The $B \to \pi K$ Puzzle Revisited

For a number of years, there has been a certain inconsistency among the measurements of the branching ratios and CP asymmetries of the four $B \to \pi K$ decays ($B^+ \to \pi^+ K^0$, $B^+ \to \pi^0 K^+$, $B^0 \to \pi^- K^+$, $B^0 \to \pi^0 K^0$). In this paper, we re-examine this $B \to \pi K$ puzzle. We find that the key unknown parameter is $|C'/T'|$, the ratio of color-suppressed and color-allowed tree amplitudes. If this ratio is large, $|C'/T'| = 0.5$, the SM can explain the data. But if it is small, $|C'/T'| = 0.2$, the SM cannot explain the $B \to \pi K$ puzzle -- new physics (NP) is needed. The two types of NP that can contribute to $B \to \pi K$ at tree level are $Z'$ bosons and diquarks. $Z'$ models can explain the puzzle if the $Z'$ couples to right-handed $u{\bar u}$ and/or $d{\bar d}$, with $g_R^{dd} \ne g_R^{uu}$. Interestingly, half of the many $Z'$ models proposed to explain the present anomalies in $b \to s \mu^+ \mu^-$ decays have the required $Z'$ couplings to $u{\bar u}$ and/or $d{\bar d}$. Such models could potentially explain both the $b \to s \mu^+ \mu^-$ anomalies and the $B \to \pi K$ puzzle. The addition of a color sextet diquark that couples to $ud$ can also explain the puzzle.


Introduction
Over the past 15 years there has been a great deal of interest in B → πK decays.There are four such processes: B + → π + K 0 (designated as +0 below), B + → π 0 K + (0+), B 0 d → π − K + (−+) and B 0 d → π 0 K 0 (00).Their amplitudes are not independent, but obey a quadrilateral isospin relation: Using these decays, nine observables have been measured: the four branching ratios, the four direct CP asymmetries A CP , and the mixing-induced indirect CP asymmetry S CP in B 0 d → π 0 K 0 .Shortly after these measurements were first made (in the early 2000s), it was noted that there was an inconsistency among them.This was referred to as the "B → πK puzzle" [1][2][3].
In Ref. [4], it was pointed out that, by performing a full fit to the data, one can quantify the discrepancy with the standard model (SM) implied by the B → πK puzzle.Ref. [4] (2004) contains the results of the first such fit, an update was done in Ref. [5] (2007), and the last fit was performed in Ref. [6] (2009).This latter study finds that "if one adds a constraint on the weak phase γ coming from independent measurements -the SM fitone finds that the fit is poor.On the other hand, it is not terrible.If one is willing to accept some deficiencies in the fit, it can be argued that the SM can explain the B → πK data."In other words, one cannot say that the B → πK puzzle suggests the presence of new physics (NP).A more correct statement would be that the measurements of B → πK decays allow for NP.
In the present paper, we update the fit using the latest experimental results.As the data has not changed enormously since 2007, we do not expect to find a different conclusion.However, the first purpose of our study is to be more precise in this conclusion.In what regions of parameter space does the SM yield a reasonable explanation of the B → πK data?And in what regions is the SM fit to the data terrible, so that NP is definitely required?
There are two types of NP that can contribute at tree level to B → πK.One is a Z boson that has a flavor-changing coupling to sb and also couples to ūu and/or dd.The other is a diquark that has db and ds couplings or ub and us couplings.The second goal of our work is to determine what precise couplings the Z or diquark must have in order to improve the fit with the B → πK data.
Finally, we make one other observation.Over the past few years, several measurements have been made that disagree with the predictions of the SM.These include (i) R K (LHCb [7]) and (ii) R K * (LHCb [8]), where R K,K * ≡ B(B +,0 → K +, * 0 µ + µ − )/B(B +,0 → K +, * 0 e + e − ), (iii) the angular distribution of B → K * µ + µ − (LHCb [9,10], Belle [11], ATLAS [12] and CMS [13]), and (iv) the branching fraction and angular distribution of B 0 s → φµ + µ − (LHCb [14,15]).Recent analyses of these discrepancies [16][17][18][19][20][21][22][23][24][25][26] combine constraints from all measurements and come to the following conclusions: (i) there is indeed a significant disagreement with the SM, somewhere in the range of 4-6σ, and (ii) the most probable explanation is that the NP primarily affects b → sµ + µ − transitions.Arguably the simplest NP explanation is that its contribution to b → sµ + µ − comes from the tree-level exchange of a Z boson that has a flavor-changing coupling to sb, and also couples to µ + µ − .Many models of this type have been proposed to explain the data.In some of these models, the Z also has couplings to ūu and/or dd.In this case, it will also contribute at tree level to B → πK decays 1 and could potentially furnish an explanation of the B → πK puzzle.If so, in such models there would be a connection between the anomalies in b → sµ + µ − and B → πK decays, which is quite intriguing.
In Sec. 2 we review the B → πK puzzle and update its status.In Sec. 3 we examine whether the SM can explain the puzzle.We find that it can if |C /T | = 0.5, but not if |C /T | = 0.2 (both values are allowed theoretically).For the case of |C /T | = 0.2, in Sec. 4 we examine the puzzle in the presence of NP.We find that the B → πK puzzle can be explained if the NP is a Z boson or a diquark.We make the connection with Z models that explain the b → sµ + µ − anomalies.We conclude in Sec. 5.

