ε

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Nuclear and Particle Physics Proceedings 300–302 (2018) 137–144 www.elsevier.com/locate/nppp

Updated Standard Model Prediction for ε /ε ∗ Hector Gisberta,∗∗, Antonio Picha a Departament

de F´ısica Te`orica, IFIC, Universitat de Val`encia – CSIC Apt. Correus 22085, E-46071 Val`encia, Spain

Abstract A recent lattice evaluation of ε /ε, finding a 2.1 σ deviation from the experimental value, has revived the old debate about a possible ε /ε anomaly. The unfounded claims of a too low Standard Model prediction are based on incorrect estimates that neglect the long-distance re-scattering of the final pions in K → 2π. In view of the current situation, we have recently updated the Standard Model calculation, including all known short- and long-distance contributions. Our result, Re (ε /ε) = (15 ± 7) · 10−4 [1], is in complete agreement with the experimental measurement. Keywords: Kaon decays, CP violation, Standard Model

1. Introduction The CP violating ratio ε /ε constitutes a fundamental test for our understanding of flavour-changing phenomena. The present experimental world average [2–10],   Re ε /ε = (16.6 ± 2.3) · 10−4 , (1) demonstrates the existence of direct CP violation in the decay transitions K 0 → 2π. On the other hand, the theoretical prediction of ε /ε has been the subject of many debates. The first nextto-leading order (NLO) calculations [11–16] obtained Standard Model (SM) values one order of magnitude smaller than (1). However, it was soon realized that the former SM predictions had missed an important ingredient: the final-state interactions (FSI) of the two emitted pions [17, 18]. Once all relevant contributions were taken into account, the theoretical prediction was found to be in good agreement with the experimental value although with a large uncertainty of non-perturbative origin [19]. ∗ Talk given at 21th International Conference in Quantum Chromodynamics (QCD 18), 2 - 6 July 2018, Montpellier - FR ∗∗ Speaker. Email addresses: [email protected] (Hector Gisbert), [email protected] (Antonio Pich)

https://doi.org/10.1016/j.nuclphysbps.2018.12.024 2405-6014/© 2019 Elsevier B.V. All rights reserved.

Lattice QCD provides a suitable tool to face nonperturbative problems. However, the lattice efforts to explain the enhancement of the ΔI = 1/2 K → 2π amplitude remained unsuccessful for many years, while attempts to estimate ε /ε were unreliable, sometimes even obtaining negative central values. The status of lattice simulations has improved considerably with the development of more sophisticated techniques and the increasing computer power. The RBC-UKQCD collaboration has achieved a successful calculation of the ΔI = 3/2 K + → π+ π0 amplitude [20–22], and has recently obtained the first statistically-significant signal of the ΔI = 1/2 enhancement [23], in good agreement with the qualitative understanding achieved long time ago with analytical techniques [24–33]. The RBC-UKQCD group has also published a first estimate of the direct CP-violation ratio, Re(ε /ε) = (1.4 ± 6.9) · 10−4 [20, 34], which exhibits a 2.1 σ deviation from the experimental value in Eq. (1). This result has brought back the old SM approaches predicting low values of ε /ε [35–37] and has triggered many new studies of possible contributions from physics beyond the SM [38–63]. However, this discrepancy cannot be taken as evidence for new physics because the same lattice simulation fails to correctly reproduce the

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138

(ππ)I phase shifts, which are a vital ingredient of the K 0 → ππ calculation and provide a quantitative test of the lattice results. While the extracted I = 2 phase shift is only 1 σ away from its physical value, the lattice analysis of Ref. [34] finds a result for δ0 which disagrees with the experimental value by 2.9 σ, a much larger discrepancy than the one quoted for ε /ε. Efforts towards a better lattice understanding of the pion dynamics are under way and improved results are expected soon [64]. Since the publication of the SM ε /ε prediction in Ref. [19], there have been a lot of improvements in • The isospin-breaking corrections [65–67], which play a very important role in ε /ε. • The quark masses, entering the large-NC determination of the penguin matrix elements, are nowadays known with much better precision [68]. • The Low-Energy Constants (LECs), that appear in the Chiral Perturbation Theory (χPT) amplitudes, are better understood [69–85]. The current situation makes mandatory to revise and update the analytical SM calculation of ε /ε [19], which is already 17 years old. In the following, we summarize the new determination of ε /ε from Ref. [1]. 2. Anatomy of ε /ε Adopting the usual isospin decomposition, the kaon decay amplitudes can be expressed as A(K 0 → π+ π− )

