Perturbative Power Counting, Lowest-Index Operators and Their Renormalization in Standard Model Effective Field Theory

0.22 MB·

by  Yi Liao

Perturbative power counting, lowest-index operators and their renormalization
in standard model effective field theory
YiLiaoa,b,c1 andXiao-DongMaa2
a SchoolofPhysics,NankaiUniversity,Tianjin300071,China
b CASKeyLaboratoryofTheoreticalPhysics,InstituteofTheoreticalPhysics,ChineseAcademyofSciences,
Beijing100190,China
c SynergeticInnovationCenterforQuantumEffectsandApplications,HunanNormalUniversity,Changsha,
Hunan410081,China
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1 Abstract
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Westudytwoaspectsofhigherdimensionaloperatorsinstandardmodeleffectivefieldtheory. Wefirstintroduce

b a perturbative power counting rule for the entries in the anomalous dimension matrix of operators with equal mass
e
dimension. The power counting is determined by the number of loops and the difference of the indices of the two
F
operators involved, which in turn is defined by assuming that all terms in the standard model Lagrangian have an
0
1 equalperturbativepower. Thenweshowthattheoperatorswiththelowestindexareuniqueateachmassdimension
d, i.e., (H†H)d/2 for even d ≥ 4, and (LTεH)C(LTεH)T(H†H)(d−5)/2 for odd d ≥ 5. Here H, L are the Higgs

]
h andleptondoublet,andε, C theantisymmetricmatrixofranktwoandthechargeconjugationmatrix,respectively.
p Therenormalizationgrouprunningoftheseoperatorscanbestudiedseparatelyfromotheroperatorsofequalmass
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p dimensionattheleadingorderinpowercounting. Wecomputetheiranomalousdimensionsatoneloopforgenerald
e andfindthattheyareenhancedquadraticallyindduetocombinatorics. Wealsomakeconnectionswithclassification
h
ofoperatorsintermsoftheirholomorphicandanti-holomorphicweights.
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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
equalmass(canonical)dimensionthatisinducedbythestandardmodel(SM)interactions. Weshowthattheleading
powerofeachentryintheanomalousdimensionmatrixisdeterminedintermsofthelooporderandthedifferenceof
indicesforthetwooperatorsinvolved. Theotherisaboutthelowest-indexoperators. Wefindthattheyareuniqueat
eachdimensionandcanberenormalizedindependentlyofotheroperatorsofequaldimensionattheleadingorderin
SM interactions. We compute their one-loop anomalous dimensions, and find that they increase quadratically with
theirdimensionduetocombinatorics.
Regarding SM as an effective field theory below the electroweak scale, the low energy effects of high scale
physicscanbeparameterizedintermsofhigherdimensionaloperators:
L =L +L +L +L +···. (1)
SMEFT 4 5 6 7
HeretheleadingtermsaretheSMLagrangian
(cid:18) (cid:19)2
1 1
L = − ∑X Xµν+(D H)†(DµH)−λ H†H− v2
4 µν µ
4 2
X
+∑Ψ¯iD/Ψ−(cid:2)Q¯Y uH˜ +Q¯Y dH+L¯Y eH+h.c.(cid:3), (2)
u d e
Ψ
where X sums over the three gauge field strengths of couplings g , and Ψ extends over the lepton and quark
1,2,3
left-handed doublets L, Q and right-handed singlets e, u, d. The Higgs field H develops the vacuum expectation
√
valuev/ 2,andH˜ =ε H∗. D istheusualgaugecovariantderivative,andY areYukawacouplingmatrices.
i ij j µ u,d,e
The higher dimensional operators, collected in L and ellipses in Eq. 1, are composed of the above SM
5,6,7
fields,andrespecttheSMgaugesymmetriesbutnotnecessarilyaccidentsymmetrieslikeleptonorbaryonnumber
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, 7, 8, 9]. If SM is augmented by sterile neutrinos below the electroweak scale, there will be
additional operators at each dimension, see Refs. [10, 11, 12, 13] for discussions on operators up to dim-7 that
involvesterileneutrinos.