The B → πK Puzzle
We begin by reviewing the B → πK puzzle.Within the diagrammatic approach [27,28], B-decay amplitudes are expressed in terms of six diagrams2 : the color-favored and colorsuppressed tree amplitudes T and C , the gluonic penguin amplitudes P tc and P uc , and the color-favored and color-suppressed electroweak penguin amplitudes P EW and P C EW .(The primes on the amplitudes indicate b → s transitions.)The B → πK decay amplitudes are given in terms of diagrams by We have explicitly written the weak-phase dependence (including the minus sign from V * tb V ts [in P tc ]), so that the diagrams contain both strong phases and the magnitudes of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.The amplitudes for the CPconjugate processes can be obtained from the above by changing the sign of the weak phase γ.
P tc and P uc are defined as follows.There are three gluonic penguin contributions P i , where i = u, c, t indicates the identity of the quark in the loop.The full penguin amplitude is then 2) The second line arises due to the unitarity of the CKM matrix (V * ub V us +V * cb V cs +V * tb V ts = 0), and we define P tc ≡ |V * tb V ts |(P t − P c ) and , where λ = 0.22 is the sine of the Cabibbo angle, It has been shown [29][30][31] that, to a good approximation, the diagrams P EW and P C EW can be related to T and C within the SM using flavor SU(3) symmetry: We refer to these as EWP-tree relations.Here, the c i are Wilson coefficients [32] and [33].Now, c 10 /c 9 = c 2 /c 1 holds to a few percent.In this limit, the above EWP-tree relations become Thus, P EW and T are roughly the same size, as are P C EW and C .Taking the above information into account, the relative sizes of all the B → πK diagrams can be roughly estimated as 31 : where λ ∼ 0.2.

Naive B → πK Puzzle
Neglecting the diagrams of O( λ2 ) in Eq. (2.5), the B → πK amplitudes become With these amplitudes, consider the direct CP asymmetries of B + → π 0 K + and B 0 d → π − K + .Such CP asymmetries are generated by the interference of two amplitudes with nonzero relative weak and strong phases.In both A 0+ and A −+ , T -P tc interference leads to a direct CP asymmetry.On the other hand, in A 0+ , P EW and T have the same strong phase (P EW ∝ T [Eq.(2.4)]), while P EW and P tc have the same weak phase (= 0), so that P EW does not contribute to the direct CP asymmetry.This means that we expect

Statistics
As noted in the Introduction, a more accurate measure of the (dis)agreement with the SM can be obtained by performing a fit to the data.From here on, in looking for SM or NP explanations of the B → πK puzzle, we will use only fits.In such fits, the B → πK amplitudes are expressed in terms of a certain number of unknown theoretical parameters.
In order to perform a fit, there must be fewer theoretical unknowns than observables.We define χ 2 as where O i are the various observables used as constraints.O exp i and ∆O i are, respectively, the experimentally-measured central values and errors.O th i are the theoretical predictions for the observables, and are functions of the unknown theoretical parameters.We use the program MINUIT [35][36][37] to find the values of the unknowns that minimize the χ 2 .
At this point, it is useful to review some basic properties of χ 2 distributions in order to establish what constitutes a good fit.The χ 2 probability distribution depends on a single parameter, n, which is the number of degrees of freedom (d.o.f.).It is given by (2.8) For n large, this becomes a normal distribution with central value n and standard deviation √ 2n.That is, in this limit the preferred value of χ 2 /d.o.f. is 1, with an error 2/n.This says that, even if we have the correct underlying theory, we still expect χ 2 /d.o.f. 1, just due to statistical fluctuations.For this reason, it is common to say that, if we find χ 2 min /d.o.f. 1 in a fit, it is acceptable.We stress that this only holds for n large -it is not justified to apply the same criterion for small values of n.One way to see this is to compute the p-value.For n large, the pvalue corresponding to χ 2 /d.o.f.= 1 is 50%.That is, a p-value of 0.5 constitutes an acceptable fit.However, here are the p-values corresponding to χ 2 /d.o.f.= 1 for smaller values of n: (2.9) These reflect the fact that, for n small, the central value of the distribution is not at χ 2 /d.o.f.= 1 -it is at smaller values.This shows that, if n = 1, though χ 2 /d.o.f.= 1 is not a bad fit, it is still somewhat less than acceptable (which corresponds to a p-value of 0.5).It also suggests that p-values are easier than χ 2 /d.o.f. for judging the goodness-offit when n is small.

True B → πK Puzzle
Taking into account the first EWP-tree relation of Eq. (2.4), the amplitudes of Eq. (2.6) depend on four unknown parameters: the magnitudes |T | and |P tc |, one relative strong phase, and the weak phase γ.In addition, the indirect CP asymmetry in B 0 d → π 0 K 0 , S CP , depends on the weak phase β.These parameters are constrained by the B → πK data of Table 1, as well as by the independent measurements of the weak phases [33]: (2.10) With more observables (11) than theoretical unknowns (5), a fit can be performed.The results are shown in Table 2. Unsurprisingly, we find a terrible fit: χ 2 min /d.o.f.= 30.9/6,corresponding to a p-value of 3.0 × 10 −5 .This can be considered the true B → πK puzzle.

