Effective Theory Approach to W-Pair Production near Threshold
aa r X i v : . [ h e p - ph ] A ug Effective Theory Approach to W -Pair Production near Threshold ∗ Christian SchwinnRWTH Aachen - Institut f¨ur Theoretische Physik ED–52056 Aachen - GermanyIn this talk, I review the effective theory approach to unstable particle production andpresent results of a calculation of the process e − e + → µ − ¯ ν µ u ¯ d X near the W -pairproduction threshold up to next-to-leading order in Γ W /M W ∼ α ∼ v . The remainingtheoretical uncertainty and the impact on the measurement of the W mass is discussed. SFB/CPP-07-44, arXiv:0708.0730 [hep-ph], August 6, 2007
The masses of particles like the top quark, the W boson or yet undiscovered particles likesupersymmetric partners can be measured precisely using threshold scans at an e − e + col-lider. In particular the error of the W mass could be reduced to 6 MeV by measuring thefour fermion production cross section near the W -pair threshold [1], provided theoreticaluncertainties are reduced well below 1%. In such precise calculations one has to treat finitewidth effects systematically and without violating gauge invariance. The next-to-leadingorder (NLO) calculations of W -pair production [2] available at LEP2 were done in the polescheme [3] and were supposed to break down near threshold. The recent computation of thecomplete NLO corrections to e − e + → f processes in the complex mass scheme [4] is validnear threshold and in the continuum, but is technically demanding and required to computeone loop six-point functions.In this talk, I report on the NLO corrections to the total cross section of the process e − e + → µ − ¯ ν µ u ¯ dX (1)near the W -pair threshold [5] obtained using effective field theory (EFT) methods [6, 7, 8].This calculation is simpler than the one of [4] and results in an almost analytical expressionof the result that allows for a detailed investigation of theoretical uncertainties. However, themethod is not easily extended to differential cross sections. Section 2 contains the leadingorder (LO) EFT description while the NLO approximation of the tree and the radiativecorrections are described in Sections 3 and 4, respectively. Results are presented in Section 5together with an estimate of the remaining theoretical uncertainties and a comparison to [4]. To provide a systematic treatment of finite width effects, in [6, 7] EFT methods were usedto expand the cross section simultaneously in the coupling constant α , the ratio Γ /M andthe virtuality of the resonant particle ( k − M ) /M , denoted collectively by δ . The modes ∗ Talk given at the International Linear Collider Workshop (LCWS/ILC07), 30 May - 3 Jun 2007, Ham-burg, Germany.
LCWS/ILC 2007 t the small scale δ (the resonance, soft or Coulomb photons, . . . ) and the external particlesare described by an effective Lagrangian L eff that contains elements of heavy quark effectivetheory or non-relativistic QED and soft-collinear effective theory (SCET) (for reviews ofthe various EFTs see e.g [10]). “Hard” fluctuations with virtualities ∼ M are not partof the EFT and are integrated out. Their effect is included in short-distance coefficientsin L eff that can be computed in fixed-order perturbation theory without resummations ofself-energies. Finite width effects are relevant for the modes at the small scale and areincorporated through complex short-distance coefficients in L eff [7, 9].It might be useful to compare the EFT approach to the pole scheme for the example ofthe production of a single resonance Φ in the inclusive process f ¯ f → X . The pole schemeprovides a decomposition of the amplitude into resonant and non-resonant pieces [3]: A ( s ) | s ∼ M = R (¯ s ) s − ¯ s + N ( s ) , (2)where both ¯ s , the complex pole of the propagator defined by ¯ s − M − Π(¯ s ) = 0, and R (¯ s ),the residue of A ( s ) at ¯ s , are gauge independent. In the EFT, it is convenient to obtain thecross section from the imaginary part of the forward-scattering amplitude that reads [7] i A ( s ) | s ∼ M = Z d x h f ¯ f | T h i O † Φ f ¯ f (0) i O f ¯ f Φ ( x ) i | f ¯ f i + h f ¯ f | i O f (0) | f ¯ f i . (3)Here O f ¯ f Φ describes the production of Φ while O f describes non-resonant contributions.The