The ultimate theoretical error on gamma from B ->D K decays

The angle gamma of the standard CKM unitarity triangle can be determined from B ->D K decays with a very small irreducible theoretical error, which is only due to second-order electroweak corrections. We study these contributions and estimate that their impact on the gamma determination is to introduce a shift |delta gamma|<~ O(10^-7), well below any present or planned future experiment.


I. INTRODUCTION
The determination of the standard CKM unitarity triangle angle γ ≡ arg(−V ud V * ub /V cd V * cb ) from B → DK and B →DK decays is theoretically extremely clean. The reason is that the B → DK transitions receive contributions only from tree operators, and none from penguin operators. Furthermore, all the relevant matrix elements can be obtained from data if enough D-decay channels are measured. The sensitivity to γ comes from the interference of b → cūs and b → ucs decay amplitudes, which have a relative weak phase γ, cf. Fig. 1. These quarklevel transitions mediate B − → D 0 K − and B − →D 0 K − decays, respectively. The D 0 and D 0 subsequently decay into a common final state f , which allows the two decay channels to interfere. Several variants of the method have been proposed, distinguished by the final state f : i) f can be a CP eigenstate such as K S π 0 and K S φ [1,2], ii) a flavor state such as K + π − and K * + ρ − [3], or iii) a multibody state such as K S π + π − , π + π − π 0 [4][5][6]. Other possibilities include the decays of neutral B mesons, B 0 and B s , [7,8], multibody B decays [9][10][11][12] and D * or D * * decays [13,14] (see also the reviews in [15] and the current combination of LHCb measurements in [16]).
The above set of methods has several sources of theoretical errors. Most of them can be reduced once more statistics becomes available. For instance, in the past the D → K S π + π − Dalitz plot needed to be modeled using a sum of Breit-Wigner resonances or using the Kmatrix formalism. Utilizing the data from entangled ψ(3770 → DD) decays measured at CLEO-c [17] and BES-III, this uncertainty can in principle be completely avoided [5]. The related error is now statistics-dominated [18,19].
Other sources of reducible uncertainties are D −D mixing and K −K mixing (for final states with K S ). Both of these effects can be included trivially by modifying the expressions for the decay amplitudes, taking meson mixing into account, and then using experimentally measured mixing parameters [20]. The effect of D−D mixing is most significant if the D decay information comes from entangled ψ(3770 → DD) decays. The shift in γ is then linear in x D , y D , giving ∆γ 2.9 • [21] (see also [22]). For flavor-tagged D decays (i.e. from D * → Dπ) the effect is quadratic in x D , y D and thus much smaller [23]. Similarly, for γ extraction from untagged B s → Dφ decays the inclusion of ∆Γ s can be important and can be achieved once ∆Γ s is well measured [24]. In the extraction of γ from B → DK, CP violation in the D system was usually neglected. Even if this assumption is relaxed, it is still possible to extract γ by appropriately modifying the expressions for the decay amplitudes (and using the fact that in the Cabibbo allowed D decays there is no direct CP violation) [16,[25][26][27][28][29].
Yet another source of reducible theory error are QED radiative corrections to the decay widths. The uncertainties from this source are expected to be below present experimental sensitivity on γ so that not much work has been done on them. Since the corrections are CP conserving they can be reabsorbed in the CP-even measured hadronic quantities and would not affect γ, as long as in the measurements the radiative corrections are treated consistently between different decay modes.
The first irreducible theory error on γ thus comes from higher-order electroweak corrections.
This error cannot be eliminated using just experimental information and may well represent the ultimate precision of the γ determination from B → DK decays. The resulting uncertainty was estimated using scaling arguments in Ref. [30] and found to be of the order of δγ/γ ∼ O(10 −6 ).
In this paper we perform a more careful analysis, and find that the induced uncertainty is in fact most probably even an order of magnitude smaller. The one-loop electroweak corrections give rise to local and nonlocal contributions. We estimate the size of the local contributions using naive factorization and obtain δγ/γ O(10 −7 ). The nonlocal contributions are more difficult to estimate, but naively one expects that they are not significantly larger than the local ones.
