TASI Lectures on Effective Field Theory and Precision Electroweak Measurements
aa r X i v : . [ h e p - ph ] J un TASI Lectures on Effective Field TheoryandPrecision Electroweak Measurements
Witold Skiba
Department of Physics, Yale University, New Haven, CT 06520
Abstract
The first part of these lectures provides a brief introduction to the concepts andtechniques of effective field theory. The second part reviews precision electroweakconstraints using effective theory methods. Several simple extensions of the StandardModel are considered as illustrations. The appendix contains some new results on theone-loop contributions of electroweak triplet scalars to the T parameter and containsa discussion of decoupling in that case. ontents S and T parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 More on the universal parameters: Y and W . . . . . . . . . . . . . . . . . . 313.3 All flavor-conserving operators . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 How the sausage is made . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A Scalar triplet contributions to the T parameter 43 A.1 Tree level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 One-loop level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References 53
Phenomena involving distinct energy, or length, scales can often be analyzed by consideringone relevant scale at a time. In most branches of physics, this is such an obvious statementthat it does not require any justification. The multipole expansion in electrodynamics isuseful because the short-distance details of charge distribution are not important when ob-served from far away. One does not worry about the sizes of planets, or their geography,when studying orbital motions in the Solar System. Similarly, the hydrogen spectrum canbe calculated quite precisely without knowing that there are quarks and gluons inside theproton.Taking advantage of scale separation in quantum field theories leads to effective fieldtheories (EFTs) [1]. Fundamentally, there is no difference in how scale separation manifestsitself in classical mechanics, electrodynamics, quantum mechanics, or quantum field theory.The effects of large energy scales, or short distance scales, are suppressed by powers of theratio of scales in the problem. This observation follows from the equations of mechanics,electrodynamics, or quantum mechanics. Calculations in field theory require extra care toensure that large energy scales decouple [2, 3].Decoupling of large energy scales in field theory seems to be complicated by the factthat integration over loop momenta involves all scales. However, this is only a superficialobstacle which is straightforward to deal with in a convenient regularization scheme, for2xample dimensional regularization. The decoupling of large energy scales takes place inrenormalizable quantum field theories whether or not EFT techniques are used. There aremany precision calculations that agree with experiments despite neglecting the effects ofheavy particles. For instance, the original calculation of the anomalous magnetic momentof the electron, by Schwinger, neglected the one-loop effects arising from weak interactions.Since the weak interactions were not understood at the time, Schwinger’s calculation includedonly the photon contribution, yet it agreed with the experiment within a few percent [4].Without decoupling, the weak gauge boson contribution would be of the same order as thephoton contribution. This would result in a significant discrepancy between theory andexperiment and QED would likely have never been established as the correct low-energytheory.The decoupling of heavy states is, of course, the reason for building high-energy acceler-ators. If quantum field theories were sensitive to all energy scales, it would be much moreuseful to increase the precision of low-energy experiments instead of building large colliders.By now, the anomalous magnetic moment of the electron is known to more than ten signif-icant digits. Calculations agree with measurements despite that the theory used for thesecalculations does not incorporate any TeV-scale dynamics, grand unification, or any notionsof quantum gravity.If decoupling of heavy scales is a generic feature of field theory, why would one considerEFTs? That depends on whether the dynamics at high energy is known and calculableor else the dynamics is either non-perturbative or unknown. If the full theory is knownand perturbative, EFTs often simplify calculations. Complex computations can be brokeninto several easier tasks. If the full theory is not known, EFTs allow one to parameterizethe unknown interactions, to estimate the magnitudes of these interactions, and to classifytheir relative importance. EFTs are applicable to both cases with the known and with theunknown high-energy dynamics because in an effective description only the relevant degreesof freedom are used. The high-energy physics is encoded indirectly though interactionsamong the light states.The first part of these lecture notes introduces the techniques of EFT. Examples of EFTsare constructed explicitly starting from theories with heavy states and perturbative interac-tions. Perturbative examples teach us how things work: how to organize power counting, howto estimate the magnitudes of terms, and how to stop worrying about non-renormalizableinteractions. Readers familiar with the concepts of EFTs are encouraged to go directly tothe discussion of precision electroweak measurements.The second part of these notes is devoted to precision electroweak measurements using anEFT approach. This is a good illustration of why using EFTs saves time. The large body ofprecision electroweak measurements can be summarized in terms of constraints on coefficientsof effective operators. In turn, one can use these coefficients to constrain extensions of theStandard Model (SM) without any need for detailed calculations of cross sections, decaywidths, etc.The topic of precision electroweak measurements consists of two major branches. Onebranch engages in comparisons of experimental data with accurate calculations in the SM [5].It provides important tests of the SM and serves as a starting point for work on extensionsof the SM. This subject is not covered in these notes. Another branch is concerned withextensions of the SM and their viability when compared with measurements. This subject is3iscussed here using EFT techniques. The effective theory applicable to precision electroweakmeasurements is well known: it is the Standard Model with higher-dimensional interactions.Since we are not yet sure if the Higgs boson exists, one could formulate an effective descriptionwith or without the Higgs boson. Only the EFT that includes the Higgs boson is discussedhere. While there are differences between the theories with and without the Higgs boson,these differences are technical instead of conceptual.These notes describe how effective theories are constructed and constrained and howto use EFT for learning about extensions of the SM. The best known example of theapplication of EFT to precision electroweak measurements are the S and T parameters. The S and T parameters capture only a subset of available precision measurements. The set ofeffective parameters can be systematically enlarged depending on the assumptions about theunderlying theory. Finally, several toy extensions of the SM are presented as an illustrationof how to constrain the masses and couplings of heavy states using the constraints on EFTs.A more complicated example of loop matching of electroweak scalar triplets is presented inthe Appendix. The one-loop results in the Appendix have not been published elsewhere. The first step in constructing EFTs is identifying the relevant degrees of freedom for themeasurements of interest. In the simplest EFTs that will be considered here, that meansthat light particles are included in the effective theory, while the heavy ones are not. Thedividing line between the light and heavy states is based on whether or not the particles canbe produced on shell at the available energies. Of course, all field theories must be effectivesince we do not know all the heavy states, for example at the Plank scale. There are manymore sophisticated uses of EFTs, for instance to heavy quark systems, non-relativistic QEDand QCD, nuclear interactions, gravitational radiation, etc. Some of these applications ofEFTs are described in the lecture notes in Refs. [6, 7, 8, 9].Formally, the heavy particles are “integrated out” of the action by performing a pathintegral over the heavy states only Z D ϕ H e i R L ( ϕ L , ϕ H ) = e i R L eff ( ϕ L ) , (1)where ϕ L , ϕ H denote the light and the heavy states. Like most calculations done in practice,integrating out is performed using Feynman diagram methods instead of path integrals. Theeffective Lagrangian can be expanded into a finite number of terms of dimension four or less,and a tower of “higher dimensional” terms, that is terms of dimension more than four L eff ( ϕ L ) = L d ≤ + X i O i Λ dim ( O i ) − , (2)where Λ is an energy scale and dim ( O i ) are the dimensions of operators O i . What is crucialfor the EFT program is that the expansion of the effective Lagrangian in Eq. (2) is into localterms in space-time. This can be done when the effective Lagrangian is applied to processesat energies lower than the masses of the heavy states ϕ H .4ecause we are dealing with weakly interacting theories, the dimension of terms in theLagrangian is determined by adding the dimensions of all fields making up a term and thedimensions of derivatives. The field dimensions are determined from the kinetic energyterms and this is often referred to as the engineering dimension. In strongly interactingtheories, the dimensions of operators often differ significantly from the sum of the constituentfield dimensions determined in free theory. In weakly interacting theories considered here,by definition, the effects of interactions are small and can be treated order by order inperturbation theory.The sum over higher dimensional operators in Eq. (2) is in principle an infinite sum.In practice, just a few terms are pertinent. Only a finite number of terms needs to bekept because the theory needs to reproduce experiments to finite accuracy and also becausethe theory can be tailored to specific processes of interest. The higher the dimension of anoperator, the smaller its contribution to low-energy observables. Hence, obtaining results to agiven accuracy requires a finite number of terms. This is the reason why non-renormalizabletheories are as good as renormalizable theories. An infinite tower of operators is truncatedand a finite number of parameters is needed for making predictions, which is exactly thesame situation as in renormalizable theories.It is a simplification to assume that different higher dimensional operators in Eq. (2) aresuppressed by the same scale Λ. Different operators can arise from exchanges of distinctheavy states that are not part of the effective theory. The scale Λ is often referred to as thecutoff of the EFT. This is a somewhat misleading term that is not to be confused with aregulator used in loop calculations, for example a momentum cutoff. Λ is related to the scalewhere the effective theory breaks down. However, dimensionless coefficients do matter. Onecould redefine Λ by absorbing dimensionless numbers into the definitions of operators. Thebreakdown scale of an EFT is a physical scale that does not depend on the convention chosenfor Λ. This scale could be estimated experimentally by measuring the energy dependence ofamplitudes at small momentum. In EFTs, amplitudes grow at high energies and exceed thelimits from unitarity at the breakdown scale. It is clear that the breakdown scale is physicalsince it corresponds to on-shell contributions from heavy states.The last remark regarding Eq. (2) is that terms in the L d ≤ Lagrangian also receive con-tributions from the heavy fields. Such contributions may not lead to observable consequencesas the coefficients of interactions in L d ≤ are determined from low-energy observables. Insome cases, the heavy fields violate symmetries that would have been present in the fullLagrangian L ( ϕ L , ϕ H = 0) if the heavy fields are neglected. Symmetry-violating effects ofheavy fields are certainly observable in L d ≤ . EFTs are based on several systematic expansions. In addition to the usual loop expansion inquantum field theory, one expands in the ratios of energy scales. There can be several scalesin the problem: the masses of heavy particles, the energy at which the experiment is done,the momentum transfer, and so on. In an EFT, one can independently keep track of powers Not all terms of a given dimension need to be kept. For example, one may be studying 2 → →
4, and may not contributeindirectly through loops to the processes of interest at a given loop order.
5f the ratio of scales and of the logarithms of scale ratios. This could be useful, especiallywhen logarithms are large. Ratios of different scales can be kept to different orders dependingon the numerical values, which is something that is nearly impossible to do without usingEFTs.When constructing an EFT one needs to be able to formally predict the magnitudes ofdifferent O i terms in the effective Lagrangian. This is referred to as power counting theterms in the Lagrangian and it allows one to predict how different terms scale with energy.In the simple EFTs discussed here, power counting is the same as dimensional analysis usingthe natural ¯ h = c = 1 units, in which [ mass ] = [ length ] − . From now on, dimensions will beexpressed in the units of [ mass ], so that energy has dimension 1, while length has dimension −
1. The Lagrangian density has dimension 4 since R L d x must be dimensionless.The dimensions of fields are determined from their kinetic energies because in weaklyinteracting theories these terms always dominate. The kinetic energy term for a scalar field, ∂ µ φ ∂ µ φ , implies that φ has dimension 1, while that of a fermion, i ψ /∂ ψ , implies that ψ hasdimension in 4 space-time dimensions.A Yukawa theory consisting of a massless fermion interacting with with two real scalarfields: a light one and a heavy one will serve as our working example. The Lagrangian ofthe high-energy theory is taken to be L = iψ /∂ ψ + 12 ( ∂ µ Φ) − M + 12 ( ∂ µ ϕ ) − m ϕ − λ ψψ Φ − η ψψϕ. (3)Let us assume that M >> m . As this is a toy example, we do not worry whether it is naturalto have a hierarchy between m and M . The Yukawa couplings are denoted as λ and η . Weneglect the potential for Φ and ϕ as it is unimportant for now.As our first example of an EFT, we will consider tree-level effects. We want to findan effective theory with only the light fields present: the fermion ψ and scalar ϕ . Theinteractions generated by the exchanges of the heavy field Ψ will be mocked up by newinteractions involving the light fields.We want to examine the ψψ → ψψ scattering process to order λ in the coupling con-stants, that is to the zeroth order in η , and keep terms to the second order in the externalmomenta. − Φ ψψ ψψ Φ12 34 12 34Figure 1: Tree-level diagrams proportional to λ that contribute to ψψ → ψψ scattering.Integrating out fields is accomplished by comparing amplitudes in the full and effectivetheories. In this example, the only amplitude we need to worry about is the ψψ → ψψ scattering amplitude. Often the “full theory” is referred to as the ultraviolet theory, andthe effective theory as the infrared theory. The UV amplitude to order λ is given by two6ree-level graphs depicted in Fig. 1 and the result is A UV = u ( p ) u ( p ) u ( p ) u ( p ) ( − iλ ) i ( p − p ) − M − { ↔ } , (4)where { ↔ } indicates interchange of p and p as required by the Fermi statistics. TheDirac structure is identical in the UV and IR theories, so we can concentrate on the propa-gator ( − iλ ) i ( p − p ) − M = i λ M − ( p − p ) M ≈ i λ M (cid:18) p − p ) M + O ( p M ) (cid:19) (5)and neglect terms higher than second order in external field momenta. The ratio p M is theexpansion parameter and we can construct an effective theory to the desired order in thisexpansion. Since the effective theory does not include the heavy scalar of mass M , it is clearthat the effective theory must break down when the scattering energy approaches M .To the zeroth order in external momenta we can reproduce the ψψ → ψψ scatteringamplitude by the four-fermion Lagrangian L p ,λ = iψ /∂ ψ + c ψψ ψψ, (6)where the coefficient of the four-fermion term includes the symmetry factor that accountsfor two factors of ψψ in the interaction. We omit all the terms that depend on the lightscalar ϕ as such terms play no role here. We will restore these terms later. The amplitudecalculated using the L p ,λ Lagrangian is A IR = u ( p ) u ( p ) u ( p ) u ( p ) ( ic ) − { ↔ } . (7)Comparing this with Eq. (5) gives c = λ M .At the next order in the momentum expansion, we can write the Lagrangian as L p ,λ = iψ /∂ ψ + λ M ψψ ψψ + d ∂ µ ψ∂ µ ψ ψψ. (8)We need to compare the scattering amplitude obtained from the effective Lagrangian L p ,λ with the amplitude in Eqs. (4) and (5). The effective Lagrangian needs to be valid bothon-shell and off-shell as we could build up more complicated diagrams from the effectiveinteractions inserting them as parts of diagrams. For the matching procedure, we can useany choice of external momenta that is convenient. When comparing the full and effectivetheories, we can choose the momenta to be either on-shell or off-shell. The external particles,in this case fermions ψ , are identical in the full and effective theories. The choice of externalmomenta has nothing to do with the UV dynamics. In other words, for any small externalmomenta the full and effective theories must be identical, thus one is allowed to makeopportunistic choices of momenta to simplify calculations.In this example, it is useful to assume that the momenta are on-shell that is p = . . . = p = 0. Therefore, the amplitude can only depend on the products of differentmomenta p i · p j with i = j . With this assumption, the effective theory needs to reproduce7he − i λ M p · p M − { ↔ } part of the amplitude in Eq. (5). The term proportional to d inthe L p ,λ Lagrangian gives the amplitude A IR = id ( p · p + p · p ) u ( p ) u ( p ) u ( p ) u ( p ) − { ↔ } . (9)The momenta p and p are assumed to be incoming, thus they contribute − ip µ , to theamplitude, while the outgoing momenta contribute + ip µ , . Conservation of momentum, p + p = p + p , implies p · p = p · p , p · p = p · p , and p · p = p · p . Hence, d = − λ M .The derivative operator with the coefficient d is not the only two-derivative term one canwrite with four fermions. For example, we could have included in the Lagrangian the term( ∂ ψ ) ψ ψψ + H . c . or included the term ∂ µ ψψ ψ∂ µ ψ . When constructing a general effectiveLagrangian it is important to consider all terms of a given order. There are four differentways to write two derivatives in the four-fermion interaction. Integration by parts impliesthat there is one relationship between the four possible terms. The term containing ∂ doesnot contribute on shell. In fact, this term can be removed from the effective Lagrangianusing equations of motion [10, 11]. We will discuss this in more detail in Sec. 2.5. Thus,there are only two independent two-derivative terms in this theory. At the tree level, onlyone of these terms turned out to be necessary to match the UV theory. So far we have focused on the fermions, but our original theory has two scalar fields. At treelevel, we have obtained the effective Lagrangian L p ,λ = iψ /∂ ψ + c ψψ ψψ + d ∂ µ ψ∂ µ ψ ψψ + 12 ( ∂ µ ϕ ) − m ϕ − η ψψϕ (10)and calculated the coefficients c and d . Parameters do not exhibit scale dependence at treelevel, but it will become clear that we calculated the effective couplings at the scale M thatis c ( µ = M ) = λ M and d ( µ = M ) = − λ M .Our next example will be computation of the ψψ → ψψ amplitude to the lowest order inthe momenta and to order λ η in the UV coupling constants. Such contribution arises at oneloop. Since loop integration generically yields factors of π ) one expects then a correctionof order η (4 π ) compared to the tree-level amplitude. This is not an accurate estimate ifthere are several scales in the problem. We will assume that m ≪ M , so the scatteringamplitude could contain large log( Mm ). In fact, in an EFT one separates logarithm-enhancedcontributions and contributions independent of large logs. The log-independent contributionsarise from matching and the log-dependent ones are accounted for by the renormalizationgroup (RG) evolution of parameters. By definition, while matching one compares theorieswith different field contents. This needs to be done using the same renormalization scale inboth theories. This so-called matching scale is usually the mass of the heavy particle thatis being integrated out. No large logarithms can arise in the process since only one scale The effective Lagrangian in Eq. (10) is not complete to order λ and p , it only contains all tree-levelterms of this order. For example, the Yukawa coupling ψψϕ receives corrections proportional to ηλ at oneloop.
