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TUM-HEP975/15,SFB/CPP-14-118,27December2014 ff Unstable-particle e ective field theory M.Beneke PhysikDepartmentT31,James-Franck-Straße1,TechnischeUniversita¨tMu¨nchen,D–85748Garching,Germany 5 1 Abstract 0 2 Unstableparticlesarenotoriousinperturbativequantumfieldtheoryforproducingsingularpropagatorsinscattering n amplitudesthatrequireregularizationbythefinitewidth. InthisreviewIdiscusstheconstructionofaneffectivefield a theoryforunstableparticles,basedonthehierarchyofscalesbetweenthemass, M,andthewidth,Γ,oftheunstable J particlethatallowsresonantprocessestobesystematicallyexpandedinpowersofthecouplingαandΓ/M,thereby 9 providinggauge-invariantapproximationsateveryorder. Iillustratethemethodwiththenext-to-leadingorderline- 2 shapeofascalarresonanceinanabeliangauge-Yukawamodel,andresultsonNLOanddominantNNLOcorrections ] to(resonantandnon-resonant)pairproductionofW-bosonsandtopquarks. h p Keywords: Unstableparticles,effectivefieldtheory,perturbativequantumfieldtheory,line-shape,topquark, - p W-boson e h [ 1. Introduction such as their unitarity on the Hilbert space built upon theone-particlestatesofonlystableparticleshavebeen 1 v TheconsistencyoftheStandardModel(SM)ofpar- answered many years ago [1]. The construction of the 0 ticle physics is tested at high-energy colliders primar- unitaryS-matrixisbasedoncertainpropertiesoftheex- 7 ilythroughtheproductionandsubsequentdecayofun- acttwo-pointfunctionoftheunstable-particlefield. Al- 3 stable particles. New particles, if discovered, are most thoughadiagrammaticinterpretationisassumed,there 7 0 likelyalsoshort-lived. Atthecurrenthigh-energyfron- is no explicit reference to a perturbation expansion in . tier all known fundamental interactions are perturba- thecouplingthatrenderstheparticleunstable. 1 0 tivelyweak, allowingforveryprecisetheoreticalcom- Sinceexacttwo-pointfunctions arenotathand, this 5 putations in principle. Nevertheless, the application of raisesthequestionofconsistent,successiveapproxima- 1 perturbationtheorytoprocesseswithunstableparticles tions. Ordinary perturbation theory in the Lagrangian v: isnotalwaysstraightforward. couplinggdoesnotwork,sincethelowest-orderprop- agator of the unstable particle leads to singularities in i Theverynotionofanunstableparticlerequiresclar- X scatteringamplitudes.Awell-knownremedyofthesin- ification. In quantum field theory the fundamental en- r gularityistheresummationofself-energycorrectionsto a tities are the fields from which the Lagrangian is con- thepropagator,whichresultsinthesubstitution structed, but the excitations of the fundamental fields maynotcorrespondtotheasymptoticparticlestatesas- 1 1 sumedinscatteringtheory,iftheyarestronglyinteract- p2−M2 → p2−M2−Π(p2). (1) ing as is the case for the quarks and gluons of QCD, or unstable with respect to decay into lighter particles. Theself-energyhasanimaginarypartoforder M2g2 ∼ Relevant cases include the electroweak gauge bosons MΓ, where Γ is the on-shell decay width of the reso- and the top quark, which although all very short-lived, nance,renderingthepropagatorlargebutfinite.“Dyson havewidthovermassratiosofafewpercent,largerthan resummation” sums a subset of singular terms of or- the accuracy of precision calculations. Principal ques- der (g2M2/[p2 − M2])n ∼ 1 (near resonance where tions related to field theories with unstable “particles” p2 ∼ MΓ)toallordersintheexpansioning2. Thispro- M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 2 cedure leaves open the question of how to identify all power-countingoffieldsandinteractionsintheeffective terms (and only these) required to achieve a specified theory leads to a systematic construction of the expan- accuracy in g2 and Γ/M. Failure to address this ques- sioninΓ/M. tionmayleadtoalackofgaugeinvarianceandunitarity The expansion of amplitudes in matching calcula- of the resummed amplitude, since these properties are tionsisperformedaroundthegauge-invariantlocation guaranteed only order-by-order in perturbation theory, M2 = M2−iMΓ (3) andfortheexactamplitude. (cid:63) Despite the fact that unstable particle fields have no of the pole in the complex p2 plane corresponding to correspondingasymptoticparticlestatesandhencetheir theresonance,whereMisidentifiedwiththepolemass, propagators should never be cut, this point is often ig- andΓwiththeon-shellwidth. Theexpansionissimilar noredinpracticeandtheparticleistreatedinFeynman to the one performed in the “pole” [4, 5] or “double- diagramandcrosssectioncalculationsasifitweresta- pole” (in pair production of resonances) [6, 7] approx- ble(“narrow-widthapproximation”). Thiscanbejusti- imation. In a certain sense, unstable-particle effective fiedwhenthewidthisverysmall,since fieldtheoryrepresentsthefield-theoreticformulationof MΓ Γ→→0πδ(p2−M2). (2) the diagrammatic pole approximation, and generalizes (p2−M2)2+M2Γ2 it to all orders in perturbation theory and beyond the leadingpowerintheΓ/Mexpansion. Afirststepinthis Thelimitholdsinthedistributionsenseandistherefore directionhadalreadybeenpresentedin[8]. validonly,ifthephase-spaceoftheunstableparticleis