The paper is organized as follows. In Section II we give a brief discussion of electroweak corrections for B → DK decays with a focus on the γ extraction. We also give numerical estimates for the shift, δγ, utilizing the analytic results of Section III, where further details of the calculation are given. Finally, we conclude in Section IV.

II. THE SHIFT IN γ FROM B → DK DUE TO ELECTROWEAK CORRECTIONS
The measurement of γ in B → DK decays is based on the interference between the treelevel b → cūs and b → ucs mediated processes, cf. Fig. 1. The sensitivity to the weak phase γ enters through the amplitude ratio where the four-fermion operators are Above we have used the short-hand notation (cb and similarly for the other quark flavors. The scale at which the Wilson coefficients are evaluated is close to the b quark mass, µ ∼ m b , with C 1 (m b ) = 1.10, and C 2 (m b ) = −0.24 at leading-log order [32], for m b (m b ) = 4.163 GeV [33] and α S (M Z ) = 0.1184 [34]. The decay amplitudes in Eq. (1) are then given at leading order in the electroweak expansion by Since the weak phases of the two contributions are different, this results in a shift δγ in the extracted value of γ.
A similar higher-order electroweak diagram contributes also to the b → ucs transition, which is given by exchanging the external u and c quarks in Fig. 2 (left). Again, the dominant contribution is effectively due to the top and bottom quark running in the loop, so that the In this case the weak phases of the LO and NLO contributions are the same to a very good approximation, so that the electroweak contributions do not induce a shift in γ.
Keeping only the local part of the box diagram, the relevant change to the effective weak Hamiltonian is very simple. The structure of the CKM coefficients in (7) and (8) is such that all the corrections relevant for the γ extraction are in the Hc u effective weak Hamiltonian The Wilson coefficients C 1,2 (µ) are the same Wilson coefficients as in Eqs. (2) and (3), while ∆C 1,2 (µ) are calculable corrections. They depend on the CKM elements and carry a weak phase γ. They therefore have a different weak phase than C 1,2 (µ), which in our phase convention are real. This introduces a shift in δγ in the extraction of the weak phase γ from B → DK decays. This shift represent the ultimate theory error on the measurement of γ.
Defining the ratio of matrix elements for the two relevant operators the shift in the ratio r B , Eq. (1), is where we expanded in the small corrections ∆C 1 , ∆C 2 to linear order. The resulting shift in the extracted value of γ is The size of the corrections ∆C 1,2 will be calculated in the next section, while here we only quote the numerical results. The unresummed result for Im(∆C 2 ), cf. Eq. (22) below, is where the error only reflects the experimental errors due to the input parameters. The results In order to obtain δγ we also need to estimate the ratio of the matrix elements, r A , in (10).
In naive factorization this ratio is where we used f D = 0.214 GeV [35], F B→K 0 (0) = 0.34 [36], f K = 0.16 GeV, F B→D 0 (0) = 1.12 [37]. In Eq. (15) we only quote the central value, since the error on this estimate is bigger than the errors on the form factors themselves. However, we do not expect the error on the estimate of r A in (15) to be bigger than a factor of a few.
Using this and setting γ = 68 • for definiteness, we obtain the estimate for the shift δγ, where to this accuracy the resummed expressions for ∆C 1,2 (with nonlocal contributions neglected) and unresummed results coincide. An uncertainty of at most an additional factor of a few can be expected on the above estimate, so that we can conclude that the ultimate theoretical error on γ measurement is safely below In the next section we derive the analytic expressions for ∆C 1,2 (µ), and then draw our conclusions in Section IV.

III. CORRECTIONS TO THE ELECTROWEAK HAMILTONIAN
In this section we consider the b → cūs box diagram, Fig. 2 (left), in detail. The results can be readily adapted to the b → ucs case by exchanging the external quarks and adjusting the CKM factors. The diagram in Fig. 2 (left) is superficially similar to the box diagrams contributing toK 0 − K 0 andB 0 (s) − B 0 (s) mixing [32], and to b → ssd, dds decays [38]. The difference is that the box diagram in Fig. 2 (left) has both up-and down-quarks running in the loop, in contrast to the case ofK 0 − K 0 andB 0 (s) − B 0 (s) mixing where both quarks in the loop are of up-type.
