The Emergence of the $ΔU=0$ Rule in Charm Physics
Yuval Grossman Stefan Schacht





yg73@cornell.edu
Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA

ss3843@cornell.edu
Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA


We discuss the implications of the 
recent discovery of CP violation in two-body SCS D decays by LHCb.
We show that the result can be explained within the SM without the need
for any large SU(3) breaking effects. It further
enables the determination of the imaginary part of the ratio of the Δ U=0 over Δ U=1 matrix elements in charm decays,
which we find to be (0.65± 0.12).
Within the standard model, the result proves the non-perturbative nature of the penguin contraction of 
tree operators in charm decays, similar to the known non-perturbative enhancement of Δ I=1/2 over Δ I=3/2 
matrix elements in kaon decays, that is, the Δ I=1/2 rule.
As a guideline for future measurements, we show how to completely
solve the most general parametrization of the D → P^+P^- system.


The Emergence of the Δ U=0 Rule in Charm Physics
    Stefan Schacht
    
================================================





§ INTRODUCTION 


In a recent spectacular result, LHCb discovered direct CP violation in
charm decays at 5.3σ <cit.>. 
The new world average of the difference of 
CP asymmetries <cit.>

    Δ a_CP^dir   ≡ 
    a_CP^dir(D^0→ K^+K^-) - a_CP^dir(D^0→π^+π^-) ,

where 

    a_CP^dir(f)    ≡|𝒜 (D^0→ f)|^2 - |𝒜(D^0→ f)|^2 
    /|𝒜(D^0→ f)|^2 + |𝒜(D^0→ f)|^2 
     ,

and which is provided by the Heavy Flavor Averaging Group (HFLAV) <cit.>, is given as <cit.>

    Δ a_CP^dir   = -0.00164±  0.00028 .


Our aim in this paper is to study the implications of this result. In
particular, working within the Standard Model (SM) and using the known values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements as input, we see how 
Eq. (<ref>) can be employed in order to extract low energy QCD quantities,
and learn from them about QCD.

The new measurement allows for the first time to determine the CKM-suppressed amplitude of singly-Cabibbo-suppressed (SCS)
charm decays that contribute a weak phase difference relative to the CKM-leading part, which leads to a non-vanishing 
CP asymmetry. 
More specifically, Δ a_CP^dir allows to determine
the imaginary part of the Δ U=0 over Δ U=1 matrix elements.

As we show, the data suggest the emergence of a Δ U=0 rule, which has
features that are similar to the known
Δ I=1/2 rule 
 in kaon physics.
This rule is the 
observation that in K →ππ the amplitude into a I=0 final state is 
enhanced by a factor ∼ 20 with respect to the one into a I=2 final 
state <cit.>. This is explained by
large non-perturbative rescattering effects. 
Analogous enhancements in charm decays have previously been discussed in 
Refs. <cit.>.
For further recent theoretical work on charm CP violation see 
Refs. <cit.>. 

In Sec. <ref> we review the completely general U-spin decomposition of the decays 
D^0→ K^+K^-, D^0→π^+π^- and D^0→ K^±π^∓. 
After that, in Sec. <ref> we show how to completely determine all U-spin parameters from data. 
Our numerical results which are based on the current measurements are given in Sec. <ref>.
In Sec. <ref> we interpret these as the emergence of a
Δ U=0 rule, and in Sec. <ref> we compare it to
the Δ I=1/2 rules in K, B and D decays. The different effect of Δ U=0 and Δ I=1/2 rules
on the phenomenology of charm and kaon decays, respectively, is discussed in Sec. <ref>. 
In Sec. <ref> we conclude.




§ MOST GENERAL AMPLITUDE DECOMPOSITION 


The Hamiltonian of SCS decays can be written as the sum 

    ℋ_eff∼Σ (1,0) - λ_b/2 (0,0) ,

where (i,j) = 𝒪^Δ U=i_Δ U_3=j,
and the appearing combination of CKM matrix elements are

    Σ   ≡V_cs^* V_us - V_cd^* V_ud/2 ,      
    -λ_b/2≡ -V_cb^* V_ub/2 = V_cs^* V_us + V_cd^* V_ud/2 ,

where numerically, |Σ| ≫ |λ_b|. 
The corresponding amplitudes have the structure

    𝒜 = Σ ( A_Σ^s - A_Σ^d ) - λ_b/2 A_b ,

where A_Σ^s, A_Σ^d and A_b contain only strong phases and we 
write also A_Σ≡ A_Σ^s - A_Σ^d .