A(K 0 → π0 π0 )

A(K + → π+ π0 )

 1  A1/2 + √ A3/2 + A5/2 2 1 = A0 eiχ0 + √ A2 eiχ2 , 2 √   = A1/2 − 2 A3/2 + A5/2 √ = A0 eiχ0 − 2 A2 eiχ2 ,   3 2 A3/2 − A5/2 = 2 3 3 + iχ+ A e 2, (2) = 2 2 =

where the complex quantities AΔI are generated by the ΔI = 12 , 32 , 52 components of the electroweak effective Hamiltonian, in the limit of isospin conservation. The A0 , A2 and A+2 amplitudes are real and positive in the CP-conserving limit. Furthermore, in the isospin limit, A0 and A2 = A+2 denote the decay amplitudes into (ππ)I states with I = 0 and 2, while the phase differences χ0 and χ2 = χ+2 are the corresponding S-wave scattering

phase shifts. From the measured K → ππ branching ratios, one gets [86]: A0 A2

=

(2.704 ± 0.001) · 10−7 GeV,

=

(1.210 ± 0.002) · 10−8 GeV,

χ 0 − χ2

=

(47.5 ± 0.9)◦ .

(3)

These numbers exhibit two important dynamical features that characterize the K → ππ decay amplitudes: 1. Strong enhancement of the isoscalar amplitude with respect to the isotensor one, the so called “ΔI = 12 rule”, ω ≡ ReA2 /ReA0 ≈ 1/22. (4) 2. The S-wave re-scattering generates a large phaseshift difference between the I = 0 and I = 2 partial waves. This implies that 50% of the A1/2 /A3/2 ratio originates from the absorptive contribution: Abs (A1/2 /A3/2 ) = 1.09 . (5) Dis (A1/2 /A3/2 ) We would see later their strong implications on ε /ε. In the presence of CP violation, the amplitudes A0 , A2 and A+2 acquire imaginary parts. To first order in CPviolating quantities,   i i(χ2 −χ0 ) ImA0 ImA2  ω − . (6) ε = −√ e ReA0 ReA2 2 Thus, ε is suppressed by the ratio ω and ε /ε is approximately real since χ2 −χ0 −π/2 ≈ 0. The CP-conserving amplitudes ReAI are in general fixed to their experimental values, given in Eq. (3), in order to reduce the theoretical uncertainty. A theoretical calculation is then only needed for ImAI . Eq. (6) contains a subtle numerical balance between the two isospin contributions, making the result very sensitive to the values of the CP-violating amplitudes. Naive estimates of ImAI result in a strong cancellation between the two terms, leading to unrealistically low values of ε /ε [11–16, 35–37], as we show in section 4. Isospin breaking effects are very important in ε /ε, due to the large ratio 1/ω [65–67]. Including isospin violation, Re(ε /ε) can be written as ⎡ emp ⎤ (0)  ε ImA2 ⎥⎥⎥ ω+ ⎢⎢⎢⎢ ImA0 ⎥⎥⎦ , (7) (1 − Ωeff ) − Re = −√ ⎢⎣ ε ReA(0) 2 |ε| ReA(0) 0 2 emp

where ImA2 contains the I = 2 contribution from the electromagnetic penguin operator, the superscript (0) denotes the isospin limit, and ω+ ≡ ReA+2 /ReA0 is directly extracted from (3). The parameter Ωeff = ΩIV − Δ0 − f5/2 = (6.0 ± 7.7) · 10−2

(8)

incorporates the computed isospin-breaking corrections [65, 66].