We will calculate the shift δγ in two ways -first by keeping only the log(m b /M W ) enhanced local contribution, but without resumming it. Subsequently we will resum this log. In the first case we will take b, t and W in the loop to be heavy and integrate them out at µ ∼ M W . In this way one obtains the local operator part of the effective field theory (EFT) with only the light quarks, u, d, s, c, and an external non-dynamical b-quark field. Keeping only the local operators in EFT is a crude approximation that does, however, suffice for our purposes -to show that the induced corrections on the γ extraction are exceedingly small. The obtained result will also give us better understanding of the correct EFT results with resummed log(m b /M W ), which we will perform next. The resummation is achieved by first integrating out t and W at µ ∼ M W and matching onto the effective theory with b, and c, s, d, u quarks. We will then evolve the Wilson coefficients down to the scale µ ∼ m b using the renormalization-group (RG).

A. The result without resummations
We first evaluate the box diagram at µ ∼ M W , treating t and b quarks as massive and u, c and d, s quarks as massless, and set all external momenta to zero (including the external bquark momentum). This will give us the local part of the EFT contributions with unresummed Wilson coefficients. Because of the double GIM mechanism, acting on both the internal upquark and down-quark lines, the leading contribution is proportional to This is easy to see by expanding the matrix element for the boxdiagram correction to the B → DK decay in terms of the quark masses, The CKM factors λ u→c The Wilson coefficient C 2 in (3) receives a similar correction but with the same weak phase as the O(G F ) term. Thus the correction does not contribute to δγ and we neglect it, cf. Eqs. (7), (8). The result of our calculation agrees with the result extracted from [39] and readŝ Note that the loop functionĈ full (x, y) vanishes if either x → 0 or y → 0. This proves that the only nonzero contribution in (18) is A 4 ∝ x t y b . In fact, it is a very good approximation to keep in this result only the log y b enhanced contribution, where the finite terms amount to an O(10%) correction. Using the values for the CKM matrix elements from the CKMfitter collaboration [31] and further input from [34], we find where the error shown is only due to the CKM elements.
The Wilson coefficientĈ(x t , y b ) contains the unresummed large logarithm log y b . The logarithm is multiplied by 2y b and would vanish in the limit of zero b quark masses. However, since the Wilson coefficientĈ(x, y) starts only at O(y b ), the term with log y b represents a large correction. In the next subsection we therefore perform a resummation of this logarithm.  The contribution proportional to y b now vanishes at the electroweak scale to the order considered. However, this contribution will be generated by mixing of two insertions of dimensionsix operators below the electroweak scale. It is therefore useful to introduce the following Hamiltonian describing the five-flavor effective theory where we used V tb V * ts = −V cb V * cs + O(λ 2 ), with λ = |V us | 0.23 (numerically, this replacement is valid up to a three-permil correction). Moreover, we denoted the usual four-quark operators by and definedQ The last two operators denoted by a tilde are formally of dimension eight because of the m 2 b factor. They have the same four-quark structure as the leading power operators Q 1,2 so that their contributions could be absorbed by redefining the Wilson coefficients C 1,2 allowing them to be complex. It is more practical, however, to keep the Wilson coefficients real and split-off explicitly the contributions to the effective Hamiltonian that carry the extra weak phase as we did in (23). Note that in the second line in Eq. (23)  Let us now look at some of the contributing terms in more detail. The sum of the two diagrams denoted by 2) in Fig. 3 yields where Q 2 Q 2 div is the common divergence of the two diagrams, which is independent of the light-quark masses. In the last step we kept only the term proportional to the factor with a weak phase, which is the only contribution entering the shift δγ. The Lorentz and color structure of Q 2 Q 2 div is the same as ofQ 2 , so that this gives the anomalous dimension of the double insertion mixing intoQ 2 . The sum of the two diagrams denoted by 1) in Fig. 3 is similar to the first case, Eq. (26), but with the replacement C 2 → C 1 , Q 2 → Q 1 . The sum of the two diagrams denoted by 3) in Fig. 3 yields and does not carry a weak phase. As such it does not contribute to δγ and can be discarded.