For the amplitudes we use the notation

    𝒜(Kπ)      ≡𝒜(D^0 → K^+π^-) , 
    
    𝒜(ππ)    ≡𝒜(D^0 →π^+π^-) , 
    
    𝒜(KK)        ≡𝒜(D^0 → K^+K^-) , 
    
    𝒜(π K)     ≡𝒜(D^0 →π^+ K^-) .

The U-spin related quartet of charm meson decays into charged final states can then be 
written as <cit.>

    (Kπ)      = V_cs V_ud^* (t_0 - 1/2 t_1  ) , 
    
    (ππ)    = -Σ^*(t_0 + s_1  + 1/2 t_2  )  
    -λ_b^*(p_0 - 1/2 p_1  ) , 
    
    (KK)        = Σ^*(t_0 - s_1  + 1/2 t_2  )  
           -λ_b^* (p_0 + 1/2 p_1  ) ,  
    
    (π K)      = V_cd V_us^* (t_0 + 1/2 t_1  ) .

The subscript of the parameters denotes the level of U-spin breaking
at which they enter.  
We write A(Kπ) and A(π K) for the Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) amplitude 
without the CKM factors, respectively.
We emphasize that the SM parametrization in Eqs. (<ref>)–(<ref>) is completely general and 
independent from U-spin considerations. 
For example, further same-sign contributions in the CF and DCS decays can be absorbed by a 
redefinition of t_0 and t_2, see Ref. <cit.>.
The meaning as a U-spin expansion only comes into play if we assume a hierarchy for the parameters according to their subscript. 

The letters used to denote the amplitudes should not be
confused with any ideas about the diagrams that generate them. That is,
the use of p_0 and t_0 is there since in some limit p_0 is dominated
by penguin diagrams and t_0 by tree diagrams. Yet, this is not always the case, and thus
it is important to keep in mind that all that we do know at this stage
is that the above is a general reparametrization of the decay
amplitudes, and that each amplitude arises at a given order in the
U-spin expansion.
In the topological interpretation of the appearing parameters, t_0 includes both tree and exchange 
diagrams, which are absorbed <cit.>. Moreover, s_1 contains the broken penguin 
and p_0 includes contributions from tree, exchange, penguin and penguin annihilation 
diagrams <cit.>.

We note that 
the U-spin parametrization is completely general when we assume no CPV
in the CF and DCS decays, which is also the case to a very good
approximation in the SM.
Beyond the SM, there can be additional amplitude contributions to 
the D^0→ K^+π^- and D^0→π^+K^- decays which come 
with a relative weak phase from CP violating new physics. We do not
discuss this case any further here.

In terms of the above amplitudes, the branching ratios are given as 

    ℬℛ(D→ P_1P_2 )    = |𝒜|^2 ×𝒫(D,P_1,P_2) ,
                                         
    𝒫(D,P_1,P_2)    = τ_D ×1/16π m_D^3√(   
    (m_D^2 - (m_P_1 - m_P_2)^2) (m_D^2 - ( m_P_1 + m_P_2)^2 )
    ) .

The direct CP asymmetries are <cit.>

    a_CP^dir   = Im(λ_b/Σ) Im(A_b/A_Σ) .





§ SOLVING THE COMPLETE U-SPIN SYSTEM  


We discuss how to extract the U-spin parameters of Eqs. (<ref>)–(<ref>) from the observables.
We are mainly interested in the ratios of parameters and less in their
absolute sizes and 
therefore we consider only quantities normalized on t_0, that is

    t̃_1 ≡t_1/t_0 ,     t̃_2 ≡t_2/t_0 ,      s̃_1 ≡s_1/t_0 ,     p̃_0 ≡p_0/t_0 ,    p̃_1 ≡p_1/t_0 .

We choose, without loss of generality, the tree amplitude t_0 to be real.
The relative phase between 𝒜(Kπ) and 𝒜(π K) is physical and can be extracted in 
experimental measurements. However, the relative phases between 𝒜(ππ), 𝒜(KK) and 
𝒜(Kπ) are unphysical, i.e. not observable on principal grounds. 
This corresponds to two additional phase choices that can be made in the
U-spin parametrization. 
Consequently, without loss of generality, we can also choose the two parameters s̃_1 and t̃_2 to be real. 
Altogether, that makes eight real parameters, that we want to extract, not counting the normalization t_0. 
Of these, four parameters are in the CKM-leading part of the amplitudes and four in the CKM-suppressed one.
In the CP limit Imλ_b → 0 we can absorb p̃_0 and p̃_1 into 
t̃_2 and s̃_1 respectively, which makes four real parameters in that limit.