The effective theory approach is minimal as it iden- integratedsufficientlyinclusively,suchthattheintegra- tifiespreciselythetermsrequiredtoachieveaspecified tioncontourinthevariablep2canbedeformedfaraway accuracy in g2, and Γ/M, and does not include more. from M2. This is not always the case. An obvious ex- This makes the calculations particularly simple. Fur- ampleistheline-shape, butalsodistributionsmaytrap thermore,theoperatorinterpretationallowsforthesum- the contour near M2. A more accurate treatment than mationoflargelogarithmsofΓ/Mthroughrenormaliza- (2)isalsorequired,whenthedesiredprecisionexceeds tiongroupequationsandanomalousdimensions. There theleading-orderapproximationinΓ/M. is a draw-back: the details of the effective theory de- Somewhat surprisingly, systematic computational pend on the inclusiveness of the observable and is not schemes to obtain approximations to scattering pro- valid locally over the entire phase-space, where some cessesinvolvingunstableparticlesinweakcouplingex- portions may involve further soft scales of order Γ in pansionsarerelativelyrecent. Thetwo,whicharegen- addition to (p2 − M2)/M. Even the prediction of the eral, are the unstable-particle effective theory and the resonanceline-shaperequiresmatchingoftheresonant complexmassscheme. (peak) region calculation within the effective theory to theoff-resonanceregioncomputedwithstandardpertur- 1.1. Unstable-particleeffectivetheory bationtheory. Thesingularityoftheunstableparticlepropagatorin- dicatessensitivitytoatimescalelargerthantheComp- 1.2. Thecomplex-massscheme ton wave-length 1/M of the particle, evidently its life- time1/Γ. ThepresenceoftwodifferentscalesΓ (cid:28) M The complex-mass scheme is an extension of the standard on-shell renormalization scheme to unstable liesintheverynatureoftheproblem,sincearesonance withΓ ∼ M wouldnotbeidentifiedassuch. Themain particles. It defines the complex mass and field renor- idea of unstable-particle effective field theory [2, 3] is malizationconstantfromthelocationandresidueofthe pole (3) of the unstable-particle propagator. The bare toexploitthishierarchyofscalesinordertosystemati- massM issplitintoarenormalizedmassandcountert- callyorganizethecalculationsinaseriesinthecoupling 0 g, and Γ/M. The short-distance scale M is integrated ermthrough out by performing standard perturbative computations M2 = M2 +δM2, (4) andthefulltheoryismatchedtoaneffectiveLagrangian 0 (cid:63) (cid:63) thatreproducesthephysicsatthescaleΓ. Theeffective and δM2 is part of the interaction Lagrangian and (cid:63) theorycontainsafieldφ ,whichdescribesaresonance treated as a perturbation. The unstable-particle prop- v withmomentum p = Mv+k,whereonlyk ∼ Γisfluc- agator i/(p2 − M2) is never infinite for physical, real (cid:63) tuating. The resonant field can interact with other soft momenta. Thecomplex-massschemewasdiscussedal- fields with momenta of order Γ, but off-shell effects at ready in [9], but it was used for the first time in a full the scale M are part of the matching coefficients. The one-loop calculation only in 2005 [10] for the process M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 3 e+e− → 4fermions(+γ) at high energies, which re- 2. Line-shapeofanunstableparticle ceivesimportantcontributionsfromtheunstableW+W− In this section, which follows [2, 3], we consider a intermediatestate. toy model that involves a massive scalar field, φ, and two fermion fields. The scalar as well as one of the Although the standard rules of perturbation theory fermionfields,ψ,(the“electron”)arechargedunderan and an expansion in the number of loops apply to the abeliangaugesymmetry,whereastheotherfermion,χ, complex-mass scheme, a re-ordering and resummation of the g2 expansion is implicit, since the propagator is (the “neutrino”) is neutral. The model allows for the oforder1/(M2g2)intheresonanceregion.Theassump- scalartodecayintoanelectron-neutrinopairthrougha Yukawa interaction. The model describes the essential tionisthatthecomplexmassinthepropagatorcaptures features of the Z-boson line-shape in the SM [13]. Its alltermsthatneedtoberesummedwhichisindeedthe Lagrangianis case(seealsonextsection). Sincetheschemeisonlya reparameterizationofthebaretheory,whichisnotmod- L = (D φ)†Dµφ−Mˆ2φ†φ+ψ¯iD(cid:54) ψ+χ¯i(cid:54)∂χ µ ified,itisobviousthatnodoublecountingoccurs.Like- 1 1 wise, gauge invariance is assured, since the split (4) is − FµνF − (∂ Aµ)2 4 µν 2ξ µ gauge-invariantandthealgebraicidentitiesthatguaran- λ tee gauge invariance are valid in the presence of com- +yφψ¯χ+y∗φ†χ¯ψ− (φ†φ)2+L , (5) ct plexparameters.Unitaritymightbeaconcern,sincethe 4 unitarityequationinvolvescomplexconjugation. How- where Mˆ denotes the renormalized mass, not neces- ever, since the bare theory is unitary, so must be the sarily the pole mass M defined by (4), L the coun- ct reparameterized one. What needs to be shown is that terterm Lagrangian, and D = ∂ − igA . We define µ µ µ the theory with the complex-mass prescription is per- α ≡ g2/(4π), α ≡ (yy∗)/(4π) (at the scale µ) and as- g y turbatively