There are also four additional diagrams, shown in Fig. 4, which lead to the mixing of double insertions into the Fierz-transformed operatorQ 1 .
To obtain the contributions of double H f =5 eff insertions to the running ofQ 1,2 we thus only need to compute the diagrams denoted by 1) and 2) in Fig. 3, with a double insertion of Q 1 and Q 2 , respectively, plus two additional diagrams with an insertion of Q 1 and then Q 2 at each of the two weak vertices, cf. Fig. 4. We expandγ i,j;k = αs 4πγ (0) i,j;k + . . ., where i, j denote the Q 1,2 insertions, and k is the labeling of theQ k operators. Extractingγ i,j;k from the one-loop divergence of the double insertion (see, for instance, [40] for details), our calculation yieldŝ with all the remaining entries either vanishing or not contributing. The initial conditions for the dimension-six Wilson coefficients are given by C 1 (µ W ) = 1, C 2 (µ W ) = 0 to leading order [32]. ExpandingC k =C (0) k +O(α s ), we findC (0) k (µ W ) = 0 at leading order. A nonvanishing value will be induced by RG running for µ < µ W , which we compute by solving where γ lk is the well-known anomalous dimension for the mixing of the Q 1,2 operators, It is advantageous to go to the diagonal basis of the current-current operators, by defining In this way Eq. (29) gets rewritten as a homogeneous equation [41], for which the standard techniques of obtaining closed expressions for the RG evolution apply. The transformed LO anomalous dimensions and the Wilson coefficients are [40] where By explicit calculation we findγ while the remaining entries are zero. Defining D + ≡ (C − ,C + /C − ) T , the renormalization-group equations forC + and C − can be combined into We obtain the corresponding solution forC − and C + by exchanging the subscripts + ↔ −.
Note that we have also included the running of the mass and the coupling constant related to the factor m 2 b /g 2 s in the definition of the operatorsQ k , given by the anomalous dimension of the quark mass γ m and the QCD beta function β. Transforming back to the original basis, we find numerically where we used α s (M Z ) = 0.1184 [34] and m b (m b ) = 4.163 GeV [33]. Note that the RG running has now also induced a nonzero correction to C 1 in (9), in contrast to the unresummed result.
We used the mathematica package "RunDec" [42] for the numerical running of the strong coupling constant.
Finally, at the bottom-quark scale we need to calculate the B → DK matrix elements using our EFT Hamiltonian (23) in order to obtain the shift δγ. This will give the leading y b behavior with resummed logarithms. We write the matrix elements suggestively as Here we expand ∆C k = 4π αs ∆C (0) k + O(1); note that in this way the artificially inserted factor of 1/g 2 s in the definition ofQ k (25) is canceled. At LO it is not necessary to compute the double insertions Q i Q j since these are loop suppressed, and therefore we effectively obtain the matching condition for the Wilson coefficients of the local operators (9) Numerically, we find |∆C 1 | = (4.5 ± 0.2) · 10 −9 , |∆C 2 | = (4.3 ± 0.2) · 10 −8 ; the errors reflect the uncertainty in the electroweak input parameters. This should be compared to the unresummed result Eq. (22). Expanding the solution of the renormalization-group equations around µ = M W and expressing G F in terms of the weak mixing angle we recover exactly the logarithm in Eq. (21):

IV. CONCLUSIONS
The determination of the SM weak phase γ from the B → DK decays has a very small irreducible theoretical error which is due to one-loop electroweak corrections. In this paper we have estimated the resulting shift in γ. Treating m b ∼ M W or resumming logs of m b /M W gives in both cases an estimated shift δγ ∼ 2 · 10 −8 , keeping only the local operator contributions at the scale µ ∼ m b . It is unlikely that the neglected non-local contributions, which come with the same CKM suppression as the local contributions, would differ from the above estimate by more than a factor of a few. We can thus safely conclude that the irreducible theoretical error on the extraction of γ from B → DK is |δγ| O(10 −7 ).