Electroweak precision pseudo-observables at the e + e − Z-resonance peak
Ievgen Dubovyk, Ayres Freitas, Janusz Gluza, Krzysztof Grzanka, Tord Riemann, Johann Usovitsch
DDESY 20-184KW 20-002
Electroweak precision pseudo-observables at the 𝒆 + 𝒆 − Z-resonance peak
Ievgen Dubovyk, 𝑎 Ayres Freitas, 𝑏 Janusz Gluza, 𝑎, ∗ Krzysztof Grzanka, 𝑎 Tord Riemann 𝑎,𝑑 and Johann Usovitsch 𝑐 𝑎 Institute of Physics, University of Silesia, Katowice, Poland 𝑏 Pittsburgh Particle physics, Astrophysics & Cosmology Center (PITT PACC),Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA 𝑐 PRISMA Cluster of Excellence, Institut für Physik,Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany 𝑑 DESY, 15738 Zeuthen, Germany
E-mail: [email protected], [email protected],[email protected], [email protected],[email protected], [email protected]
Phenomenologically relevant electroweak precision pseudo-observables related to Z-boson physicsare discussed in the context of the strong experimental demands of future 𝑒 + 𝑒 − colliders. The recentcompletion of two-loop Z-boson results is summarized and a prospect for the 3-loop StandardModel calculation of the Z-boson decay pseudo-observable is given. ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] D ec lectroweak precision pseudo-observables at the 𝑒 + 𝑒 − Z-resonance peak
Janusz GluzaOne of the exciting activities in searching for non-standard effects in particle physics is theprecision study of the 𝑍 − -boson decay in 𝑒 + 𝑒 − collisions. Electron-positron collisions form the 𝑍 resonance at center-of-mass energies around 91 GeV. This process was instrumental in the LEP era,leading to the detailed knowledge of crucial parts of the Standard Model (SM) [1, 2]. Up to 5 × 𝑍 -boson decays are planned to be observed at the 𝑍 -boson resonance with the FCC-ee collider [3, 4],while it would be about one order of magnitude less at the CEPC [5]. These statistics are about sixorders of magnitude larger than at LEP and may lead to very accurate experimental measurementsof the so-called Electro-Weak Pseudo-Observables (EWPOs), if the systematic experimental errorscan be hold appropriately small. In turn, this means that theoretical predictions must also be veryexact, of the order of 3- to 4-loop QCD and EW effects [6]. This level of accuracy and potentialdistortions from the SM predictions will put stringent limits on theory scenarios beyond the SMwith New Physics virtual particles and interactions. A substantial step in this direction of accuracywithin the SM was a recent calculation of the most difficult massive bosonic two-loop contributionsto the 𝑍 -boson decay [7–9]. In this way, the Standard Model electroweak two-loop corrections arecompleted. The focus can be directed now on the next, NNNLO order of loop calculations. Theircontributions will be necessary in order to meet the anticipated experimental accuracies.Tab. 1 shows the results of higher order contributions to the Z-boson decay partial widths.Tab. 2 summarizes the estimation of the errors connected with unknown higher order corrections.For other EWPOs like sin 𝜃 ℓ eff , sin 𝜃 𝑏 eff , branching ratios, and the hadronic cross section at theZ-resonance, see [8–10]. The total error for Γ Z in Tab. 2 amounts to 0.4 MeV, which is at thelevel of the CEPC accuracy ( 0 . complete perturbation theorycalculations of Feynman integrals beyond one loop. For this reason, numerical integration methods Γ 𝑖 [MeV] Γ 𝑒 Γ 𝜈 Γ 𝑑 Γ 𝑢 Γ 𝑏 Γ Z Born 81.142 160.096 371.141 292.445 369.562 2420.19
O ( 𝛼 ) O ( 𝛼𝛼 s ) O ( 𝛼 t 𝛼 , 𝛼 t 𝛼 , 𝛼 𝛼 s , 𝛼 ) O ( 𝑁 𝑓 𝛼 ) O ( 𝑁 𝑓 𝛼 ) O ( 𝛼 ) Table 1:
Contributions of different perturbative orders to the partial and total 𝑍 widths. A fixed value of 𝑀 W has been used as input, instead of 𝐺 𝜇 . The 𝑁 𝑓 and 𝑁 𝑓 refer to corrections with one and two closedfermion loops, respectively, whereas 𝛼 denotes contributions without closed fermion loops. Furthermore, 𝛼 t and 𝛼 s are scale-dependent strong couplings. Table from [8]. lectroweak precision pseudo-observables at the 𝑒 + 𝑒 − Z-resonance peak
Janusz GluzaObservable 𝛼𝛼 𝛼𝛼 𝛼 𝛼 s 𝛼 Total Γ 𝑒,𝜇,𝜏 [MeV] 0.008 0.001 0.010 0.013 0.018 Γ 𝜈 [MeV] 0.008 0.001 0.008 0.011 0.016 Γ 𝑢,𝑐 [MeV] 0.025 0.004 0.08 0.07 0.11 Γ 𝑑,𝑠 [MeV] 0.016 0.003 0.06 0.05 0.08 Γ 𝑏 [MeV] 0.11 0.02 0.13 0.06 0.18 Γ Z [MeV] 0.23 0.035 0.21 0.20 0.4 Table 2:
Leading unknown higher-order corrections and their estimated order of magnitude for variouspseudo-observables. The different orders always correspond to missing higher orders beyond the knownapproximations in the limit of a large top Yukawa coupling. The last column gives the total theory errorobtained by adding the individual orders in quadrature. Table taken from [8]. are presently the most promising, if not the only, avenues for addressing those challenges. Analyticaltechniques are expected to be important in many respects, but numerical integration methods haveadvantages when increasing the number of masses and momentum scales. Fortunately, there hasbeen impressive progress in recent years in this direction [6]. In 2014 the only advanced automaticnumerical two-loop method was sector decomposition (SD). However, the corresponding softwarewas not sufficiently developed to evaluate the complete set of Feynman integrals for the massiveelectroweak bosonic two-loop corrections to the Z-boson decay with the desired high precision(aiming at eight digits per integral). The task could be completed successfully with a substantialdevelopment of a competing numerical approach, based on Mellin-Barnes (MB) representations ofFeynman integrals [10]. These calculations are challenging due to the numerical role of particlemasses 𝑀 𝑍 , 𝑀 𝑊 , 𝑚 𝑡 , 𝑀 𝐻 , leading to (i) an enormous number of contributions, ranging from tens tohundreds of thousands of diagrams (at 3-loops), and (ii) the occurrence of up to four dimensionlessparameters in Minkowskian kinematics (at 𝑠 = 𝑀 𝑍 ) with intricate threshold and on-shell effectswhere contour deformation fails. In tackling more loops or legs, merging both the MB- and SD-methods in numerical calculations, was the key for solving the complete massive SM two-loopcase. We illustrate recent advances for multi-loop calculations applied to the Z-boson precisioncalculations using both methods.The non-trivial diagrams which we will discuss are gathered in Fig. 1. The MB representationfor the non-planar diagram on the left hand side is four dimensional. In this case, results obtainedfor the constant parts of the 𝜖 -expansion with different methods and programs in the Euclideanregion are, for ( 𝑝 + 𝑝 ) = − 𝑚 = − − . − . − . − . − . lectroweak precision pseudo-observables at the 𝑒 + 𝑒 − Z-resonance peak
Janusz Gluza - p1 - p2 [ k1,0 ][ k1 + p1 + p2,0 ] p1 [ k1 - k2,0 ][ k1 - k2 + p1,m ] p2 [ k2,0 ][ k2 + p2,0 ] ex11 ex22ex3 3 45 - p1 - p2 [ k1 + p1 + p2,0 ][ k1,0 ] p1 [ k2,m ][ k2 + p1,0 ] p2 [ k1 + p1,m ][ k1 - k2,0 ] ex11ex2 2 ex3345 Figure 1:
Left: Non-planar vertex with one massive crossed line. Right: Planar vertex with a finite partin the 𝜖 expansion represented by the single 3-dimensional MB integral of Eqn. (3). Figures generated by PlanarityTest [11, 12]. Both vertices are special cases for which analytical solutions are available.