SM Fits
In the previous section, we saw that the SM cannot account for the B → πK data if the small diagrams C , P uc and P C EW are neglected.Thus, in order to test whether the data can be explained by the SM, the small diagrams must be included in the fit.With the EWP-tree relations of Eq. (2.3), there are only four independent diagrams in the B → πK amplitudes4 : T , C , P tc and P uc .This corresponds to 8 unknown parameters: four magnitudes of diagrams, three relative strong phases, and the weak phase γ. (As before, the value of β, which is required for S CP , can be constrained from independent measurements.This is done in all fits.)We therefore have more observables (9) than theoretical unknowns (8), so that a fit can be done.Additional constraints can come from the independent measurement of γ and/or theoretical input.In this section we perform various SM fits in order to determine under what circumstances the SM can explain the B → πK puzzle.
Regarding the strong phases, these are mainly generated by QCD rescattering from another diagram with the same CKM matrix elements.As an example, take P c in Eq. (2.2).It can arise directly via a gluonic penguin amplitude with a c quark in the loop, or it can be generated by rescattering from the tree operator b → scc: Only the rescattered contribution has a substantial strong phase.Now, the tree diagram is much larger than the penguin diagram.But rescattering comes with a cost: the rescattered penguin is only about 5-10% as large as the tree.Indeed it is of the same order as (P c ) dir .The net effect is that (P c ) tot can have a sizeable strong phase.On the other hand, the strong phase of T can only arise due to self-rescattering.Since this self-rescattered amplitude is only about 5-10% as large as the original amplitude, the strong phase of T is expected to be small.The bottom line is that δ T should be small, while the other strong phases can be large.In the fits we adopt the convention that δ T = 0. (In any case, only relative strong phases are measurable.)

All diagrams free, constraint on γ added
In this fit, we keep all diagrams and allow their values to vary, but constrain γ by including the independent measurement of Eq. (2.10) (β is always constrained in this way.)The results of the fit are given in Table 3.We see that the p-value is only 17%, which is below the 50% required for an acceptable fit.More importantly, the best fit has |C /T | = 0.75±0.32,which is considerably larger than the estimate in Eq. (2.5).
SM fit (2): χ 2 /d.o.f.= 3.5/2, p-value = 0.17 Parameter Best-fit value γ (72.0 ± 5.8 3. Ref. [45] does NNLO calculations within perturbative QCD (pQCD), and finds |C /T | = 0.53.We note that a range of values is not given, which suggests that this result is for a specific choice of the theoretical parameters.It is not clear what the smallest allowed value of |C /T | is within pQCD.
It therefore appears that |C /T | can be as large as ∼ 0.5.But it may also be the case that |C /T | is quite a bit smaller.In light of this, below we repeat the SM fit, taking as theoretical input |C /T | = 0.2 or 0.5.
There is one more thing.In Eq. (2.5), it is estimated that |P uc /P tc | = O( λ2 ).However, it has been argued that |P uc | is actually even smaller: |P uc /P tc | = O( λ3 ) [46], so that it can be neglected, to a good approximation.Indeed, in Table 3 P uc is the smallest diagram.Therefore, from now on we will also add the theoretical input P uc = 0. Note that this is the most favorable assumption for the SM: it increases the d.o.f., and hence the p-value, of a fit.If we find that a particular SM fit is poor, it would be even worse had we allowed P uc to vary.We now perform the SM fit, but with the theoretical constraints |C /T | = 0.5 and P uc = 0.The results of the fit are given in Table 5.We find χ 2 min /d.o.f.= 4.9/5, for a p-value of 43%, which is an acceptable fit.We therefore conclude that, if |C /T | = 0.5, there is no B → πK puzzle -the data can be explained by the SM.It is interesting to compare the results of this fit with those in Table 3.By χ 2 min , this fit is worse, since (χ 2 min ) Table 5 > (χ 2 min ) Table 3 .On the other hand, by p-value it is better.The reason is that the number of d.o.f. has changed, so that (χ 2 min /d.o.f.) Table 5 < (χ 2 min /d.o.f.) Table 3 .This type of behavior often occurs when the number of d.o.f. is small, making it difficult to judge which of two fits is truly better.
Note that the strong phase δ C is produced mainly by rescattering from T .Thus, for a large value of |C /T |, δ C should be on the small side.And indeed, from Table 5, we see that δ C is not that large.Now, in this fit γ is constrained by its independently-measured value.However, this is not necessary -γ can be treated as an unknown parameter and extracted from the measurements of B → πK decays.Indeed, if the SM explains the B → πK data, we would expect the extracted value of γ to be the same as that measured in tree-level decays.We investigate this in the following subsection.
3.4 |C /T | = 0.5, P uc = 0, γ free Here we repeat the fit of the previous subsection, but remove the constraint from the independent measurement of γ.The results are given in Table 6.By χ 2 min , this fit is better than that of Table 5.However, by p-value it is worse.This is a result of the different d.o.f. in the two fits.More importantly, it is found that γ = (51.2± 5.1) • , which deviates from its measured value of (72.1 ± 5.8) • by 2.7σ.So this is a reason not to be entirely satisfied that the SM explains the B → πK puzzle, even if |C /T | = 0.5.

