A complete update of ε ′ /ε in the Standard Model
AA complete update of ε (cid:48) / ε in the Standard Model V. Cirigliano
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USAE-mail: [email protected]
H. Gisbert ∗ Fakultät Physik, TU Dortmund,Otto-Hahn-Str.4, D-44221 Dortmund, GermanyE-mail: [email protected]
A. Pich
Departament de Física Teòrica, IFIC, CSIC — Universitat de ValènciaEdifici d’Instituts de Paterna, Apt. Correus 22085, E-46071 València, SpainE-mail: [email protected]
A. Rodríguez-Sánchez
Department of Astronomy and Theoretical Physics, Lund University, Sölvegatan 14A, SE 223-62Lund, SwedenE-mail: [email protected]
The recent release of improved lattice data has revived again the interest on precise theoreticalcalculations of the direct CP-violation ratio ε (cid:48) / ε . We present a complete update of the Stan-dard Model prediction [1,2], including a new re-analysis of isospin-breaking corrections whichare of vital importance in the theoretical determination of this observable. The Standard Modelprediction, Re ( ε (cid:48) / ε ) = ( ± ) · − , turns out to be in good agreement with the experimentalmeasurement. European Physical Society Conference on High Energy Physics - EPS-HEP2019 -10-17 July, 2019Ghent, Belgium ∗ Speaker. c (cid:13) 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 ] N ov complete update of ε (cid:48) / ε in the Standard Model H. Gisbert
1. Introduction
The matter-antimatter asymmetry in our Universe requires the violation of the CP symmetry.Although it has been observed in B , D and K systems, the amount of CP violation in the StandardModel (SM) is too low to reproduce the observed asymmetry, hence new sources of CP violationare needed to explain this large imbalance. The CP-violating ratio ε (cid:48) / ε represents a fundamen-tal test for our understanding of this phenomena. In the SM, this observable is proportional tothose Cabibbo-Kobayashi-Maskawa (CKM) matrix elements that account for the violation of thissymmetry, therefore any new source of CP violation should have a direct impact on this ratio.The different sources of CP violation in K decays are parametrized by ε (cid:48) and ε , which arerelated with the branching ratios of the K L and K S decays into two pions, η + − ≡ A ( K L → π + π − ) A ( K S → π + π − ) = ε + ε (cid:48) , η ≡ A ( K L → π π ) A ( K S → π π ) = ε − ε (cid:48) . (1.1)The dominant effect from CP violation in K mixing is contained in ε , and its experimental value isa per-mill effect | ε | = ( . ± . ) · − [3]. In the case of ε (cid:48) , which depends on the differencebetween η + − and η , the effect is tinier. Its experimental average [4, 5]Re ( ε (cid:48) / ε ) exp = ( . ± . ) · − , (1.2)clearly demonstrates the existence of direct CP violation in K decays. In addition, its small sizemakes it particularly sensitive to new sources of CP violation, providing a formidable way to searchfor physics beyond the SM.
2. The fingerprints of K → ππ decays In this section, we explore the dynamical features of K → ππ decays, taking into account theexperimental data. This goal requires to adopt the usual isospin decomposition of the physicalamplitudes [6] A [ K → π + π − ] = A e i χ + √ A e i χ = A / + √ ( A / + A / ) , A [ K → π π ] = A e i χ − √ A e i χ = A / − √ ( A / + A / ) , (2.1) A [ K + → π + π ] = A + e i χ + = (cid:18) A / − A / (cid:19) , where A / ≡ A e i χ , A / + A / ≡ A e i χ and A / − ( / ) A / ≡ A + e i χ + . In the isospin limit, A and A = A + are the decay amplitudes into ( ππ ) , states, and χ I can be identified with theS-wave ππ scattering phase shifts δ I . In the CP-conserving limit, the amplitudes A I are real andpositive. Using Eqs. (2.1) and the measured K → ππ branching ratio, one obtains [7] A = ( . ± . ) · − GeV , A = ( . ± . ) · − GeV , (2.2) χ − χ = ( . ± . ) ◦ . (2.3)When CP violation is turned on, the amplitudes A , and A + acquire imaginary parts, and ε (cid:48) can bewritten to first order in CP violation as ε (cid:48) = − i √ i ( χ − χ ) ω (cid:20) Im A Re A − Im A Re A (cid:21) = − i √ i ( χ − χ ) ω Im A Re A (cid:18) − ω Im A Im A (cid:19) . (2.4)1 complete update of ε (cid:48) / ε in the Standard Model H. Gisbert
Taking into account Eqs. (2.2) and (2.3) together with Eq. (2.4), we can easily study the impact ofthe K → ππ dynamical properties on ε (cid:48) : • Eqs. (2.2) exhibit the well-known “ ∆ I = / ω − ≡ Re A Re A ≈ , (2.5)which directly implies a strong suppression of ε (cid:48) . In addition, any small isospin-breakingcorrection to the ratio Im A / Im A is enhanced by the factor ω − in Eq. (2.4). • Furthermore, Eq. (2.3) shows that the S-wave ππ re-scattering generates a large phase-shiftdifference between the I = I = ( A / / A / ) ≈ Dis ( A / / A / ) . (2.6)Thus, the absorptive contribution to this ratio is of the same size as the dispersive one. A goodtheoretical control of both contributions is then mandatory to obtain a reliable prediction forRe ( ε (cid:48) / ε ) . • The presence of absorptive contributions is a direct consequence of unitarity, which becomesspecially relevant for the isoscalar amplitude A / ≡ A e i δ = Dis ( A / ) + i Abs ( A / ) .Using the known value of the I = δ = ( . ± . ) ◦ [9], one immediatelyobtains A ≡ | A / | = Dis ( A / ) (cid:112) + tan δ ≈ . × Dis ( A / ) . (2.7)Therefore, the absorptive contribution increases the numerical size of A by 30%. • The absorptive amplitudes are generated by intermediate on-shell pions, through the Feynman-diagram topology depicted in Figure 1. The dispersive and absorptive loop contributions arerelated by analyticity. A large absorptive contribution implies a large dispersive loop correc-tion.
