Physics beyond the standard model can affect top-quark physics indirectly. We
describe the effective field theory approach to describing such physics, and
contrast it with the vertex-function approach that has been pursued previously.
We argue that the effective field theory approach has many fundamental
advantages and is also simpler.Comment: 7 pages, 5 figures. To appear in the Proceedings of the International
  Workshop on Top Quark Physics (top2010), Bruges, Belgium, May 31 - June 4,
  201

Willenbrock, Scott

Zhang, Cen

English

arXiv

Physics beyond the standard model can affect top-quark physics indirectly. We describe the effective field theory approach to describing such physics, and contrast it with the vertex-function approach that has been pursued previously. We argue that the effective field theory approach has many fundamental advantages and is also simpler

Zhang, C.

Willenbrock, S.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2010-10678-9Colloquia: TOP2010IL NUOVO CIMENTO Vol. 33 C, N. 4 Luglio-Agosto 2010Effective field theory for top quark physicsC. Zhang and S. WillenbrockDepartment of Physics, University of Illinois at Urbana-Champaign1110 West Green Street, Urbana, IL 61801, USA(ricevuto il 21 Luglio 2010; approvato il 21 Luglio 2010; pubblicato online il 26 Ottobre 2010)Summary. — Physics beyond the standard model can affect top-quark physicsindirectly. We describe the effective field theory approach to describing such physics,and contrast it with the vertex-function approach that has been pursued previously.We argue that the effective field theory approach has many fundamental advantagesand is also simpler.PACS 14.65.Ha – Top quarks.PACS 11.15.-q – Gauge field theories.There are two broad categories of approaches to physics beyond the standard model(SM). The first is to look for the new physics directly via the production of new particles.The second is to look for new interactions of the known particles of the SM. Here weaddress the question of the best way to formalize the latter approach [1].An example of the two approaches is displayed in fig. 1, in the context of a Z ′ boson.At energies above the mass of the Z ′, one observes the new particle directly. At energiesbelow the mass of the Z ′, one observes new interactions of SM fermions mediated by theexchange of a virtual Z ′ boson. At energies much less than the Z ′ mass, this appears asan effective four-fermion interaction.Let us attach a coupling g to the Z ′ interaction with the SM fermions, and includethe Z ′ propagator, proportional to (p2−M2)−1. At energies much less than the Z ′ mass,we can neglect the momentum of the Z ′ in the propagator. The effective four-fermioninteraction is thus of strength g2/M2, and the theory is described by the Lagrangian(1) Leff = LSM + g2M2ψ¯ψψ¯ψ.We can generalize the above discussion with the help of dimensional analysis. Recallthat in units where h¯ = c = 1, the fields of the SM have mass dimensionsdimAμ = 1,(2a)dimφ = 1,(2b)dimψ = 3/2,(2c)c© Societa` Italiana di Fisica 285286 C. ZHANG and S. WILLENBROCKFig. 1. – At energies greater than the Z′ mass, one observes the new particle directly. At energiesbelow the Z′ mass, one observes its effects on SM particles indirectly.where Aμ is a gauge field, φ is the Higgs field, and ψ is a fermion field. Every term inLSM is of dimension four or less, while the new four-fermion interaction is of dimensionsix. This makes sense, because it has a coefficient with a dimension of two inverse powersof mass.The generalization of eq. (1) is now clear:(3) Leff = LSM +∑iciΛ2Oi,where the sum is over all dimension-six operators (that is, products of SM fields). Thecoefficient ci is the coupling of the SM fields to the new physics and Λ is the scale at whichthe new physics resides. This construction, called an effective field theory, is completelygeneral and independent of the details of the new physics, whether it be new particles,extra dimensions, strings, etc. [2].The bad news is that there are nearly 60 dimension-six operators, not even takinginto consideration three generations of fermions [3-5]. The good news is that only a fewof these operators affect top quark physics. Let us look at an example.The decay of the top quark in the SM is shown in fig. 2. The W boson goes on to decayto leptons or light quarks, and we will assume this is described to a good approximationby the SM, since this vertex is well tested. The Wtb vertex is much less constrained, sowe seek a dimension-six operator that affects the decay of the top quark.Fig. 2. – Top quark decay. The W boson goes on to decay to a pair of quarks or leptons.EFFECTIVE FIELD THEORY FOR TOP QUARK PHYSICS 287In the SM, the branching ratio of the top quark to zero helicity, negative helicity(left-handed), and positive helicity (right-handed) W bosons isF0 =m2tm2t + 2M2W≈ 0.7,(4a)FL =2M2Wm2t + 2M2W≈ 0.3,(4b)FR ≈ 0,(4c)where I have neglected the b quark mass, which is a good approximation. Of the nearly60 