Summary
If |C /T | = 0.2, we have found that the SM cannot explain the B → πK puzzle.Even in the most optimistic scenario, where P uc is neglected, the p-value is only 3%, constituting a poor fit.On the other hand, if |C /T | = 0.5 and P uc = 0, the p-value is 43%, which is acceptable.There is still an aspect of this fit that is not entirely satisfactory, but on the whole we find that, in this case, the SM can explain the B → πK data.
We therefore conclude that it is the size of |C /T | that determines whether or not there is truly a B → πK puzzle.If |C /T | = 0.5, which is theoretically its maximally-allowed value, the SM can explain the B → πK data.However, if |C /T | = 0.2, which is towards the lower end of its theoretically-allowed range, then the SM cannot explain the data, and NP is required.In this case, it is natural to investigate what type of NP is required.We do this in the next section.

NP Fits
In this section we assume |C /T | = 0.2 and examine whether the B → πK puzzle can be explained with the addition of new physics.

Model-independent formalism
In the general approach of Ref. [47], the NP operators that contribute to the B → πK amplitudes take the form O ij,q N P ∼ sΓ i b qΓ j q (q = u, d), where Γ i,j represent Lorentz structures, and color indices are suppressed.The NP contributions to B → πK are encoded in the matrix elements πK| O ij,q N P |B .In general, each matrix element has its own NP weak and strong phases.Now, above we noted that strong phases are basically generated by QCD rescattering from diagrams with the same CKM matrix elements.We then argued that the strong phase of T is expected to be very small since it is due to self-rescattering.For the same reason, all NP strong phases are also small, and can be neglected.In this case, many NP matrix elements can be combined into a single NP amplitude, with a single weak phase: Here the strong phase is zero.There are two classes of such NP amplitudes, differing only in their color structure: sα Γ i b α qβ Γ j q β and sα Γ i b β qβ Γ j q α (q = u, d).They are denoted A ,q e iΦ q and A C,q e iΦ C q , respectively [48].Here, Φ q and Φ C q are the NP weak phases.In general, A ,q = A C,q and Φ q = Φ C q .Note that, despite the "color-suppressed" index C, the matrix elements A C,q e iΦ C q are not necessarily smaller than A ,q e iΦ q .There are therefore four NP matrix elements that contribute to B → πK decays.However, only three combinations appear in the amplitudes: A ,comb e iΦ ≡ −A ,u e iΦ u + A ,d e iΦ d , A C,u e iΦ C u , and A C,d e iΦ C d [48].The B → πK amplitudes can now be written in terms of the SM diagrams and these NP matrix elements.Here we neglect the small SM diagram P uc : In Ref. [6], a different set of NP operators is defined: In order, these imply the inclusion of NP in the color-allowed electroweak penguin, the gluonic penguin, and the color-suppressed electroweak penguin amplitudes.Of course, the two sets of NP operators, {A ,comb e iΦ , A C,u e iΦ C u , A C,d e iΦ C d } and {P EW,N P , P N P , P C EW,N P }, are equivalent.Depending on the situation, one set or the other may be used.
In the most general case, there are three independent NP operators that contribute to B → πK decays.Although their strong phases are negligible, they each have their own weak phase.There are therefore 12 parameters in the B → πK amplitudes: 6 magnitudes of diagrams, 2 relative strong phases, 3 NP weak phases, and the CKM phase γ.As before, the CKM phase β also contributes to the indirect CP asymmetry in B 0 d → π 0 K 0 , so that there are a total of 13 unknown parameters.However, there are only 12 constraints: the 9 B → πK observables, the independent measurements of β and γ, and the theoretical input |C /T | = 0.2.With more unknowns than constraints, a fit cannot be done.However, this may be improved in the context of a specific model.There may be fewer unknown parameters, or they may not all be independent.We examine this possibility in the following subsections.

Z models
For B → πK, the relevant decay is b → sqq (q = u, d).This can occur via the tree-level exchange of a Z that has a flavor-changing coupling to sb and also couples to qq.The exact form of the sbZ coupling is unimportant, and for the light quarks, all four currents g qq L(R) qγ µ P L(R) q are possible.Assuming the weak and mass eigenstates are the same (i.e., we neglect the off-diagonal terms in the CKM matrix), the only thing that is certain is that g dd L = g uu L due to SU (2) L symmetry.Still, how the Z couples to dd and ūu has direct consequences for the NP operators: 1. Suppose that the Z couples only to left-handed dd and ūu, i.e., g dd R = g uu R = 0. Since g dd L = g uu L , the NP operators A ,d e iΦ d and A ,u e iΦ u are equal, as are A C,d e iΦ C d and A C,u e iΦ C u .In this case, P EW,N P = P C EW,N P = 0 in Eq. (4.3); the only nonzero NP operator is P N P .This also holds if the Z couples vectorially to dd and ūu, in which case 2. On the other hand, if g dd R and g uu R are nonzero, but g uu R = −2g dd R , then P N P = 0, but P EW,N P and P C EW,N P are nonzero.
3. Alternatively, if only g dd R is nonzero, A ,comb e iΦ and A C,d e iΦ C d are nonzero (equivalently, P EW,N P is nonzero and P N P e iΦ P = −(2/3)P C EW,N P e iΦ C EW ).
4. Similarly, if only g uu R is nonzero, A ,comb e iΦ and A C,u e iΦ C u are nonzero (equivalently, P EW,N P is nonzero and P N P e iΦ P = (1/3)P C EW,N P e iΦ C EW ).
5. For all other choices of couplings, all three NP operators are nonzero.
The point is that, although we consider a specific NP model, it has many variations, so that its study includes a number of different NP scenarios.Below we consider each of the cases above (which are identified by their number in the list).Note also that, for this particular kind of NP, we do naively expect the color-suppressed operators to be smaller than the color-allowed ones.
The four-fermion operator corresponding to b → sqq is proportional to g bs L g qq L(R) .Note that g qq L(R) must be real, since the light-quark current qγ µ P L(R) q is self-conjugate.However, g bs L can be complex, i.e., it can contain a weak phase.This is the only source of CP violation in the model and, as it appears in all NP operators, the weak phases of all operators are equal, i.e., there is only a single NP weak phase.That is, } and Φ EW = Φ P = Φ C EW ≡ Φ for {P EW,N P , P N P , P C EW,N P }.Now, Z models with a flavor-changing coupling to sb also contribute to B 0 s -B0 s mixing.The larger g bs L is, the more Z models contribute to -and receive constraints from -this mixing.In particular, the phase of B 0 s -B0 s mixing has been measured to be quite small: ϕ ccs s = −0.030± 0.033 [34].Thus, if g bs L is big, its phase, which is the NP weak phase Φ in B → πK, must be small.Φ can be large only if g bs L is small, which means g qq L(R) is big.(This type of argument was first made in Ref. [49].)