The eight parameters can be extracted from eight observables that can be used to completely determine them.
Additional observables can then be used in order to overconstrain the system. 
We divide the eight observables that we use to determine the system into four categories:

(i) Branching ratio measurements (3 observables) <cit.>. 
They are used to calculate the squared matrix elements.  We neglect the tiny effects of
order |λ_b/Σ| and we get 

    | A_Σ(KK)|^2    = 
    ℬ( D^0→ K^+K^-)) / |Σ|^2 𝒫(D^0,K^+,K^-)  , 
    
    | A_Σ( ππ)|^2    = 
    ℬ( D^0→π^+π^-) / |Σ|^2 𝒫(D^0,π^+,π^-)  , 
    
    | A( Kπ)|^2    = 
    ℬ( D^0→ K^+π^-) / |V_cs V_ud^*|^2 𝒫(D^0, K^+, π^-)  , 
    
    | A(π K)|^2    = 
    ℬ( D^0→ K^-π^+) / |V_cd V_us^*|^2 𝒫(D^0,K^-,π^+ )    .

We consider three ratios of combinations of the four branching ratios, which are 

    R_Kπ   ≡| A(Kπ)|^2 - 
    | A(π K)|^2
    /| A(Kπ) |^2 + 
    | A(π K) |^2
     , 
    
    R_KK,ππ   ≡| A(KK)|^2 - 
    | A(ππ)|^2
    /| A(KK)|^2 + 
    | A(ππ)|^2
     ,  
    
    R_KK,ππ,Kπ   ≡| A( KK )|^2
    + | A( ππ)  |^2
    - | A( Kπ)  |^2
    - | A( π K) |^2
    /| A( KK) |^2
    + | A( ππ)  |^2
    + | A( K π)  |^2
    + | A( π K) |^2
     .


(ii) Strong phase which does not require CP violation (1 observable). The relative strong phase between CF and DCS decay modes

    δ_Kπ   ≡arg( 𝒜(D^0→ K^-π^+)/𝒜(D^0→ K^-π^+))
      = arg(𝒜(D^0→ K^+π^-)/𝒜(D^0→ K^-π^+))

can be obtained from time-dependent 
measurements <cit.> 
or correlated D^0 D^0 decays <cit.> 
at a charm-τ factory.  

(iii) Integrated direct CP asymmetries (2 observables). In
particular we use <cit.>

    Δ a_CP^dir   ≡ a_CP^dir(D^0→ K^+K^-) - a_CP^dir(D^0→π^+π^-) , 
    
    Σ a_CP^dir   ≡ a_CP^dir(D^0→ K^+K^-) + a_CP^dir(D^0→π^+π^-) .


(iv) Strong phases that require CP violation (2 observables) 
<cit.>. 
These are the relative phases of the amplitudes of a D^0 and D^0
going into one of the CP eigenstates. They are proportional to CPV
effects and thus very hard to extract. In particular,

    δ_KK   ≡arg(𝒜(D^0→ K^+K^-)/𝒜(D^0→ K^+K^-)) ,     
    δ_ππ≡arg(𝒜(D^0→π^+π^-)/𝒜(D^0→π^+π^-))  .

These can be obtained from time-dependent measurements or measurements
of correlated D^0D^0 pairs. 

In principle, using the above observables the system Eqs. (<ref>)–(<ref>) 
is exactly solvable as long as the data is very precise. 
In the CP limit the branching ratio measurements (i) and the strong phase (ii) are sufficient to determine
t̃_1, t̃_2 and s̃_1, which are the complete set of independent parameters in this limit. 

For our parameter extraction with current data, we expand the observables to first nonvanishing 
order in the U-spin expansion. We measure the power counting of that
expansion with a generic parameter ε, which, for nominal U-spin
breaking effects is expected to be ε∼ 25%.
All of the explicit results that we give below have the nice feature that the parameters can be extracted from 
them up to relative corrections of order
𝒪(ε^2). Below it is understood that we neglect
all effects of that order.

In terms of our parameters the ratios of branching ratios are given as

    R_Kπ   = - Re(t̃_1)  , 
    
    R_KK,ππ   = - 2 s̃_1 , 
    
    R_KK,ππ,Kπ   =  1/2( s̃_1^2 - 1/4|t̃_1|^2 +
    t̃_2      ) .

By inserting the expressions for R_Kπ and R_KK,ππ into Eq. (<ref>) we can solve the 
above equations for the independent parameter combinations. The result
up to 𝒪(ε^2) is

    Re( t̃_1 )    = - R_Kπ , 
    s̃_1    = -1/2 R_KK,ππ , 
    
    -1/4(Im t̃_1)^2 
    + t̃_2     =  2 R_KK,ππ,Kπ - 1/4R_KK,ππ^2 + 1/4 R_Kπ^2   .