unitary in the sense that unitarity violation sumeα ∼α ∼α,andα ≡λ/(4π)∼α2/(4π). g y λ in any given order in the loop expansion are of higher The line-shape is the totally inclusive cross section orderintheexpansionparameters(countingΓ/M ∼g2). fortheprocess This point was demonstrated explicitly at one-loop for fermion-fermionscatteringthroughavector-bosonres- ν¯(q)+e−(p)→ X (6) onance[11],andingeneralin[12]. as a function of s ≡ (p+q)2, which can be calculated fromtheimaginarypartoftheforwardscatteringampli- The complex-mass scheme is conceptually straight- tudeT(s).1 Inparticular,weareinterestedintheregion forward. It does not require separate treatments of the resonanceandoff-resonanceregions,andcaneasilybe s≈ M2,ormorepreciselys−M2 ∼ MΓ∼αM2 (cid:28) M2, where we expect an enhancement of the cross section appliedtokinematicdistributions. Comparedtotheef- due to the resonant production of the scalar. Defining fective field theory method the scheme does not make use (explicitly) of expansions in Γ/M and hence does thedimensionlessvariable not simplify the problem as much as possible in prin- s−Mˆ2 Γ ciple. The difficulty of the calculation is equivalent to δ≡ Mˆ2 ∼ M, (7) the corresponding standard loop calculation with the the cross section far away from the resonance can be additionalcomplicationofloopintegralswithcomplex expandeding2intheusualmanneraccordingto masses. Thisisnotapracticalproblemattheone-loop order,makingthecomplex-massschemethemethodof σ=g4f (δ)+g6f (δ)+.... (8) 1 2 choiceforautomatednext-to-leadingordercalculations. Ontheotherhand,calculationsbeyondthisorderwould Ateveryorder,thecoefficient fn(δ)isafunctionofthe presentlybedifficultandtheresummationoflogarithms variable δ ∼ 1. On the other hand, near resonance we lnM/Γcannotbeperformed. mayexploitδ(cid:28)1toexpandtheamplitudeinδ. Atthe sametime,asg2/δ∼1sinceΓ∼ Mg2,sometermsmust In the following I do not discuss the complex-mass be summed to all orders. A systematic approximation schemefurther, butfocusonunstable-particleeffective theory. I use the line-shape of a resonance to illustrate the framework and the discuss results on pair produc- 1The total cross section of process (6) is not infrared finite for masslesselectronsduetoaninitial-statecollinearsingularity,which tion of W-bosons and top quarks near threshold which has to be absorbed into the electron distribution function. In what (Ibelieve)benefitparticularlyfromthismethod. followsitisunderstoodthatthissingularityissubtractedminimally. M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 4 totheline-shapeintheresonanceregionthereforetakes theform (cid:88)(cid:32)g2(cid:33)n σ ∼ ×{1(LO);g2,δ(NLO),...} δ n = h (g2/δ)+g2h (g2/δ)+... (9) 1 2 withnon-trivialfunctionsh (g2/δ)ateveryorderinthe n reorganized expansion. The effective theory identifies Figure 1: Reduced diagram topologies in 2 → 2 scattering near resonance.Left:resonantscattering.Right:non-resonantscattering. therelevanttermsandconstructstheexpansion(9). 2.1. Relevantmodesandreducedscatteringdiagrams forallmasslessfieldsoftheoriginalLagrangian. Here The effective theory is based on the hierarchy of n± are two light-like vectors with n+ · n− = 2, n− is scalesΓ (cid:28) M. Inafirststepweintegrateouthardmo- thedirectionoftheelectronfour-momentum,and p⊥ is mentak∼ M. Theeffectivetheorywillthennotcontain transverseton−andn+.2 any longer dynamical hard modes since their effect is The space-time picture of the kinematically allowed included in the coefficients of the operators. The hard processes is very simple and the corresponding re- effects are associated with what is usually called fac- duced diagram topologies are shown in Figure 1 for torizablecorrections,whereastheeffectsofthedynam- theforward-scatteringamplitude. Theleftdiagramde- ical modes correspond to the non-factorizable correc- scribes the production of the resonance through a hard tions [8]. On the level of Feynman diagrams, the hard process,representedintheeffectivetheorybysomelo- contributioncanbeidentifieddirectlyusingthemethod cal operator O(pk), and its subsequent propagation over ofregionstoseparateloopintegralsintovariouscontri- distancesoforder1/Γ. Theresonance(doubleline)can butions[14].Thehardpartisobtainedbyexpandingthe interactwithsoftfluctuations. Theinitial-stateelectron full-theoryintegrandinδ. legcanbedressedwithcollinearcorrections. However, The modes to be described by the effective La- collinear modes cannot be exchanged across the dou- grangian correspond to kinematically allowed scatter- ble line, since this would not leave enough energy to ing processes with virtualities much smaller than M2. producethescalarnearresonance. Theprocessjustde- Particleswithmassesabove MΓarenolongerpresent, scribedisrepresentedintheeffectivetheorybythefirst except for the unstable particle, which by construction lineofthemasterformula is close to mass-shell. To account for this, we write (cid:88)(cid:90) the momentum of the scalar particle as P = Mˆv + k, iT = d4x(cid:104)νe|T(iO(pk)(0)iO(pl)(x))|νe(cid:105) where the velocity vector v satisfies v2 = 1 and the k,l (cid:88) residual momentum k scales as Mδ ∼ Γ. In analogy + (cid:104)νe|iO(k)(0)|νe(cid:105). (11) nr toheavy-quarkeffectivetheory(HQET)weremovethe k rapidspatialvariatione−iMˆv·xfromtheφfieldanddefine φv(x) ≡ eiMˆv·xP+φ(x),whereP+ projectsontothepos- fortheforward-scatteringamplitude. The scattering may also occur without the produc- itive frequencypart toensurethat φ is apure destruc- v tionfield. Afieldwithmomentumfluctuationsk ∼ Γis tion of the scalar near its mass-shell (right diagram in Figure 1). In the present toy theory this still requires calleda“soft”field. Thus,forthesoftscalarfieldφ we v an intermediate scalar line, since the neutrino has only have P2 − Mˆ2 ∼ M2δ. This remains true if the scalar Yukawa interactions. The scalar may be off-shell, be- particleinteractswithasoftgaugebosonwithmomen- tumMδ,sotheeffectiveLagrangianshouldcontainsoft cause the electron has radiated an energetic (hard or (s)fieldsforeverymasslessfieldofthefulltheory. Theunstableparticleisproducedinthescatteringof 2Inthegeneralcaseseveraltypesofcollinearmodesarerequired, on-shell particles with large energy of order M. These oneforeachdirectiondefinedbyenergeticparticlesintheinitialand can remain near mass-shell by radiating further ener- final state. For the inclusive line-shape we calculate the forward- geticparticlesintheirdirectionofflight. Theeffective scatteringamplitude,sonodirectionisdistinguishedinthefinalstate. Wethenneedtwosetsofcollinearmodes,oneforthedirectionofthe Lagrangian must therefore also contain hard-collinear incomingelectron,labelledby“c1”(oroftensimply“c”),theother (c1)modeswithmomentumscaling forthedirectionoftheincomingneutrino(labelled“c2”). Sincethe neutrinoiselectricallyneutral, thecollinearfieldsψc2, Ac2 andχc1 n+p∼ M, p⊥ ∼ Mδ1/2, n−p∼ Mδ (10) appearonlyinhighlysuppressedterms,sowecanignorethemhere. M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 5 collinear)photonbeforeithitstheneutrino. Inthiscase agatornearresonancecanbewrittenas theinvariantmassofthecollidingelectron-neutrinosys- iR iR tem is of order M2 but not near M2, producing a non- φ = φ . (13) resonant scalar. In the effective theory this process is P2−M2 2Mˆv·k+k2−(M2 −Mˆ2) (cid:63) (cid:63) representedbyalocalfour-fermionoperatorO(k),with- nr We now define the matching coefficient ∆ ≡ (M2 − out φv fields. In general, non-resonant scattering in- Mˆ2)/Mˆ. There are two solutions to P2 = M2, one(cid:63)of cludes all “background processes”, which produce one (cid:63) whichisirrelevantsinceitscalesasv·k ∼ Mˆ. Forthe of the final states under consideration. This topology otherwefind doesnotinvolvearesonantheavyscalar, andbothsoft and collinear fields can be exchanged across the dia- (cid:113) v·k = −Mˆ + Mˆ2+Mˆ∆−k2 gram. The matrix elements in (11) are understood to (cid:62) beevaluatedwiththeeffectiveLagrangian. ∆ ∆2+4k2 = − (cid:62) +O(δ3), (14) 2 8Mˆ 2.2. ConstructionoftheeffectiveLagrangian where we expanded in δ in the second line, using ∆ ∼ We divide the effective Lagrangian into three parts. kg(cid:62)2i,∼wMe dδe.dEuxcpeatnhdeinbgili∆ne=ar(cid:80)tei=rm1∆s(ii)nin(1to2)tefrrmomsothfeorddies-r Roughlyspeaking,thefirst,LHSET,describestheheavy persionrelation(14). Gaugeinvarianceoftheeffective scalar field near mass-shell and its interaction with the Lagrangian implies that the leading soft-photon inter- gauge field. The second part, L , describes energetic ± actions can be obtained from the bilinear terms by re- fermionsandtheirinteractionswiththegaugefield. Fi- placing∂ → D . Thegaugeinvarianceofthematching nally,thethirdpart,LintcontainsthelocaloperatorsO(pk) coefficientfolloswsfromtheinvarianceoftheunstable- andO(nkr)responsiblefortheproductionoftheresonance particlepoleM(cid:63). andoff-shellprocesses.Inthefollowing,wewritedown In the underlying theory the full renormalized prop- alltermsneededforanext-to-leadingorder(NLO)cal- agator of the unstable particle is given by i(s − Mˆ2 − culationoftheline-shape. Π(s))−1, where −iΠ(s) corresponds to the amputated The soft Lagrangian LHSET is an extension of the 1PI graphs including counterterms. Comparing this to HQETLagrangian[15]toa(herescalar)particlewhose (13)andexpandingΠ(s)around Mˆ2 andinthenumber mass-shellisdefinedbythecomplexpolelocation(4). ofloopsintheformΠ(s)= Mˆ2 (cid:80) δlΠ(k,l),whereitis k,l The residual mass term which is usually set to zero in understoodthatΠ(k,l) ∼g2k,weobtain HQETbychoosing M tobethepolemassoftheheavy (cid:16) (cid:17) quark, is now necessarily non-vanishing and complex. ∆= Mˆ Π(1,0)+Mˆ Π(2,0)+Π(1,1)Π(1,0) +.... (15) Therelevanttermsare Π(1,0) and Π(2,0) + Π(1,1)Π(1,0) (but not Π(2,0) and Π(1,1) (cid:32) ∆(1)(cid:33) L = 2Mˆφ† iv·D − φ separately) are infrared-finite, which justifies the inter- HSET v s 2 v pretation of ∆ as a short-distance coefficient. Explicit (cid:32)(iD )2 [∆(1)]2 ∆(2)(cid:33) results for ∆(1) and ∆(2) in the MS and pole renormal- +2Mˆφ†v 2sM,ˆ(cid:62) + 8Mˆ − 2 φv izationschemecanbefoundin[3]. Hereweonlynote thatinthepolescheme(Mˆ ≡ M),wehave∆ = −iΓ,in 1 − F Fµν+ψ¯ iD(cid:54) ψ +χ¯ i(cid:54)∂χ , (12) whichcasetheresidual“mass”ispurelyimaginaryand 