Case (5): all three NP operators nonzero
We begin with the general case, in which all three NP operators are nonzero.Since all three NP weak phases are equal in the Z model, there are 11 unknown parameters.However, there are 12 constraints -the 9 B → πK observables, the independent measurements of β and γ, and the theoretical input |C /T | = 0.2 -so a fit can be done.
The results are shown in Table 7 (left-hand table ).With χ 2 /d.o.f.= 0.41/1 and a p-value of 52%, this is an excellent fit.However, there is a serious problem: the best fit has P C EW,N P /P EW,N P = 16, whereas, in the Z model, the color-suppressed NP operators are expected to be smaller than the color-allowed ones.We therefore conclude that this result cannot arise within a Z NP model.NP fit ( 5 To focus on a possibility consistent with a Z NP model, we impose an additional (theoretical) constraint: |P C EW,N P /P EW,N P | = 0.3.The results of this fit are shown in Table 7 (right-hand table).Here χ 2 /d.o.f.= 1.85/2, for a p-value of 40%, which is a good fit.Of course, this good p-value is a consequence of the fact that, with the constraint, the d.o.f. has increased from 1 to 2. If we consider instead d.o.f = 1 (which essentially corresponds to the region in the space of the fit of Table 7 (left-hand table) with |P C EW,N P /P EW,N P | = 0.3), the p-value is 17%.Although far from an excellent fit, it is still better than that of the SM.
In the above fits, all NP operators are allowed.In the following, we examine whether a better fit can be found if the model contains only a subset of the operators.Clearly the minimum χ 2 will be larger than that found above, but since the d.o.f. will also be larger, a larger p-value may be found, indicative of a better fit.

Case (1): only P N P nonzero
The case where only P N P is nonzero arises when the Z couplings to right-handed d and u quarks obey g dd R = g uu R (these can both vanish or be nonzero).Since g dd L = g uu L by weak isospin invariance, A ,d e iΦ d = A ,u e iΦ u and A C,d e iΦ C d = A C,u e iΦ C u , so that P EW,N P = P C EW,N P = 0.In model-building terms, this corresponds to the case where the Z couples only to left-handed dd and ūu, or where it couples vectorially to these quark pairs.Assuming that only P N P is nonzero, we note that all four B → πK amplitudes of Eq. ( 4 With χ 2 min /d.o.f.= 3.5/1 and a p-value of 6%, this is a poor fit, despite the addition of NP.But we can understand what's going on here.We have used Re(z), Im(z) and Im(z ) as unknown parameters.Referring to Eq. (4.4), we see that, if P N P = 0, Im(z) = Im(z ).But this is essentially what is found in Table 8.That is, with this particular type of NP, we cannot do better than the SM.(This was also the conclusion of Ref. [6].)Indeed, comparing with Table 3, we see that the results are very similar.In particular, the χ 2 min is identical (the p-values are different due to the different d.o.f.).
We therefore see that the only way for this NP to improve on the SM is if the Z couplings to right-handed d and u quarks obey g dd R = g uu R .