dimension-six operators, there is only one that affects these branching ratios:(5) OtW = (q¯σμντ It)φ˜W Iμν + h.c.,where W Iμν is the SU(2) field-strength tensor, φ is the Higgs doublet, t is the right-chiraltop quark, and q is the left-chiral quark doublet containing top and bottom. The matrixσμν = i2 [γμ, γν ] is a tensor constructed from Dirac matrices, and τ I are the SU(2) Paulimatrices.The presence of the Higgs doublet field deserves special attention. We are assumingthat the new physics, at scale Λ, is above the scale of electroweak symmetry breaking,which in the SM is the Higgs boson mass. Thus the Higgs doublet is part of the theorybelow Λ, and the dimension-six operators must respect the unbroken SU(2) × U(1)Ysymmetry of the SM.When the Higgs field acquires a vacuum-expectation value, the dimension-six operatorOtW yields the effective interaction(6) Leff = LSM + CtWΛ2 v(b¯σμν(1 + γ5)t)∂μW−ν + h.c.This unfamiliar interaction is not present in the SM. Its effect is to modify the top-quarkbranching ratios:F0 =m2tm2t + 2m2W− 4√2CtW v2Λ2mtmW (m2t −m2W )(m2t + 2m2W )2,(7a)FL =2m2Wm2t + 2m2W+4√2CtW v2Λ2mtmW (m2t −m2W )(m2t + 2m2W )2,(7b)FR = 0.(7c)By measuring these branching ratios, we may extract (or place a bound on) the coefficientCtW /Λ2.Let us now turn to another physical process involving the top quark: single topproduction, shown in fig. 3. In addition to the operator OtW , two other operators affectthe single-top cross section [6]:O(3)φq = i(φ†τ IDμφ)(q¯γμτ Iq) + h.c.,(8a)O(1,3)qq = (q¯iγμτIqj)(q¯γμτ Iq),(8b)288 C. ZHANG and S. WILLENBROCKFig. 3. – Single-top production via the electroweak interaction. The diagrams as shown corre-spond to s-channel production; t-channel production corresponds to the same diagrams turnedon their sides.where Dμ is the gauge-covariant derivative, and the superscripts on the quark doubletfields indicate that they are from the first two generations (i, j = 1, 2). The superscript(3) on the operators indicates that they contain SU(2) triplet currents, and the addi-tional superscript (1) indicates that the currents are color singlets. When the Higgs fieldacquires a vacuum-expectation value, the operator O(3)φq yields(9) Leff = LSM +C(3)φq gv22√2Λ2(b¯γμ(1− γ5)t)W−μ + h.c.This is identical in form to the V − A coupling of the SM. Thus this operator simplyrenormalizes the SM charged-current interaction. The operator O(1,3)qq is a four-fermioninteraction that couples light quarks to third-generation quarks, and thus contributes tosingle-top production as shown in fig. 3(c).To extract (or bound) the coefficients of the three operators OtW , O(3)φq , and O(1,3)qq , thefollowing strategy can be employed. First, we use top decay to extract CtW /Λ2. Then weuse single-top production to extract C(3)φq /Λ2 and C(1,3)qq /Λ2. The latter coefficient enterswith the opposite sign in s-channel and t-channel single-top production, so it is useful tomeasure the two processes separately. In practice, one might fit all three coefficients toall three processes simultaneously.Let us compare this program with the program that has been followed up until now,which I will refer to as the vertex-function approach. In this approach, one writes downthe most general vertex function, consistent with Lorentz invariance, coupling a virtualW boson to on-shell top and bottom quarks [7-11]:(10) ΓμWtb = −g√2Vtb(γμ[fL1 PL + fR1 PR]− iσμνMW(pt − pb)ν [fL2 PL + fR2 PR]),where PL,R = (1 ∓ γ5)/2. The four form factors fL1 , fR1 , fL2 , fR2 are arbitrary functionsof the virtuality of the W boson, Q2. The vertex function is sometimes expressed as aLagrangian,(11) LWtb = g√2Vtb(W−μ b¯γμ[fL1 PL + fR1 PR]t−1MW∂νW−μ b¯σμν [fL2 PL + fR2 PR]t)EFFECTIVE FIELD THEORY FOR TOP QUARK PHYSICS 289but this is misleading, because it is not an effective Lagrangian in the modern sensedescribed above. In addition, the coefficients fL1 , fR1 , fL2 , fR2 in the Lagrangian cannotbe functions of the W boson virtuality, because the Lagrangian is expressed in positionspace, not momentum space.In the effective field theory approach, the form factors of the vertex function arederived from the dimension-six operators. One findsfL1 = 1 + C(3)φqv2Λ2,(12a)fR2 =√2CtWv2Λ2.(12b)As mentioned above, the operator O(3)φq simply renormalizes the V −A coupling of the SM,as shown in eq. (9). The operator OtW contributes to the right-handed tensor coupling,as shown in eq. (6). The other two form factors, fR1 and fL2 , contribute to physicalprocesses only at order O(mb) and O(1/Λ4), and are hence suppressed. We neglect themthroughout. Thus, in contrast to the vertex-function approach, the effective field theoryapproach provides a rationale for neglecting some of the form factors, and also justifiessetting the other form factors to constants independent of Q2.Let us compare the effective field theory approach and the vertex-function approachin more detail. There are several aspects to the comparison:– The effective field theory approach is well motivated and provides guidance as tothe most likely places to observe the indirect effects of physics beyond the standardmodel. The vertex-function approach does not share either of these features.