8s involved. The logs of the matching scale divided by a low-energy scale must be identicalin the two theories since the two theories are designed to be identical at low energies. Wewill illustrate loop matching in the next section. It is very useful that one can compute thematching and running contributions independently. This can be done at different orders inperturbation theory as dictated by the magnitudes of couplings and ratios of scales.In our effective theory described in Eq. (10) we need to find the RG equation for theLagrangian parameters. For concreteness, let us assume we want to know the amplitude atthe scale m . Since we will be interested in the momentum independent part of the amplitude,we can neglect the term proportional to d . By dimensional analysis, the amplitude we areafter must be proportional to λ η π M . The two-derivative term will always be proportionalto M , so it has to be suppressed by m M compared to the leading term arising from thenon-derivative term. This reasoning only holds if one uses a mass-independent regulator,like dimensional regularization with minimal subtraction. In dimensional regularization, therenormalization scale µ only appears in logs.In less suitable regularization schemes, the two-derivative term could contribute as muchas the non-derivative term as the extra power of M could become Λ M , where Λ is the regu-larization scale. With the natural choice Λ ≈ M , the two-derivative term is not suppressedat all. Since the same argument holds for terms with more and more derivatives, all termswould contribute exactly the same and the momentum expansion would be pointless. Thisis, for example, how hard momentum cut off and Pauli-Villars regulators behave. Such reg-ulators do their job, but they needlessly complicate power counting. From now on, we willonly be using dimensional regularization.To calculate the RG running of the coefficient c , we need to obtain the relevant Z factors.First, we need the fermion self energy diagram p p + kk = ( − iη ) Z d d k (2 π ) d i ( /k + /p )( k + p ) ik − m = η Z d d l (2 π ) d Z dx /l + (1 − x ) /p [ l − ∆ ] = iη (4 π ) ǫ (cid:18)Z dx (1 − x ) /p (cid:19) + finite = iη /p π ) ǫ + finite , (11)where we used Feynman parameters to combine the denominators and shifted the loopmomentum l = k + xp . We then used the standard result for loop integrals and expanded d = 4 − ǫ . Only the ǫ pole is kept as the finite term does not enter the RG calculation.The second part of the calculation involves computing loop corrections to the four-fermionvertex. There are six diagrams with a scalar exchange because there are six different pairingsof the external lines. The diagrams are depicted in Fig. 2 and there are two diagrams ineach of the three topologies. All of these diagrams are logarithmically divergent in the UV,so we can neglect the external momenta and masses if we are interested in the divergentparts. The divergent terms must be local and therefore be analytic in the external momenta.Extracting positive powers of momenta from a diagram reduces its degree of divergence whichis apparent from dimensional analysis. Diagrams (a) in Fig. 2 are the most straightforwardto deal with and the divergent part is easy to extract2( − iη ) ic Z d d k (2 π ) d i/kk i/kk ik = − cη Z d d k (2 π ) d k = − icη (4 π ) ǫ + finite . (12)9a) (b) (c)Figure 2: Diagrams contributing to the renormalization of the four-fermion interaction. Thedashed lines represent the light scalar ϕ . The four-fermion vertices are represented by thekinks on the fermion lines. The fermion lines do not touch even though the interaction ispoint-like. This is not due to limited graphic skills of the author, but rather to illustrate thefermion number flow through the vertices.We did not mention the cross diagrams here, denoted { ↔ } in the previous section,since they go along for the ride, but they participate in every step. Diagrams (b) in Fig. 2require more care as the loop integral involves two different fermion lines. To keep track ofthis we indicate the external spinors and abbreviate u ( p i ) = u i . The result is2( − iη ) ic Z d d k (2 π ) d u i/kk u u − i/kk u ik = icη π ) ǫ u γ µ u u γ µ u + finite . (13)This divergent contribution is canceled by diagrams (c) in Fig. 2 because one of the momen-tum lines carries the opposite sign2( − iη ) ic Z d d k (2 π ) d u i/kk u u i/kk u ik . (14)If the divergent parts of the diagrams (b) and (c) did not cancel this would lead to operatormixing which often takes place among operators with the same dimensions. We will illustratethis shortly.To calculate the RG equations (RGEs) we can consider just the fermion part of theLagrangian in Eq. (10) and neglect the derivative term proportional to d . We can thinkof the original Lagrangian as being expressed in terms of the bare fields and bare couplingconstants and rescale ψ = p Z ψ ψ and c = cµ ǫ Z c . As usual in dimensional regularization,the mass dimensions of the fields depend on the dimension of space-time. In d = 4 − ǫ , thefermion dimension is [ ψ ] = − ǫ and [ L ] = 4 − ǫ . We explicitly compensate for this changefrom the usual 4 space-time dimensions by including the factor µ ǫ in the interaction term.10his way, the coupling c does not alter its dimension when d = 4 − ǫ . The Lagrangian isthen L p ,λ η log = iψ /∂ ψ + c ψ ψ ψ ψ = iZ ψ ψ /∂ ψ + c Z c Z ψ µ ǫ ψψ ψψ = iψ /∂ ψ + µ ǫ c ψψ ψψ + i ( Z ψ − ψ /∂ ψ + µ ǫ c Z c Z ψ − ψψ ψψ, (15)where in the last line we separated the counterterms. We can read off the counterterms fromEqs. (11) and (12) by insisting that the counterterms cancel the divergences we calculatedpreviously. Z ψ − − η π ) ǫ and c ( Z c Z ψ −
1) = 2 cη (4 π ) ǫ , (16)where we used the minimal subtraction (MS) prescription and hence retained only the ǫ poles. Comparing the two equations in (16), we obtain Z c = 1 + η (4 π ) ǫ .The standard way of computing RGEs is to use the fact that the bare quantities do notdepend on the renormalization scale0 = µ ddµ c = µ ddµ ( cµ ǫ Z c ) = β c µ ǫ Z c + 2 ǫcµ ǫ Z c + cµ ǫ µ ddµ Z c , (17)where β c ≡ µ dcdµ . We have µ ddµ Z c = π ) ηβ η ǫ . Just like we had to compensate for the di-mension of c , the renormalized coupling η needs an extra factor of µ ǫ to remain dimensionlessin the space-time where d = 4 − ǫ . Repeating the same manipulations we used in Eq. (17),we obtain β η = − ǫη − η d log Z η d log µ . Keeping the derivative of Z η would give us a term that is ofhigher order in η as for any Z factor the scale dependence comes from the couplings. Thus,we keep only the first term, β η = − ǫη , and get µ ddµ Z c = − η (4 π ) . Finally, β c = 6 η (4 π ) c. (18)We can now complete our task and compute the low-energy coupling, and thus thescattering amplitude, to the leading log order c ( m ) = c ( M ) − η (4 π ) c log (cid:18) Mm (cid:19) = λ M (cid:20) − η (4 π ) log (cid:18) Mm (cid:19)(cid:21) . (19)Of course, at this point it requires little extra work to re-sum the logarithms by solving theRGEs. First, one needs to solve for the running of η . We will not compute it in detail here,but β η = η (4 π ) . Solving this equation gives1 η ( µ ) − η ( µ ) = 10(4 π ) log µ µ . (20)Putting the µ dependence of η from Eq. (20) into Eq. (18) and performing the integral yields c ( m ) = C ( M ) (cid:18) η ( m ) η ( M ) (cid:19) , (21)11hich agrees with Eq. (19) to the linear order in log (cid:0) Mm (cid:1) .It is worth pointing out that the Yukawa interaction in the full theory, η ψψϕ , receivescorrections from the exchanges of both the light and the heavy scalars. Hence, the betafunction β η receives contributions proportional to η and ηλ . In fact, β η = η (4 π ) + ηλ (4 π ) . Thebeta function has no dependence on the mass of the heavy scalar nor on the renormalizationscale, so one might be tempted to use this beta function at any renormalization scale. Forexample, this would imply that heavy particles contribute to the running of η at energyscales much smaller than their mass. Clearly, this is unphysical. If this was true, we couldcount all the electrically charged particles even as heavy as the Planck scale by measuringthe charge at two energy scales, for example by scattering at the center of mass energiesequal to the electron mass and equal to the Z mass. The fact that the beta function hasno dependence on the renormalization scale is characteristic of mass-independent regulators,like dimensional regularization coupled with minimal subtraction. When using dimensionalregularization, heavy particles need to be removed from the theory to get physical answersfor the beta function. This is yet another reason why dimensional regularization goes handin hand with the EFT approach.The contribution from the heavy scalar, proportional to ηλ , is absent in the effective the-ory since the heavy scalar was removed from the theory. However, diagrams that reproducethe exchanges of the heavy scalar do exist in the effective theory. Such diagrams are propor-tional to cη and arise from the four-fermion vertex corrections to the Yukawa interaction.Since c is proportional to M , the dimensionless β η must be suppressed by m ψ M . We assumedthat m ψ = 0, so the cη contribution vanishes. Integrating out the heavy scalar changed theYukawa β η function in a step-wise manner while going from the full theory to the effectivetheory. Calculations of the beta function performed using a mass-dependent regulator yield asmooth transition from one asymptotic value of the beta function to another, see for exampleRef. [6]. However, the predictions for physical quantities are not regulator dependent.It is interesting that lack of renormalizability of the four-fermion interaction never playedany role in our calculation. Our calculation would have looked identical if we wanted to obtainthe RGE for the electric charge in QED. The number of pertinent terms in the Lagrangianwas finite since we were interested in a finite order in the momentum expansion.The diagrams we calculated to obtain Eq. (19) are in a one-to-one correspondence withthe diagrams in the full theory. These are depicted in Fig. 3. Even though the EFT calcu-lation may seem complicated, typically the EFT diagrams are simpler to compute as fewerpropagators are involved. Also, computing the divergent parts of diagrams is much easierthan computing the finite parts. In the full theory, one would need to calculate the finiteparts of the box diagrams which can involve complicated integrals over Feynman parameters.There is one additional complication that is common in any field theory, not just in anEFT. When we integrated out the heavy scalar in the previous section, the only momentum-independent operator that is generated at tree level is ψψ ψψ . This is not the only four-fermion operator with no derivatives. There are other operators with the same field contentand the same dimension, for example ψγ µ ψ ψγ µ ψ . Suppose that we integrated out a massivevector field with mass M and that our effective theory is instead L p ,V = iψ /∂ ψ + c V ψγ µ ψ ψγ µ ψ + 12 ( ∂ µ ϕ ) − m ϕ − η ψψϕ. (22)12a) (b) (c)Figure 3: The full theory analogs of the diagrams in Fig. 2. The thicker dashed lines withshorter dashes represent Φ, while the thinner ones with longer dashes represent ϕ .We could ask the same question about low-energy scattering in this theory, that is ask aboutthe RG evolution of the coefficient c V . The contributions from the ϕ exchanges are identicalto those depicted in Fig. 2. The only difference is that the four-fermion vertex contains the γ µ matrices. Diagrams (a) give a divergent contribution to the ψγ µ ψ ψγ µ ψ operator. However,the sum of the divergent parts of diagrams (b) and (c) is not proportional to the originaloperator, but instead proportional to ψσ µν ψ ψσ µν ψ , where σ µν = i [ γ µ , γ ν ]. This means thatunder RG evolution these two operators mix. The two operators have the same dimensions,field content, and symmetry properties thus loop corrections can turn one operator intoanother.To put it differently, it is not consistent to just keep a single four-fermion operator inthe effective Lagrangian in Eq. (22) at one loop. The theory needs to be supplemented sincethere needs to be an additional counterterm to absorb the divergence. At one loop it isenough to consider L p ,V T = iψ /∂ ψ + c V ψγ µ ψ ψγ µ ψ + c T ψσ µν ψ ψσ µν ψ + 12 ( ∂ µ ϕ ) − m ϕ − η ψψϕ, (23)but one expects that at higher loop orders all four-fermion operators are needed. Since weassumed that the operator proportional to c V was generated by a heavy vector field at treelevel, we know that in our effective theory c T ( µ = M ) = 0 and c V ( µ = M ) = 0. At lowenergies, both coefficients will be nonzero.We do not want to provide the calculation of the beta functions for the coefficients c V and c T in great detail. This calculation is completely analogous to the one for β c . The vectoroperator induces divergent contributions to itself and to the tensor operator, while the tensoroperator only generates a divergent contribution for the vector operator. The coefficients ofthe two counterterms are c V ( Z V Z ψ −
1) = η (4 π ) ( − c V + 6 c T ) 1 ǫ , (24) c T ( Z T Z ψ −
1) = η (4 π ) c V ǫ , (25)13here we introduced separate Z factors for each operator since each requires a counterterm.These Z factors imply that the beta functions are β c V = 12 c T η (4 π ) and β c T = 2 ( c T + c V ) η (4 π ) . (26)This result may look surprising when compared with Eqs. (24) and (25). The differencein the structures of the divergences and the beta functions comes from the wave functionrenormalization encoded in Z ψ . It is easy to solve the RGEs in Eq. (24) by treating them asone matrix equation µ ddµ (cid:18) c V c T (cid:19) = 2 η (4 π ) (cid:18) (cid:19) (cid:18) c V c T (cid:19) . (27)The eigenvectors of this matrix satisfy uncoupled RGEs and they correspond to the combi-nations of operators that do not mix under one-loop renormalization. Construction of effective theories is a systematic process. We saw how RGEs can account foreach ratio of scales, and we now increase the accuracy of matching calculations. To improveour ψψ → ψψ scattering calculation we compute matching coefficients to one-loop order.As an example, we examine terms proportional to λ . This calculation illustrates severalimportant points about matching calculations.Our starting point is again the full theory with two scalars, described in Eq. (3). Sincewe are only interested in the heavy scalar field, we can neglect the light scalar for the timebeing and consider L = iψ /∂ ψ − σψψ + 12 ( ∂ µ Φ) − M − λ ψψ Φ + terms that depend on ϕ. (28)We added a small mass, σ , for the fermion to avoid possible IR divergences and also to beable to obtain a nonzero answer for terms proportional to M .(a) (b) (c) (d)Figure 4: Diagrams in the full theory to order λ . Diagram (d) stands in for two diagramsthat differ only by the placement of the loop.The diagrams that contribute to the scattering at one loop are illustrated in Fig. 4. As wedid before, we will focus on the momentum-independent part of the amplitude and we will14ot explicitly write the terms related by exchange of external fermions. The first diagramgives ( a ) = ( − iλ ) Z d d k (2 π ) d u i ( /k + σ ) k − σ u u i i ( − /k + σ ) k − σ u i ( k − M ) = λ (cid:20) − u γ α u u γ β u Z d d k (2 π ) d k α k β ( k − σ ) ( k − M ) + u u u u Z d d k (2 π ) d σ ( k − σ ) ( k − M ) (cid:21) . (29)The loop integrals are straightforward to evaluate using Feynman parameterization1( k − σ ) ( k − M ) = 6 Z dx x (1 − x )( k − xM − (1 − x ) σ ) . (30)The final result for diagram (a) is( a ) F = iλ (4 π ) (cid:20) U V Z dx x (1 − x ) xM + (1 − x ) σ + σ U S Z dx x (1 − x )( xM + (1 − x ) σ ) (cid:21) = iλ (4 π ) (cid:20) U V (cid:18) M + σ M (3 − M σ )) (cid:19) + U S σ M (log( M σ ) − (cid:21) + . . . , (31)where we abbreviated U S = u u u u , U V = u γ α u u γ α u , and in the last line omittedterms of order M and higher. The subscript F stands for the full theory, We will denotethe corresponding amplitudes in the effective theory with the subscript E . The cross boxamplitude (b) is nearly identical, except for the sign of the momentum in one of the fermionpropagators( b ) F = iλ (4 π ) (cid:20) − U V (cid:18) M + σ M (3 − M σ )) (cid:19) + U S σ M (log( M σ ) − (cid:21) + . . . . (32)Diagrams (c) and (d) are even simpler to evaluate, but they are divergent.