4 sµν s s s s s s coincideswiththeon-shellwidth. where ψs (χs) denotes the soft electron (neutrino) field EachterminLHSET canbeassignedascalingpower andthecovariantderivativeDs ≡ ∂−igAs includesthe in δ. Since Ds ∼ k ∼ Γ ∼ Mδ and ∆(1) ∼ Mg2 ∼ softphotonfield. Furthermore, aµ ≡ aµ −(v·a)vµ for Mδ,bothtermsinthefirstlineof(12)areofequalsize (cid:62) any vector. The only non-trivial short-distance match- and leading terms. The unstable-particle propagator is ingcoefficientsinthisexpressionarethequantities∆(i) therefore tobedefinedbelow. i The bilinear terms in the soft scalar field φv are de- 2Mˆ(v·k−∆(1)/2), (16) terminedbytherequirementthatL reproducesthe HSET two-pointfunctionofthescalarinthefulltheoryclose which corresponds to a fixed-width prescription. The toresonance. Denotingthecomplexpoleofthepropa- linearityofthepropagatorinthe(residual)momentum gatorbyM2 andtheresidueatthepolebyR ,theprop- makes calculations in the effective theory particularly (cid:63) φ M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 6 simple. The fact that only ∆(1) appears in the leadingorder Lagrangian proves that only the two-point func- tionintheoriginaltheoryneedstoberesummedbyin- cluding the one-loop self-energy into the unperturbed Lagrangian. No higher-point functions require resummation, which is intuitively obvious, since the origin Figure2: Scalarself-energycorrectiontotheforward-scatteringam- of the long-distance scale is associated with a single- plitude. particleeffect,thelife-timeoftheresonance. In momentum space the propagator (16) of the φ (cid:82) v atleadingpower. fieldscalesas1/δ. Hence, because d4k countsasδ4, Withtheexternal-collinearmodeswecanimplement the soft scalar field φ (x) scales as δ3/2. It follows that v the production and non-resonant sources as interaction the terms in the second line of (12) scale as δ5. Being termsinL . AtNLOtherelevanttermsread suppressed by one power in δ or g2 relative to the first int line, theymustbeincludedonlyinacalculationofthe L = Cyφ ψ¯ χ +Cy∗φ†χ¯ ψ alisneδ-sahnadptehweistohfNt fLeOrmpiorencifiseioldns. sFcianlaellays,sδi3n/c2e, tAhµsestcearmless int + D yMˆy∗2v (cid:0)ψn¯−n−nχ+n+(cid:1)(cid:0)χ¯n+ψnv−(cid:1)n,+ n− (18) inthelastlineof(12)scaleasδ4 andrepresentleading whereC =1+O(α)andDarethematchingcoefficients. interactionsamongthesoft,masslessmodes.Byadding The two lines correspond to the two reduced diagram furthertermstheLagrangiancanbeimprovedtoanyac- topologies in Figure 1. We note that the effective La- curacydesired. Next,weturntotheconstructionoftheeffectiveLa- grangianisnotmanifestlyhermitian, sinceitdescribes thedecayofthescalar. Nevertheless,itgeneratesauni- grangian, L , associated with the energetic fermions. ± tarytimeevolution,sinceitreproducesbyconstruction Theinteractionsofcollinearmodeswiththemselvesand the unitary underlying theory to the specified order in with soft modes are described within soft-collinear ef- theexpansioninδ. fective theory (SCET) [16, 17, 18, 19]. The coupling The external fields scale as δ3/2. Thus, an insertion of collinear modes to the scalar field φ , and among (cid:82) collinear fields with different directionsvproduces off- ofaφψχoperatorresultsin d4xφvψ¯n−χn+ ∼δ1/2. The shell fluctuations, which are not part of the effective forward-scatteringamplituderequirestwoinsertionsof this operator. Accounting for the scaling of the exter- Lagrangian. The momenta associated with generic nal state (cid:104)ν¯e−| ∼ δ−1, we find T(0) ∼ g2/δ for the am- collinearfieldsψ andχ¯ donotadduptoamomen- c1 c2 tum of the form P = Mv + k. This kinematic con- plitude at leading order, which is the expected result. Thefour-fermionoperatorissuppressedinδandresults straint is implemented by adding the production and non-resonant operators, O(k) and O(k), respectively, as in a contribution of order g2 to T. Thus, to compute p nr the NLO correction T(1) we need C(1), the O(g2) con- external “sources” for the specific process. The line- tributiontothematchingcoefficientC,while Disonly shapeisthengivenbythecorrelationfunction(11). needed at tree level. The matching coefficients are ob- Alternatively, the dynamical hard-collinear modes tainedfromthehardcontributionstothecorresponding can be integrated out in a second matching step, in on-shell three- and four-point functions in the full the- whichthecollinearfunctions(labelled“C”inFigure1) ory. I refer to [3] for the precise matching equation as appearasmatchingcoefficientsof(non-local)operators. wellastheexplicitresults. The new effective Lagrangian contains an “external- collinear” electron mode with momentum Mˆn /2+k, − 2.3. Examplediagram which describes the remaining soft fluctuations k ∼ δ It is instructive to discuss how the self-energy cor- around the fixed large component. Similar to the reso- rectiontotheintermediatescalarinthefulltheory, see nance field, we extract the large component and define ψn−(x) ≡ eiMˆ/2(n−x)P+ψc1(x),whereP+ projectsonthe Ffiirgsturseep2a,riasterepthreeshenarteddainndthseofetffceocntitvreibduetisocnrispttihoen.oWnee- positivefrequencypartofψ . Addingthecorrespond- c1 loopself-energy,Π(s)=Π (s)+Π (s),andthenexpand ingfieldwithn−andn+exchangedfortheneutrino,the the hard part Π (s) = Mˆ2h(cid:80) δlΠ(s1,l). The soft part is softinteractionsoftheexternal-collinearfieldaregiven h l reproducedbytheeffectivetheoryself-energy. Thefirst by termΠ(1,0) inthehardexpansionisgauge-invariantand L± =ψ¯n−in−Ds(cid:54)n2+ ψn− +χ¯n+in+∂(cid:54)n2− χn+. (17) creoandtyribreulteevsatnot∆to(1t)haesledaisdciunsgs-eodrdbeerfloirnee.