Cases (3), (2), (4)
We begin with case (3), where only g dd R is nonzero.Now P EW,N P is nonzero and P N P = −(2/3)P C EW,N P ).We also impose the constraint |P C EW,N P /P EW,N P | = 0.3.The results of the fit for this case are shown in Table 9.Here the p-value is 30%, which is not bad (and is far better than that of the SM).
NP fit (3): χ 2 /d.o.f.= 3.67/3, p-value = 0.   [39].Taking central values, we have This particular Z has no couplings to leptons, i.e., it is leptophobic.The experimental limits on such Z bosons are very weak.However, suppose they were stronger: say M Z > 1 TeV is required.The above constraint can still be satisfied with such values of M Z , while keeping perturbative couplings.The only additional constraint comes from B 0 s -B0 s mixing.The formalism is described in Ref. [49], to which we refer the reader for details.Briefly, in the presence of SM and NP contributions, B 0 s -B0 s mixing is due to the operator where (Recall that g bs L = |g bs L |e iΦ .)The B 0 s -B0 s mixing parameters are given by Here, f Bs BBs = 270±16 MeV.The experimental measurements of the mixing parameters yield [34] ∆M exp s = 17.757 ± 0.021 ps −1 , while the SM predictions are With all of this, we can obtain an upper bound on the NP contribution.From Eq. (4.7), we have We allow the experimental quantities to vary by ±2σ, while f Bs BBs varies within its theoretical range.This leads to (A similar result can be found in Ref. [50].) Combining this with Eq. (4.5), we have With this condition, all the constraints can be satisfied.Therefore this Z can indeed explain the B → πK puzzle.
The results for cases (2) (g dd R and g uu R are nonzero, but g uu R = −2g dd R ) and (4) (only g uu R is nonzero) are similar.The fit for case (2) has a p-value of 28%, while that for case (4) has p-value = 26%.We therefore conclude that a Z model can explain the B → πK puzzle, but it is necessary that the Z couple to right-handed d and/or u quarks, with g dd R = g uu R .
4.2.4Z b → sµ + µ − models and b → sqq As noted in the introduction, there are currently several measurements that disagree with the predictions of the SM.These include (i) R K (LHCb [7]) and (ii) R K * (LHCb [8]), (iii) the angular distribution of B → K * µ + µ − (LHCb [9,10], Belle [11], ATLAS [12] and CMS [13]), especially the observable P 5 [51], and (iv) the branching fraction and angular distribution of B 0 s → φµ + µ − (LHCb [14,15]).The discrepancies in R K and R K * are quite clean and are at the level of 2.2-2.6σ.On the other hand, the discrepancies in B → K * µ + µ − and B 0 s → φµ + µ − have some amount of theoretical input.Depending on how one treats the hadronic uncertainties, the disagreement with the SM is in the 2.5-4σ range.
Of these measurements, the most recent is that of R K * .Following its announcement, a number of papers appeared [16][17][18][19][20][21][22][23][24][25][26] computing the size of the discrepancy with the SM, and determining the general properties of the NP required to explain the results.Combining constraints from all measurements, the general consensus is that there is indeed a significant disagreement with the SM, somewhere in the range of 4-6σ (this large range is due to the fact that different groups deal with the theoretical uncertainties in different ways).In order to account for all measurements, the most probable explanation is that the NP primarily affects b → sµ + µ − transitions.
Arguably the simplest NP explanation is that the contribution to b → sµ + µ − arises due to the tree-level exchange of a Z boson.Here the Z has a flavor-changing coupling to sb, and it also couples to μµ.Many models of this type have been proposed to explain the data.In some of these models, the Z also has couplings to ūu and/or dd, so that it could potentially furnish an explanation of the B → πK puzzle.If so, in such models there would be a connection between the anomalies in b → sµ + µ − and B → πK decays, which is quite intriguing.
Many Z models have been proposed to explain the b → sµ + µ − anomalies .(Note that Refs.[58][59][60][61] all discuss the 3-3-1 model.Here the Z couples equally to e + e − and µ + µ − , so this model cannot explain R K ( * ) .)The question is: are there models in which the Z couples to right-handed d and/or u quarks, with g dd R = g uu R ?A survey of the models reveals the following: • Models in which the Z couplings to uū or d d either vanish or are very small include Refs.[52,75,77,82,[85][86][87][88].
• Models in which the Z has vectorlike couplings to uū and d d include Refs.[53,64,68,73,74,76].In this case the Z model cannot explain the B → πK puzzle, see Sec. 4.2.2.
• Many models have been proposed in which the Z couples to LH quarks, but not RH quarks.A significant fraction of these can be easily modified to allow the Z to couple to RH quarks.These include Refs.[54-57, 62, 63, 67, 69-72, 78, 80, 83].
There are a total of 34 Z models.Of these, 17 have, or can be modified to have, the Z coupling to right-handed uū and/or d d, with g dd R = g uu R .These models can therefore potentially also explain the B → πK puzzle.
It is necessary to check other constraints.In order to reproduce the b → sµ + µ − data, the Z mass and couplings must satisfy For C µµ 9 (NP) = −C µµ 10 (NP) = −0.8[49], we require (4.15) The Z will also contribute to ν µ N → ν µ N µ + µ − (neutrino trident production).Measurements of this process lead to the constraint [49] For example, in Ref. [24], the b → sµ + µ − data were analyzed in the context of Z models.Good fits were found for M Z = 1 TeV, with g µµ L 0.5 and g bs L −2.5 × 10 −3 .From Eq. (4.13), this implies that the Z couplings to right-handed uū and/or d d must be on the large side, O(1).
However, there is another, more important process to consider.A Z that couples to both quarks and muons can be detected at the LHC via pp → Z → µ + µ − .If the Z couples only to sb, its production cross section may be small enough to escape detection (for example, see Ref. [77]).On the other hand, if the Z couples to dd and/or ūu, it will be produced plentifully in pp collisions.Recently, using the 2015 and 2016 data at √ s = 13 TeV, the ATLAS Collaboration searched for high-mass resonances decaying into dileptons, but found nothing [89].They put an upper limit on the product of the production cross section and decay branching ratio, converting this to a lower limit on M Z > ∼ 4 TeV for the Z models analyzed.We expect that this limit applies to our Z .
The conclusion is that some of the Z models proposed to explain the b → sµ + µ − data may also explain the B → πK puzzle.The Z must be somewhat massive, with M Z > ∼ 4 TeV.On the other hand, its mass cannot be much larger than this lower limit, as the couplings would become nonperturbative in this case.So perhaps such a Z will be observed at the LHC in the coming years.