Nowweconsiderpowercountingintheanomalousdimensionmatrixγ ofhigherdimensionaloperatorsdueto
SMinteractions. Werestrictourselvesinthisworktothemixingofoperatorswithequalmassdimension. Sincethe
power counting is additive, it is natural to assign an index of power counting χ[O] to the operator O which in turn
isasumoftheindicesfortheelementsinvolvedinO. Forthepurposeofpowercounting,wedenotegasageneric
couplinginSM.SupposeaneffectiveinteractionCO inL isdressedbySMinteractionsatn-loopstoinduce
i i SMEFT
an effective interaction, ∆ O (no sum over j), involving the operator O of equal dimension. Including the SM
ji j j
n-loop factor ofg2n yields the power counting of ∆ to be 2n+χ[O]−χ[O ]. As ∆ O contributes a counterterm
ji i j ji j
to the effective interactionC O and thus the running ofC , we obtain the power counting for the entry γ in the
j j j ji
anomalousdimensionmatrix
χ[γ ]=2n+χ[O]−χ[O ]. (3)
ji i j
Theissuenowbecomesdefininganindexforoperatorsuptoaconstant.
SinceweareconcernedwithoverallpowercountinginSMinteractions,itisplausibletoassumethatalltermsin
L havethesameindexofpowercounting. Asimilarphilosophywastakenpreviouslytodefinechiraldimensions
4
2
index 3 2 2 2 2 2 1 0
index γ X3 X2H2 Ψ¯ΨHX H4D2 Ψ¯ΨH2D Ψ¯2Ψ2 Ψ¯ΨH3 H6
ij
3 X3 g2 g1 g1 g1 g1 g1 0 0
2 X2H2 g3 g2 g2 g2 g2 g2 g1 0
2 Ψ¯ΨHX g3 g2 g2 g2 g2 g2 g1 0
2 H4D2 g3 g2 g2 g2 g2 g2 g1 0
2 Ψ¯ΨH2D g3 g2 g2 g2 g2 g2 g1 0
2 Ψ¯2Ψ2 g3 g2 g2 g2 g2 g2 g1 0
1 Ψ¯ΨH3 g4 g3 g3 g3 g3 g3 g2 g1
0 H6 g5 g4 g4 g4 g4 g4 g3 g2
Table1: Indicesofpowercountingfordim-6operatorsandpowercountingoftheiranomalousdimensionmatrixat
oneloop.
index 3 3 2 2 2 1
index γ Ψ2H2D2 Ψ¯Ψ3D Ψ2H2X Ψ2H3D Ψ¯Ψ3H Ψ2H4
ij
3 Ψ2H2D2 g2 g2 g1 g1 g1 0
3 Ψ¯Ψ3D g2 g2 g1 g1 g1 0
2 Ψ2H2X g3 g3 g2 g2 g2 g1
2 Ψ2H3D g3 g3 g2 g2 g2 g1
2 Ψ¯Ψ3H g3 g3 g2 g2 g2 g1
1 Ψ2H4 g4 g4 g3 g3 g3 g2
Table2: SimilartoTable1butfordim-7operators.
in chiral perturbation theory involving chiral fermions coupled to electromagnetism [14, 15, 16, 17]. Denoting
arbitrarily
χ[H]=x, χ[λ]=2y, (4)
sothatχ[L ]=4x+2y,itisstraightforwardtodeterminetheindicesofothercomponentsinL :
4 4
3 1
χ[Ψ]= x+ y, χ[X ]=2x+y, χ[D ]=x+y, χ[g ]=χ[Y]=y. (5)
µν µ 1,2,3
2 2
It is evident that the x term actually counts canonical dimension. Since we are concerned with renormalization
mixingofoperatorswithequaldimension, thepowercountingfortheiranomalousdimensionmatrixdependsonly
onthedifferenceoftheindicesaccordingtoEq.(3). Wecanthuschoosetoworkwithx=0andy=1, sothatthe
nonvanishingindicesforpowercountingare
1
χ[Ψ]= , χ[X ]=1, χ[D ]=1, χ[g ]=χ[Y]=1, χ[λ]=2. (6)
µν µ 1,2,3
2
Thelowestindexthatanoperatorcouldhaveiszerointhisconvention. Notethatthescalarself-couplingλ counts
asasquaredgaugecoupling,justasaquarticgaugeinteraction.