Figure 1:
One-loop contribution to K → ππ with its absorptive cut in red.
3. Current estimate of ε (cid:48) / ε from lattice QCD In 2015, the RBC-UKQCD collaboration published their first estimate of ε (cid:48) / ε [10, 11]:Re ( ε (cid:48) / ε ) lattice = ( . ± . ) · − . (3.1)2 complete update of ε (cid:48) / ε in the Standard Model H. Gisbert
This result is consistent with zero and shows a clear discrepancy of 2.1 σ with the experimentalvalue given in Eq. (1.2). The disagreement has triggered many analyses of possible new physicscontributions in order to explain the apparent anomaly. However, one should realize the technicallimitations of this lattice estimate. For example, the phase shifts δ , play a crucial role in the latticedetermination of ε (cid:48) / ε and provide a quantitative test of the obtained result. While the extracted I = σ away from its experimental value, the RBC-UKQCD collaborationfinds a I = σ . This discrepancy ismuch larger than the one exhibited by their ε (cid:48) / ε result.Therefore, it is still premature to derive strong implications from the 2015 RBC-UKQCDlattice data, since the important effects of ππ re-scattering are still not well reproduced in the I = ε (cid:48) / ε anomaly, from groups using analytical methods [14],are based on simplified calculations which either use the RBC-UKQCD matrix elements (withsomewhat smaller uncertainties) or adopt model-dependent K → ππ amplitudes without any ab-sorptive components, missing completely the important ππ re-scattering corrections.
4. Multi-scale framework
Due to the presence of widely separated mass scales ( M π < M K (cid:28) M W ), the theoretical de-scription of the K → ππ decays requires the use of two different effective field theories (EFTs).Above the electroweak scale M W , all flavour-changing processes are described in terms of quarks,leptons and gauge bosons. We can apply the renormalization group equations and the operatorproduct expansion to go down to low-energy scales ( ∼ ∆ S = L ∆ S = = − G F √ V ud V ∗ us ∑ i = C i ( µ ) Q i ( µ ) , (4.1)which is a sum of local four-fermion operators that are weighted by the Wilson coefficients, C i ( µ ) = z i ( µ ) + τ y i ( µ ) . The dependence on the CKM matrix elements is carried by the global V ud V ∗ us factor and the parameter τ ≡ − V td V ∗ ts / ( V ud V ∗ us ) that contains the CP-violating phase. Theinformation on the heavy masses has been absorbed into the Wilson coefficients C i ( µ ) , which areknown at next-to-leading-order (NLO) [16–19]. Some next-to-next-to-leading-order (NNLO) cor-rections [20, 21] are already known and efforts towards a complete calculation at the NNLO arecurrently under way [22].Below the resonance region where the physics of study is defined in terms of Goldstonebosons ( π , K , η ), one can use symmetry considerations in order to build an EFT valid in thisnon-perturbative regime. Chiral Perturbation Theory ( χ PT) provides a formidable theoreticalframework to describe the pseudoscalar-octet dynamics as a perturbative expansion in powers ofmomenta and quark masses over the chiral symmetry-breaking scale Λ χ . Using the chiral sym-metry, we can build all the allowed operators with exactly the same symmetry properties as theshort-distance Lagrangian (4.1). To lowest order, the chiral realization of L ∆ S = contains threeoperators L ∆ S = = G L + G L + G g ewk L ewk , (4.2)3 complete update of ε (cid:48) / ε in the Standard Model H. Gisbert with their respective low-energy couplings (LECs) G , G and G g ewk encoding all quantuminformation from high-energy scales [23]. The determination of these LECs requires to performa matching between both Lagrangians (4.1) and (4.2) in a common region of validity. However,performing consistently this matching is a very challenging task that still remains unsolved. Thelarge- N C limit provides a partial solution to this problem. In this limit, the T-product of two colour-singlet currents factorizes and, since we have a well-known representation for these currents in χ PT, the matching can be done at leading order in the 1 / N C expansion. It is important to remindthat the missing NLO contributions to the matching are not enhanced by any large logarithms.