( c ) F = − iλ (4 π ) σ M U S (cid:20) ǫ + 3 log( µ σ ) + 1 (cid:21) + . . . , (33)where ǫ = ǫ − γ + log(4 π ). µ is the regularization scale and it enters since coupling λ carriesa factor of µ ǫ in dimensional regularization. The four Yukawa couplings give λ µ ǫ . However, µ ǫ should be factored out of the calculation to give the proper dimension of the four-fermioncoupling, while the remaining µ ǫ is expanded for small ǫ and yields log( µ ). In the followingexpression a factor of two is included to account for two diagrams( d ) F = − iλ (4 π ) M U S (cid:20) ǫ + 1 + log( µ M ) + σ M (cid:18) − M σ ) (cid:19)(cid:21) + . . . . (34)The sum of all of these contributions is( a + . . . + d ) F = 2 iλ U S (4 π ) M (cid:20) − ǫ − − log( µ M ) + σ M (cid:18) − ǫ − µ σ ) − M σ ) (cid:19)(cid:21) . (35)15a) (b) (c) (d)Figure 5: Diagrams in the effective theory to order c . Diagram (d) stands in for two diagramsthat are related by an upside-down reflection. As we drew in Fig. 2, the four-fermion verticesare not exactly point-like, so one can follow each fermion line.We also need the fermion two-point function in order to calculate the wave function renor-malization in the effective theory. The calculation is identical to that in Eq. (11). We need thefinite part as well. The amplitude linear in momentum is i/p λ π ) (cid:16) ǫ + log( µ M ) + + . . . (cid:17) .It is time to calculate in the effective theory. The effective theory has a four-fermioninteraction that was induced at tree level. Again, we neglect the light scalar ϕ as it does notplay any role in our calculation. The effective Lagrangian is L = izψ /∂ ψ − σψψ + c ψψ ψψ. (36)We established that at tree level, c = λ M , but do not yet want to substitute the actualvalue of c as not to confuse the calculations in the full and effective theories. To match theamplitudes we also need to compute one-loop scattering amplitude in the effective theory.The two-point amplitude for the fermion kinetic energy vanishes in the effective theory. Thefour-point diagrams in the effective theory are depicted in Fig. 5. Diagrams in an effectivetheory have typically higher degrees of UV divergence as they contain fewer propagators.For example, diagram ( a ) E is quadratically divergent, while ( a ) F is finite. This is not anobstacle. We simply regulate each diagram using dimensional regularization.Exactly like in the full theory, the fermion propagators in diagrams ( a ) E and ( b ) E haveopposite signs of momentum, thus the terms proportional to U V cancel. The parts propor-tional to U S are the same and the sum of these diagrams is( a + b ) E = 2 ic σ (4 π ) U S (cid:20) ǫ + log( µ σ ) (cid:21) + . . . . (37)If one was careless with drawing these diagrams, one might think that there is a closedfermion loop and assign an extra minus sign. However, the way of drawing the effectiveinteractions in Fig. 5 makes it clear that the fermion line goes around the loop withoutactually closing. Diagram ( c ) E is identical to its counterpart in the full theory. Since weare after the momentum-independent part of the amplitude, the heavy scalar propagators in( c ) F were simply equal to − iM . Therefore,( c ) E = − ic σ (4 π ) U S (cid:20) ǫ + 3 log( µ σ ) + 1 (cid:21) + . . . . (38)16s in the full theory, ( d ) E includes a factor of two for two diagrams( d ) E = 2 ic σ (4 π ) U S (cid:20) ǫ + 3 log( µ σ ) + 1 (cid:21) + . . . . (39)The sum of these diagrams is( a + . . . + d ) E = − ic σ (4 π ) U S (cid:20) ǫ + 2 log( µ σ ) + 1 (cid:21) . (40)Of course, we should set c = λ M at this point.Before we compare the results let us make two important observations. There are severallogs in the amplitudes. In the full theory, log( µ M ), log( µ σ ) and log( M σ ) appear, while inthe effective theory only log( µ σ ) shows up. Interestingly, comparing the full and effectivetheories diagram by diagram, the corresponding coefficients in front of log( σ ) are identical.This means that log( σ ) drops out of the difference between the full and effective theories solog( σ ) never appears in the matching coefficients. It had to be this way. We already arguedthat the two theories are identical in the IR, so non-analytic terms depending on the lightfields must be the same. This would hold for all other quantities in the low-energy theory,for instance for terms that depend on the external momenta. This correspondence betweenlogs of low-energy quantities does not have to happen, in general, diagram by diagram, but ithas to hold for the entire calculation. This provides a useful check on matching calculations.When the full and effective theory are compared, the only log that turns up is the log( µ M ).This is good news as it means that there is only one scale in the matching calculation andwe can minimize the logs by setting µ = M .The ǫ poles are different in the full and effective theories as the effective theory diagramsare more divergent. We simply add appropriate counterterms in the full and the effectivetheories to cancel the divergences. The counterterms in the two theories are not related. Wecompare the renormalized, or physical, scattering amplitudes and make sure they are equal.We are going to use the M S prescription and the counterterms will cancel just the ǫ poles.It is clear that since the counterterms differ on the two sides, the coefficients in the effectivetheory depend on the choice of regulator. Of course, physical quantities will not depend onthe regulator.Setting µ = M , the difference between Eqs. (35) and (40) gives c ( µ = M ) = λ M − λ (4 π ) M − λ σ (4 π ) M . (41)To reproduce the two-point function in the full theory we set z = 1 + λ π ) in the M S prescription since there are no contributions in the effective theory. To obtain physicalscattering amplitude, the fermion field needs to be canonically normalized by rescaling √ zψ → ψ canonical . This rescaling gives an additional contribution to the λ (4 π ) M term inthe scattering amplitude from the product of the tree-level contribution and the wave func-tion renormalization factor. Without further analysis, it is not obvious that it is consistentto keep the last term in the expression for c ( µ = M ). One would have to examine if thereare any other terms proportional to M that were neglected. For example, the momentum-dependent operator proportional to d in Eq. (8) could give a contribution of the same order17hen the RG running in the effective theory is included. Such contribution would be propor-tional to λ η σ (4 π ) M log( M m ). There can also be contributions to the fermion two-point functionarising in the full theory from the heavy scalar exchange. We were originally interested ina theory with massless fermions which means that σ = 0. It was a useful detour to do thematching calculation including the M terms as various logs and UV divergences do not fullyshow up in this example at the M order.We calculated the scattering amplitudes arising from the exchanges of the heavy scalar.In the calculation of the ψψ → ψψ scattering cross section, both amplitudes coming fromthe exchanges of the heavy and light scalars have to be added. These amplitudes dependon different coupling constants, but they can be difficult to disentangle experimentally sincethe measurements are done at low energies. The amplitude associated with the heavy scalaris measurable only if the mass and the coupling of the light scalar can be inferred. This canbe accomplished, for example, if the light scalar can be produced on-shell in the s channel.Near the resonance corresponding to the light scalar, the scattering amplitude is dominatedby the light scalar and its mass and coupling can be determined. Once the couplings of thelight scalar are established, one could deduce the amplitude associated with the heavy scalarby subtracting the amplitude with the light scalar exchange. If the heavy and light statesdid not have identical spins one could distinguish their contributions more easily as theywould give different angular dependence of the scattering cross section. Integrating out a fermion in the Yukawa theory emphasizes several important points. Weare going to study the same “full” Lagrangian again, but this time assume that the fermionis heavy and the scalar ϕ remains light L = iψ /∂ ψ − M ψψ + 12 ( ∂ µ ϕ ) − m ϕ − η ψψϕ, (42)where M ≫ m . We will integrate out ψ and keep ϕ in the effective theory. As we did earlier,we have neglected the potential for ϕ assuming that it is zero. There are no tree-leveldiagrams involving fermions ψ in the internal lines only. We are going to examine diagramswith two scalars and four scalars for illustration purposes. The diagrams resemble thoseof the Coleman-Weinberg effective potential calculation, but we do not necessarily neglectexternal momenta. The momentum dependence could be of interest. The two point functiongives = ( − − iηµ ǫ ) Z d d k (2 π ) d i Tr[( /k + /p + M )( /k + M )][( k + p ) − M ]( k − M )= − iη (4 π ) (cid:20) ( 3 ǫ + 1 + 3 log( µ M ))( M − p p − p M + . . . (cid:21) , (43)where we truncated the momentum expansion at order p . The four-point amplitude, to thelowest order in momentum is = − iη (4 π ) (cid:20)
3( 1 ǫ + log( µ M )) − . . . (cid:21) . (44)18here are no logarithms involving m or p in Eqs. (43) and (44). Our effective theoryat the tree-level contains a free scalar field only, so in that effective theory there are nointeractions and no loop diagrams. Thus, logarithms involving m or p do not appearbecause they could not be reproduced in the effective theory. Setting µ = M and choosingthe counterterms to cancel the ǫ poles we can read off the matching coefficients in the scalartheory L = (1 − η π ) ) ( ∂ µ ϕ ) − ( m + 4 η M (4 π ) ) ϕ η π ) M ( ∂ ϕ ) η (4 π ) ϕ
4! + . . . (45)To obtain physical scattering amplitudes one needs to absorb the 1 − η π ) factor in thescalar kinetic energy, so the field is canonically normalized. The scalar effective Lagrangianin Eq. (45) is by no means a consistent approximation. For example, we did not calculate thetadpole diagram and did not calculate the diagram with three scalar fields. Such diagramsdo not vanish since the Yukawa interaction is not symmetric under ϕ → − ϕ . There are nonew features in those calculations so we skipped them.The scalar mass term, m + η M (4 π ) , contains a contribution from the heavy fermion. Ifthe sum m + η M (4 π ) is small compared to η M (4 π ) one calls the scalar “light” compared tothe heavy mass scale M . This requires a cancellation between m and η M (4 π ) . Cancellationhappens when the two terms are of opposite signs and close in magnitude, yet their originsare unrelated. No symmetry of the theory can relate the tree-level and the loop-level terms.If there was a symmetry that ensured the tree-level and loop contributions are equal inmagnitude and opposite in sign, then small breaking of such symmetry could make makethe sum m + η M (4 π ) small. But no symmetry is present in our Lagrangian. This is why lightscalars require a tuning of different terms unless there is a mechanism protecting the massterm, for example the shift symmetry or supersymmetry.The sensitivity of the scalar mass term to the heavy scales is often referred to as thequadratic divergence of the scalar mass term. When one uses mass-dependent regulators,the mass terms for scalar fields receive corrections proportional to Λ (4 π ) . Having light scalarsmakes fine tuning necessary to cancel the large regulator contribution. There are no quadraticdivergences in dimensional regularization, but the fine tuning of scalar masses is just thesame. In dimensional regularization, the scalar mass is quadratically sensitive to heavyparticle masses. This is a much more intuitive result compared to the statement about anunphysical regulator. Fine tuning of scalar masses would not be necessary in dimensionalregularization if there were no heavy particles. For example, if the Standard Model (SM) wasa complete theory there would be no fine tuning associated with the Higgs mass. Perhapsthe SM is a complete theory valid even beyond the grand unification scale, but there isgravity and we expect Planck-scale particles in any theory of quantum gravity. Anotherterm used for the fine tuning of the Higgs mass in the SM is the hierarchy problem. Havinga large hierarchy between the Higgs mass and other large scales requires fine tuning, unlessthe Higgs mass is protected by symmetry.It is apparent from our calculation that radiative corrections generate all terms allowed bysymmetries. Even if zero at tree level, there is no reason to assume that the potential for thescalar field vanishes. The potential is generated radiatively. We obtained nonzero potentialin the effective theory when we integrated out a heavy fermion. However, generation of19erms by radiative corrections is not at all particular to effective theory. The RG evolutionin the full theory would do the same. We saw another example of this in Sec. 2.2, wherean operator absent at one scale was generated radiatively. Therefore, having terms smallerthan the sizes of radiative corrections requires fine tuning. A theory with all coefficientswhose magnitudes are not substantially altered by radiative corrections is called technicallynatural. Technical naturalness does not require that all parameters are of the same order,it only implies that none of the parameters receives radiative corrections that significantlyexceed its magnitude. As our calculation demonstrated, a light scalar that is not protectedby symmetry is not technically natural.Naturalness is a stronger criterion. Dirac’s naturalness condition is that all dimensionlesscoefficients are of order one and the dimensionful parameters are of the same magnitude [12].A weaker naturalness criterion, due to ’t Hooft, is that small parameters are natural if settinga small parameter to zero enhances the symmetry of the theory [13]. Technical naturalnessis yet a weaker requirement. The relative sizes of terms are dictated by the relative sizesof radiative corrections and not necessarily by symmetries, although symmetries obviouslyaffect the magnitudes of radiative corrections. Technical naturalness has to do with howperturbative field theory works. After determining the light field content and power counting of an EFT one turns to enumer-ating higher-dimensional operators. It turns out that not all operators are independent aslong as one considers S -matrix elements with one insertion of higher-dimensional operators.Let us consider again an effective theory of a single scalar field theory that we discussed inthe previous section. Suppose one is interested in the following effective Lagrangian L ϕ = 12 ( ∂ µ ϕ ) − m ϕ − η ϕ − c ϕ + c ϕ ∂ ϕ, (46)where both coefficients c and c are coefficients of operators of dimension 6. We performa field redefinition ϕ → ϕ ′ + c ϕ ′ in the Lagrangian in Eq. (46). Field redefinitions do notalter the S matrix as long as h ϕ | ϕ ′ | i 6 = 0, where | ϕ i is a one-particle state created by thefield ϕ . In other words, ϕ ′ is an interpolating field for the single-particle state | ϕ i . Thisis guaranteed by the LSZ reduction formula which picks out the poles corresponding to thephysical external states in the scattering amplitude.Under the ϕ → ϕ ′ + c ϕ ′ redefinition L ϕ → ( ∂ µ ϕ ′ ) − c ϕ ′ ∂ ϕ ′ − m ϕ ′ − c m ϕ ′ − η ϕ ′ − η c ϕ ′ − c ϕ ′ + c ϕ ′ ∂ ϕ ′ + . . . = ( ∂ µ ϕ ′ ) − m ϕ ′ − ( η
4! + c m ) ϕ ′ − ( c + ηc
3! ) ϕ ′ + . . . , (47)where we omitted terms quadratic in the coefficients c , . This field redefinition removed the ϕ ∂ ϕ term and converted it into the ϕ term. Field redefinitions are equivalent to using thelowest oder equations of motions to find redundancies among higher dimensional operators.The equation of motion following from the Lagrangian in Eq. (46) is ∂ ϕ = − m ϕ − η ϕ .20ubstituting the derivative part of the ϕ ∂ ϕ operator with the equation of motion gives L D> = − c ϕ + c ϕ ∂ ϕ → − c ϕ + c ϕ ( − m ϕ − η ϕ ) = − ( c + ηc