We can now associate an index of power counting χ[O] to a higher dimensional operator O by simply adding
up the indices of its components according to Eq. (6). The entry γ in the anomalous dimension matrix for the set
ji
of operators O due to SM interactions at n-loops has the index of power counting shown in Eq. (3) in terms of
k √
a generic coupling g, which denotes g , Y , and λ. Our results for dim-6 and dim-7 operators are shown
1,2,3 e,u,d
in Table 1 and Table 2 respectively. The one-loop γ matrix for dim-6 operators has been computed in a series of
papers [18, 19, 20, 21, 22, 23, 24], and is consistent with power counting in Table 1. The γ submatrix for baryon
3
number violating dim-7 operators is available recently [5], and also matches power counting in Table 2. Note that
someentriesinthetablesmayactuallyvanishduetostructuresofone-loopFeynmandiagramsornonrenormalization
theorem [25, 26, 27]. Since at least one vertex of SM interactions is involved in one-loop diagrams, γ counts as g1
or higher. This explains the presence of zero in the last two columns of the tables. The power counting in the
explicitresultofone-loopγ matrixfordim-6operatorshasalsobeenexplainedinRef.[28]usingtheargumentsof
naivedimensionalanalysis[29]thatrescaleoperatorsforthandbackbyfactorsofcouplingsandpowersof4π. Our
analysisaboveismorestraightforwardandassumesonlytheuniformapplicationofSMperturbationtheory.
With the above definition 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 blocks (H, Ψ, D , X ) for higher dimensional operators only H has a vanishing index. This means that
µ µν
itshouldappearasmanytimesaspossibleinthelowest-indexoperatorsforagivenmassdimensiond. Ford even,
thisiseasytofigureout,i.e.,
Od = (H†H)d/2. (7)
H
For d odd, additional building blocks must be introduced. In the absence of fermions, X and D have to appear
µν µ
at least twice due to Lorentz invariance, which costs no less than two units of index. And in addition, this cannot
yieldanoperatorofodddimension. Thecheapestpossiblewaywouldbetointroducetwofermionfieldsinascalar
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 3, we arrive at the unique
operatoratoddd dimension,
Od pr = (cid:2)(LTεH)C(LTεH)T(cid:3)(H†H)(d−5)/2, (8)
LH p r
where p, r are lepton flavor indices. This is the generalized dim-d Weinberg operator for neutrino mass whose
uniquenesswasestablishedpreviouslyinRef.[30]usingYoungtableau.
The lowest-index operators are of interest because their renormalization running under SM interactions is gov-
ernedattheleadingorderbytheirownanomalousdimensions;i.e.,theyareonlyrenormalizedatthenext-to-leading
order by higher-index operators of the same canonical dimension. This is evident from Eq. (3) and the last row in
Tables1and2. Theuniquenessofthelowest-indexoperatorsateachdimensionfurthersimplifiestheconsideration
oftheirrenormalizationrunning,whichwillbetakenupintheremainingpartofthiswork. Beforethat,wemakea
connectiontoclassificationofoperatorsintermsoftheirholomorphicandanti-holomorphicweightsω, ω¯ [25,27].
Theclaimisthatourlowest-indexoperatorsOd, Od arealsotheoneswiththelargestweights,i.e.,bothoftheirω
H LH
andω¯ arethelargestamongoperatorsofagivencanonicaldimension. Toshowthis, weintroducesomenotations.
WedenoteΨtobeleft-handedfermionfields,i.e.,L, Q, eC, uC, dC,andΨ¯ theright-handedones,andXµν =Xµν∓
±
(i/2)εµνρσX . Thepairofweightshasthevalues(ω,ω¯)=(1,1), (1,1), (3/2,1/2), (1/2,3/2), (0,0), (0,2), (2,0)
ρσ
for the building blocks of operators, H, H†, Ψ, Ψ¯, D, X , X , respectively. The weights (ω(Od),ω¯(Od)) of an
− +
operatorOd ofdimensiond arethesumofthecorrespondingweightsofitscomponents:
1
ω(Od) = n +n + (n +3n )+2n =d−(n +n +2n )≤d, (9)
H H† 2 Ψ¯ Ψ X+ Ψ¯ D X−
1
ω¯(Od) = n +n + (3n +n )+2n =d−(n +n +2n )≤d, (10)
H H† 2 Ψ¯ Ψ X− Ψ D X+
wheren denotesthepowerofthecomponentBappearinginOd. Thelargestω andω¯ thatanoperatorcouldhaveis
B
thusitscanonicaldimension. Fordeven,thisiseasytorealizebysendingn =n =n =n =0,i.e.,theoperator
X± D Ψ Ψ¯
withthehighestweightsisthelowest-indexoperatorOd madeuppurelyoftheHiggsfield. Ford odd,itisknown
H
3Thebilinearform(L¯e)mustcoupletoanoddtotalnumberofH†andH resultinginanevendim-doperator. Thebilinear(ee)requires
fourmorepowersofH thanH†tobalancehypercharge,whichthencannotbemadeweakisospininvariant. Thisleavestheonlypossibility
asshown.