Diquarks
Another NP particle that can contribute to b → sqq (q = u, d) at tree level is a diquark [90].Diquarks are scalar particles that couple to two quarks 5 .Quarks are 3s under SU (3) C , so that the diquark must transform as a 3 (triplet) or a 6 (sextet).Since diquark couplings are fermion-number-violating, the diquarks couple to two quarks of the same chirality.That is, they couple to , where q L = (u L , d L ) is an SU (2) L doublet, and i, j are flavor indices.There are therefore a total of 8 different types of diquark.These are listed in Table 10.
The triplets are antisymmetric in color, while sextets are symmetric.This implies that their couplings are also respectively antisymmetric and symmetric in flavor [90].This has important consequences for flavor-changing neutral currents (FCNCs).Consider the diquarks that couple to ds (diquarks I, II, V II and V III).In principle, they contribute at tree level to K 0 -K0 mixing via the t-channel exchange of a diquark.This involves the transitions d → d and s → s, i.e., the flavor indices of the couplings have i = j.However, because the couplings of the triplet diquarks are antisymmetric in flavor, these couplings vanish for these diquarks.That is, while sextet diquarks can generate ∆F = 2 FCNCs, triplet diquarks cannot [90].Thus, the measurements of neutral meson mixing constrain diquarks I, V and V II to be very massive, so that their effects in other low-energy processes are negligible.However, there are no similar constraints on diquarks II, III, IV , V I and V III.
Diquarks contribute at tree level to b → sqq (q = u, d) via b → qD * (→ sq).From this, we see that diquark V I does not contribute to B → πK.Recall that in Sec.4.1 we noted that there are four NP matrix elements that contribute to B → πK decays: A ,q e iΦ q and A C,q e iΦ C q (q = u, d).The color-allowed and color-suppressed matrix elements use the operators sα Γ i b α qβ Γ j q β and sα Γ i b β qβ Γ j q α , respectively.Now consider diquark V III.It contributes to A ,d e iΦ .This same type of behavior holds for all diquarks, except that diquark III is a sextet, which is symmetric, so that A C,u e iΦ C u = +A ,u e iΦ u .The four diquarks that contribute to B → πK are II, III, IV and V III.These have the following properties: • II: decays to q i L q j L , which includes u i L u j L , u i L d j L and d i L d j L .This implies that the amplitudes for b → sūu and b → s dd are equal, so that A ,d = A ,u and A C,d = A C,u .We then have P EW,N P = P C EW,N P = 0; the only nonzero NP operator is P N P .(We also have A C,q e iΦ C q = −A ,q e iΦ q , but this is not important.) With this information, we can perform fits for the four diquark models.

Diquark II
The scenario where P N P is the only nonzero NP operator was examined in Sec.4.2.2.There it was found that the fit was no better than that of the SM, and that the B → πK puzzle could not be explained.

Diquark III
The results of the fit with diquark III are shown in In order to get a sense of what values are required for the diquark mass and couplings, we proceed as in Sec.4.2.3:we compare P EW,N P and T .Here, however, the form factors for the two diagrams are different, so this comparison will only give a rough idea.We have This is similar to what was found for Z models.

Diquark IV
The results of the fit with diquark IV are shown in

Diquark V III
The results of the fit with diquark V III are shown in Table 13.The best fit has χ 2 /d.o.f.= 11.6/3, for a p-value of 0.9%.This is a very poor fit, worse than that of the SM.This diquark cannot explain the B → πK puzzle.

Summary
In the previous section, we saw that, if |C /T | = 0.2, the SM cannot explain the B → πK puzzle.The two types of NP that can contribute to B → πK at tree level are Z bosons and diquarks.In this section, we found that either NP model can explain the puzzle if |C /T | = 0.2.For Z models, the Z must couple to right-handed uū and/or d d, with g dd R = g uu R .Half of the Z models proposed to explain the b → sµ + µ − anomalies have the required Z couplings to uū and/or d d.As for diquarks, the only one that works is a color 6 that couples to ud.For both NP models, the fits have p-values in the range 25