– The effective field theory approach incorporates the known SU(3)C × SU(2) ×U(1)Y symmetry of the SM. The vertex-function approach turns its back on gaugesymmetry.– The effective field theory approach includes contact interactions (such as O(1,3)qq ) aswell as interactions that contribute to vertices. The vertex-function approach onlyconcerns itself with the latter.– The effective field theory approach is valid regardless of whether the top and bottomquarks are real or virtual. The vertex-function approach assumes that the top andbottom quarks are on shell. If they are off shell, one must add additional formfactors.– In the effective field theory approach, one can calculate radiative corrections as-sociated with SM vertices as well as with those arising from the dimension-sixoperators. There is no unambiguous way to calculate radiative corrections in thevertex-function approach.This last point may be the most important of all. We aspire to make contact betweendirect top-quark measurements and precision electroweak measurements, where the topquark enters in loops. Because the effective field theory is renormalizable in the modernsense, it is possible to carry this program out. All divergences that arise from dimension-six operators can be absorbed in the coefficients of other dimension-six operators. Soalthough the theory is not renormalizable in the old-fashioned sense, it allows for theunambiguous calculation of radiative corrections to O(1/Λ2).290 C. ZHANG and S. WILLENBROCKFig. 4. – Associated production of a single top quark and a W boson.Let us pick up where we left off and ask what other dimension-six operators can beprobed by top-quark physics. We show in fig. 4 another single-top process, where thetop quark is produced in association with a W boson. This involves two of the sameoperators, OtW and O(3)φq , that we encountered in s- and t-channel single top production,along with a new operator:(13) OtG = (q¯σμνλAt)φ˜GAμν + h.c.,where λA is an SU(3) matrix and GAμν is the gluon field-strength tensor. This operatoris the strong-interaction analogue of OtW . The four-quark operator O(1,3)qq does not con-tribute to this process. By measuring Wt associated production, together with the otherweak interaction processes discussed above, we can extract (or bound) the coefficientCtG/Λ2.Finally, let us consider top-quark pair production. At the LHC, this is dominated bythe subprocess gg → tt¯, as shown in fig. 5. This is influenced by the operator OtG thatwe just discussed, but also by the operatorsOG = fABCGAνμ GBρν GCμρ ,(14a)OφG =12φ†φGAμνGAμν .(14b)The operator OG affects the triple-gluon vertex. Although it does not involve the topquark directly, bounds on this operator from other processes are rather weak, so the bestmeasurement (or bound) may come from top-quark pair production [12]. The operatorOφG couples the Higgs field directly to a pair of gluons, and will be measured (or bounded)by Higgs production [13]. Other operators are probed by qq¯ → tt¯ [12]; I leave that toanother presenter [14].EFFECTIVE FIELD THEORY FOR TOP QUARK PHYSICS 291Fig. 5. – Top-quark pair production via a gluon-sluon collision.ConclusionsThe time is ripe to abandon the vertex-function approach and to adopt the moremodern and systematic approach of effective field theory for top-quark physics. Althougheffective field theories have been around for a long time and are widely used in particlephysics, nuclear physics, condensed matter physics, and elsewhere, they have been slowto be adopted in the context of anomalous couplings. While I was already convinced ofthe timeliness of the effective field theory approach prior to this workshop, my convictionwas further solidified on my arrival day when I noticed a popular Belgian beer, Leffe,that beckons us to embrace effective Lagrangians.∗ ∗ ∗This work was supported in part by the U. S. Department of Energy under contractNo. DE-FG02-91ER40677.REFERENCES[1] Zhang C. and Willenbrock S., arXiv: 1008.3869 [hep-ph].[2] Weinberg S., Physica A, 96 (1979) 327.[3] Leung C. N., Love S. T. and Rao S., Z. Phys. C, 31 (1986) 433.[4] Buchmuller W. and Wyler D., Nucl. Phys. B, 268 (1986) 621.[5] Aguilar-Saavedra J. A., Nucl. Phys. B, 812 (2009) 181 [arXiv:0811.3842 [hep-ph]].[6] Cao Q. H., Wudka J. and Yuan C. P., Phys. Lett. B, 658 (2007) 50 [arXiv:0704.2809[hep-ph]].[7] Kane G. L., Ladinsky G. A. and Yuan C. P., Phys. Rev. D, 45 (1992) 124.[8] Boos E., Dudko L. and Ohl T., Eur. Phys. J. C, 11 (1999) 473 [arXiv:hep-ph/9903215].[9] Chen C. R., Larios F. and Yuan C. P., Phys. 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Effective field theory for top quark physics

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Effective Field Theory for Top Quark Physics

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