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3. Theoretical framework The physical origin of ε /ε is at the electroweak scale where all the flavour-changing processes are described in terms of quarks and gauge bosons. Due to the presence of very different mass scales (Mπ < MK MW ), the gluonic corrections to the K → ππ amplitudes are amplified with large logarithms that can be summed up using the Operator Product Expansion (OPE) and the renormalization group equations (RGEs), all the way down to scales μ < mc . Finally, one gets an effective Lagrangian defined in the three-flavour theory [87], =1 LΔS eff

10  GF ∗ = − √ Vud Vus Ci (μ) Qi (μ) , 2 i=1

Energy

Fields

Effective Theory

MW

W, Z, γ, Ga τ, μ, e, νi t, b, c, s, d, u

Standard Model

OPE ? γ, Ga ; μ, e, νi s, d, u


γ ; μ, e, νi π, K, η

(9)

=1 = G8 L8 + G27 L27 + G8 gewk Lewk . (10) LΔS 2 =1 determines the K 0 → ππ amplitudes at Thus, LΔS 2 2 O(p ) in terms of the LECs G8 , G27 and G8 gewk . A first-principle computation of these three LECs requires to perform a matching between the short-distance and effective Lagrangians in Eqs. (9) and (10). This can be easily done in the limit of an infinite number of quark colours, where the four-quark operators factorize into currents with well-known chiral realizations. Since the large-NC limit is only applied in the matching between the two effective field theories, the only missing contributions are 1/NC corrections that are not enhanced by any large logarithms. Figure 1 shows schematically the chain of effective theories entering the analysis of the kaon decay dynamics.

N =3

f =1,2 LQCD , LΔS eff

NC → ∞ ? mK

which is a sum of local operators weighted by shortdistance coefficients Ci (μ) that depend on the heavy masses (μ > M) and CKM parameters. The Wilson coefficients Ci (μ) are known at NLO [88–91]. This inn cludes all corrections of O(αns tn ) and O(αn+1 s t ), where t ≡ log (M1 /M2 ) refers to the logarithm of any ratio of heavy mass scales M1 , M2 ≥ μ. Some next-to-next-toleading-order (NNLO) corrections are already known [92, 93] and efforts towards a complete short-distance calculation at the NNLO are currently under way [94]. Below the resonance region, where perturbation theory no longer works, we can use symmetry considerations to define another effective field theory in terms of the QCD Goldstone bosons (π, K, η). χPT describes the pseudoscalar octet dynamics through a perturbative expansion in powers of momenta and quark masses over the chiral symmetry breaking scale Λχ ∼ 1 GeV [95– 97]. At lowest order, the most general effective bosonic Lagrangian with the same SU(3)L ⊗ SU(3)R transforma=1 tion properties as LΔS contains three terms [98]: eff

139

χPT

Figure 1: Evolution from MW to the kaon mass scale.

4. Strong cancellation in simplified analysis The CP-odd amplitudes in Eq. (7) are dominated by the penguin operators Q6 and Q8 , due to their chiral enhancement. Taking only into account these two operators and ignoring all others in the estimation of ImAI , one finds [99, 100]: √ , (11) ImA0 |Q6 = X6 4 2 (F K − Fπ ) B(1/2) 6