3! ) ϕ − c m ϕ , (48)which agrees with Eq. (47).One might worry that this a tree-level result only. Perhaps the cleanest argument showingthat this is true for any amplitude can be given using path integrals, see Sec. 12 in Ref. [10]and also Refs. [11, 14]. One can show that given a Lagrangian containing a higher dimensionaloperator with a part proportional to the equations of motion L = L D ≤ + c F ( ϕ ) δ L D ≤ δϕ , (49)all correlation functions of the form h ϕ ( x ) . . . ϕ ( x n ) F ( ϕ ( y )) δ L D ≤ δϕ ( y ) i vanish. η ηc c ∂ ∂ (a) (b)Figure 6: Diagrams with one non-derivative quartic interaction and one quartic interactioncontaining ∂ . Diagrams (a) and (b) differ only by the placement of the derivative term. Indiagram (a) the derivative acts on the internal line and shrinks the propagator to a point,while in diagram (b) the derivative acts on any of the external lines.These results can also be obtained diagrammatically. The diagrams in Fig. 6 show thesix-point amplitude arising from one insertion of c ϕ ∂ ϕ . Diagams (a) and (b) differ onlyby the placement of the second derivative. The derivative is associated with the internal linein diagram (a), while in diagram (b) with one of the external lines. The amplitude is A ( a ) = ( − iη ) ik − m ( − ic k )3! = − iηc − iη ( − ic m ik − m , (50)where the momentum dependence of the interaction vertex was used to partially cancel thepropagator by writing k = k − m + m . The two terms on the right-hand side of Eq. (50)have different interpretation. The first term has no propagator, so it represents a local six-point interaction. This is a modification of the ϕ interaction and its coefficient is the sameas the one in Eq. (48) even though it may not be apparent at first. When comparing theamplitudes one needs to keep track of the multiplicity factors. The ϕ interaction comeswith the 6! symmetry factor, while there are (cid:18) (cid:19) choices of the external lines in Fig. 6(a).The term with the propagator on the right-hand side of Eq. (50) together with diagram(b) in Fig. 6 give the modification of the ϕ interaction in Eq. (48). Diagram (b) is associatedwith the 3! · · ϕ term in Eq. (48).21 .6 Summary We have constructed several effective theories so far. It is a good moment to pause andreview the observations we made. To construct an EFT one needs to identify the light fieldsand their symmetries, and needs to establish a power counting scheme. If the full theory isknown then an EFT is derived perturbatively as a chain of matching calculations interlacedby RG evolutions. Each heavy particle is integrated out and new effective theory matched tothe previous one, resulting in a tower of effective field theories. Consecutive ratios of scalesare accounted for by the RG evolution.This is a systematic procedure which can be carried out to the desired order in the loopexpansion. Matching is done order by order in the loop expansion. When two theoriesare compared at a given loop order, the lower order results are included in the matching.For example, in Sect. 2.3 we calculated loop diagrams in the effective theory including theeffective interaction we obtained at the tree level. At each order in the loop expansion, theeffective theory valid below a mass threshold is amended to match the results valid justabove that threshold. Matching calculations do not depend on any light scales and if logsappear in the matching calculations, these have to be logs of the matching scale divided bythe renormalization scale. Such logs can be easily minimized to avoid spoiling perturbativeexpansion. The two theories that are matched across a heavy threshold have in generaldifferent UV divergences and therefore different counterterms.EFTs naturally contain higher-dimensional operators and are therefore non-renormal-izable. In practice, this is of no consequence since the number of operators, and thereforethe number of parameters determined from experiment, is finite. To preserve power countingand maintain consistent expansion in the inverse of large mass scales one needs to employa mass-independent regulator, for instance dimensional regularization. Consequently, therenormalization scale only appears in dimensionless ratios inside logarithms and so it doesnot alter power counting. Contributions from the heavy fields do not automatically decouplewhen using dimensional regularization, thus decoupling should be carried out explicitly byconstructing effective theories.Large logarithms arise from the RG running only as one relates parameters of the theoryat different renormalization scales. The field content of the theory does not change whileits parameters are RG evolved. However, distinct operators of the same dimension can mixwith one another. The RG running and matching are completely independent and can bedone at unrelated orders in perturbation theory. The magnitudes of coupling constants andthe ratios of scales dictate the relative sizes of different contributions and dictate to whatorders in perturbation theory one needs to calculate. A commonly repeated phrase is thattwo-loop running requires one-loop matching. This is true when the logarithms are verylarge, for example in grand unified theories. The log( M GUT M weak ) is almost as large as (4 π ) , sothe logarithm compensates the loop suppression factor. This is not the case for smaller ratiosof scales.The contributions of the heavy particles to an effective Lagrangian appear in both renor-malizable terms and in higher dimensional terms. For the renormalizable terms, the con-tributions from heavy fields are often unobservable as the coefficients of the renormalizableterms are determined from low-energy experiments. The contributions of the heavy fieldssimply redefine the coefficients that were determined from experiments instead of being pre-22icted by the theory. The coefficients of the higher-dimensional operators are suppressed byinverse powers of the heavy masses. As one increases the masses of the heavy particles, theireffects diminish. This is the observation originally made in Ref. [3]. This typical situationis referred to as the decoupling of heavy fields.Counterexamples of “non-decoupling” behavior are rare and easy to understand. Thesuppression of higher-dimensional operators can be overcome by large dimensionless coeffi-cients. Suppose that the coefficient of a higher-dimensional operator is proportional to h M ,where h is a dimensionless coupling constant. If h and M are proportional to each other,then taking M → ∞ does not bring h M to 0. Instead, h M can be finite in the M → ∞ limit.This happens naturally in theories with spontaneous symmetry breaking. For example, thefermion Yukawa couplings in the SM are proportional to the fermion masses divided by theHiggs vacuum expectation value. We are going to see examples of non-decoupling in Sec 3.1.The non-decoupling examples should be regarded with some degree of caution. When M islarge, the dimensionless coupling h must be large as well. Thus the non-decoupling result,that is a nonzero limit for h M as M → ∞ , is not in the realm of perturbation theory. Formasses M small enough that the corresponding value of h is perturbative, there is no fall offof h M with increasing M and such results are trustworthy.When the high-energy theory is not known, or it is not perturbative, one still benefitsfrom constructing an EFT. One can power count the operators and then enumerate thepertinent operators to the desired order. One cannot calculate the coefficients, but onecan estimate them. In a perturbative theory, explicit examples tell us what magnitudesof coefficients to expect at any order of the loop expansion. In strongly coupled QCD-liketheories, or in supersymmetric theories, one estimates coefficients differently, see for exampleRefs. [15, 16]. A common task for anyone interested in extensions of the SM is making sure that the pro-posed hypothetical particles and their interactions are consistent with current experimentalknowledge. The sheer size of the Particle Data Book [17] suggests that the amount of avail-able data is vast. A small subset of accurate data, consisting of a few dozen observables onflavor diagonal processes involving the electroweak W and Z gauge bosons, is referred to asthe precision electroweak measurements. The accuracy of the measurements in this set is atthe 1% level or better. We will describe the precision electroweak (PEW) measurements inSec. 3.4.This common task of analyzing SM extensions and comparing with experiments is inprinciple straightforward. One needs to calculate all the observables, including the contri-butions of the proposed new particles, and needs to make sure that the results agree withthe experiments within errors. In practice, this can be quite tedious. When the new parti-cles are heavy compared to the energies at which the PEW measurements were made, onecan integrate the new particles out and construct an effective theory in terms of the SMfields only [18, 19]. The PEW experiments can be used to constrain the coefficients of theeffective theory. This can be, and has been, done once for all, or at least until there is newdata and the bounds need to be updated. Various SM extensions can be constrained by23omparing with the bounds on the effective coefficients instead of comparing to the exper-imental data. The EFT approach in this case is simply a time and effort saver, as directcontact with experimental quantities can be done only once when constraining coefficientsof higher-dimensional operators. Constraints on the effective operators can be used to con-strain masses and couplings of proposed particles. Integrating out fields is much less timeconsuming than computing numerous cross sections and decay widths.The PEW measurements contain some low-energy data, observables at the Z pole, andLEP2 data on e + e − scattering at various CM energies between the Z mass and 209 GeV.Particles heavier than a few hundred GeV could not have been produced directly in theseexperiments, so we can accurately capture the effects of such particles using effective theory.The field content of the effective theory is the same as the SM field content. We know allthe light fields and their symmetries, except for the sector responsible for EW symmetrybreaking. We are going to assume that EW symmetry is broken by the Higgs doubletand construct the effective theory accordingly. Of course, it is possible to make a differentassumption—that there is no Higgs boson and the EW symmetry is nonlinearly realized. Inthat case there would be no Higgs doublet in the effective theory, but just the three eatenGoldstone bosons, see Refs. [20, 21]. However, the logic of applying effective theory in thetwo cases is completely identical, so we only concentrate on one of them. It is worth notingthat the SM with a light Higgs boson fits the experimental data very well, suggesting thatthe alternative is much less likely.Given a Lagrangian for an extension of the SM we want to construct the effective La-grangian L ( ϕ SM ; χ BSM ) −→ L eff = L SM ( ϕ SM ) + X i a i O i ( ϕ SM ) , (51)where we collectively denoted the SM fields as ϕ SM and the heavy fields as χ BSM . Allthe information about the original Lagrangian and its parameters is now encoded in thecoefficients a i of the higher-dimensional operators O i . The operators O i are independent ofany hypothetical SM extension because they are constructed from the known SM fields. Wewill discuss various O i that are important for PEW measurements in the following sections.One can find two different approaches in the literature to constructing effective theoriesfor PEW observables. The difference between the two approaches is in the treatment of theEW gauge sector. In one approach, an EFT is constructed in terms of the gauge boson masseigenstates—the γ , Z , and W bosons. In the other approach, an effective theory is expressedin terms of the SU (2) L × U (1) Y gauge multiplets, A iµ and B µ . Of course, actual calculations ofany experimental quantity are done in terms of the mass eigenstates. In the EFT approach,one avoids carrying out these calculations anyway. However, when one expands around theHiggs vacuum expectation value (vev) one completely looses all information about the gaugesymmetry and the constraints it imposes. For our goal, that is for constraining heavy fieldswith masses above the Higgs vev, using the full might of EW gauge symmetry is a muchbetter choice. The EW symmetry is broken by the Higgs doublet at scales lower than themasses of particles that we integrate out to obtain an effective theory. The interactions inany extension of the SM must obey the SU (2) L × U (1) Y gauge invariance, so we shouldimpose this symmetry on our effective Lagrangian.To stress this point further, let us compare the coefficients of two similar operators written24n terms of the W and Z bosons. ( A ) : W + µ W − µ W + ν W − ν , (52)( B ) : Z µ Z µ Z ν Z ν . (53)Both operators have the same dimension and the same Lorentz structure. Operator ( A ) ispresent in the SM in the non-Abelian part of the gauge field strength A iµν A iµν and has acoefficient of order one. However, operator ( B ) is absent in the SM and can only arise from agauge-invariant operator of a very high dimension, thus its coefficient is strongly suppressedin any theory with a light Higgs. This information is simply lost when one does not usegauge invariance. If one cannot reliably estimate coefficients of operators then the effectivetheory is useless as it cannot be made systematic.From now on, all operators will be explicitly SU (2) L × U (1) Y gauge invariant and builtout of quarks, leptons, gauge and Higgs fields. All the operators we are going to discuss areof dimension 6. There is only one gauge invariant operator of dimension 5 consistent withgauge invariance and it gives the Majorana mass for the neutrinos. The neutrino mass isinconsequential for PEW measurements. Thus, the interesting operators start at dimension6 and given the agreement of the SM with data we do not need operators of dimension 8, orhigher. S and T parameters There is a special class of dimension-6 operators that arises in many extensions of the SM.We are going to analyze this class of operators in this section and the next one as well.These are the operators that do not contain any fermion fields. Such operators originatewhenever heavy fields directly couple only to the SM gauge fields and the Higgs doublet. Weare going to refer to such operators as “universal” because they universally affect all quarksand leptons through fermion couplings to the SM gauge fields. Sometimes such operatorsare referred to as “oblique.”It is easy to enumerate all dimension-6 operators containing the gauge and the Higgsfields only. The operator ( H † H ) , where H denotes the Higgs doublet, is an example. Thisoperator is not constrained by the current data, as we have not yet observed the Higgsboson. It alters the Higgs potential, but without knowing the Higgs mass and its couplingswe have no information on operators like ( H † H ) . Here are another two operators that arenot constrained by the present data: H † H D µ H † D µ H and H † H A iµν A iµν . Since there areno experiments involving Higgs particles, operators involving the Higgs doublet are sensitiveto the Higgs vev only. After electroweak symmetry breaking, the two operators we justmentioned renormalize dimension-4 terms that are already present in the SM: the Higgskinetic energy and the kinetic energies of the W and Z bosons, respectively.Two important, and very tightly constrained experimentally, universal operators are O S = H † σ i HA iµν B µν , (54) O T = (cid:12)(cid:12) H † D µ H (cid:12)(cid:12) , (55) Even in theories without a light Higgs, in which the electroweak symmetry is nonlinearly realized, thereis still information about the SU (2) L × U (1) Y gauge invariance. Such effective theories can also be writtenin terms of gauge eigenstates. σ i are the Pauli matrices, meanwhile B µν and A iµν are the U (1) Y and SU (2) L fieldstrengths, respectively. The operator O S introduces kinetic mixing between B µ and A µ whenthe vev is substituted for H . The second operator, O T , violates the custodial symmetry. Thecustodial symmetry guarantees the tree-level relation between the W and Z masses, M W = M Z cos θ w , where θ w is the weak mixing angle. After substituting h H i in O T , O T ∝ Z µ Z µ while there is no corresponding contribution to the W mass.The custodial symmetry can be made explicit by combining the Higgs doublet H with˜ H = iσ H ∗ into a two-by-two matrix Ω = (cid:16) ˜ H, H (cid:17) , see for example Ref. [22] for more details.The SM Higgs Lagrangian L Higgs = 12 tr (cid:0) D µ Ω † D µ Ω (cid:1) − V (cid:0) tr(Ω † Ω) (cid:1) (56)is invariant under SU (2) L × SU (2) R transformations that act Ω → L Ω R † . The Higgs vevbreaks SU (2) L × SU (2) R to its diagonal subgroup which is called the custodial SU (2) c .The custodial symmetry is responsible for the relation M W = M Z cos θ w . The operator O T is contained in the operator tr(Ω † D µ Ω σ ) tr( D µ Ω † Ω σ ) that does not preserve SU (2) L × SU (2) R , but only preserves its SU (2) L × U (1) Y subgroup.For the time being, we want to consider the SM Lagrangian amended by the two higher-dimensional operators in Eqs. (54) and (55): L = L SM + a S O S + a T O T . (57)We called these operators O S and O T because there is a one-to-one correspondence betweenthese operators and the S and T parameters of Peskin and Takeuchi [23] , see also Ref. [24]for earlier work on this topic. The S and T parameters are related to the coefficients a S and a T in Eq. (57) as follows S = 4 scv α a S and T = − v α a T , (58)where v is the Higgs vev, s = sin θ w , c = cos θ w , and α is the fine structure constant.The coefficients a S and a T should be evaluated at the renormalization scale equal to theelectroweak scale. In practice, scale dependence is often too tiny to be of any relevance.The experimentally allowed range of the S and T parameters is shown in Fig. 7. This isa key figure for understanding the EFT approach to constraints on new physics from PEWmeasurements. The colored regions are allowed at the 1 σ confidence level. The regionsindicate the values of the operator coefficients that are consistent with data. What is crucialis that Fig. 7 incorporates all the relevant experimental data simultaneously. This is oftenreferred to as global analysis of PEW measurements. The relevant data are combined intoone statistical likelihood function from which bounds on masses an couplings of hypotheticalnew particles are determined. The global analysis provides more stringent constraints thanconsidering a few independent experiments and it also takes into account the correlationsbetween experimental data. There are three parameters introduced in Ref. [23]: S , T , and U . The U parameter corresponds to adimension-8 operator in a theory with a light Higgs boson. All three parameters are on equal footing intheories in which the electroweak symmetry is nonlinearly realized. S -1.00-0.75-0.50-0.250.000.250.500.751.00 T all: M H = 117 GeVall: M H = 340 GeVall: M H = 1000 GeV Γ Z , σ had , R l , R q asymmetriesM W ν scattering Q W E 158
Figure 7: Combined constraints on the S and T parameters. This figure is reproducedfrom the review by J. Erler and P. Langacker in Ref. [17]. Different contours correspond todifferent assumed values of the Higgs mass and are all at the 1 σ (39%) confidence level. TheHiggs mass dependence is discussed in Sect. 3.5.The global analysis, that includes all data and correlations, is possible using the EFTmethods. All of the data is included in bounding the effective parameters S and T . One needsto consider the two-dimensional allowed range for S and T instead of the independent boundson these parameters. When S and T are bounded independently, one of the parameters isvaried while the other one is set to zero. This only gives bounds along the S = 0 and T = 0axes of Fig. 7, and the corresponding limits are S = − . ± .
09 and T = 0 . ± .
