Yi Liao(廖益)? and Xiao-Dong Ma(馬小東)?

1School of Physics,Nankai University,Tianjin 300071,China

2CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190,China

3Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University,Changsha 410081,China

We study in this short paper two general aspects in standard model effective field theory(SMEFT).One is a power counting rule in perturbation theory for anomalous dimension matrix of higher dimensional operators with equal mass(canonical)dimension that is induced by the standard model(SM)interactions.We show that the leading power of each entry in the anomalous dimension matrix is determined in terms of the loop order and the difference of indices for the two operators involved.The other is about the lowest-index operators.We find that they are unique at each dimension and can be renormalized independently of other operators of equal dimension at the leading order in SM interactions.We compute their oneloop anomalous dimensions,and find that they increase quadratically with their dimension due to combinatorics.

Regarding SM as an effective field theory below the electroweak scale,the low energy effects of high scale physics can be parameterized in terms of higher dimensional operators:

whereXsums over the three gauge field strengths of couplingsg1,2,3,and Ψ extends over the lepton and quark lefthanded doubletsL,Qand right-handed singletse,u,d.The Higgs fieldHdevelops the vacuum expectation valueis the usual gauge covariant derivative,andYu,d,eare Yukawa coupling matrices.

The higher dimensional operators,collected inL5,6,7and ellipses in Eq.(1),are composed of the above SM fields,and respect the SM gauge symmetries but not necessarily accident symmetries like lepton or baryon number conservation.They are generated from high scale physics by integrating out heavy degrees of freedom,with their Wilson coefficients naturally suppressed by powers of certain high scale.It is thus consistent to leave aside those Wilson coefficients when we do power counting for their renormalization running effects due to SM interactions.The higher dimensional operators start at dimension- five(dim-5),which turns out to be unique.[1]The complete and independent list of dim-6 and dim-7 operators has been constructed in Refs.[2–3]and[4–5],respectively.The number of operators increases horribly fast with their dimension;for discussions on dim-8 operators and beyond,see recent papers.[6?9]If SM is augmented by sterile neutrinos below the electroweak scale,there will be additional operators at each dimension,see Refs.[10–13]for discussions on operators up to dim-7 that involve sterile neutrinos.

Now we consider power counting in the anomalous dimension matrixγof higher dimensional operators due to SM interactions.We restrict ourselves in this work to the mixing of operators with equal mass dimension,because this is the leading renormalization effect due to SM interactions that is not suppressed by a high scale.Since the power counting is additive,it is natural to assign an index of power countingto the operator,which in turn is a sum of the indices for the elements involved in.For the purpose of power counting,we denotegas a generic coupling in SM.Suppose an effective interactioninLSMEFTis dressed by SM interactions atn-loops to induce an effective interaction,(no sum overj),involving the operatorof equal dimension.The SMn-loop factor ofg2nis shared by the difference of the indices of the operatorsand the induced ultraviolate divergent coefficientcontributes a counterterm to the effective interactionfrom whichγjiis determined for the running ofCj,we obtain the power counting for the entryγjiin the anomalous dimension matrix

The issue now becomes de fining an index for operators up to a constant,χ[O],which could be understood as an intrinsic power counting of SM couplings for the operator

Since we are concerned with overall power counting in SM interactions,it is plausible to treat all terms inL4on the same footing by assuming an equal index of perturbative power counting when the kinetic terms have been canonically normalized.A similar argument was assumed previously in chiral perturbation theory involving chiral fermions coupled to electromagnetism.[14?17]Denoting generically

It is evident that thexterm actually counts canonical dimension and theycounts the power ofg.Since we are concerned with renormalization mixing of operators with equal dimension,the power counting for their anomalous dimension matrix depends only on theyterm according to Eq.(3).Although ourχ[γij]does not depend onx,we find it most convenient to work withx=0 andy=1,so that the nonvanishing indices for power counting are

The lowest index that an operator could have is zero in this convention.Using a differentxamounts to shifting the indices of all fields and derivatives by a multiplier of their mass dimensions without changingχ[γij],and choosingy=1 simply fits the usual convention that all gauge and Yukawa couplings count asg1while the scalar selfcouplingλcounts as a quartic gauge couplingg2.

We can now associate an index of power countingχ[O]to a higher dimensional operatorOby simply adding up the indices of its components according to Eq.(6).The entryγjiin the anomalous dimension matrix for the set of operatorsOkdue to SM interactions atn-loops has the index of power counting shown in Eq.(3)in terms of a generic couplingg,which denotesOur results for dim-6 and dim-7 operators are shown in Tables 1 and 2 respectively.The one-loopγmatrix for dim-6 operators has been computed in a series of papers,[18?24]and is consistent with power counting in Table 1.Theγsubmatrix for baryon number violating dim-7 operators is available recently,[5]and also matches power counting in Table 2.Note that some entries in the tables may actually vanish due to structures of one-loop Feynman diagrams or nonrenormalization theorem.[25?27]Since at least one vertex of SM interactions is involved in one-loop diagrams,γcounts asg1or higher.This explains the presence of zero in the last two columns of the tables.The power counting in the explicit result of oneloopγmatrix for dim-6 operators has also been explained in Ref.[28]using the arguments of naive dimensional analysis developed for strong dynamics[29]that rescale operators forth and back by factors of couplings and powers of 4π.Our analysis above is more straightforward and assumes only the uniform application of SM perturbation theory.