-sThhaipset.eTrmheisneaxl-t M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 7 term Π(1,1) cancels one of the adjacent scalar propagators, such that the self-energy correction merges with the local production vertex. Π(1,1) is gauge-dependent. Thegaugedependencecancelswiththevertexdiagram to produce a gauge-independent NLO hard-matching coefficient C(1). Continuing in this way, we find that Π(1,2) contributes to the one-loop matching coefficient D(1) ofthefour-fermionoperator(cid:0)ψ¯ χ (cid:1)(cid:0)χ¯ ψ (cid:1),be- n− n+ n+ n− causethescalarpropagatorstotheleftandrightareboth cancelled. The contribution is again required to obtained a gauge-invariant one-loop matching coefficient Figure3: Hard(upper)andsoft(lowerdiagrams)contributionsto [3],thoughitisalreadyaNNLOtermfortheline-shape. T(1). This example illustrates the power of the effective fieldtheorymethod. Itautomaticallybreaksadiagram resultin into different pieces and organizes them into gauge-   itnhveaLriaagnrtanogbijaenctas.lloTwhseopnoewtoeri-dceonutniftiyntgheastseormcisatreedlewvaitnht iTs(1) =iT(0)× (4gπ2)2 4L2−4L+ 56π2 (21) for a specified accuracy before any explicit calculation needstobeperformed. with L = ln(−2D/µ). The partonic line-shape is ob- tainedaftersubtractingtheinitial-statecollinearsingu- larityandtakingtheimaginarypart. Thepartonicline- 2.4. Line-shapeatnext-to-leadingorder shapemustthenbeconvolutedwiththeelectrondistri- Our goal is to carry out this programme for the butionfunction. forward-scatteringamplitudeT(0)+T(1)atNLO,where Wenotethesimplicityoftheresult,whichisaconse- T(0) sums up all terms that scale as (g2/δ)n ∼ 1 and quence ofthe fact thatthe complete calculationis bro- T(1) contains all terms that are suppressed by an addi- ken into separate single-scale calculations by factoriz- tional power of g2 or δ. At leading order there is only ingthehardandsoftregions.TheNLOcorrectionleads onediagram,involvingtwothree-pointverticesandone to a distortion of the line-shape relative to the Breit- resonantscalarpropagator. Weget Wignerform,whichinnon-inclusivesituationscande- pend on the final state. Fitting a measured line-shape −iyy∗ iT(0) = [u¯(p)v(q)][v¯(q)u(p)], (19) totheBreit-Wignerformratherthanthetrueshapepre- 2MˆD dictedbytheoreticalcalculationsleadstoerrorsinmass √ wherewedefinedD≡ s−Mˆ −∆(1)/2. Theinclusive determinations. Inthepresenttoymodel, choosingthe line-shape is related to T(0) by σ = ImT(0)/s through pole mass M = 100 GeV (such that the MS mass is Mˆ = 98.8GeVatLOand Mˆ = 99.1GeVatNLO)and theopticaltheorem.TheaboveexpressiongivesaBreit- √ couplings g2/(4π) = |y|2/(4π) = 0.1 to mimick the pa- Wignerdistributionin s. In the effective theory there are three classes of rametersofelectroweakgaugebosons, theerrorwould diagrams that contribute to T(1), corresponding to beoforder100MeV. Figure4showstheleading-orderpartonicline-shape hard, hard-collinear and soft contributions. The hard- intheeffectivetheoryandthetree-level(orderα2)cross collinearcorrectionstotheexternallinesleadtoscale- section off resonance in the full theory. The two re- less integrals and vanish. The hard corrections consist ofapropagatorinsertion[∆(1)]2/4−Mˆ∆(2),aproduction sultsagreeinanintermediateregionwherebothcalcu- vertex insertion C(1), and a four-point vertex diagram lationsarevalid. Thisal√lowstoobtainaconsistentLO duetothe(ψ¯χ)(χ¯ψ)operatorinL ,asshownintheup- result for all values of s. The figure also shows the int NLO line-shape for the numerical values given above. per diagrams of Figure 3. The sum of these diagrams In order to obtain an improved NLO result in the en- reads √ tireregionof s,theNLOline-shapewouldhavetobe   iTh(1) =iT(0)×2C(1)−[8∆D(1M)]ˆ2+∆2D(2)−2DMˆ . (20) mfualltcthheeodrtyo. the NLO off-resonance cross section in the ThemethoddiscussedheremakesNNLOline-shape The soft-photon one-loop corrections (lower set of di- calculations in 2 → 2 scattering possible with present agrams in Figure 3) computed in the effective theory techniques. An outline of such a calculation has been M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 8 Coulomb force is generated by QCD, while the decay stilloccursthroughtheelectroweakinteraction. Count- ing α ∼ α2 , we now find α /v ∼ 1. Diagrammati- s EW s cally,ladderdiagramsthatcontaintheseenhancedterms must be summed to all orders in perturbation theory, which generates toponium bound-states in the spectral functions. Since the characteristic energy near threshold E ∼ Mv2 is of order Γ, the bound-states appear as broad resonances, of which only the first one leaves a distinctivefeatureinthett¯crosssection[21,22]. In