matching coefficients of these operators are gauge independent since they are computedfrom on-shell scattering amplitudes in the underlying theory, where for unstable particles“on-shell” implies k = ¯ s . The structure of (3) is similar to (2), but the EFT providesa field theoretic definition of the several terms. Higher order corrections to the matchingcoefficients correspond to the factorizable corrections in the pole scheme. Loop correctionsto the matrix elements in the EFT correspond to the non-factorizable corrections [6].Turning to W -pair production near threshold, the appropriate effective Lagrangian todescribe the two non-relativistic W bosons with k − M W ∼ M W v ∼ M W δ is given by [8] L NRQED = X a = ∓ " Ω † ia iD + ~D M W − ∆2 ! Ω ia + Ω † ia ( ~D − M W ∆) M W Ω ia (4)with the matching coefficient [7] ∆ ≡ (¯ s − M W ) /M W . If M W is the pole mass, this becomes∆ = − i Γ W . The fields Ω i ± ≡ √ M W W i ± describe the three physical polarizations of the W s; the unphysical modes are not part of the EFT [8]. The covariant derivative D µ Ω i ± ≡ ( ∂ µ ∓ ieA µ )Ω i ± includes interactions with those photon fluctuations that keep the virtualitiesof the Ωs at the order δ . These are soft photons with ( q , ~q ) ∼ ( δ, δ ) and potential (Coulomb)photons with ( q , ~q ) ∼ ( δ, √ δ ). Collinear photons are also part of the EFT but do notcontribute at NLO. The Lagrangian (4) reproduces the expansion of the resummed transverse W propagator in δ , as can be seen by writing the W four-momenta as k µ = M W v µ + r µ with v µ ≡ (1 ,~
0) and a potential residual momentum ( r , | ~r | ) ∼ M W ( v , v ) ∼ ( δ, √ δ ): ik − M W − Π WT ( k ) (cid:18) − g µν + k µ k ν k (cid:19) ⇒ i ( − g µν + v µ v ν )2 M W ( r − ~r M W + ∆2 ) . (5) LCWS/ILC 2007 igher orders in the expansion of the propagator are reproduced by the higher order kineticterms in (4) and residue factors included in the production operators [5].The production of a pair of non-relativistic W bosons is described by the operator [8] O (0) p = παs w M W (cid:0) ¯ e c ,L (cid:0) γ i n j + γ j n i (cid:1) e c ,L (cid:1) (cid:16) Ω † i − Ω † j + (cid:17) (6)that is determined from the on-shell tree-level scattering amplitude e − e + → W + W − : (cid:1) eν e We W + (cid:2) ee γ/Z WW ⇒ (cid:3) ee ΩΩ O (0) p . (7)At threshold, only the t -channel diagram and the e − L e + R helicity contribute at leading orderin δ . Similar to (3), the LO e − e + forward-scattering amplitude in the EFT is given by theexpectation value of a time ordered product of the operators (6), evaluated using (4): i A (0) = Z d x h e − e + | T[ i O (0) † p (0) i O (0) p ( x )] | e − e + i = (cid:4) ΩΩ O (0) p O † (0) p . (8)One estimates A (0) ∼ α √ δ , noting that each Ω propagator (5) contributes δ − and countingthe potential loop integral as dk d k i ∼ δ / . The total cross section for the process (1) isobtained from appropriate cuts of A (0) , where cutting an Ω ± line has to be interpreted ascutting the self-energies resummed in the EFT propagator. At LO, the cuts contributing tothe flavour-specific final state are correctly extracted by multiplying the imaginary part of A (0) by the leading-order branching fractions. In terms of E = √ s − M W one obtains [5] σ (0) ( e − e + → µ − ¯ ν µ u ¯ d ) = πα s w s Im " − i r − E + i Γ W M W . (9) Some parts of the NLO EFT calculation of the process (1) are included in a Born calculationin the full theory with a fixed width prescription. One contribution arises from four-electronoperators O ( k )4 e analogous to those in (3). Their matching coefficients C ( k )4 e are obtained fromthe forward-scattering amplitude in the full electroweak theory. The leading imaginary partsof C ( k )4 e are of order α and arise from cut two-loop diagrams corresponding to all squared treediagrams of the processes e − e + → W − u ¯ d and e − e + → W + µ − ¯ ν µ , calculated in dimensionalregularization without self-energy resummations, but expanded near threshold: (cid:5) e We W eeu ¯ d + (cid:6) ee γ/Z uW eW ed ¯ d + (cid:7) ee γ/Z u W eed ¯ d W + · · · ⇒ (cid:8) ee ee Im O (1 / f (10)Since these corrections to the amplitude are of order α , and counting α ∼ δ , they aresuppressed by δ / compared to A (0) ∼ α δ / and are denoted as ” √ NLO” corrections.
