Electroweak effects in the extraction of the CKM angle $\gamma$ from $B \to D \pi$ decays

The angle $\gamma$ of the standard CKM unitarity triangle can be determined from tree-level $B$-meson decays essentially without hadronic uncertainties. We calculate the second-order electroweak corrections for the $B \to D \pi$ modes and show that their impact on the determination of $\gamma$ could be enhanced by an accidental cancellation of poorly known hadronic matrix elements. However, we do not expect the resulting shift in $\gamma$ to exceed $\big|\delta \gamma^{D\pi} /\gamma \big| \lesssim {\mathcal O}\big(10^{-4}\big)$.


Introduction
The Cabibbo-Kobayashi-Maskawa (CKM) angle γ ≡ arg(−V ud V * ub /V cd V * cb ) can be extracted from B → DK and B → Dπ decays that receive contributions only from tree operators [1]. The absence of penguin contributions and the fact that all relevant hadronic matrix elements can be obtained from data makes this determination theoretically extremely clean, thus providing a standard candle for the search for physics beyond the standard model (SM).
Whereas in most analyses γ has been extracted only from the B → DK modes, the LHCb collaboration recently included also the B → Dπ modes in their full combination [17,18]. The sensitivity to γ of these modes is smaller than that of the B → DK modes, due to a smaller interference term; this effect is, however, partially compensated by the larger B → Dπ branching ratio.
The extraction of γ from tree-level decays suffers from various uncertainties. Some of them can be reduced once more statistics becomes available, for instance, those related to a Dalitz-plot analysis [6,[19][20][21]. Other sources of reducible uncertainties are D −D mixing and, for final states with a K S , also K −K mixing. Both of these effects can be taken into account by measuring the mixing parameters and appropriately modifying the expressions for the decay amplitudes [22][23][24][25]. In a similar manner, the effects of nonzero ∆Γ s can be included into the γ extraction from untagged B s → Dφ decays [26]. It is also possible to allow for CP violation in the D-meson decays [17,[27][28][29][30]. The effects of CP violation in kaon mixing have recently been discussed in [31]. Finally, the impact on γ of new-physics contributions to tree-level Wilson coefficients has been estimated in [32].
As shown in [33], the first irreducible theory error on the determination of γ arises from higher-order electroweak corrections. It has been calculated for the B → DK modes, resulting in an upper bound on the shift in γ of δγ DK /γ O(10 −7 ) [33]. The shift due to electroweak corrections for the extraction of γ from the B → Dπ modes has not yet been computed; we close the gap in this letter.
The main difference between the B → Dπ and the B → DK modes lies in their CKM structure. Consequently, as we will see later, the effect of the electroweak corrections for the B → Dπ modes could potentially be much larger than for the B → DK modes, due to an approximate cancellation of hadronic matrix elements. However, we do not expect the final shift in γ to exceed δγ Dπ /γ O 10 −4 without some accidental fine tuning -well below the precision of any current or future measurement.
This letter is organized as follows. In Sec. 2 we calculate the electroweak corrections to the relevant Wilson coefficients and estimate the resulting shift in γ in Sec. 3. We conclude in Sec. 4.

Calculation of the electroweak corrections
We calculate the shift in γ due to electroweak corrections in close analogy to the procedure in Ref. [33]. The sensitivity of the B → Dπ modes to γ enters through the amplitude ratio where r Dπ which involve the usual four-fermion operators defined by Here Electroweak corrections, of the order of O(G 2 F ), to the amplitudes will induce a shift δγ Dπ in the extracted value of γ if the O(G F ) and O(G 2 F ) contributions differ in their weak phase. As argued in [33], the only second-order weak corrections to (1) and (6) that need to be considered are those arising from W box diagrams that have a different CKM structure than the corresponding tree amplitude, see for the box diagram. They differ in their weak phases and thus lead to a shift in the extracted value of γ.
The b → cūd transition receives a similar correction (right diagrams in Fig. 1 and Fig. 2 for the box diagram. The effects of these diagrams are CKM suppressed with respect to the previous contribution by two orders of magnitude and can be safely neglected. To a very good approximation, the only effect of the box diagrams is thus a correction to the Wilson coefficients in the effective Hamiltonian (3). Keeping only the local parts of the box diagrams we can write The Wilson coefficients C 1,2 (µ) are the same as in Eqs. (2) and (3) To get a first estimate of the size of the effect we will perform a matching , and we find for the shift ∆C 2 of the Wilson coefficient C 2 in Eq. (7) with the CKM angle β ≡ arg(−V cd V * cb /V td V * tb ) and the loop function The result of our calculation agrees with the corresponding loop function extracted from [37]. In this first estimate, the shift of the Wilson coefficient C 1 is zero. Using the input from [36] we find where the shown error is dominated by the uncertainty on the CKM elements V tb , V cb , and V td .
The loop functionĈ(x t , y b ) is dominated by the term proportional to log y b : where the subleading terms amount to a 10% correction. In order to capture also the leading QCD corrections, we now refine our analysis and perform a resummation of the terms proportional to log y b to all orders in the strong coupling constant. To achieve this, we first match the SM to the effective theory below the scale µ W = O(M W ), where the top quark and the heavy gauge bosons are integrated out, but the bottom quark is still a dynamical degree of freedom.
In fact, the matching correction at µ W vanishes to leading order. However, the renormalization-group (RG) running will generate this term at the bottom-

