464 6 0.018d 3 1012 s21, to infer the experimentalvalue of Ssxtd and, therefore, to set a limit on the dgbLcontribution. The explicit dependence on the quark-mixing parameters can be resolved by putting together theconstraints from DMB0d , «K , and Gsb ! udyGsb ! cd.VOLUME 78, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 14 APRIL 1997Using the Wolfenstein parametrization [13] of the quark-mixing matrix, one hasVtdlVcb­qs1 2 rd2 1 h2­s1.21 6 0.09dpSsxtd185 MeVphB sp2 fBpBB d­s1.2110.5020.30dpSsxtd, (19)hfs1 2 rdA2h2 Ssxtd 1 P0gA2BK ­ 0.226 , (20)VublVcb­qr2 1 h2 ­ 0.36 6 0.09 . (21)We have taken l ; jVusj ­ 0.2205 6 0.0018, jVcbj ;Al2 ­ 0.040 6 0.003, and jVub jyjVcbj ­ 0.08 6 0.02.The numerical factor on the right-hand side of Eq. (19)should be understood as an allowed range, because theerror is dominated by the large theoretical uncertaintiesin the hadronic matrix element of the DB ­ 2 operator;it corresponds to [14,15] phB sp2 fBpBB d ­ s185 645d MeV . In Eq. (20), h2 ­ 0.57 6 0.01 is the short-distance QCD correction [16], while P0 ­ 0.31 6 0.02takes into account the charm contributions [14]. For theDS ­ 2 hadronic matrix element we have chosen therange [15] BK ­ 0.6 6 0.2.Both the circle (19) and the hyperbola (20) depend onthe value of Ssxtd. The intersection of the two circles(19) and (21) restricts Ssxtd to be in the range 0.39 ,jSsxtdj , 9.7. The request of simultaneous intersectionwith the hyperbola eK imposes a further constraint.Since a positive value of BK is obtained by all presentcalculations and SsxtdSM . 0, the SM implies a positivevalue for h. In our case, the constraint that the totalSsxtd ­ SsxtdSM 1 dSsxtd is positive does not exist andthis opens the possibility of solutions also with h , 0;however, this would imply a huge correction dSsxtd.Taking h . 0, the three curves (bands) intersect ifSsxtd . Ssxtdmin ­ 1.0.The minimum value of Ssxtd is reached for V maxcd ,BmaxK , and jVubyVcb jmax. Taking a more conservative60.14 error in Eq. (21) (corresponding to jVubyVcbj ­0.08 6 0.03) would result in Ssxtdmin ­ 0.8.The shift in gbL required by Rb [Eq. (15)] and relation(18) implydS ­ 22.0 6 1.1 , (22)i.e., 20.7 , Ssxtd , 1.5. Thus, the present experimentalmeasurements of Rb and the low-energy constraints fromthe usual unitarity triangle fits are compatible with theintroduction of the operator O11.From Eqs. (18) and (15) and the constraint S $ Smin ­1, we can see that the maximum (positive) value of dNPbVallowed by low-energy physics isdNPbV , 0.01 ,which is even stronger than the values obtained by thepresent direct measurements of Rb [Eq. (15)]. For L ,1 TeV , this translates into an O s10%d upper bound ona11; this is comparable to the present limits for thosecouplings which contribute to the oblique corrections atthe one-loop level.We acknowledge interesting discussions with MishaBilenky, Domenec Espriu, and Joaquim Matias. Oneof us (D. C.) is indebted to the Spanish Ministry ofEducation and Science for a postdoctoral fellowship. Thiswork has been supported by Grant No. AEN-96y1718 ofCICYT, Spain.[1] T. Appelquist, “Gauge Theories and Experiments at HighEnergies,” edited by K. C. Brower and D.G. Suther-land, Scottish University Summer School in Physics,St. Andrews; T. Appelquist and C. Bernard, Phys. Rev.D 22, 200 (1980).[2] A. C. Longhitano, Phys. Rev. D 22, 1166 (1980); Nucl.Phys. B188, 188 (1981).[3] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964(1990); Phys. Rev. D 46, 381 (1992); W. Marciano andJ. Rosner, Phys. Rev. Lett. 65, 2963 (1990).[4] G. Altarelli and R. Barbieri, Phys. Lett. B 253, 161 (1990);G. Altarelli, R. Barbieri, and J. Jadach, Nucl. Phys. B369,3 (1992).[5] See, for instance, F. Feruglio, Int. J. Mod. Phys. A 8, 4937(1993).[6] R. D. Peccei and X. Zhang, Nucl. Phys. B337, 269 (1990);R. D. Peccei, S. Peris, and X. Zhang, Nucl. Phys. B349,305 (1991).[7] W. Buchmüller and D. Wyler, Nucl. Phys. B268, 621(1986); C. T. Hill and X. Zhang, Phys. Rev. D 51,3563 (1995); J. Bernabeu, D. Comelli, A. Pich, andA. Santamaria (to be published).[8] I. Tomalin, Proceedings of the ICHEP 96 Conference,Warsaw, 1996 (to be published).[9] The LEP Electroweak Working Group and the SLD HeavyFlavor Group, Report No. LEPEWWGy96-02, 1996 (tobe published).[10] A. Akundov, D. Bardin, and T. Riemann, Nucl. Phys.B276, 1 (1986); J. Bernabéu, A. Pich, and A. Santamaria,Phys. Lett. B 200, 569 (1988); Nucl. Phys. B363, 326(1991); W. Beenakker and W. Hollik, Z. Phys. C 40, 141(1988).[11] G. Buchalla, A. J. Buras, and M.E. Lautenbacher, Rev.Mod. Phys. 68, 1125 (1996).[12] L. Gibbons, Proceedings of the 28th International Con-ference on High Energy Physics, Warsaw, 1996 (to bepublished).[13] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).[14] A. J. Buras, Proceedings of the 28th International Confer-ence on High Energy Physics, Warsaw, 1996, Report No.hep-phy9610461 (to be published).[15] A. Pich and J. Prades, Phys. Lett. B 346, 342 (1995).[16] A. J. Buras, M. Jamin, and P. H. Weisz, Nucl. Phys. B347,491 (1990).2905

Physical Review Letters

Hard mt Corrections as a Probe of the Symmetry Breaking Sector

Hard $m_t$ Corrections as a Probe of the Symmetry Breaking Sector

Hard $m_t$ Corrections as a Probe of the Symmetry Breaking Sector

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Hard mtm_tmt​ Corrections as a Probe of the Symmetry Breaking Sector

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Hard $m_t$ Corrections as a Probe of the Symmetry Breaking Sector