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that all operators in SMEFT necessarily involve fermion fields [31], with the minimal choice being n +n =2.
Ψ Ψ¯
Thiscanbearrangedbychoosingn =2, n =n =n =0resultingintheoperatorOd ofthehighestweights
Ψ X± D Ψ¯ LH
(d,d−2),orbychoosinginsteadn =2asitsHermitianconjugateOd†. Thealternativechoicen =n =1would
Ψ¯ LH Ψ Ψ¯
requireafactorofDduetoLorentzsymmetry,whichreducesω (orω¯)bytwounitscomparedwithOd (orOd†).
LH LH
This establishes the claim. As a side remark, the above equations together with Lorentz symmetry also imply that
theoperatorsateven(odd)dimensionhaveeven(odd)holomorphicandanti-holomorphicweights.
Now we compute the anomalous dimensions at leading order for the lowest-index operators Od at even dim-
H
d and Od pr for odd dim-d in Eqs. (7,8). The Feynman diagrams shown in Figs. 1 and 2 are for O6 and O7 pr
LH H LH
respectively. At higher dimensions one has to be careful with combinatorics due to powers of H†H involved in
the operators. We perform the calculation in dimensional regularization and minimal subtraction scheme and in
the general R gauge. The cancelation of the ξ parameters in the final answer then serves as a useful check. The
ξ
renormalization group equations for the Wilson coefficients of the above two operators are, at leading order in
perturbationtheory,
(cid:20) (cid:21)
d 3 9
16π2µ Cd = 3d2λ− dg2− dg2+dW Cd, (11)
dµ H 4 1 4 2 H H
(cid:20) (cid:21)
d 3 3
16π2µ Cd pr = (3d2−18d+19)λ− (d−5)g2− (3d−11)g2+(d−3)W Cd pr
dµ LH 4 1 4 2 H LH
3(cid:104) (cid:105)
− (YY†) Cd vr+(YY†) Cd pv , (12)
2 e e vp LH e e vr LH
whereW =Tr[3(Y†Y )+3(Y†Y )+(Y†Y )]comesfromfieldstrengthrenormalizationofH.
H u u d d e e
Wemakesomefinalcommentsontheaboveresult. ThetermsintheanomalousdimensionsduetotheHiggsself-
couplingλ increasequadraticallywithcanonicaldimensiondduetocombinatorics,makingrenormalizationrunning
effects significantly more and more important for higher dimensional operators. The Yukawa terms in Eq. (12) are
independent of d because the lepton field L cannot connect to (H†H)(d−5)/2 to yield a nonvanishing contribution
due to weak isospin symmetry. The large numerical factor in the λ term forC6 was observed previously in [21],
H
andourleadingorderresultsindeedmatchthatwork. Includingasymmetryfactorof1/2intheλ termofEq.(11)
that appears in graphs (4)-(5) in Fig. 1 at d =4, our result also applies to renormalization of the λ coupling and is
consistent with [32] upon noting different conventions for λ. The renormalization of the Weinberg operator O5 pr
LH
wasfinallygiveninRef.[33]andcorrespondstographs(1)-(5)inFig.2. Ourresultatd =5isconsistentwiththat
work again after taking into account different conventions for λ. The λ term of the γ function for Od pr increases
LH
significantlywithd forthefirsttwooperatorsinparticular,from4λ atd=5to40λ atd=7.
Acknowledgement
ThisworkwassupportedinpartbytheGrantsNo. NSFC-11025525,No. NSFC-11575089andbytheCASCenter
forExcellenceinParticlePhysics(CCEPP).
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