Conclusions
There are four B → πK decays -B + → π + K 0 , B + → π 0 K + , B 0 d → π − K + and B 0 d → π 0 K 0 -whose amplitudes obey a quadrilateral isospin relation.In the early 2000s, their branching ratios and CP asymmetries (direct and indirect) were measured, and it was noted that there was a tension between the measurements and the SM.This was referred to as the "B → πK puzzle."Over the years, a number of analyses were done, attempting to quantify the seriousness of the puzzle, and to identify the type of new physics that can ameliorate the problem.
In the present paper, we perform an update of the B → πK puzzle by doing fits to the data using a diagrammatic decomposition of the B → πK amplitudes.We find that the key unknown parameter is |C /T |, the ratio of color-suppressed and color-allowed tree amplitudes.Theoretically, this ratio is predicted to be 0.15 < ∼ |C /T | < ∼ 0.5.If it is large, |C /T | = 0.5, we find that the SM can explain the data: the fit has a p-value of 43% (an excellent fit has p-value = 50%).On the other hand, if it is small, |C /T | = 0.2, the fit has a p-value of 4%, which is poor.Our conclusion is that, if |C /T | is small, the SM cannot explain the B → πK puzzle -NP is needed.
The two types of NP that can contribute to B → πK at tree level are Z bosons and diquarks.For the case of |C /T | = 0.2, we examine whether the B → πK puzzle can be explained with the inclusion of such NP.For both types of NP, the answer is yes.
In the case of Z models, the decay b → sqq (q = u, d) is produced via the tree-level exchange of a Z that couples to sb and to qq.We find that, if the Z couples only to left-handed qq, things do not work.The B → πK puzzle can be explained only if the Z couples to right-handed uū and/or d d, with g dd R = g uu R -the p-values of the fits are in the range 25-40%.
This particular NP solution is intriguing because there are currently anomalies involving the process b → sµ + µ − that can also be explained by the addition of a Z .We find that, of all the Z models proposed for the b → sµ + µ − anomalies, half have the required Z couplings to uū and/or d d.Such models could potentially explain both the b → sµ + µ − anomalies and the B → πK puzzle.
Turning to diquarks, there are eight different types.Taking into account constraints from other processes, particularly ∆F = 2 FCNCs, there is only one diquark that works.It is a color 6 that couples to ud.It contribute at tree level to b → sūu via b → ūD * (→ su).Its fit has a p-value of 26%.
χ 2 min /d.o.f. and best-fit values of unknown parameters in amplitudes of Eq. (2.1).Constraints: B → πK data, measurements of β and γ.Even though the fit is only fair, it is clear that the SM fit prefers a large value of |C /T |.But this raises the question: what does theory predict for |C /T |? 1.In Ref. [39], B → πK decays were analyzed in the context of QCD factorization (QCDf).The various NLO contributions were computed for three different values of the renormalization scale, µ = m b /2, m b and 2m b .In all three cases it was found that |C /T | 0.2. 2. The NNLO corrections within QCDf have been considered in Refs.[40-43].Including these corrections, it is found [44] that 0.13 ≤ |C /T | ≤ 0.43, with a central value of |C /T | = 0.23, very near its NLO value.

3. 2
|C /T | = 0.2, P uc = 0, constraint on γ added We now perform the same fit as above, but add the theoretical constraints |C /T | = 0.2 and P uc = 0.The results of the fit are given in

3. 3
|C /T | = 0.5, P uc = 0, constraints on γ added One important question is: what values of the Z mass and couplings are required to explain the B → πK puzzle?This can be deduced from Table 9.The SM T diagram involves the tree-level decay b → ūW + * (→ us = K + ).The NP P EW,N P diagram looks very similar -we have the tree-level decay b → sZ * (→ d d = π 0 / √ 2).Within factorization, the SM and NP diagrams involve A πK ≡ F B→π 0 (0)f K and A Kπ ≡ F B→K 0 (0)f π , respectively, where F B→K,π 0 (0) are form factors and f π,K are decay constants.The hadronic factors are similar in size: |A Kπ /A πK | = 0.9 ± 0.1 d and A C,d e iΦ C d .However, it is straightforward to see that these are not independent.The b → sqq transition involves b α → qβ D * αβ .The virtual D * then decays equally to s β d α and s α d β .But the first decay creates the color-allowed operator, the second the color-suppressed operator.And since the triplet is antisymmetric, there is a relative minus sign.That is, we have A C,d e iΦ C d = −A ,d e iΦ d , or Φ C d = Φ d and A C,d = −A ,d

Table 4 .
The situation is better than in Table2, but we still have a poor fit: χ 2 min /d.o.f.= 12.1/5, corresponding to a p-value of 3%.This demonstrates conclusively that, if |C /T | = 0.2, the B → πK puzzle cannot be explained by the SM.

Table 8 .
.2) contain the following combination: P tc − P N P e iΦ P .Writing P tc = |P tc |e iδ P tc , this contains the four quantities |P tc |, δ P tc , P N P and Φ P .However, here these are not all independent.One can see this by noting that the combinations that appear in B and B decays are:|P tc |e iδ P tc − P N P e iΦ P ≡ z ,|P tc |e iδ P tc − P N P e −iΦ P ≡ z .(4.4)z and z are complex numbers; their four real and imaginary parts can be written in terms of the four theoretical parameters.However, it is clear from the above expressions that Re(z) = Re(z ).In order to take this into account, we use Re(z), Im(z) and Im(z ) as unknown parameters in the fit.The results are shown in Table8.χ 2 min /d.o.f. and best-fit values of unknown parameters [z and z are defined in Eq. (4.4)] for the Z model where NP is present in B → πK, but only P N P is nonzero.Constraints: B → πK data, measurements of β and γ.

Table 9 .
χ 2 min /d.o.f. and best-fit values of unknown parameters for the Z model where only g dd R is nonzero.Constraints: B → πK data, measurements of β and γ, |C /T | = 0.2, |P C EW,N P /P EW,N P | = 0.3.

Table 11 .
The best fit has χ 2 /d.o.f.= 4.0/3, for a p-value of 26%.Like the Z model of Sec.4.2.3, the fit is not bad (and is far better than the SM).It could explain the B → πK puzzle.

Table 11 .
χ 2 min /d.o.f. and best-fit values of unknown parameters for the model of diquark III.Constraints: B → πK data, measurements of β and γ, |C /T | = 0.2.

Table 12 .
The best fit has χ 2 /d.o.f.= 5.4/3, for a p-value of 15%.This is only so-so.Given that the best-fit values of the ratios |T /P tc | and |P EW,N P /P tc | are both somewhat larger than expected, we conclude that this diquark does not provide a good explanation of the B → πK puzzle.