09 [17].It is clear that Fig. 7 contains a lot more information. Suppose that an extension of theSM predicts nonzero values of S and T depending, for the sake of argument, on one freeparameter. The allowed range of this free parameter depends on how S and T are correlated.If S and T happen to vary along the elongated part of Fig. 7 the allowed range could be quitelarge. If S and T happen to lie along the thin part of the allowed region, the range could bequite small. This information would not be available if one considered one effective parameterat a time by restricting the other one to be zero. Considering simultaneous bounds on S and T is equivalent to using the likelihood function directly from the data and the EFT providessimply an intermediate step of the calculation. In Sections 3.2 and 3.3 we are going to studyeffective Lagrangians, very much like the one in Eq. (57), with more effective operators, butthe logic of the approach will be exactly the same.Provided with the bounds in Fig. 7 one simply needs to match an extension of the SMto Eq. (57). We will consider here a hypothetical fourth family of quarks as an example. Wewill call such new quarks B and T and assume that they have the same SU (2) L × U (1) Y L new = iQ L /DQ L + iT R /DT R + iB R /DB R − h y T Q L ˜ HT R + y B Q L HB R + H.c. i , (59)where Q L = (cid:18) TB (cid:19) L is the left-handed SU (2) doublet and y T,B are the Yukawa couplings.Given that h H i = (cid:18) v √ (cid:19) , the quark masses are M B,T = v √ y B,T . Since the new quarks, B and T , do not couple directly to the SM fermions, the operators induced by integratingout these quarks are necessary universal. The Yukawa part of the quark Lagrangian canbe rewritten using the matrix representation of the Higgs field, Ω, by combining the right-handed fields into a doublet y T Q L ˜ HT R + y B Q L HB R + H.c. = y T + y B Q L Ω Q R + y T − y B Q L Ω σ Q R + H.c. (60)Due to the presence of σ in the term proportional to y T − y B , that term violates the custodialsymmetry, so we can expect contributions to the T parameter whenever y T = y B . A iµ B µ (a) (b)Figure 8: Fermion contributions to the operators O S (a) and O T (b). The dashed linesrepresent the Higgs doublet.Before we plunge into calculations we can estimate how the S and T parameters dependon the quark masses. The one-loop diagrams are depicted in Fig. 8. Assuming that M B = M T and therefore y B = y T , the contribution to the S parameter can be estimated from diagram(a) in Fig. 8 to be a S ∼ N c (4 π ) gg ′ y M ≈ N c gg ′ (4 π ) y y v = N c gg ′ (4 π ) v , (61)where N c = 3 is the number of colors, while g and g ′ are the SU (2) L and U (1) Y gaugecouplings, respectively. The external lines consist of two gauge fields and two Higgs fields,hence the diagram is proportional to the square of the Yukawa coupling and to the g and g ′ gauge couplings. This is an example of a non-decoupling result as a S is constant for largequark mass M . Using Eq. (58), we expect that S ∼ N c π for large M . This is the situationwe mentioned in Sec. 2.6 where dimensionless coefficients compensate for mass suppression.The T parameter is even more interesting. Let us assume that M T ≫ M B so that only the T quark runs in the loop in Fig. 8(b). The estimate for this diagram is a T ∼ N c (4 π ) y T M T ≈ N c (4 π ) M T v (62)28nd thus T ∼ N c π M T v . Since four powers of the Yukawa coupling are needed to generate O T ,it is not surprising that the T parameter grows as M T . If we did not take into account thefull SU (2) L × U (1) Y symmetry, the Higgs Yukawa couplings would have been absorbed intoquark masses and it would be difficult to understand Eqs. (61) and (62).The actual calculation is easy, but there is one complication. We have been treating thequarks as massive while they only obtain masses when the theory is expanded around theHiggs vev. Chiral quarks are not truly massive fields, so we need a small trick. We aregoing to match the theories in Eqs. (57) and (59) with the Higgs background turned on.We will compare Eqs. (57) and (59) as a function of the Higgs vev [25]. We do not need tokeep any external Higgs fields and only keep the external gauge bosons. The Higgs vev willappear implicitly in the masses of the quarks. This is quite a unique complication that doesnot happen for fields with genuine mass terms, for example vector quarks. When the Higgsbackground is turned on, the calculation is very similar to the one done in the broken theory.However, we do not need to express the gauge fields in terms of the mass eigenstates. T B T BT B B TA µ A ν A µ A ν A µ A ν A µ A ν + − − Figure 9: Diagrams that contribute to O T in the Higgs background. The Higgs vev isincorporated into the masses of the quarks in this calculation.Expanding O T in Eq. 55 around the Higgs vev gives (cid:12)(cid:12) H † D µ H (cid:12)(cid:12) = v g ( A µ ) + . . . , wherewe omitted terms with the B µ field and terms with derivatives. The relevant diagramsare shown in Fig. 9 and they can be calculated at zero external momentum. We need tosubtract diagrams with two external A µ bosons because the diagrams with A µ ’s contributeto both the O T operator and to an overall, custodial symmetry preserving, gauge boson massrenormalization. The operators that preserve custodial symmetry have equal coefficients ofterms proportional to ( A µ ) and ( A µ ) . TTA µ A ν = − N c (cid:18) igµ ǫ (cid:19) Z d d k (2 π ) d i tr[ γ µ P L ( /k + M T ) γ ν P L ( /k + M T )]( k − M T ) = iN c g M T g µν (4 π ) (cid:18) ǫ + ln( µ M T ) (cid:19) , (63)where P L = − γ . The diagram with the B quark in the loop gives the same answer, exceptfor the M T → M B replacement. The two diagrams with external A µ bosons are identical29nd can be evaluated as TBA µ A ν = − N c (cid:18) igµ ǫ (cid:19) Z d d k (2 π ) d i tr[ γ µ P L ( /k + M T ) γ ν P L ( /k + M B )]( k − M T )( k − M B )= iN c g g µν π ) Z dx ( xM T + (1 − x ) M B ) (cid:20) ǫ + ln( µ xM T + (1 − x ) M B ) (cid:21) = iN c g g µν π ) M T + M B ǫ + M T ln( µ M T ) − M B ln( µ M B ) + M T − M B M T − M B ) . (64)When combining the four diagrams in Fig. 9, the divergent parts of Eqs. (63) and (64) cancel.In a renormalizable theory there cannot be any divergences for higher-dimensional operators,as divergences would indicate need for new counterterms and would spoil renormalizability.The remaining, finite, part gives the T parameter when the amplitude is compared with O T expanded around the Higgs vev and the relation in Eq. (58) is used T = − N c v α (4 π ) M T M B ln( M T M B ) − M T + M B M T − M B . (65)As we anticipated, for large M T , T ∝ M T v [26]. Moreover, it is easy to check that T → M B → M T which is consistent with the argument based on custodial symmetry.Another example of a field that contributes to the T parameter is a scalar that transformsin the three-dimensional representation of SU (2) L . We postpone the discussion of tripletscalars to Appendix A. Integrating out the triplet at tree level is not more involved than theexamples presented in this section. Obtaining one-loop results is more tedious and it wouldtake too much space here, hence the triplet example is presented in the appendix.To calculate the quark doublet contribution to the S parameter we expand O S aroundthe Higgs vev, O S = − v A µν B µν + . . . . There are four diagrams that contribute, these areshown on the left-hand sides of Eqs. (66) through (69). We assume that the quark doublethas hypercharge Y such that D µ = ∂ µ − ig σ i A iµ − ig ′ Y B µ to make our result general. Fora genuine fourth generation quark doublet, Y = . In order to simplify this calculationfurther, we calculate the diagrams mixing A µ and B ν and only keep terms proportional to p g µν . TA µ B ν P L P R = − igg ′ N c (4 π ) (cid:18) Y + 12 (cid:19) p g µν . . . , (66) BA µ B ν P L P R = igg ′ N c (4 π ) (cid:18) Y − (cid:19) p g µν . . . , (67) TA µ B ν P L P L = − igg ′ N c (4 π ) Y p g µν (cid:18) ǫ + ln( µ M T ) − (cid:19) + . . . , (68) BA µ B ν P L P L = igg ′ N c (4 π ) Y p g µν (cid:18) ǫ + ln( µ M B ) − (cid:19) + . . . , (69)30here we omitted all terms that do not depend on the momentum as p g µν . Summing thefour diagrams in Eqs. (66) through (69) and matching to the expansion of O S gives v a S = gg ′ N c π ) (1 + 2 Y log( M B M T )) . (70)Using the conversion factor (58) between a S and S yields S = N c π (1 + 2 Y log( M B M T )) . (71)Indeed, the S parameter does not depend on the quark mass when M B = M T , which is anexample of non-decoupling. Y and W A systematic study of all operators of dimension 6 shows that O S and O T are not theonly operators that can be called universal [27]. There are two more operators that can beconstructed out of the gauge fields only O Y = 12 ( ∂ ρ B µν ) , (72) O W = 12 ( D ρ A iµν ) . (73)These operators are clearly of the same dimension as O S and O T and just as important. Itturns out that there are no more universal operators of dimension 6 that are bound by thecurrent data. The effective Lagrangian L = L SM + a S O S + a T O T + a Y O Y + a W O W (74)contains all the universal operators for which PEW constrains exist.It is useful to rewrite O Y using the Bianchi identity ∂ ρ B µν + ∂ µ B νρ + ∂ ν B ρµ = 0 O Y = 12 ( − ∂ ρ B µν ∂ µ B νρ − ∂ ρ B µν ∂ ν B ρµ ) = ( ∂ µ B µν ) , (75)where the last equality is obtained by integrating by parts and using the antisymmetry ofthe field strength. Similarly, O W = ( D µ A iµν ) . (76)These forms are often more suitable for calculations.As an example of applicability of this formalism we consider a U (1) extension of the SM.Suppose that the SM gauge symmetry is extended to [ SU (3) c × SU (2) L × U (1) Y ] × U (1) ′ such that the Lagrangian is L = L SM −
14 ( B ′ µν ) + κ B µν B ′ µν + L (Φ) , (77)where L (Φ) is a scalar field Lagrangian that spontaneously breaks the U (1) ′ symmetry. Thedetails of L (Φ) are not relevant and we will assume that as a result of symmetry breaking the31auge field B ′ µ acquires mass M . We have assumed that the new sector communicates withthe SM only through the kinetic mixing with the hypercharge U (1) Y which would certainlybe the case if SM fields do not carry any charges under U (1) ′ . There could be heavy particlesthat carry both the SM and U (1) ′ quantum numbers. Such particles would induce kineticmixing between B µ and B ′ µ .To constrain this new theory we need to calculate one diagram only. The Lagrangian inEq. (77) gives a tree-level contribution to O Y . B B ′ B = iκ ( p g µα − p µ p α ) − i ( g αβ − p α p β M ) p − M iκ ( p g βν − p β p ν )= iκ p p − M ( p g µν − p µ p ν ) ≈ − iκ M p ( p g µν − p µ p ν ) . (78)We compare this result with the Feynman rule for the operator O Y . Writing O Y explicitlyin terms of the B µ gauge field and derivatives O Y = ( ∂ µ B µν ) = ( ∂ B ν ) − ∂ B ν ) ∂ ν ∂ ρ B ρ + ( ∂ ν ∂ µ B µ )( ∂ ν ∂ ρ B ρ ) , (79)yields the amplitude with one insertion of a Y O Y ia Y ( g µν p − p p µ p ν + p p µ p ν ) = 2 ia Y p ( p g µν − p µ p ν ) . (80)Comparing Eqs. (78) and (80) gives a Y = − κ M . (81)Ref. [27] contains combined bounds on the coefficients of the four universal operators, includ-ing the bounds on a Y . Obtaining the bounds on the U (1) extension of the SM was certainlya straightforward exercise, yet it is not a simplified toy example. Many extension of the SMcontain extra U (1) gauge symmetries and such extensions are studied in the literature, seefor instance Ref. [28]. There are many extensions in which the heavy fields couple directly to the SM fermions. Insuch extensions integrating out the heavy fields yields not only the universal operators thatare included in Eq. (74) but yields other operators as well. We now turn to examine a largera class of operators that will enable us to constrain a wide range of SM extensions.A complete list of all baryon and lepton number conserving operators of dimension 6 inthe SM is provided in Ref. [18]. The equations of motion were used to eliminate redundantoperators and there are still 80 operators listed in Ref. [18] even with the assumption thatthere is only one family of quarks and leptons. We will use the notation of Ref. [18] for thenames of the operators.In most of this section we will follow the analysis in Ref. [29]. There are several symmetryassumptions one can make to focus on the operators that are relevant to PEW measurements.The most important assumption is about flavor and CP violation. It is likely that the flavor32tructure in the SM is generated at a much higher scale than the EW symmetry breakingscale. Current constraints on flavor and CP violation expressed as bounds on coefficients ofdimension 6 operators point to suppression scales of order 10 to 10 TeV. Such stringentconstraints can be inferred, for example, from the K − K mass difference or from the limitson the µ → eγ decay. It is then reasonable to assume that that the electroweak symmetrybreaking is independent of flavor physics. It is possible to lower the scale of new flavorphysics, for example by assuming the minimal flavor violation structure of new physics [30],but we will assume that the EW and flavor scales are well separated. We will concentrateon operators that have nothing to do with flavor, but that can be relevant for modificationsof the electroweak symmetry-breaking sector.The SM has a large flavor symmetry when the Yukawa couplings are neglected. Thekinetic energy terms for the fermions do not distinguish fields of different flavors. Thus, forthree families of fermions with the same charge assignment there is a U (3) symmetry. Forinstance, if we denote by u the triplet of the right-handed up, charm, and top quarks thenthe kinetic terms for the right-handed up quarks are invariant under U (3). Suppressing theflavor indices, we have a U (3) symmetry under which q → U q q, u → U u u, d → U d d, l → U l l, e → U e e, (82)where q denotes the left-handed quarks, u and d the right-handed quarks, l the left-handedleptons, and e the right handed leptons. We will assume that the operators of interest obeythe U (3) flavor symmetry.Imposing the U (3) symmetry and CP conservation on the 80 operators in Ref. [18]reduces the list to 52 operators. At this step, operators that change fermion chirality areeliminated since fermions of different chiralities transform independently under the U (3)flavor symmetry.It is only worthwhile to consider operators that are well constrained by the data, aspoorly constrained operators contribute little to constraints on hypothetical new particles.The bounds on some operators are very mild. This is the case for operators that only affectQCD processes for which experimental precision does not match that of PEW measurements.For example, four-fermion quark operators, or the f abc G aµν G bνρ G cρµ operator, are rather poorlyconstrained, where G aµν is the gluon field strength. We therefore study operators that eithercontain some SU (2) L × U (1) Y gauge fields, or contain some leptons. This reduces the numberof operators to 34.Of the 34 remaining operators, 6 are not observable at present, as they renormalizeexisting terms in the SM Lagrangian when the Higgs field is replaced by its vev. We saw afew examples of such operators in Sect. 3.1. Finally, 7 operators satisfy all assumptions wemade so far, but are nevertheless very poorly constrained by the available data. All 7 areoperators of the form iψγ µ D ν ψF µν , where ψ represents SM fermions. The interference termsbetween such operators and the SM contributions vanish, except at the Z pole. However,at the Z pole the interference term is suppressed by Γ Z M Z . Since the interference terms withthe SM vanish for such operators, the amplitude square is proportional to the square of theoperator coefficient which would be of the same order as an interference term of a dimension The flavor symmetry assumption can be relaxed, for example to single out the third generation, seeRef. [31]. In some models, the third generation is integral to EW symmetry breaking.
33 operator with the SM. Thus, it would not be consistent to keep the operators of the form iψγ µ D ν ψF µν while we otherwise have truncated the expansion at dimension 6.We are left with 21 operators that can be divided into 4 classes.1. Two universal operators O S and O T . (These are, respectively, called O W B and O H inRef. [18].)2. 11 four-fermion operators O sll = 12 ( lγ µ l )( lγ µ l ) , O tll = 12 ( lγ µ σ a l )( lγ µ σ a l ) ,O slq = ( lγ µ l )( qγ µ q ) , O tlq = ( lγ µ σ a l )( qγ µ σ a q ) ,O le = ( lγ µ l )( eγ µ e ) , O qe = ( qγ µ q )( eγ µ e ) ,O lu = ( lγ µ l )( uγ µ u ) , O ld = ( lγ µ l )( dγ µ d ) ,O ee = 12 ( eγ µ e )( eγ µ e ) , O eu = ( eγ µ e )( uγ µ u ) , O ed = ( eγ µ e )( dγ µ d ) .