the following I review results for W and top pair production nearthreshold obtained withinthe effective field theory approach, leaving out all the technical de- Figure4: Theline-shape(inGeV−2)intheeffectivetheoryatLO (lightgrey/magentadashed)andNLO(lightgrey/magenta)andthe tailsthatcanbefoundintheoriginalpapers. LO cross section off resonance in the full theory (dark grey/blue dashed)asafunctionofthecentre-of-massenergy(inGeV).Figure 3.1. W bosons from[2]. This subsection summarizes results from [23, 24]. Weconsidertheprocesse−e+ →µ−ν¯ ud¯Xwithcentre- √ µ given in [3], though no complete calculation has been of-massenergy s = 160...170GeV,whereitisdomi- performedtodate. natedbyaW+W−intermediatestatenearthresholdwith subsequent semi-hadronic decay. The inclusive cross section is extracted from specific cuts of the forward 3. Pairproductionnearthreshold amplitude I reviewed in some detail the case of the line-shape, 1 since it serves well to illustrate the general framework σˆ = s ImA(e−e+ →e−e+)|µ−ν¯µud¯, (23) ofunstable-particleeffectivetheory. However,someof the more interesting results concern pair production of whichalsoincludesdiagramswithonlyasingleinternal unstable particles, specifically the W bosons and top W line. We perform a “QCD-style” calculation of the quarks, near threshold. In e+e− collisions very precise “partonic” cross section σˆ with massless electrons in measurementofthemassesoftheseparticlescanbeob- the MS scheme, and convolute it with the MS electron tainedfromathresholdscan. distributionfunction: The threshold dynamics is determined by the inter- (cid:90) 1 playofthestrength oftheelectromagnetic(W bosons) σ(s)= dx1dx2 fe/e(x1) fe/e(x2)σˆ(x1x2s). (24) 0 orcolour(topquarks)Coulombforceandthesizeofthe decaywidthoftheparticle. Thesmallparametersare The MS electron distribution function depends on m , √ e Γ √ butnoton s,M,Γ. δ≡ , v2 ≡( s−[2M+iΓ])/M, (22) In the effective field theory (EFT) the W bosons are M describedbytwonon-relativisticthree-vectorfieldsΩi, a and the coupling α = g2/(4π). For W bosons, Γ ∼ wherea = ±referstothechargeoftheW. TheHSET W M α and therefore the effective strength of the Lagrangian relevant to a single (scalar) unstable parti- W EW √ Coulombforceisα /v∼ δ(cid:28)1.Thisleadstoanen- cle is replaced by the PNRQED Lagrangian [25], gen- em hancement,buttheCoulombforceisneverO(1),andno eralized to the case of an unstable vector particle. The resummationisneeded[20]. TherapiddecayoftheW relevanttermsare bosonpreventstheformationofanyvisibleW+W−resonance.Thesituationisdifferentfortopquarks,sincethe LPNRQED =(cid:88)a=∓Ω†aiiD0s + 2M(cid:126)∂2W − ∆2Ωia+Ω†ai ((cid:126)∂2−8MMW3W∆)2 Ωia+(cid:90) d3(cid:126)r (cid:104)Ω†−iΩi−(cid:105)(x+(cid:126)r)(cid:18)−αrem(cid:19)(cid:104)Ω†+jΩ+j(cid:105)(x).(25) M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 9 Σfb @ D 600 EFTLO @ D 500 EFT@(cid:143)!N!!!!!LOD EFTNLO @ D 400 exactBorn 300 Figure6: One-loopdiagramforthehard-matchingcoefficient(left) 200 andasoftNLOcontributiontotheforward-scatteringamplitudein theeffectivetheory(right). 100 (cid:143)!s!!!GeV 156 158 160 162 164 166 168 170 @ D While constructing an expansion of the Born cross sectionwhenanexact,numericalresultisreadilyavail- Figure5: SuccessiveEFTapproximations: LO(long-dashed/blue), able,appearsasanunnecessarycomplication,thecom- N1/2LO (dash-dotted/red) and NLO (short-dashed/green). The putation of the NLO radiative correction in unstable- solid/black curve is the full Born result computed with Whizard/ particleeffectivetheory[23]isremarkablesimplecom- CompHep.TheN3/2LOEFTapproximationisindistinguishablefrom thefullBornresultonthescaleofthisplot.Figurefrom[23]. paredtothecorresponding calculationinthecomplex- massscheme[10,28]. Themostcomplicatedpartisthe computationoftheNLOmatchingcoefficientoftheop- The master formula for the forward amplitude A co- erator(26),which,however,isastandardone-loopcal- incides with (11), but the production and non-resonant culation. ArepresentativediagramisshowninFigure6 operatorsarenowoftheform left. The diagram on the right displays a soft, “non- factorizable”NLOcorrectiontothetwo-pointfunction (cid:16) (cid:17)(cid:16) (cid:17) O(pk) =C(pk) e¯c2,L/Rγ[inj]ec1,L/R Ω†−iΩ†+j , (26) of production operators in (11), and results again in a simple expression, similar to (21). A comparison with O(k) =C(k)(e¯ Γ e )(e¯ Γ e ), (27) the complex-mass scheme calculation and the double- nr nr c1 1 c2 c2 2 c1 poleapproximation(DPA),includingQCDcorrections with Γ , Γ Dirac matrices, a[ibj] ≡ aibj +ajbi, and(cid:126)n and initial-state radiation is given in the following ta- 1 2 ble.4 Thenumericaldifferenceof1%betweentheEFT theunit-vectorinthedirectionoftheincomingelectron and full ee4f results is presumably in part due to the three-momentum. N3/2LO correction associated with the NLO matching Due to the 1/v enhancement of electromagnetic coefficient of O(k), which is implicitly contained in the Coulombexchange,thesystematicexpansionofAgoes nr √ NLOfullee4fcalculation. in powers of δ. Also, the non-resonant term appears as such a “N1/2LO” correction, since