LCWS/ILC 2007 he second class of contributions arises from production-operator and propagator correc-tions . Performing the tree-level matching (7) up to order ∼ v and v leads to higher orderproduction operators O (1 / p and O (1) p . The operators O (1 / p like (cid:0) ¯ e L γ j e L (cid:1) (cid:0) Ω i − ( − i ) D j Ω i + (cid:1) are given in [8]. At NLO one needs diagrams with two insertions of an O (1 / p operator, oneinsertion of an O (1) p operator and insertions of kinetic corrections from (4): i A (1)born = (cid:9) ΩΩ O (1 / p O † (1 / p + (cid:10) ΩΩ O (1) p O † (0) p + (cid:11) ( ~k − M ∆) O (0) p O † (0) p Equivalently one can directly expand the spin averaged squared matrix elements [5].
156 158 160 162 164 166 168 170 (cid:143)!!!! s @ GeV D Σ @ fb D exact BornEFT @ NLO D EFT @(cid:143)!!!!!!
N LO D EFT @ LO D Figure 1: Convergence of EFT approximations tothe born cross section from WhizardAs seen in Figure 1, the EFT ap-proximations converge to the full Bornresult but it turns out that a partial in-clusion of N / LO corrections is requiredto get an agreement of ∼ .
1% at 170GeV and ∼
10% at 155 GeV [5]. Forhigher-order initial state radiation (ISR)improvement by a convolution with radi-ator functions, one needs σ Born at ener-gies far below threshold, where the EFTis not valid. For the numerical results inSection 5 the ISR-improved Born crosssection from Whizard [11] was used, butone could also match the EFT to the fulltheory below, say, √ s = 155 GeV. The radiative corrections needed up to NLO are given by higher order calculations of shortdistance coefficients and by loop calculations in the EFT. Counting the QCD couplingconstant as α s ∼ α ew ∼ δ , the corrections to Γ W up to order α ew α s ( √ NLO), α ew and α ew α s (NLO) have to be included. The flavour-specific NLO decay corrections are correctlytaken into account by multiplying the imaginary part of the LO forward-scattering amplitudewith the one-loop corrected branching ratios. For the NLO renormalization of the productionoperator (6) one has to calculate the one-loop corrections to the on-shell scattering e − e + → W + W − at leading order in the non-relativistic expansion: (cid:12) e eZe γ W WW + (cid:13) e ν e e γ WW + Æ e ν e e Wν e e WW + · · · ⇒ (cid:15) ee ΩΩ C (1) p O (0) p Due to the threshold kinematics, many of the 180 one-loop diagrams do not contribute,consistent with the vanishing of the tree-level s -channel diagrams at leading order in v . In LCWS/ILC 2007 erms of a finite coefficient c (1 , fin) p given in [5], the matching coefficient reads C (1) p = α π "(cid:18) − ε − ε (cid:19) (cid:18) − M W µ (cid:19) − ε + c (1 , fin) p . (11)The first and second Coulomb correction arise from the exchange of potential photons.Their magnitude can be estimated counting the loop-integral measure in the potential regionas d q ∼ δ / , the Ω propagator and the potential photon propagator i/ | ~q | as δ − . Onefinds that single Coulomb exchange is a √ NLO correction compared to the LO amplitude: (cid:16) γ ∼ α Z d k d q | ~q | δ − ∼ α ∼ A (0) √ δ (12)At threshold the one-photon exchange is of the order of 5% of the LO amplitude whiletwo-photon exchange is only a few-permille correction [12] and no resummation is necessary. Soft photon corrections