In the Minkowskian region, with ( 𝑝 + 𝑝 ) = 𝑚 = − . − . · 𝑖 MBnumerics [7, 17] : − . − . · 𝑖 MB ( Cuhre ) : − . − . · 𝑖 SecDec : big error [ ] , − . − i · . [ ] , − . − i · . [ ] pySecDec + rescaling : − . − 𝑖 · . [ ] (2)The SecDec group discussed this integral in [16]. Using the splitting method the reported result is − . − 𝑖 · .
1. For pySecDec with quasi-Monte Carlo integration (
QMC ) [18] and using rescaling for10 generated points, the accuracy is much better. Such integral is relatively easy for the MB method,because it includes only one massive propagator. The result for MB ( Cuhre ) has been obtained withthe MB.m options: MaxPoints 10 , AccuracyGoal 8, PrecisionGoal 8. It took about 5 minutes on amoderate laptop.Another interesting case is the planar scalar integral in Fig. 1, right.The MB representation for the constant term of this diagram is three-dimensional: 𝐼 = ( 𝜋𝑖 ) 𝑠 𝑖 ∞− ∫ − 𝑖 ∞− 𝑑𝑧 𝑖 ∞− ∫ − 𝑖 ∞− 𝑑𝑧 𝑖 ∞− ∫ − 𝑖 ∞− 𝑑𝑧 (cid:18) 𝑚 − 𝑠 (cid:19) 𝑧 Γ (− − 𝑧 ) Γ ( + 𝑧 ) Γ (− − 𝑧 ) Γ (− 𝑧 ) Γ ( + 𝑧 − 𝑧 ) Γ ( + 𝑧 ) Γ (− 𝑧 ) Γ (− 𝑧 + 𝑧 ) Γ (− 𝑧 + 𝑧 )/ Γ (− 𝑧 ) Γ ( − 𝑧 ) Γ ( − 𝑧 + 𝑧 ) . (3)The diagram has also an analytical solution [19] which makes it ideal for a non-trivial comparison ofdifferent numerical techniques. Numerical results for Eq. 3 are presented in Tab. 3 for 𝑠 = 𝑚 = CUHRE routine,
VEGAS routine [21, 22],
QMC . The
QMC quasi-MC or
VEGAS
Monte Carlo methods surpass
CUHRE for higher dimensional integrals. The
QMC library seems to be especially suitable for thenumerical integration of MB integrals in the Minkowskian region. It will be tested in more detailat the 3-loop level. The new
Vegas+ package [23] will be also studied.4 lectroweak precision pseudo-observables at the 𝑒 + 𝑒 − Z-resonance peak
Janusz Gluza AS − . + . 𝑖 MB − . + . 𝑖 Cuhre, 10 , 10 − MB − . + . 𝑖 Vegas, 10 , 10 − MB − . + . 𝑖 QMC, 10 , 10 − MB − . + . 𝑖 QMC, 10 , 10 − Table 3:
Numerical results for Eq. 3 with 𝑠 = 𝑚 = AS - analytical solution. For details on different MB integration routines and transformations of the infinite integration region used, see [20]. Table taken fromthere, shortened. In summary, there is substantial progress in the numerical treatment of multi-loop Feynmanintegral calculations with MB and SecDec , approaching now the massive 3-loop diagrams. Thetechniques presented here can be extended for the computation of massive three-loop electroweakFeynman integrals needed for Z-peak physics. It is also worth mentioning that the differentialequations method [24, 25] and the quoted
IBP reductions are rapidly developing [26, 27]. Theyare expected to be very helpful, if not decisive for solving complete sets of integrals, as the thirdnumerical method in the forthcoming three-loop studies. Based on initial work in this direction wesee no showstoppers for this specific technical task, and even though much additional work will beneeded to assemble them into phenomenological results, this goal also appears within reach in theforeseeable future.
Acknowledgments.
The work of
A.F. is supported in part by the National Science Foundation under grant PHY-1820760.
J.U. received funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme under grant agreement no. 647356(CutLoops). The work is also supported in part by the Polish National Science Centre under grantno. 2017/25/B/ST2/01987 and COST Action CA16201 PARTICLEFACE.
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