3. 7 operators that are products of the Higgs current with various fermion currents O sHl = i ( H † D µ H )( lγ µ l ) + H . c ., O tHl = i ( H † σ a D µ H )( lγ µ σ a l ) + H . c .,O sHq = i ( H † D µ H )( qγ µ q ) + H . c ., O tHq = i ( H † σ a D µ H )( qγ µ σ a q ) + H . c .,O Hu = i ( H † D µ H )( uγ µ u ) + H . c ., O Hd = i ( H † D µ H )( dγ µ d ) + H . c .,O He = i ( H † D µ H )( eγ µ e ) + H . c . . When the vev is substituted for the Higgs doublet, these operators modify the couplingsof the Z and W to the fermions.4. One operator that alters the cubic gauge boson couplings O W = ǫ abc W aνµ W bλν W cµλ . Note that the operators O Y and O W discussed in the previous section are not on the list.Eqs. (75) and (76) make it clear that these operators can be easily re-expressed using theequations of motion for the gauge fields, for example ∂ µ B µν = j νY , where j Y is the hyperchargecurrent that consists of the fermion and Higgs contributions. The square of the current canbe written in terms of O T , four-fermion operators, and the operators of the form O Hψ . Moredetails on the use of equations of motion are contained in Refs. [18, 32]. Of course, if theheavy fields couple to the gauge and Higgs bosons only, it is much more straightforward todeal with the set of four universal operators described in Secs. 3.1 and 3.2. If the couplingsare not universal, it is better to avoid O Y and O W in favor of the operator basis presentedin this section because O Y and O W are four-derivative operators. Matching is more messywhen one needs to evaluate diagrams to the fourth order in external momenta.The effective theory we will consider now is L = L SM + X i =1 a i O i , (83)348.27% 95.45% 99.73%1 1 4 92 2.29 6.18 11.83 3.53 8.02 14.25 5.89 11.3 18.210 11.5 18.6 26.921 23.5 33.1 43.5Table 1: Increments of the χ distribution depending on the number of free parametersand on confidence levels. Confidence levels are listed in the top row, while the number ofdegrees of freedom in the leftmost column. The “allowed” values of χ are those for which χ ≤ χ (“best fit”) + ∆.where O i stand for the operators enumerated in this section. As we did before, to constrainan extension of the SM one matches the new theory to the effective Lagrangian (83). Withmore than two parameters, it is difficult to visualize the experimentally allowed space of thecoefficients a i . We will discuss how the constraints on a i are obtained in Sec. 3.5. Briefly,each relevant observable X α is computed as a function of the SM couplings, collectivelydenoted g SM , and the coefficients a i X α ( g SM , a i ) = X SMα ( g SM ) + a i X iα + a i a j X ijα , (84)where X iα is the interference term between SM and operator O i and X ijα are the productsof the amplitudes containing an insertion of O i and an insertion of O j . As we mentionedearlier, terms quadratic in a i can be neglected because these would be equivalent, by powercounting, to the interference of dimension-8 terms with the SM amplitudes. By comparingwith experimental data, a χ distribution is constructed χ ( a i ) = X α ( X expα − X α ( a i )) σ α = χ min + X i,j =1 ( a i − ˆ a i ) M ij ( a j − ˆ a j ) , (85)where the last equation follows because χ ( a i ) is quadratic in a i . This is because we onlykept the linear terms in a i in Eq. (84). The sum over α runs over all different observablequantities, X expα are the measured values of the observable, while σ α are the correspondingerrors. In practice, one also needs to take into account correlations between measurements,but this does not change the fact that χ ( a i ) is quadratic in a i .It is worth stressing that the matrix M ij in Eq. (85) and the coefficients ˆ a i , for which χ is minimized, are constants determined from experiments. The allowed region in the spaceof coefficients is a 21-dimensional ellipsoid centered at ˆ a i whose axes are determined by thematrix M ij . Eq. (85) is an analog of the S − T plot in Fig. 7. The S − T plot is obtainedwhen all coefficients, except a S and a T , are set to zero.By matching, the operator coefficients a i are calculated in terms of the masses andcouplings of the heavy fields. The allowed range of the parameters is then determined byfinding the minimum of χ and accepting the values of the underlying parameters for which χ ≤ χ (“best fit”) + ∆, where ∆ is determined by the desired confidence level and by the35umber of free parameters. Table 1 shows the values of ∆ for several confidence levels andseveral numbers of free parameters. In general, χ (“best fit”) ≥ χ min , but χ (“best fit”) isless than the SM value χ ( a i = 0).Eq. (85) allows one to constrain arbitrary linear combinations of operators O i instead ofjust constraining each coefficient independently one at a time. As we already discussed inSec. 3.1, this is necessary for implementing a global analysis in the EFT approach. Oncethe heavy fields are integrated out, the operator coefficients a i are given in terms of theunderlying parameters. The coefficients a i are determined by the same couplings and massesof the heavy states in the full theory, so these coefficients are typically not independent.In the remainder of this section we are going to consider two sample extensions of the SMand integrate out the heavy fields to further illustrate how one obtains the coefficients a i andthus how one constrains new theories. As the first example, suppose that the EW sector ofthe SM is extended to have the SU (2) × SU (2) × U (1) Y gauge group. The SU (2) × SU (2) group is spontaneously broken to its diagonal subgroup, that is to SU (2) L . Moreover, weassume that the SM fermions are charged under the SU (2) group, while the SM Higgs bosonis charged under SU (2) so that the couplings are L = g A i µ j iµψ + g A i µ j iµH , (86)where j iµψ = q σ i γ µ q + l σ i γ µ l is the SU (2) fermion current, while j iµH = iH † σ i D µ H − i ( D µ H † ) σ i H is the SU (2) Higgs current. When the product SU (2) × SU (2) group isbroken to the diagonal SU (2) L , the SU (2) L coupling constant is given by g = g g p g + g and g = g s H = g c H , (87)where we introduced the sine and cosine of the mixing angle between the gauge couplings,denoted s H and c H , respectively. One linear combination of the vector bosons, W iH = c H A i − s H A i , becomes massive, while the orthogonal combination gives the A i bosons ofthe SU (2) L . The SU (2) × SU (2) gauge bosons can be expressed as A i = c H W iH + s H A i and A i = c H A i − s H W iH . Diagrams representing tree-level exchanges of W H are shown inFig. 10. Integrating out W iH gives L = − g c H M ( j iµψ ) + g g s H c H M j iψµ j iµH − g s H M ( j iµH ) , (88)where M is the mass of the heavy vector bosons. Since the operator ( j iµH ) does not breakthe custodial symmetry, it does not contain a piece proportional to O T . O T is the onlyoperator on our list with the same field content as ( j iµH ) that is containing just the Higgsfields and derivatives. If there is no O T in ( j iµH ) we can neglect this term because ( j iµH ) must correspond to unobservable, or poorly constrained, operators. This can be checked byan explicit calculation. The other two products of currents give a tlq = a tll = − g c H s H M and a tHl = a tHq = g M . (89)36 ψ ψH HH Figure 10: Diagrams with exchanges of heavy vector bosons that give products of the fermionand Higgs currents.Our next example is an additional vector-like doublet of quarks. We choose the left-handed doublet Q to have the same hypercharge as the SM quark doublets, so that thequarks can mix. The Lagrangian for the heavy quarks is L = − M QQ − ( λ d QdH + λ u Qu ˜ H + H.c. ) , (90)where the mass term is allowed because both the right- and left-handed components of Q have the same quantum numbers. The relevant diagram is shown in Fig. 11(a). Since thisdiagram will match to the O Hd operator we need to extract the amplitude proportional to onepower of the external momentum. The corresponding amplitude for the external d quarks is A = ( − iλ d ) u ( p ) P L i ( /p + M ) p − M P R u ( p ) , (91)where the d quarks are by assumption right-handed, so the projection operators pick out the /p part of the propagator in Eq. (91). The momentum flowing through the internal line is p = p + p = p + p = p + p + p + p . However, the external quarks are massless, so /p u = 0and u/p = 0. Comparing this result with the amplitude from an insertion of O Hd we obtain a Hd = λ d M . (92)Obtaining the amplitude with external u quarks is just as simple, but one needs to convertthe current written in terms of ˜ H to the current written in terms of H . This results in anextra minus sign compared to Eq. (92) a Hu = − λ u M . (93)(a) (b) Hd, u Q
Figure 11: Heavy doublet contributions to O Hd and O Hu .It is worth pointing out that when we matched the UV amplitude to the operators O Hd and O Hu we only took into account the partial derivative part of the Higgs current.37hese operators also have a part proportional to the gauge fields. This part arises from thediagram in Fig. 11(b). We could have calculated either diagram (a) or (b), but since thetwo are related by gauge invariance it was enough to calculate one of them. Extracting theamplitude with the gauge fields allows one to neglect all external momenta A = ( − iλ d ) u d i ( /p + M ) p − M ( i g σ i /A i + i g ′ /B ) i ( /p + M ) p − M u d ≈ λ d M u d ( i g σ i /A i + i g ′ /B ) u d , (94)which is even simpler than the previous calculation. This agrees with Eq. (92) when either theamplitude with an external B µ or with an external A iµ are compared with the correspondingterms in O Hd .An interesting exercise is checking the results in Eqs. (92) and (93) directly by diago-nalizing the quark mass terms in Eq. (90). The light mass eigenstates are mixtures of the“original” SM right-handed quarks d and u with the right-handed part of Q . Since Q hasdifferent quantum numbers than u and d , the light quarks couple differently to the Z bosoncompared to the ordinary SM quarks. The modifications of the Z couplings can be com-pared, and have to agree, with those given by the operators O Hd and O Hu . Refs. [33] containseveral further examples of various applications of this formalism for constraining interestingextensions of the SM. The PEW constraints arise from data gathered by many different experiments. For thepurpose of this discussion we divide the data into four categories. We briefly review the typesof data in this section and discuss which operators are sensitive to different measurements.The four types of measurements are1. Z -pole observables gathered by the experiments at LEP1 and at SLAC. These includethe Z mass M Z , the Z width Γ Z , branching ratios of the Z into quarks and leptons,forward-backward asymmetries, and left-right asymmetries depending on the beampolarizations. The Z -pole measurements achieved very high statistics and typicallythese measurements are the most relevant for PEW constraints. However, not alloperators can be constrained by the Z -pole data.2. W mass. We single out this measurement because of its high accuracy and also becauseit is obtained by both the Tevatron and LEP2 experiments. Due to its accuracy, thismeasurement puts very stringent constraints on several operators.3. LEP2 measurements. These include measurements of e + e − → ψψ scattering at thecenter of mass energies above the Z mass as well as e + e − → W + W − scattering. Thedata that is used includes a combination of total cross sections, asymmetries, anddifferential scattering cross sections in a few channels.4. Low-energy observables. This class encompasses many diverse experiments. The twomost precisely measured quantities are the QED fine structure constant α and theFermi coupling G F . There is a lot of data on neutrino scattering, both deep inelasticscattering of ν µ on nucleons, and neutrino-electron scattering. Measurements of atomic38arity violation constitute the next set of measurements. These are usually reportedin terms of an effective weak charge of the nucleus, for example Q W ( Cs ) or Q W ( T l ).The nuclei in which the highest precision has been achieved are cesium and thallium,but there are also measurements of atomic parity violation in lead and bismuth. Otherexperiments include Moller, that is e − e − → e − e − , scattering and the measurements ofthe muon anomalous magnetic moment.No data from hadron colliders, other than the W mass measurement are included in this list.There are many processes which would be useful for constraining effective operators. Forexample, jet production cross sections probe quark four-fermion interactions. The accuracyof such measurements, due to poor knowledge of the parton distribution functions and limitedprecision of hadronic measurements, is much smaller than the accuracy of the measurementsthat are considered PEW observables.In the electroweak sector, the SM has three undetermined parameters that is the gaugecoupling constants g and g ′ and the electroweak vev v . Three most precisely measuredquantities, α , G F , and M Z , are used to determine the parameters of the SM. These threemeasurements cannot be therefore used to constrain new physics. As we will discuss in thenext section, the precision of the measurements requires one-loop electroweak calculationsin the SM that depend on the top quark mass. Even though the top quark mass is known,it has not been measured as accurately as other parameters of the SM. The uncertainty in m top is sometimes important for comparisons of the SM with experiment and needs to beincluded in the estimates of errors.Looking back at Eq. (85), it is clear that experimental uncertainties determine the sizeof the allowed region in the space of coefficients a i that is encoded in the matrix M ij . Thequadratic dependence on a i is solely determined by the uncertainties. The central values ofthe coefficients, denoted ˆ a i in Eq. (85), are determined by the differences between the centralvalues of measurements and the SM predictions. It is important that many measurementsare correlated instead of being independent. The expression for χ in Eq. (85) assumes thatthe experimental quantities are independent, so Eq. (85) needs to be modified to includecorrelations χ ( a i ) = X α,β ( X expα − X α ( a i )) (cid:0) σ αβ (cid:1) − ( X expα − X α ( a i )) , (95)where the error matrix σ αβ can be expressed in terms of the correlation matrix ρ αβ and thestandard deviations as σ αβ = σ α ρ αβ σ β . Correlations are particularly prominent among the Z -pole measurements [34] and among LEP2 measurements. The differential cross sectionsat LEP2 measured at different energies are correlated. As the title suggests, not everyone may be interested in reading this section. In a way, thatis the point of the EFT approach. The bounds on the coefficients of operators have alreadybeen extracted and the details how it was done are not that important. One can constraintheir favorite model without ever being concerned with the actual experimental data.Foremost, to constrain new physics one needs accurate SM calculations. This is a topicthat we will not discuss in these notes. The precision of measurements generally requires39ne-loop electroweak corrections and often higher-oder QCD corrections. The electroweakcorrections depend on the masses of the SM particles, including the unknown Higgs mass.Thus, the predictions are always shown with a chosen reference value for the Higgs mass, asillustrated in Fig. 7. Since the couplings of the Higgs to the light fermions are tiny, only theuniversal parameters are sensitive to the Higgs mass. The leading dependence of S and T on the Higgs mass is logarithmic [23]∆ S ≈ π log M h M h,ref ! and ∆ T ≈ − πc log M h M h,ref ! . (96)It is this dependence that gives indirect estimates of the Higgs mass in the SM.To constrain the coefficients of operators we use the interference terms between the SMand the effective operators. The experimental accuracy of PEW measurements is compa-rable to the one-loop electroweak corrections. Thus, the suppression of higher dimensionaloperators is of the same order. When computing the interference terms electroweak loopcorrections can be neglected, as the product of suppression of higher-dimensional terms withthe electroweak loop suppression is much smaller than the experimental accuracy.Operator(s) shift M W Z-pole ν AP V ψψ W + W − O S α, M Z √ √ √ √ √ O T M Z O tll G F √ √ O sll , O le √ √ O ee √ O slq , O tlq , O lu , O ld √ √ √ O eq , O eu , O ed √ √ O thl G F √ √ √ √ √ O shl , O he √ √ √ √ √ O hu , O hd , O shq , O thq √ √ √ √ O W √ Table 2: Measurements affected by different operators. The abbreviations used for thetypes of measurements are: ν for neutrino scattering experiments, APV for atomic parityviolation, ψψ for e + e − → ψψ at LEP2, and W + W − for e + e − → W + W − at LEP2. The checkmarks, √ , indicate “direct” corrections only. The operators that shift input parameters aremarked in the “shift” column by indicating the affected input quantity.We now examine two examples of how constraints on the coefficients of effective operatorsare obtained. We consider the operators O T and O ee . These examples illustrate two distinctpossibilities. The operator O T does not directly contribute to any observables used forconstraining new physics. There are no diagrams with an insertion of O T that give rise toscattering or decay widths, etc. Instead, O T contributes to the Z mass. Since the Z massdetermines the SM input parameters all SM predictions will be altered when O T is present.We calculate the “shifts” in the values of the input parameters to the linear order in a T because we are interested in the interference terms