correspond to two-loop diagrams in the EFT containing a photonwith momentum ( q , | ~q | ) ∼ ( δ, δ ). They give rise to O ( α ) corrections as can be seen froma power-counting argument similar to the one for Coulomb-exchange but counting the soft-photon propagator − i/q as δ − and the soft loop-integral as δ . In agreement with gaugeinvariance arguments and earlier calculations [13], the sum of all diagrams where a softphoton couples to an Ω line vanishes. The only remaining diagrams give (cid:17) + (cid:18) = 4 π α s w M W απ Z d d r (2 π ) d η − η + "(cid:18) ε + 512 π (cid:19) (cid:18) − η − µ (cid:19) − ε (13)with η − = r − | ~r | M W + i Γ (0) and η + = E − r − | ~r | M W + i Γ (0) . The ǫ − poles cancel between (13)and diagrams with an insertion of the NLO production operator (11) while the remaining ǫ − poles proportional to (2 log ( η − /M W ) + 3 /
2) are discussed below.
The radiative corrections in Section 4 were calculated for m e = 0 so the result is not infraredsafe. It should be convoluted with electron distribution functions in the MS scheme afterminimal subtraction of the IR poles. However, the available distribution functions assume m e as IR regulator. Our result can be converted to this scheme by adding contributionsfrom the hard-collinear region where k µ ∼ M W , k ∼ m e , and the soft-collinear region where k µ ∼ Γ W , k ∼ m e Γ W M W . These cancel the ǫ -poles but introduce large logs of M W /m e : σ (1) ( s ) = α s w s Im ( ( − q − E + i Γ W M W (cid:18) (cid:18) − E + i Γ W ) M W (cid:19) ln (cid:18) M W m e (cid:19) − (cid:18) M W m e (cid:19) + Re h c (1 , fin) p i + π (cid:19)) + ∆ σ (1) Coulomb + ∆ σ (1) decay . (14) LCWS/ILC 2007 t this stage, one can compare to the results of [4] for the strict O ( α ew ) correctionswithout higher order ISR improvement, σ (161GeV) = 105 . σ (170GeV) =377 . σ EFT (161GeV) = 104 . σ EFT (170GeV) =373 . . − σ Born . Compared to thelarge correction from ISR improvement of σ Born alone (blue/dashed), the size of the genuineradiative correction is about +8%.
160 162 164 166 168 170 (cid:143)!!!! s @ GeV D - - - - - - ∆Σ (cid:144) Σ Born @ % D NLONLO H ISR - tree L Born H ISR L Figure 2: Size of the relative NLO cor-rections for different treatments of ISRThe largest remaining uncertainty is due to thetreatment of ISR that is accurate only at leading-log level. It is formally equivalent to improve only σ Born by higher order ISR [2, 4], but not the radia-tive corrections. The results of this approach areshown in the red (dash-dotted) line in Figure 2 anddiffer by almost 2% at threshold from the treat-ment discussed above. This translates to an un-certainty of δM W ∼
31 MeV [5]. The remainingtheory uncertainty comes from the uncalculatedN / LO corrections in the EFT. The O ( α ) correc-tions to the the four-electron operators (10) leadto an estimated uncertainty of δM W ∼ δM W ∼ M W measurement since the largest remaininguncertainties can be eliminated by an improved treatment of ISR and with input of the fullfour fermion calculation. Acknowledgments
I thank M. Beneke, P. Falgari, A. Signer and G. Zanderighi for the collaboration on [5] andfor comments on the manuscript. I acknowledge support by the DFG SFB/TR 9.
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