where Xi ≡

− X8 2 Fπ B(3/2) , (12) 8   2 MK2 A2 λ5 η yi (μ) md (μ)+m , F K is the (μ) s

=

ImA2 |Q8 GF √ 2

and B(3/2) kaon decay constant and the factors B(1/2) 8 6 parametrize the deviations of the true hadronic matrix elements from their large-NC approximations. Notice, in the definition of Xi , that the short-distance scale of y6 (μ) and y8 (μ) is cancelled by the scale dependence of the quark masses since, at NC → ∞, the only elements of the anomalous dimension matrix γi j that survive are γ66 and γ88 [101]. This nice scale cancellation illustrates how the product of two colour-singlet quark currents factorizes and, in addition, that the product mq qq ¯ is renormalization-scale invariant. Introducing this rough prediction of the CP-odd amplitudes in Eq. (7), one obtains  ε   (3/2) (1 ) . (13) − Ω − 0.48 B Re ≈ 2.2·10−3 B(1/2) eff 8 6 ε With B(1/2) = B(3/2) = 1 (large-NC values) and Ωeff = 8 6 0.06, the naive estimation gives Re(ε /ε) ≈ 1.0 · 10−3 as the order of magnitude for the SM prediction. An interesting observation is the delicate cancellation among the different terms in (13) which makes the final number very sensitive to the chosen inputs [102–106]. With the

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values adopted in Ref. [37], B(1/2) = 0.57, B(3/2) = 0.76 8 6 and Ωeff = 0.15, one obtains Re(ε /ε) ≈ 2.6 · 10−4 , which is an order of magnitude smaller and in clear conflict with the experimental value in Eq. (1). However, these simplified estimates neglect completely the strong FSI present in K 0 → ππ decays, and miss the very large absorptive contribution giving rise to the measured phase-shift difference. The two relevant decay amplitudes get large logarithmic corrections from pion loops [17–19] that can be rigorously calculated using the usual χPT methods. They turn out to be positive for A0 |Q6 , while negative for A2 |Q8 . Consequently, the numerical cancellation in Eq. (13) disappears and one gets a sizeable enhancement of the SM prediction for ε /ε, in good agreement with its experimental value. 5. K → ππ amplitudes in χPT With the Lagrangian (10), the kaon decay amplitudes are easily obtained at O(p2 ) through a simple perturbative calculation in χPT. One only needs to consider tree=1 . level Feynman diagrams with one insertion of LΔS 2 Assuming isospin conservation, the AΔI amplitudes defined in Eq. (2) are given by [19, 66]    2 √ A1/2 = − 2 G8 F MK2 − Mπ2 − F 2 e2 gewk 3 √   2 G27 F MK2 − Mπ2 , − 9   2 10 A3/2 = − G27 F MK2 − Mπ2 + G8 F 3 e2 gewk , 9 3 A5/2 = 0 . (14) From the measured amplitudes in Eq. (3), one immediately obtains the tree-level determinations g8 = 5.0 and g27 = 0.25 for the octet and 27-plet chiral couplings, respectively, with the normalization G8,27 ≡ ∗ − G√F2 Vud Vus g8,27 . The large numerical difference between these two LECs√just reflects the smallness of the measured ratio ω ≈ 5 2 g27 /(9 g8 ). At LO in χPT, the phase shifts are predicted to be zero, because they are generated through loop diagrams with ππ absorptive cuts. Figure 2 displays the only oneloop topology contributing to the absorptive amplitudes. The large value of the measured phase-shift difference in Eq. (3) indicates a very large absorptive contribution. Analyticity relates the absorptive and dispersive parts of the one-loop diagram, which implies that the dispersive correction is also very large. A proper calculation of chiral loop corrections is then compulsory in order to obtain a reliable prediction for ε /ε.

π K0 π Figure 2: One-loop K → ππ topology with an absorptive cut.

The incorrect estimates, claiming small SM values of ε /ε [35–37], are flawed because they totally ignore the presence of absorptive cuts. They are based on (modeldependent) real K → ππ amplitudes that fail to comply with the experimental constraint in Eq. (5). At the NLO in χPT, the AΔI amplitudes can be written in the form   (g) 2 2 − e F g A AΔI = − G8 Fπ (MK2 − Mπ2 ) A(8) π ewk ΔI ΔI − G27 Fπ (MK2 − Mπ2 ) A(27) ΔI ,

(15)

(27) where A(8) ΔI and AΔI represent the octet and 27-plet components, and A(g) ΔI contains the electroweak penguin contributions. Moreover, these quantities can be further decomposed as   (X) (X) (X) A(X) = a A + Δ A 1 + Δ (16) L C ΔI ΔI ΔI ΔI , (X) with a(X) ΔI the tree-level contributions, ΔL AΔI the one(X) loop chiral corrections and ΔC AΔI the NLO local corrections at O(p4 ). The numerical values of the different (X) A(X) 1/2 and A3/2 components are displayed in tables 1 and 2, respectively.