With the above de finition of the index of power counting for an operator,we make an interesting observation that the operator with the lowest index is unique at each mass dimension.To show this,we notice that out of the building blocksfor higher dimensional operators onlyHhas a vanishing index.This means that it should appear as many times as possible in the lowestindex operators for a given mass dimensiond.Fordeven,this is easy to figure out,i.e.,

These operators represent a correction to the SM scalar potential from high scale physics,and could impact the vacuum properties.Fordodd,additional building blocks must be introduced.In the absence of fermions,XμνandDμhave to appear at least twice due to Lorentz invariance,which costs no less than two units of index.And in addition,this cannot yield an operator of odd dimension.The cheapest possible way would be to introduce two fermion fields in a scalar bilinear form on top of the Higgs fields,resulting in an operator of index unity.It turns out that gauge symmetries require the fermions to be leptons.Sorting out the quantum numbers of lepton fields,§§The bilinear formmust couple to an odd total number of H?and H thus resulting in an even dim-d operator.The bilinear(ee)requires four more powers of H than H?to balance hypercharge,which then cannot be made weak isospin invariant.This leaves the only possibility as shown.we arrive at the unique operator at oddddimension,

wherep,rare lepton fl avor indices.This is the generalized dim-dWeinberg operator for Majorana neutrino mass whose uniqueness was established previously in Ref.[30]using Young tableau.

Table 1 Indices of power counting for dim-6 operators and power counting of their anomalous dimension matrix at one loop.

Table 2 Similar to Table 1 but for dim-7 operators.

The lowest-index operators are of interest because their renormalization running under SM interactions is governed at the leading order by their own anomalous dimensions;i.e.,they are only renormalized at the nextto-leading order by higher-index operators of the same canonical dimension.This is evident from Eq.(3)and the last row in Tables 1 and 2.The uniqueness of the lowest-index operators at each dimension further simpliif es the consideration of their renormalization running,which will be taken up in the remaining part of this work.Before that,we make a connection to classi fication of operators in terms of their holomorphic and antiholomorphic weightsThe weights are de fined asfor an operatorO,wheren(O)is the minimal number of particles for on-shell amplitudes that the operatorOcan generate andh(O)the total helicity of the operator.The claim is that our lowest-index operatorsare also the ones with the largest weights,i.e.,both of theirωandare the largest among operators of a given canonical dimension. To show this,we introduce some notations.We denote Ψ to be left-handed fermion fields,i.e.,L,Q,eC,uC,dC,andthe right-handed ones,andThe pair of weights has the values(3/2,1/2),(1/2,3/2),(0,0),(0,2),(2,0)for the building blocks of operators,respectively.The weightsof an operatorOdof dimensiondare the sum of the corresponding weights of its components:

wherenBdenotes the power of the componentBappearing inOd.The largestωandthat an operator could have is thus its canonical dimension.Fordeven,this is easy to realize by sendingi.e.,the operator with the highest weights is the lowestindex operatormade up purely of the Higgs field.Fordodd,it is known that all operators in SMEFT necessarily involve fermion fields,[31]with the minimal choice beingThis can be arranged by choosingresulting in the operatorof the highest weights(d,d?2),or by choosing insteadas its Hermitian conjugateThe alternative choicewould require a factor ofDdue to Lorentz symmetry,which reducesω(orˉω)by two units compared withThis establishes the claim.As a side remark,the above equations together with Lorentz symmetry also imply that the operators at even(odd)dimension have even(odd)holomorphic and anti-holomorphic weights.

Now we compute the anomalous dimensions at leading order for the lowest-index operatorsat even dim-dandfor odd dim-din Eqs.(7)–(8).The Feynman diagrams shown in Figs.1 and 2 are forandrespectively.At higher dimensions one has to be careful with combinatorics due to powers ofinvolved in the operators.We perform the calculation in dimensional regularization and minimal subtraction scheme and in the generalRξgauge.The cancelation of theξparameters in the final answer then serves as a useful check.The renormalization group equations for the Wilson coefficients of the above two operators are,at leading order in perturbation theory,

We make some final comments on the above result.The terms in the anomalous dimensions due to the Higgs self-couplingλincrease quadratically with canonical dimensionddue to combinatorics,making renormalization running effects signi ficantly more and more important for higher dimensional operators.The Yukawa terms in Eq.(12)are independent ofdbecause the lepton fieldLcannot connect toto yield a nonvanishing contribution due to weak isospin symmetry.The large numerical factor in theλterm forwas observed previously in Ref.[21],and our leading order results indeed match that work.Including a symmetry factor of 1/2 in theλterm of Eq.(11)that appears in graphs(d)–(e)in Fig.1 atd=4,our result also applies to renormalization of theλcoupling and is consistent with Ref.[32]upon noting different conventions forλ.The renormalization of the Weinberg operatorwas finally given in Ref.[33]and corresponds to graphs(a)–(e)in Fig.2.Our result atd=5 is consistent with that work again after taking into account different conventions forλ.Theλterm of theγfunction forincreases signi ficantly withdfor the first two operators in particular,from 4λatd=5 to 40λatd=7.

Fig.1 One-loop Feynman diagrams for self-renormalization of the operatorshown as a grey square.The wavy(dashed)line represents gauge(scalar) fields.The arrows indicate the fl ow of hypercharge.

Fig.2 Similar to Fig.1 but for the operatorThe arrow on the solid line stands for lepton number fl ow.For d≥9 the incoming scalars to the grey square in the loop of graphs(b)and(e)can also be outgoing.

In summary,we have provided a simple perturbative power counting for renormalization effects of higher dimensional operators due to SM interactions in the framework of SMEFT.In the course of our analysis we introduced an index that parametrizes the perturbative order of operators.We found that the lowest-index operators are unique at each mass dimension,and that their renormalization running under SM interactions is determined at leading perturbative order by their own anomalous dimensions.We computed the anomalous dimensions of those operators for any mass dimension and found that they increase quadratically with their mass dimension.This will be useful in the study of effective scalar potential and generation of tiny Majorana neutrino masses in the framework of SMEFT.

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