the lead- √ σ(e−e+→µ−ν¯µud¯X)(fb) ingimaginarypartsofC(k)areproportionaltoα3,while s[GeV] Born(SM) EFT fullee4f DPA √ nr A∼α2 δ.3 161 107.06(4) 117.38(4) 118.77(8) 115.48(7) √ 170 381.0(2) 399.9(2) 404.5(2) 401.8(2) TheEFTconstructsanexpansioninΓ/M and( s− 2M)/M of the full theory Born cross section. Before We can now estimate the theoretical uncertainty in turningtoradiativecorrectionsitisinstructivetocom- the W mass determination from a threshold scan. Fig- paresuccessivetermsinthisexpansiontothefullBorn ure 7 shows κ = σ(s,M + δM )/σ(s,M ) for W W W result computed numerically (using Whizard [26] and M = 80.377GeV and different values of δM as W W CompHep [27]). This is shown in Figure 5. The LO function of the cms energy. The relative change in non-relativisticapproximationoverestimatesthetruere- the cross section is shown as dashed lines for δM = W sult. The N1/2LO non-resonant correction yields a ±15,±30,±45MeV. The shape of these curves shows (nearly) constant, negative term and provides alr√eady thatthesensitivityofthecrosssection√totheW massis goodagreementclosetothenominalthresholdat s≈ largestaroundthenominalthreshold s≈161GeV,as √ 161 GeV. To extend the approximation in a wider re- expected,andrapidlydecreasesforlarger s.(Theloss gionaroundthethreshold,itisnecessarytoincludeall insensitivityispartiallycompensatedbyalargercross termsuptoN3/2LO. section,implyingsmallerstatisticalerrorsoftheantici- patedexperimentaldata.) Theshadedareasprovidean √ 3Thefactor δarisesfromtheleading-orderEFTmatrixelement andcorrespondstothephase-spacesuppressionnearthreshold. 4The“fullee4f”columnreferstotheerratumof[28]. M.Beneke,Unstable-particleeffectivefieldtheory(˜2015)1–14 10 ΚΚ -45MeV 11..0022 -30MeV ISR 11..0011 -15MeV (cid:143)(cid:143)!!ss!!!!!!GGeeVV 116600 116622 116644 116666 116688 117700 @@ DD +15MeV 00..9999 +30MeV Figure8:SomeNNLOdiagramsthatcountasN3/2LOintheδexpan- sionandtheirEFTrepresentation(upperline). 00..9988 +45MeV Σ@fbD 150 Loosecuts 125 Figure7:W-massdependenceofthetotalcrosssection.Allthecross sectionsarenormalizedtoσ(s,MW =80.377GeV). Seetextforex- 100 planations.Figurefrom[23]. 75 50 estimate of the uncertainty from uncalculated N3/2LO Tightcuts 25 terms. The inner band is associated with the interference of single Coulomb exchange with one-loop hard 0 MW(cid:144)L 1 1.5 2 3 5 7 10 or soft corrections, which are genuine NNLO correc- √ tionsinotherschemes. Theouterbandaccountsforthe Figure9:ComparisonoftheBorncrosssectioninthefullSMat s= 161 GeV computed with WHIZARD (red dots) with the effective- non-resonant term already mentioned above. Finally, theoryresultfortheloose-cutimplementation(dashedbluecurve)and thelinemarked“ISR”estimatesambiguitiesintheim- thetight-cutimplementation(solidblackcurve).Figurefrom[24]. plementation of initial-state radiation. This represents thelargestcurrentuncertainty. Inordertoobtainacom- petitive determination of M , one eventually needs a taininganexpansioninthepower-countingparameterδ W more accurate computation of the electron distribution is not available and probably difficult to achieve. The function. specific case of invariant-mass cuts |M2 − M2| < Λ2 fifj W Since this is not a fundamental problem and since on the W-decay products has been considered in [24]. the full theory NLO ee4f calculation is available, the TheimplementationdependsonhowΛscaleswiththe accuracy of the theoretical prediction is limited by the parameter δ. For loose cuts, Λ ∼ M . Since by as- W N3/2LO terms in the δ expansion, which correspond sumption the virtualities in the EFT are at most of or- totwo-loopcorrections(inthecomplex-massscheme). der MΓ ∼ Mδ, the loose cut does not affect the EFT SomeofthediagramstogetherwiththeirEFTrepresen- diagrams. However, thehard-matchingcoefficientsare tation are shown in Figure 8. These consist of mixed modifiedandacquireadependenceonΛinadditionto hard-Coulomb corrections (first column), interference theothershort-distancescales. Thesituationisreversed √ of Coulomb exchange with soft and collinear radiative fortightcutswithΛ ∼ MΓ ∼ M δ. Thetightcutcuts corrections (2nd and 3rd column, respectively), and a into the (approximate) Breit-Wigner distribution of the correctiontotheelectromagneticCoulombpotentialit- single-W invariantmassdistributionandthereforemust self. Thesegenuinehigher-ordercorrectionshavebeen beappliedtothecalculationoftheEFTloopintegrals. computed[24]andwerefoundtobebelow0.5%,lead- Ontheotherhand,iteliminatesoff-shellcontributions, ingtoshiftsofW massoflessthan5MeV. andhencetheshort-distancecoefficientC(k) ofthenon- nr Up to now, we considered the total cross section for resonantfour-electronoperator(27)vanishes. Figure9 the flavour-specific final state µ−ν¯ ud¯X. Since experi- shows good agreement of the effective-theory calcula- µ mentallycertaincutsmustbeapplied,itwouldbedesir- tion of the cut Born cross section with the numerical abletocomputedirectlythecutcrosssectionintheEFT. resultfromWHIZARDintheregionswheretherespec- A framework to implement arbitrary cuts while main- tiveloose/tight-cutcountingruleisappropriate.