X 8 g 27

a(X) ΔL A(X) 1/2 1/2 √ 2 0.27 + 0.47 i √ 2 2 √3 2 9

+ [ΔC A(X) 1/2 ]

− [ΔC A(X) 1/2 ]

0.01 ± 0.05

0.02 ± 0.05

0.27 + 0.47 i −0.19 ± 0.01 −0.19 ± 0.01 1.03 + 0.47 i

0.01 ± 0.63

0.01 ± 0.63

Table 1: Numerical predictions for the A1/2 components. The local ]+ ) and CP-odd ([ΔC A(X) ]− ) NLO correction to the CP-even ([ΔC A(X) 1/2 1/2 amplitudes is only different in the octet case.

X

a(X) 3/2

ΔL A(X) 3/2

ΔC A(X) 3/2

g

2 3 10 9

−0.50 − 0.21 i

−0.19 ± 0.19

−0.04 − 0.21 i

0.01 ± 0.05

27

Table 2: Numerical predictions for the A3/2 components.

The absorptive chiral corrections are large and positive for the ΔI = 1/2 amplitudes and much smaller and

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negative for ΔI = 3/2. Furthermore, they do not depend on the chiral renormalization scale νχ . Besides, table 1 shows a huge dispersive one-loop correction to the A(27) 1/2 amplitude. However, since Im(g27 ) = 0, the 27-plet components do not contribute to the CP-odd amplitudes and, therefore, do not introduce any uncertainty in the final numerical value of ImA0 . The relevant NLO loop corrections for ε /ε are (g) ΔL A(8) 1/2 and ΔL A3/2 . The first one generates a sigΔL A(8) 1/2 |

≈ 1.35, nificant enhancement of ImA0 , |1 + emp while the second one produces a suppression in ImA2 , |1 + ΔL A(g) 3/2 | ≈ 0.54. Consequently, the numerical cancellation between the I = 0 and I = 2 terms in Eq. (13) is completely destroyed by the chiral loop corrections. Tables 1 and 2 show also the numerical predictions for the NLO local corrections ΔC A(X) ΔI , which have been estimated in the large-NC limit. The dependence with νχ (absent at large NC ) is our main source of uncertainty. In order to estimate the errors, we have varied νχ between 0.6 and 1 GeV in the corresponding loop contributions ΔL A(X) ΔI . In addition, we have taken the uncertainty associated to the short-distance scale varying μ between Mρ and mc , but the impact on the ΔC A(X) ΔI corrections is negligible compared with the νχ uncertainty. The most − significant local corrections for ε /ε are [ΔC A(8) 1/2 ] and ΔC A(g) 3/2 ; nevertheless, they are much smaller than the loop contributions. 6. The SM prediction for ε /ε Taking into account all computed corrections in Eq. (7), our SM prediction for ε /ε is     Re ε /ε = 15 ± 2μ ± 2ms ± 2Ωeff ± 61/NC × 10−4 = (15 ± 7) × 10−4 .