5. Isospin-breaking corrections to ε (cid:48) / ε Eqs. (2.4) and (2.5) exhibit the important role of isospin-breaking effects in ε (cid:48) / ε . Includingthese corrections, Re ( ε (cid:48) / ε ) can be written as [6, 24]Re (cid:16) ε (cid:48) ε (cid:17) = − ω + √ | ε | (cid:34) Im A ( ) Re A ( ) ( − Ω eff ) − Im A emp2 Re A ( ) (cid:35) , (5.1)where the superscript ( ) denotes the isospin limit, Im A emp2 contains the I = ω + ≡ Re A + / Re A . The parameter Ω eff contains theisospin-breaking corrections. Implementing the current improvements on the inputs that enter inthis parameter, we have updated the Ω eff prediction with the result [1] Ω eff = ( . + . − . ) · − , (5.2)which agrees within errors with the previous determination [6, 24] but has a larger central value.
6. Strong cancellation in simplified analyses
The CP-odd amplitudes Im A , are mainly dominated by ( V − A ) × ( V + A ) operators becausethey have a chiral enhancement that can be easily estimated in the large- N C limit. Due to thesize of y i ( µ ) , it is a good numerical approximation to consider only Q and Q and neglect thecontributions to Im A , from other operators. With this rough estimation, one obtains [2]Re ( ε (cid:48) / ε ) ≈ . · − (cid:110) B ( / ) ( − Ω eff ) − . B ( / ) (cid:111) , (6.1)where B ( / ) and B ( / ) parametrize the deviations of the true hadronic matrix elements from theirlarge- N C approximations B ( / ) = B ( / ) =
1, which do not include any absorptive contribution.Taking Ω eff = .
11 [1], Eq. (6.1) gives Re ( ε (cid:48) / ε ) ≈ . · − at N C → ∞ ; the same order of mag-nitude as its experimental value in Eq. (1.2). In contrast, with the values adopted in Ref. [14], B ( / ) = . B ( / ) = .
76 and Ω eff = .
15, one gets Re ( ε (cid:48) / ε ) ≈ . · − , one order of magni-tude smaller than (1.2). Clearly, with this choice of B , parameters, the simplified approximationin Eq. (6.1) suffers a strong cancellation between the different contributions.We can go one step further and include naively the chiral loop corrections [6, 25–27] (Fig-ure 1). These contributions are mainly dominated by ∆ L A ( ) / and ∆ L A ( g ) / , which imply the follow-ing shifts, B ( / ) → | + ∆ L A ( ) / | B ( / ) ≈ . B ( / ) and B ( / ) → | + ∆ L A ( g ) / | B ( / ) ≈ . B ( / ) ,4 complete update of ε (cid:48) / ε in the Standard Model H. Gisbert
Set-up B ( / ) B ( / ) Re ( ε (cid:48) / ε ) Large N C + FSI I = , . · − Large N C | I = + LQCD I = + FSI I = . · − LQCD + FSI I = ∗ . · − † Table 1:
Naive estimates of Re ( ε (cid:48) / ε ) , including some final-state interactions (FSI) in Eq. (6.1). in Eq. (6.1). With this shifts in mind, we can again estimate Re ( ε (cid:48) / ε ) for different setups, see Ta-ble 1. We can observe that the chiral loop corrections destroy the strong numerical cancellation inEq. (6.1), yielding results of the same order of magnitude as the experimental measurement.
7. Standard Model prediction for ε (cid:48) / ε in χ PT With the theoretical framework presented in Section 4, which includes all four-fermion opera-tors (not only Q and Q ), the full 1-loop χ PT contributions and the new updated isospin-breakingcorrections given by Eq. (5.2), our SM prediction for Re ( ε (cid:48) / ε ) [1],Re (cid:0) ε (cid:48) / ε (cid:1) = (cid:0) . ± . γ ± . LECs ± . / N C (cid:1) · − = ( ± ) · − , (7.1)is in excellent agreement with the experimental world average in Eq. (1.2). Eq. (7.1) displays thethree different sources of uncertainty in Re ( ε (cid:48) / ε ) . The first error reflects the choice of schemefor γ . The second error originates from the input values of the strong LECs L , , . The last errorparametrizes our ignorance about 1 / N C -suppressed contributions in the matching region whichhave been estimated very conservatively through the variation of µ SD and ν χ in the intervals [ . , . ] GeV and [ . , ] GeV, respectively. Further details can be found in Ref. [1].
Acknowledgements
We want to thank the organizers for their effort to make this conference such a successful event.This work has been supported in part by the Spanish Government and ERDF funds from the EUCommission [grant FPA2017-84445-P], the Generalitat Valenciana [grant Prometeo/2017/053], theSpanish Centro de Excelencia Severo Ochoa Programme [grant SEV-2014-0398], the Swedish Re-search Council [grants 2015-04089 and 2016-05996] and by the European Research Council (ERC)under the EU Horizon 2020 research and innovation programme (grant 668679). The work of H.G.is supported by a FPI doctoral contract [BES-2015-073138], funded by the Spanish Ministry ofEconomy, Industry and Competitiveness and the Bundesministerium für Bildung und Forschung(BMBF). V.C. acknowledges support by the US DOE Office of Nuclear Physics.
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