only. The operators O S , O tll , and O tHl O S , O tll , or O tHl are present, the SM input parametersneed to be shifted and insertions of these operators considered in the scattering amplitudes.A lucid explanation of how to account for the shifts in the SM inputs is contained in Ref. [35].Table 2, adopted from Ref. [29], shows which operators contribute to different experiments, orcontribute to the shifts of the input parameters, and in turn which operators are constrainedby which measurements.In this section we deal with cross sections and decay rates, thus we need to use the gaugeboson mass eigenstates. Expanding O T around the Higgs vev we obtain O T = m Z v Z µ + . . . .Hence if a T O T is present in the effective Lagrangian there is an extra contribution to the Z mass. Below the EW symmetry breaking scale L SM + a T O T ⊃ − A µν A µν − Z µν Z µν + ˆ m Z (1 + γ )2 Z µ Z µ − eA µ j µem − esc Z µ j νNC , (97)where we only included the photon and the Z kinetic terms and their couplings to thecurrents, while γ = a T v . We are going to consider the electric charge, the Z mass, and theweak mixing angle as the input parameters, and we abbreviate s = sin θ w , c = cos θ w . Theseparameters are equivalent to g , g ′ , and v . In the absence of higher dimensional operatorsone would extract the values e , s , and m Z from the measurements of α , M Z , and G F . When a T O T is added to the Lagrangian, one deduces instead the values ˆ e , ˆ s , and ˆ m Z . We usethe lower-case m Z for the Lagrangian parameter, while the physical value of the Z mass isdenoted M Z .Reading off from Eq. (97) we obtain e = ˆ e, m Z = ˆ m Z (1 + γ ) , and 8 G F √ e m Z s c = ˆ e ˆ m Z ˆ s ˆ c , (98)where the expression for G F can be taken as our definition of s and c . Solving these equationsto the linear order in γ givesˆ e = e, ˆ m Z = m Z (1 − γ ) , ˆ s = s (1 + γc c − s ) , and ˆ c = c (1 − γs c − s ) . (99)For every observable, for example the Z width into fermions ψ Γ( Z → ψψ ), one can takethe corresponding tree-level expression in terms of the input parameters and calculate itschange due to the shift in the input parameters in Eq. (99). At the tree level,Γ( Z → ψψ ) = M Z π (cid:0) g V + g A (cid:1) , (100)where g V = ˆ e ˆ s ˆ c ( T − Q ˆ s ) and g A = − ˆ e ˆ s ˆ c T . Meanwhile, T denotes the third componentof the SU (2) L generator, that is ± , and Q the electric charge of the fermion. CombiningEqs. (99) and (100) gives the change of Γ due to O T : δ Γ( Z → ψψ ) = − a T v M Z π e s c (cid:20) ( T − Qs )( T + Q s c − s ) + T (cid:21) . (101)41 , Z Figure 12: The SM diagram and four-fermion contribution to e + e − → µ + µ − .Of course, such a calculation needs to be repeated for every observable before χ in Eq. (95)can be calculated. For instance, M W = ˆ m Z ˆ c = m z (1 − γ ) c (1 − γ s c − s ) = m z c (1 − γ c c − s ) + O ( γ ) (102)so that δM W = − a T v c M Z c − s . In the equation for the predicted change of the W mass,denoted δM W , we replaced m Z with M Z . This is justified because the difference between m Z with M Z is given by loop effects. Loop corrections can be neglected when multipliedby the small parameter a T v . The four operators that shift the input parameters, O S , O T , O tll , and O tHl have the most stringent bounds on their coefficients among all the operatorsconsidered here. This happens because shifts of the input parameters affect all observables,so all measurements are statistically combined when obtaining bounds.Let us briefly examine the operator O ee that contributes directly to some observables anddoes not shift the input parameters. O ee = 12 ( eγ µ e )( eγ µ e ) = 12 ( eγ µ e + µγ µ µ + τ γ µ τ ) , (103)where e denotes at first a U (3) triplet of the right-handed leptons and then denotes justthe electron right-handed field. Hopefully, this abuse of notation will not be confusing. Allthe fields are right-handed, so there are implicit chirality projectors in the equation above.Suppose we are interested in the e + e − → µ + µ − scattering. The operator O ee has a verysimple structure and one needs to calculate the interference between the SM graph and thefour-fermion interaction. The Feynman diagrams are depicted in Fig. 12. The amplitude forthe O ee operator is simply A ee = ia ee uγ µ u uγ µ u, (104)where a ee is the coefficient of O ee and u ’s indicate Dirac spinors for the external electronsand muons. The Z boson exchange amplitude is proportional to A Z ∝ (cid:18) igc (cid:19) − ik − M Z + i Γ Z M Z (cid:18) g µν − k µ k ν M Z (cid:19) uγ µ u uγ ν u. (105)We are not going to do this straightforward calculation in detail, but want to point somethingout. At the Z pole, the factor multiplying the spinors in Eq. (105) is real. However,the analogous factor in Eq. (104) is imaginary, so the interference of the two amplitudesvanishes. This is general: four-fermion operators are not significantly constrained by the42-pole measurements. Of course, there is a photon exchange diagram as well, but it issuppressed by the photon propagator and therefore small. The four-fermion operators areconstrained by the low-energy observables and by LEP2 data. A Scalar triplet contributions to the T parameter Scalars that transform in the triplet representation of SU (2) L are a common ingredient ofmany extensions of the SM. Triplet scalars contribute to the T parameter because theyviolate the custodial symmetry if they acquire a vev. In this section we will integrate outscalar triplets at the tree and one-loop levels. One of the reasons for the discussion at theone-loop order are claims in the literature that the effects of triplets on the T parameterdo not decouple when the triplet mass is very large [36, 37]. This is difficult to understandbased on power counting. We discuss the power counting at the end of Sec. A.1 and againat the beginning of Sec. A.2 before we describe the loop calculations. However, we do nothave an answer as to why the result obtained here and the results in Refs. [36, 37] disagreequalitatively.We first calculate the tree-level contribution of triplets to the T parameter. We obtainthe coefficient of the T operator in several different ways in Sec. A.1. The method that mayseem the least straightforward at tree level will turn out to be useful in loop calculations. InSec. A.2 we calculate one-loop matching coefficients, but do not calculate one-loop runningof the T operator in the effective theory. While the RG contributions can be numericallysignificant, it is clear that such contributions cannot alter decoupling. The one-loop RGlogs multiply the tree-level contribution, so the decoupling of the tree-level result impliesdecoupling of the RG-corrected contribution. A.1 Tree level
Scalar triplets, like any other scalars that are not in the doublet representation of the SU (2) L ,violate the custodial symmetry if they acquire a vev. Thus, we are interested in the tripletcontributions to O T . The triplet can only obtain a vev if its hypercharge is either 0 or ± U (1) EM . We will use ϕ a to denote thetriplet with hypercharge 0 and φ a to denote the one with hypercharge -1. The correspondingLagrangians, including the couplings to the SM Higgs, are L = 12 D µ ϕ a D µ ϕ a − M ϕ a ) + κ H † σ a Hϕ a , (106) L ± = ( D µ φ a ) ∗ D µ φ a − M | φ a | + κ (cid:16) ˜ H † σ a Hφ a + H.c. (cid:17) , (107)where all other couplings not explicitly written in these Lagrangians are not relevant for ourcalculation. The covariant derivatives acting on the the triplets are D µ ϕ a = ∂ µ ϕ a + gǫ abc A bµ ϕ c and D µ φ a = ∂ µ φ a + gǫ abc A bµ φ c + ig ′ B µ φ a . The coupling constant κ has mass dimension 1since it is the coefficient of a cubic scalar interaction. When H obtains a vev, the cubicterms proportional to κ become linear terms for the triplet thus forcing a triplet vev. In theUV theory, one should not be concerned with what happens at low-energies that is with a43ev for a light field. One simply integrates out the triplet which induces the operator O T . O T reproduces the custodial symmetry breaking effects of either h ϕ a i or h φ a i . p p p p (a) (b)Figure 13: Triplet contributions to O T . The external dashed lines represent the Higgsdoublet, while the internal dashed line represents the heavy triplet.Fig. 13(a) depicts tree-level triplet exchange that gives O T . This amplitude needs to beevaluated to the second order in the external momenta since there are no interesting termswithout derivatives. The kinematic part of the amplitude arising from the ϕ a exchange,neglecting for the moment the σ a matrices, is A = ( iκ ) ip − M ≈ iκ M (cid:18) p M (cid:19) = iκ M (cid:18) p p + p p M (cid:19) , (108)where the last equality follows from p = p + p = − p − p and from assuming that p = . . . = p = 0. All of the external momenta are assumed to be incoming. Themomentum-dependent part of amplitude in Eq. (108) corresponds to the amplitude obtainedfrom D µ H † σ a D µ H H † σ a H which can be rewritten using the completeness relation for thePauli matrices to produce O T and other uninteresting operators of dimension 6. Finally, a (0) T = − κ M . (109)Integrating out φ a does not give the same result because the amplitude in Eq. (108) corre-sponds to the operator D µ ˜ H † σ a D µ H H † σ a ˜ H + H.c. , which gives a ( ± T = 4 κ M . (110)As we observed before, gauge invariance ensures that diagram (b) in Fig. 13 reproducesthe gauge field dependent part of O T even though we only matched the part without anyexternal gauge fields. We can also use that amplitude to derive a T . This way of matching theeffective theory will turn out to be very useful in the next section. Expanding the covariantderivatives in O T gives O T = (cid:12)(cid:12) H † ∂ µ H (cid:12)(cid:12) + g ′ B µ H † H ) + g A iµ A jµ H † σ i HH † σ j H + . . . , (111)where we omitted terms linear in the gauge fields. Expressing the Higgs doublet in compo-nents H = (cid:18) H H (cid:19) O T = | H ∂ µ H | + | H ∂ µ H | + g ′ B µ (cid:0) | H | + . . . (cid:1) + g ( A µ ) H ∗ H + H H ∗ ) + . . . (112)44e notice that B µ couples to | H | while ( A µ ) does not. It is not enough to extract thecoefficient of the term B µ | H | to obtain O T since there are other operators of dimension 6that contain this term, for example D µ H † D µ HH † H . However, all operators containing fourHiggs and two gauge fields that do not violate the custodial symmetry have equal coefficientsfor the terms B µ | H | and ( A µ ) | H | . Thus, we will extract the difference between theamplitudes depicted in Fig. 14. This difference is proportional to the T parameter a T = 12 ( c B − c A ) , (113)where appropriate powers of the gauge couplings and ig µν have been absorbed into thedefinitions of c B and c A , as described in Fig. 14. Let us test this method on the tree-leveltriplet contributions. The diagram in Fig. 13(b) gives for the hypercharge 0 triplet c (0) A = 4 κ M . (114)Since ϕ a has no hypercharge, c (0) B = 0 and Eq. (114) agrees with Eq. (109). Analogouscomputation for the charged triplet yields Eq. (110). H ∗ H H H ∗ B µ B ν ig µν g ′ c B H ∗ H H H ∗ A µ A ν ig µν g c A Figure 14: The amplitudes that define the coefficients c B and c A .Yet another way of obtaining Eqs. (109) and (110) is by matching the coefficient of O T in the background of the Higgs field. Expanding the Lagrangians (106) and (107) aroundthe Higgs vev gives a linear term for the triplet field. The linear term forces a vev for thetriplet, which in turn gives extra contributions to the masses of the gauge bosons. One needsto compare the mass terms for the gauge bosons with the gauge boson masses arising from O T in the Higgs background discriminating against contributions from other operators ofdimension 6 that do not violate the custodial symmetry. This can be done, for example, bycalculating the difference between the mass terms for A µ and A µ , which we used in Sec. 3.1.Note that Eqs. (109) and (110) exhibit decoupling even if κ ∝ M . It is certainly naturalto assume that parameters of mass dimension 1 scale proportionately to large masses in thetheory. Here, one can assume that κ ∝ M . Even with such a scaling, one does not expectnon-decoupling effects of higher-dimensional operators similar to the non-decoupling we ob-served when dimensionless quantities scale proportionately to large masses. In perturbationtheory, the amplitudes depend on positive powers of the couplings. Thus, whenever coupling45onstants have positive mass dimensions, the coefficients of higher-dimensional operatorsmust be suppressed by a power of the heavy particle masses larger than the dimension of thecoupling constants. Obviously, this argument has nothing to do with tree-level perturbationtheory. In the next section, we are going to examine two types of one-loop contributions tothe T parameter. One contribution is proportional to κ M and another one proportional to κ M . Both contributions vanish in the limit κ ∝ M → ∞ . A.2 One-loop level
We now turn to the one-loop contributions of the scalar triplets. We are going to discuss theeffects of the hypercharge-neutral triplet only, but there is no qualitative distinction betweenthe charged and the neutral cases. We will not present a complete analysis of all one-loopeffects. We will calculate certain classes of diagrams chosen such that it is clear that in theeffective theory the triplet contributions to the T parameter decouple.As usual in an effective theory, log-enhanced contributions come from RG running andterms without large logs arise from matching. We matched the theory with the tripletto the SM and found that a T = − κ M at tree level. We will omit the superscript (0) for a T since we will only deal with the neutral triplet in this section. There are two types ofdiagrams that correct O T at one loop: gauge boson exchanges and Higgs quartic interactions.Schematically, these give either a T ∼ g (4 π ) κ M log( m h M ) or a T ∼ λ (4 π ) κ M log( m h M ), where m h is the Higgs mass and λ is the Higgs quartic coupling constant. Neglecting the massesof the SM fields, compared to M , the dimensionless couplings in the SM cannot alter theproportionality of a T to κ M through the RG running. Hence, it is clear that the log-enhancedterms decouple in the limit κ ∝ M → ∞ . Moreover, there is no contribution to the runningof O T from two insertions of O T when the masses of the SM fields are neglected. Twoinsertions of O T in the effective theory yield a coefficient proportional to κ M , which couldgive O T only when multiplied by the mass squared of a SM field, for example it could give a T ∼ m h κ M . This term is additionally suppressed by m h M compared to the terms we willconsider next. The tree-level result is also modified by the Higgs wave function renormalization due tothe triplet exchange. Straightforward calculation gives (1+ κ π ) M ) D µ H † D µ H for the Higgskinetic energy in the effective theory. This gives another contribution of order κ M withoutany log enhancement.We will now discuss two cases of matching contributions. To gain experience with lesscomplex calculations first, we will start with diagrams that give a T ∼ λ (4 π ) κ M . Then wecompute terms proportional to π ) κ M . The corresponding diagrams are shown in Fig. 15.For discussion of decoupling, diagrams (3), (4), and (5) in Fig. 15 are the most interesting.These diagrams have the highest power of the cubic coupling κ one can get at one loop,so one expects that these are the most important when κ is large. However, since the T There is also an RG contribution of order a T ∼ π ) κ M log( m h M ) arising from one insertion of O T and one insertion of κ M ( H † H ) that one also obtains from tree-level matching. This contribution is notdistinguishable at low energies from a T ∼ λ (4 π ) κ M log( Mm h ) since both terms arise from the same Higgsquartic coupling. T parameter at orders κ M λ (top) and κ M (bot-tom). The short dashed lines represent massive triplet fields, while the lines with long dashesrepresent the Higgs doublet.parameter corresponds to an operator of dimension 6, these diagrams are proportional to κ M . Of course, this dimensional argument is not particular to the one-loop approximation.To perform one-loop matching we will not work directly with the diagrams in Fig. 15,but instead extract the coefficients of the terms B µ | H | and ( A µ ) | H | . This is the secondmethod of calculating the T parameter we used in Sec. A.1. Extracting the coefficientof (cid:12)(cid:12) H † ∂ µ H (cid:12)(cid:12) is actually more difficult because it depends on the momenta of the externalstates. Keeping external momenta makes loop calculations more complicated. An additionalcomplication is that all the diagrams in Fig. 15 are IR divergent. This means that one cannotsimply expand the propagators around the zero values of external momenta and then retainterms quadratic in those momenta.To extract the coefficients of the terms B µ | H | and ( A µ ) | H | we attach two gaugebosons in all possible ways to the internal lines of the diagrams in Fig. 15 and set all theexternal momenta to zero. The loop integrals are much simpler to compute, but the priceof this approach is proliferation of diagrams. The diagrams with different ways of attachingthe gauge bosons are depicted in Fig. 16. The diagrams in Fig. 16 correspond to differentways of attaching gauge bosons to diagram (3) in Fig. 15. Of course, we consider all possibleways of attaching two external gauge bosons to the remaining diagrams in Fig. 15. Theseare completely analogous to the ones drawn in Fig. 16, except that for diagrams (1) and (2)in Fig. 15 there is no corresponding diagram (f) because the Higgs quartic vertex containsno gauge bosons. Fig. 16 does not show all possible permutations of attaching photons, butrepresentative diagrams. For example, diagram (a) represents two diagrams where a pair ofgauge bosons is attached to either of the two internal Higgs lines. Diagram (b) representsthree diagrams in which two gauge bosons are attached to either of the two Higgs lines orone gauge boson is attached to each line, etc.The diagrams in Fig. 16 are still IR divergent. The IR divergences must be matched bythe loop diagrams in the effective theory using the matching coefficients obtained at treelevel. The effective theory diagrams are shown in Fig. 17. We will compare diagrams in the47a) (b)(d) (c)(e) (f)Figure 16: Diagrams with two gauge bosons obtained from diagram (3) in Fig. 15.(a) (b) (c) (d)Figure 17: Diagrams in the effective theory. The dashed lines represent the Higgs doublet.full theory with the corresponding diagrams in the effective theory to make sure that theIR divergences match. Diagrams (a) through (d) in the full theory correspond to diagrams(a) through (d) in the effective theory, respectively. The full theory diagrams (e) and (f)are finite in the IR. The cancellations of IR divergences happens diagram by diagram, so wecheck this in every case. Both diagrams (c) and (e) in the full theory correspond to diagram(c) in the effective theory, but since diagram (e) is finite we evaluate it it separately.The full theory diagrams involve integrals of the form I n,m = Z d d k (2 π ) d k ) n ( k − M ) m = Z d d k (2 π ) d Z dx Γ( n + m )(1 − x ) n − x m − Γ( n )Γ( m )( k − xM ) n + m = i ( − n + m (4 π ) − ǫ ( M ) n + m − ǫ Γ( n + m − ǫ )Γ( n )Γ( m ) Z dx (1 − x ) n − x m − x n + m − ǫ = i ( − n + m (4 π ) − ǫ ( M ) n + m − ǫ Γ( n + m − ǫ )Γ(2 − n − ǫ )Γ( m )Γ(2 − ǫ ) , (115)where in the last line we performed the Feynman parameter integral in d = 4 − ǫ dimensions48sing the standard Euler beta function integral. The IR divergences of the integrals with n ≥ ǫ →