(17)

The first uncertainty has been estimated by varying the short-distance renormalization scale μ between Mρ and mc . The second error shows the sensitivity to the strange quark mass, within its allowed range, while the third one displays the uncertainty from the isospin-breaking parameter Ωeff . The last error is our dominant source of uncertainty and reflects our ignorance about 1/NC suppressed contributions that we have missed in the matching process. In figure 3, we plot the prediction for ε /ε as function of the χPT coupling L5 , which clearly shows a strong dependence on this parameter. The experimental 1 σ range is indicated by the horizontal band, while

141

30

SM

25 20

Exp

15 10 5 LLatt 5

1.0

1.2

1.4

1.6

1.8

Figure 3: SM prediction for ε /ε as function of L5 (red dashed line) with 1 σ errors (oblique band). The horizontal blue band displays the experimentally measured value with 1 σ error bars. The dashed vertical line shows the current lattice determination of L5r (Mρ ).

the dashed vertical lines display the current lattice determination of L5r (Mρ ). The measured value of ε /ε is nicely reproduced with the preferred lattice inputs.

7. Final remarks Our SM prediction for ε /ε is in perfect agreement with the measured experimental value. We have shown the important role of FSI in K 0 → ππ. When ππ rescattering corrections are taken into account, the numerical cancellation between the Q6 and Q8 terms in Eq. (13) is completely destroyed because of the positive enhancement of the Q6 amplitude and the negative suppression of the Q8 contribution. Once these important corrections are included, the contributions from emp other four-quark operators to ImA(0) be0 and ImA2 come numerically less relevant, since the cancellation is no longer operative. The claims [35–37] of a flavour anomaly in ε /ε originate in naive approximations that overlook the important role of pion chiral loops. These incorrect estimates are using simplified ansatzs for the K → ππ amplitudes, without any absorptive contributions, in complete disagreement with the strong experimental evidence of a very large phase shift difference. The recent lattice results look quite encouraging, since it is the first time that a clear signal of the ΔI = 1/2 enhancement seems to emerge from lattice data [23]. RBC-UKQCD has obtained a first numerical estimate of ε /ε with a quite small central value, 2.1 σ lower than the experimental measurement. However, the same lattice simulation finds a (ππ)I=0 phase shift 2.9 σ away from its physical value, which indicates that these results are still in a very premature stage. Substantial

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improvements, with much larger statistics and a better control of the final pion dynamics, are expected soon. Our prediction of ε /ε agrees well with the measured value, providing a qualitative confirmation of the SM mechanism of CP violation. Although the theoretical error is still large, improvements can be achieved in the next years via a combination of analytical calculations, numerical simulations and data analyses. For instance, • A computation of the Wilson coefficients at NNLO is currently been performed [94]. • The isospin-breaking effects are vital for a correct ε /ε prediction. A complete re-analysis with updated inputs is currently under way [107]. • The O(e2 p0 ) coupling g8 gewk can be expressed as a dispersive integral over the hadronic vector and axial-vector spectral functions. The τ decay data can then be used to perform a direct determination of this LEC. A new phenomenological analysis is close to being finalized [108]. • The dominant χPT corrections originate from large chiral logarithms. A reliable estimate of higher-order contributions should be feasible either through explicit two-loop calculations or with dispersive techniques [17–19, 109, 110]. • A matching calculation of the weak LECs at NLO in 1/NC remains a very challenging task. A fresh view to previous attempts [24–33, 102–104, 111, 112], with a modern perspective, could suggest new ways to face this unsolved problem. • Lattice QCD simulations are expected to provide new improved data on K 0 → ππ transitions in the next years [64, 113]. Combined with appropriate χPT techniques, a better control of systematic uncertainties could be achieved. Acknowledgements We want to thank the organizers for their effort to make this conference such a successful event. We also thank Vincenzo Cirigliano, Gerhard Ecker, Elvira G´amiz, Helmut Neufeld, Jorge Portol´es and Antonio Rodr´ıguez S´anchez for useful discussions. This work has been supported in part by the Spanish State Research Agency and ERDF funds from the EU Commission [Grants FPA2017-84445-P and FPA2014-53631C2-1-P], by Generalitat Valenciana [Grant Prometeo/2017/053] and by the Spanish Centro de Excelencia Severo Ochoa Programme [Grant SEV-2014-0398].

The work of H.G. is supported by a FPI doctoral contract [BES-2015-073138], funded by the Spanish State Research Agency.

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