via bilocal insertions of the effective Hamiltonian
Here, Z andẐ are the renormalization constants for the local and bilocal insertions, respectively. The first line in Eq. (12) contains the four-quark operators obtained by integrating out the W and Z bosons. We denote them by The second line in Eq. (12) contains the operators They arise as counterterms to the bilocal insertions and are thus formally of dimension eight; this is made explicit by the m 2 b prefactor. These operators have the same four-quark structure as the leading-power operators Q 1,2 . We neglect the six-quark operators which arise from integrating out the W boson and the top quark, as they are suppressed by an additional factor of 1/M 2 W . To arrive at the CKM structure of the second line in Eq. (12) we note first that the two diagrams in Fig. 3 (right) have exactly the same phase as the corresponding tree-level diagram, so we can drop them. For the remaining diagrams we use the unitarity relation V ub V * ud + V cb V * cd = −V tb V * td , combining pairs of diagrams with internal up and charm quarks as shown in Fig. 3 and 4, and then factor out the tree-level coefficient V ub V * cd . The relevant diagrams in Fig. 3 and 4 yield the following mixing (we usê γ i,j;k = 2Ẑ i,j;k and expandγ i,j;k = αs 4πγ (0) i,j;k + . . ., where i, j denote the Q 1,2 insertions, and k is the label of theQ k operators): and employing "RunDec" [38] for the numerical running of the strong coupling constant, is Finally, at the bottom-quark scale we need to match the matrix elements of the two Hamiltonians (12) and (3). This will yield the leading y b behaviour with resummed logarithms. We write the matrix elements as where we expand ∆C k = 4π αs ∆C (0) k + . . . in such a way that the artificially inserted factor of 1/g 2 s in the definition ofQ k (14) is canceled. In Eq. (17) we have dropped the double insertions Q i Q j as they enter at higher order in α s and need not be calculated in our approximation. Therefore, we effectively obtain the matching condition for the Wilson coefficients of the local operators (7) in the form Numerically, we find ∆C 1 = −(1.14 ± 0.10) · 10 −8 × e iβ , ∆C 2 = −(1.09 ± 0.09) · 10 −7 × e iβ ; (19) the quoted errors reflect the uncertainty in the electroweak input parameters.
This should be compared to the unresummed result Eq. (10): we see that, indeed, the RG running has induced a nonzero correction to the Wilson coefficient (7). Moreover, also C 1 gets a small correction, in contrast to the unresummed result. As a check of our calculation we expand the solution of the RG equations about µ = M W and recover exactly the logarithm in Eq. (11), where we dropped the CKM factors.

The induced shift in γ
The imaginary part of the shift in the Wilson coefficients calculated in the previous two sections induces a shift in γ via a modification of the ratio r Dπ B , Eq. (1): where we expanded in the small corrections ∆C 1 , ∆C 2 to linear order. The resulting shift in the extracted value of γ is To estimate its size we need to evaluate the amplitude ratio r A , defined as Keeping in mind that the D meson is much heavier than the pion we see that both numerator and denominator in r A are suppressed by powers of Λ QCD /m b [39]. Using color counting and neglecting annihilation topologies yields r A ∼ N c = 3 as a naive estimate, with large uncertainties. A crude numerical estimate treating both final-state particles as light [40] and using an asymmetric D-meson wave function [39] suggests that the annihilation contribution is indeed negligible and that r A ≈ 1.
Interestingly, for a value of r A ≈ 4.6 the two terms in the denominators in Eq. (22) cancel each other, so that the electroweak correction to the ratio r Dπ B could, in principle, become arbitrarily large. The reason, of course, is that this cancellation would imply the vanishing of r Dπ B . Ignoring differences in the matrix elements related to the replacement of pions by kaons, this would also imply the vanishing of the ratio r DK B , in contradiction to the measured value (cf. the discussion below Eq. (1)). A complete cancellation can thus be safely excluded, although a more quantitative statement is difficult to obtain. To be conservative we will take r A = 4.5 for our estimate of δγ Dπ . Using sin 2β = 0.682 [36] we then obtain δγ Dπ 9.7 · 10 −6 (unresummed) , δγ Dπ 9.2 · 10 −6 (resummed) .
Large uncertainties are associated with these numbers due to the poorly known value of r A and missing nonlocal contributions, but it seems very unlikely that the shift in γ exceeds δγ Dπ /γ 10 −4 . Note that for values of r A 3 the shift δγ Dπ /γ drops below 10 −6 . On the other hand, considerable fine tuning would be required for an almost complete cancellation of the denominators in Eq. (22). For instance, to find |δγ Dπ /γ| larger than 10 −3 would require a tuning of r A of the order of 10 −4 .

Summary and Conclusion
The determination of the CKM phase γ from tree-level decays is theoretically exceptionally clean, as all necessary branching fractions and amplitude ratios can be obtained from experimental data. In the SM, the only shift in γ is induced by electroweak corrections to the effective Hamiltonian that carry a weak phase relative to the leading contributions. In this letter we have estimated the shift for the extraction of γ from the B → Dπ decay modes. We calculated the electroweak corrections in two ways, first integrating out the bottom quark together with the top quark and the W boson, then also summing leading QCD logs of m b /M W in a two-step matching procedure.
Interestingly, the different CKM structure compared to the B → DK modes could lead to a moderately large shift in γ via an approximate cancellation of hadronic matrix elements. Whereas these matrix elements are hard to estimate, we find that without large accidental fine tuning the expected shift in γ is very unlikely to exceed δγ Dπ /γ 10 −4 .
A better estimate of the hadronic matrix elements seems worthwile and could reduce this uncertainty.