0, Γ(2 − n − ǫ ) is divergent for n ≥
2. Thepoles of Γ(2 − n − ǫ ) = Γ( d/ − n ) occur in d = 2 n, n − , n − , . . . , which is characteristicof an IR divergence. We will also need I µνn,m = Z d d k (2 π ) d k µ k ν ( k ) n ( k − M ) m = i ( − n + m − g µν (4 π ) − ǫ ( M ) n + m − ǫ Γ( n + m − ǫ )Γ(3 − n − ǫ )2Γ( m )Γ(3 − ǫ ) , (116)which is IR divergent when n ≥
3. The expressions in Eqs. (115) and (116) apply only when m > I n, and I µνn, vanish in dimensional regularization since there is no massscale to make up for the dimension of the integral. However, I , and I µν , appear in the fulland effective theories and these integrals contain both the IR and UV divergences. Sinceboth the IR and UV divergences manifest as ǫ poles the two divergences cancel for I , and I µν , in dimensional regularization. To show explicitly that the IR divergences are identicalin the full and effective theories we rewrite I , = Z d d k (2 π ) d k − M ( k ) ( k − M ) = I , − M I , = i [Γ( ǫ )Γ(1 − ǫ ) + Γ( − ǫ )Γ(1 + ǫ )](4 π ) − ǫ ( M ) ǫ Γ(2 − ǫ ) . (117)Of course, Γ( ǫ )Γ(1 − ǫ ) + Γ( − ǫ )Γ(1 + ǫ ) = 0 which can be shown by multiplying by ǫ andusing z Γ( z ) = Γ( z + 1). However, by rewriting the integral we separated the UV and IRdivergences which are encoded in Γ( ǫ ) and Γ( − ǫ ), respectively. Similarly, we can rewrite I µν , = I µν , − M I µν , = ig µν (4 π ) − ǫ ( M ) ǫ Γ( ǫ )Γ(1 − ǫ ) + Γ( − ǫ )Γ(1 + ǫ )2Γ(3 − ǫ ) . (118)For n >
2, rewriting I n, using the trick described above does not yield anything usefulbecause the integrals are UV convergent. Thus, dimensional regularization sets the IRdivergence to zero.We are almost ready to do the calculation, except that in the effective theory we need allterms of order κ M and κ M . We have calculated the coefficient of O T in the previous section,but neglected all other operators. Integrating out ϕ a at tree level gives L eff = c H † H ) + c (cid:20) | H † D µ H | + 14 D H † HH † H + 14 H † D HH † H − D µ H † D µ HH † H (cid:21) , (119)where each derivative acts only on the field immediately next to it and not on all the fieldsto the right of the derivative. The coefficients are c = 2 κ M and c = − κ M . The firstterm in Eq. (119) is of the same form as the ordinary Higgs quartic coupling in the SM.The coefficient of the quartic term in the effective theory is − λ + c , where λ is the quarticcoupling in the full theory above M . Our convention for the quartic coupling is such that,at tree level, V ( H ) = λ ( H † H − v ) . It may seem odd that in the effective theory we careabout terms that do not violate the custodial symmetry, for example D µ H † D µ HH † H . Inthe following calculation we will be extracting coefficients of all operators with four Higgsfields and two gauge bosons, and not just the coefficient of O T . If we only cared about thecancellation of IR divergences for O T we may not need to keep all of the operators in the49ffective theory. However, it is a very useful consistency check on the calculation to be ableto show cancellation of IR divergences for individual diagrams.With the integrals in Eqs. (115) through (118) at hand, the problem is reduced to com-binatorics. We will show a couple of examples in detail and then present the results. Toprovide further checks we calculate separately the amplitudes depending on the flow of thescalar field number. Since the Higgs field is complex, we can assign arrows indicating thedirection of the flow of the scalar field. We will separate diagrams in which the arrows onthe Higgs lines in Fig. 16 are in the same direction from the ones in which the arrows arein the opposite directions. We will denote the amplitudes in the full theory by F and inthe effective theory by E adding the superscripts −→→ and −→← to indicate the arrow directions.The subscripts will indicate the topology of the diagram, as shown in Figs. 16 and 17, andthe type of the gauge fields: either B µ or A µ .As our first example, we compute diagram (1) in Fig. 15 with the Higgs lines in the samedirection and two B µ fields coupling at the same point to the Higgs line, as represented indiagram (a) in Fig. 16. F −→→ B ( a ) = 4( iκ ) ( − iλ ) g ′ ig µν Z d d k (2 π ) d i ( k ) ik − M = − κ λg ′ g µν I , , (120)where the factor of four comes from exchanging the two external lines on the left due toBose statistics and from two possible directions for the arrows. The reversal of the arrowdirections corresponds to exchanging the external H fields with the H ∗ ’s. The remainingfactors are the coupling constants for the vertices and the propagators, where we set all theexternal momenta to zero. Note that diagram (2) in Fig. 16 is identically zero when thearrow directions are parallel because ϕ a couples to H and H † . In the effective theory, thecorresponding diagram gives E −→→ B ( a ) = 2( ic )( − iλ ) g ′ ig µν Z d d k (2 π ) d i ( k ) ( − k ) = − c λg ′ g µν I , , (121)where the factor of two is due to the reversal of arrow directions, or equivalently due toexchanging the c and λ interaction vertices. The factor − k arises from the two-derivativeterms in Eq. (119). The IR divergent part of the difference F −→→ B ( a ) − E −→→ B ( a ) is proportional toΓ(2 + ǫ )Γ( − − ǫ ) + Γ(1 + ǫ )Γ( − ǫ ) = (1 + ǫ )Γ(1 + ǫ )Γ( − − ǫ ) + Γ(1 + ǫ )Γ( − ǫ )= − Γ(1 + ǫ )Γ( − ǫ ) + Γ(1 + ǫ )Γ( − ǫ ) = 0 . In this case the IR divergent terms cancel exactly, but in some cases the difference betweenthe diagrams is finite. Since B µ does not couple to ϕ a , diagrams (c) through (f) are absent.These diagrams vanish in the effective theory because there is no term proportional to B µ inthe effective Lagrangian in Eq. (119). This is expected since the effective Lagrangian comesfrom integrating out ϕ a , but is not apparent as the covariant derivatives in (119) contain the B µ field.As the second detailed example, we compute F −→← A ( c ) and F −→← A ( c ) for the κ M λ contributionsthat is diagrams (1) and (2) in Fig. 15. Both diagrams in the full theory contribute, and sincediagram (2) is IR divergent it needs to be accounted for to ensure cancellation of divergences. F −→← A ( c ) = − κ λg g µν I , − − κ λg g µν M I , , (122)50here the factors of four are from exchanges of the external H lines and exchanges of H ∗ ’s.The remaining factors come from the couplings. In diagram (1), the triplet components canbe either ϕ or ϕ , which is responsible for the 2 + 1 factor. In diagram (2), the factor of2 − H ’s and H ’s exchanged in the loop. In each of these diagrams, 2 ± | H | and the | H H | couplings in theHiggs quartic term. In the effective theory there is only one diagram. To calculate it oneneeds to extract the coefficient of | H | ( A µ ) in the effective Lagrangian (119), which thengives E −→← A ( c ) = 8 c λg g µν I , . (123)The difference between the IR divergent parts of the full and effective theory amplitudes isproportional to − ǫ )Γ( − ǫ ) − Γ(2 + ǫ )Γ( − ǫ ) + 4Γ(1 + ǫ )Γ( − ǫ ) = − ǫ Γ(1 + ǫ )Γ( − ǫ )= Γ(1 + ǫ )Γ(1 − ǫ ) , (124)which is finite when ǫ → F −→→ E −→→ F −→← E −→← B ( a ) − κ λg µν I , − c λg µν I , − κ λg µν I , − c λg µν I , B ( b ) 4 κ λI µν , c λI µν , κ λI µν , c λI µν , A ( a ) − κ λg µν I , − c λg µν I , − κ λg µν I , − c λg µν I , A ( b ) 8 κ λI µν , c λI µν , κ λI µν , c λI µν , A ( c ) − κ λg µν I , c λg µν I , − κ λg µν (3 I , + M I , ) 8 c λI , A ( d ) 16 κ λI µν , − c λI µν , κ λI µν , − c λI µν , A ( e ) 16 κ λI µν , − κ λI µν , − B ( a ) − κ g µν I , c c g µν I , − κ g µν (5 I , + M I , ) 4 c c I , B ( b ) 4 κ I µν , − c c I µν , κ (5 I µν , + M I µν , ) − c c I µν , A ( a ) − κ g µν I , c c g µν I , − κ g µν (5 I , + M I , ) 4 c c I , A ( b ) +8 κ I µν , − c c I µν , κ (7 I µν , + M I µν , ) − c c I µν , A ( c ) − κ g µν I , − c c g µν I , κ g µν ( − I , + M I , + M I , ) − c c g µν I , A ( d ) 32 κ I µν , c c I µν , κ I µν , c c I µν , A ( e ) 32 κ I µν , − κ I µν , − A ( f ) 32 κ I µν , − κ I µν , − Table 3: The amplitudes corresponding to the diagrams in Figs. 16 and 17. The rowscorrespond to different ways of attaching gauge boson lines as shown in the figures. B and A indicate the external gauge fields: either B µ or A µ , respectively. To save space, the gaugecouplings are omitted. The diagrams with the B µ fields are proportional to g ′ , while theones with A µ to g . The top part of this table lists the amplitudes proportional to κ λM ,while the bottom part proportional to κ M . The columns give the full and effective theoryamplitudes with either parallel or antiparallel Higgs lines.The complete answer for all diagrams is presented in Table 3. The IR divergences cancelin each row of the table between the two corresponding amplitudes, as we already described51n the previous examples. Altogether, there are 24 cancellations of IR divergences thatprovide consistency checks on this calculation. The full theory diagrams (e) and (f) areindeed IR finite, and there are no corresponding effective theory diagrams.We can now extract the matching coefficients by calculating the differences between thefull and effective theories. The coefficients c B and c A defined in Fig. 14 are c B = κ λ (4 π ) M (cid:18) ǫ + 192 (cid:19) + κ (4 π ) M (cid:18) − ǫ − (cid:19) , (125) c A = κ λ (4 π ) M (cid:18) ǫ + 252 (cid:19) + κ (4 π ) M (cid:18) − ǫ − (cid:19) , (126)which finally gives a T = − κ λ (4 π ) M − κ (4 π ) M + 6 κ (4 π ) M , (127)where the last term comes from the wave function renormalization of the tree-level term. Inobtaining Eq. (127) we absorbed the ǫ poles into counterterms using the M S prescription.These poles can be used to calculate the running of the T operator in the effective theory.The renormalization scale has been set to M , so that the logarithms of µM are absent.The numerical coefficients in Eq. (127) are not crucial for us. This calculation provideda thorough illustration of the methods we discussed in these notes. What is interesting isthat the one-loop result exhibits decoupling in the limit κ ∝ M → ∞ . There was no otherpossibility in the effective theory since this is guaranteed by power counting even withoutdoing an explicit calculation. One might wonder if the effective theory reproduces properlythe full theory. The cancellation of the IR divergences among various terms in Table 3provides convincing evidence that it does. The results in Refs. [36, 37] that motivated thiscalculation were obtained in the EW broken phase without using EFT methods. It is unlikelythat the non-decoupling observed in Refs. [36, 37] is a result of an algebraic error. Oneplausible reason may be the triplet correction to the Higgs mass term, which is proportionalto κ (4 π ) . (This is another example of the quadratic sensitivity of the Higgs mass to the heavyscales, even though the diagram with the triplet exchange is only logarithmically divergent.)This contribution might creep into the Higgs vev calculation, but should be cancelled whenthe calculation is expressed in terms of the physical Higgs mass. Unfortunately, we have nofirm argument as to why the two approaches disagree. Acknowledgments
These notes are based on five lectures given at TASI during the summer of 2009. One of theselectures reviewed the Standard Model and since this topic is covered almost every summer,see for example Ref. [22], it is omitted here. I am grateful to the TASI organizers, especiallyCsaba Cs´aki, Tom DeGrand, and K.T. Mahantappa, for a well designed and smoothly runprogram. I very much enjoyed lively reception of these lectures by the TASI participants.I am indebted to Walter Goldberger for discussions and comments on the manuscript,and to Zuhair Khandker for carefully inspecting the calculations and comments on themanuscript. This work was supported in part by the US Department of Energy undergrant DE-FG02-92ER-40704. 52 eferences [1] S. Weinberg, Physica A , 327 (1979).[2] K. G. Wilson and J. B. Kogut, Phys. Rept. (1974) 75.[3] T. Appelquist and J. Carazzone, Phys. Rev. D , 2856 (1975).[4] J. S. Schwinger, Phys. Rev. , 416 (1948); P. Kusch and H. M. Foley, Phys. Rev. ,250 (1948).[5] A. Sirlin, Phys. Rev. D , 971 (1980); G. Passarino and M. J. G. Veltman, Nucl. Phys.B , 151 (1979); W. F. L. Hollik, Fortsch. Phys. , 165 (1990); J. Erler and P.Langacker in C. Amsler et al. [Particle Data Group], Phys. Lett. B , 1 (2008), andreferences therein.[6] A. V. Manohar, “Effective field theories,” arXiv:hep-ph/9606222.[7] I. Z. Rothstein, “TASI lectures on effective field theories,” arXiv:hep-ph/0308266.[8] D. B. Kaplan, “Effective field theories,” arXiv:nucl-th/9506035.[9] W. D. Goldberger, “Les Houches lectures on effective field theories and gravitationalradiation,” arXiv:hep-ph/0701129; W. D. Goldberger and I. Z. Rothstein, Phys. Rev.D , 104029 (2006) [arXiv:hep-th/0409156].[10] H. D. Politzer, Nucl. Phys. B , 349 (1980).[11] H. Georgi, Nucl. Phys. B , 339 (1991).[12] P. A. M. Dirac, Nature , 323 (1937); Proc. Roy. Soc. Lond. A , 199 (1938).[13] G. ’t Hooft, “Naturalness, Chiral Symmetry, And Spontaneous Chiral Symmetry Break-ing,” NATO Adv. Study Inst. Ser. B Phys. , 135 (1980).[14] C. Arzt, Phys. Lett. B , 189 (1995) [arXiv:hep-ph/9304230].[15] A. Manohar and H. Georgi, Nucl. Phys. B , 189 (1984).[16] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B , 301 (1997)[arXiv:hep-ph/9706275].[17] C. Amsler et al. [Particle Data Group], Phys. Lett. B , 1 (2008).[18] W. Buchmuller and D. Wyler, Nucl. Phys. B , 621 (1986).[19] B. Grinstein and M. B. Wise, Phys. Lett. B , 326 (1991).[20] T. Appelquist and C. W. Bernard, Phys. Rev. D , 200 (1980); A. C. Longhitano,Phys. Rev. D , 1166 (1980); Nucl. Phys. B , 118 (1981).[21] J. Wudka, Int. J. Mod. Phys. A , 2301 (1994) [arXiv:hep-ph/9406205].5322] S. Willenbrock, “Symmetries of the standard model,” arXiv:hep-ph/0410370.[23] M. E. Peskin and T. Takeuchi, Phys. Rev. D , 381 (1992).[24] M. Golden and L. Randall, Nucl. Phys. B , 3 (1991); B. Holdom and J. Terning,Phys. Lett. B , 88 (1990); M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. , 964(1990); G. Altarelli and R. Barbieri, Phys. Lett. B , 161 (1991).[25] M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl. Phys. , 711 (1979) [Yad. Fiz. , 1368 (1979)].[26] A. G. Cohen, H. Georgi and B. Grinstein, Nucl. Phys. B , 61 (1984); M. B. Einhorn,D. R. T. Jones and M. J. G. Veltman, Nucl. Phys. B , 146 (1981).[27] R. Barbieri, A. Pomarol, R. Rattazzi and A. Strumia, Nucl. Phys. B , 127 (2004)[arXiv:hep-ph/0405040].[28] K. S. Babu, C. F. Kolda and J. March-Russell, Phys. Rev. D , 6788 (1998)[arXiv:hep-ph/9710441].[29] Z. Han and W. Skiba, Phys. Rev. D , 075009 (2005) [arXiv:hep-ph/0412166].[30] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B , 155 (2002)[arXiv:hep-ph/0207036].[31] Z. Han, Phys. Rev. D , 015005 (2006) [arXiv:hep-ph/0510125]; Z. Han, AIP Conf.Proc. , 435 (2007) [arXiv:hep-ph/0610302].[32] C. Grojean, W. Skiba and J. Terning, Phys. Rev. D , 075008 (2006)[arXiv:hep-ph/0602154].[33] Z. Han and W. Skiba, Phys. Rev. D , 035005 (2005) [arXiv:hep-ph/0506206];M. S. Carena, E. Ponton, J. Santiago and C. E. M. Wagner, Phys. Rev. D , 035006(2007) [arXiv:hep-ph/0701055]; S. Mert Aybat and J. Santiago, Phys. Rev. D , 035005(2009) [arXiv:0905.3032 [hep-ph]].[34] [LEP Collaboration and . . . ], “A Combination of preliminary electroweak measurementsand constraints on the standard model,” arXiv:hep-ex/0312023.[35] C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, Phys. Rev. D ,6115 (1994) [arXiv:hep-ph/9312291].[36] M. C. Chen and S. Dawson, Phys. Rev. D , 015003 (2004) [arXiv:hep-ph/0311032];M. C. Chen, S. Dawson and T. Krupovnickas, Int. J. Mod. Phys. A , 4045 (2006)[arXiv:hep-ph/0504286], Phys. Rev. D , 035001 (2006) [arXiv:hep-ph/0604102].[37] P. H. Chankowski, S. Pokorski and J. Wagner, Eur. Phys. J. C50