We are then able to determine t̃_1 with Eq. (<ref>) and the strong phase between the CF and 
DCS mode, see also Ref. <cit.>,

    δ_Kπ   = arg(- 1-1/2t̃_1 /1+1/2t̃_1 )
           = -Im (t̃_1) ,

where in the last step we neglect terms of relative order of
ε^2.

After that we can  determine s̃_1 and t̃_2 from Eqs. (<ref>) and (<ref>), respectively.   
The sum and difference of the integrated direct CP asymmetries can be used together 
with the phases δ_KK and δ_ππ to  determine p̃_0 and p̃_1.
We have 

    Δ a_CP^dir   = Im(λ_b/Σ) ×  
    4 Im(p̃_0 ) 
    
     ,

and

    Σ a_CP^dir =
      2 Im(λ_b/Σ) 
    ×[ 
    2 Im(p̃_0 ) s̃_1 +
         Im(p̃_1) ]  .

Note that also Δ a_CP^dir and Σ a_CP^dir share the feature of 
corrections entering only at the relative order 𝒪(ε^2) compared to the leading result.
The measurement of Δ a_CP^dir is basically a direct measurement of Im p̃_0, 

    Im p̃_0    =  1/4 Im(λ_b/Σ)Δ a_CP^dir .


The phases δ_KK and δ_ππ give (see e.g. Ref. <cit.>)

    Re(A_b(D^0→ K^+K^-)/A_Σ(D^0→ K^+K^-)) - Re(A_b(D^0→π^+π^-)/A_Σ(D^0→π^+π^-))     =
    4 Re(p̃_0)  ,

and

    Re(A_b(D^0→ K^+K^-)/A_Σ(D^0→ K^+K^-)) + Re(A_b(D^0→π^+π^-)/A_Σ(D^0→π^+π^-))    =
    2 Re(2 p̃_0 s̃_1 + p̃_1 ) 
       = 2 [ 2 Re(p̃_0) s̃_1 + Re(p̃_1)]  .

As s̃_1 is already in principle determined from the other observables, this gives us then the full information on p̃_0 and p̃_1.

As the observables δ_KK and δ_ππ are the hardest
to measure, we are not providing here the explicit 
relation of Eq. (<ref>) and Eq. (<ref>) to these observables, acknowledging just that the
corresponding parameter combinations can be determined from these in a straight forward way. 

Taking everything into account, we conclude that the above system of eight
observables for eight parameters can completely be solved. This is done
where the values of the CKM elements are used as inputs.
We emphasize that in principle with correlated double-tag measurements at a future charm-tau factory <cit.>
we could even overconstrain the system.




§ NUMERICAL RESULTS  


We use the formalism introduced in Sec. <ref> now with the currently available measurements.
As not all of the observables have yet been measured, we cannot
determine all of the U-spin parameters. Yet, we use the ones that we do
have data on to get useful information on some of them.



  * Using Gaussian error propagation without taking into account correlations, from the branching ratio 
measurements <cit.>

    ℬℛ(D^0→ K^+K^-)        = (3.97± 0.07) · 10^-3 , 
    
    ℬℛ(D^0→π^+π^-)    = (1.407± 0.025)· 10^-3 , 
    
    ℬℛ(D^0→ K^+π^- )     = (1.366 ± 0.028) · 10^-4 , 
    
    ℬℛ(D^0→ K^- π^+ )     = (3.89± 0.04)· 10^-2 ,

we obtain the normalized combinations

    R_Kπ   = -0.11 ± 0.01 ,   
    
    R_KK,ππ   = 0.534 ± 0.009 ,  
    
    R_KK,ππ,Kπ   = 0.071 ± 0.009 .


  * The strong phase between DCS and CF mode for the scenario of no CP violation in the DCS mode is <cit.>

    δ_Kπ   = (8.6^+9.1_-9.7) ^∘ .


  * The world average of Δ a_CP^dir is given in Eq. (<ref>).

  * The sum of CP asymmetries Σ a_CP^dir in which CP violation has not yet been observed. 
In order to get an estimate we use the HFLAV averages for the single measurements of the 
CP asymmetries <cit.> 

    A_CP(D^0→π^+π^-)    =  0.0000 ± 0.0015 , 
    
    A_CP(D^0→ K^+K^-)        = -0.0016 ± 0.0012 ,

and subtract the contribution from indirect charm CP violation a_CP^ind = (0.028 ± 0.026)% <cit.>.
We obtain 

    Σ a_CP^dir   = A_CP(D^0→ K^+K^-) +  A_CP(D^0→π^+π^-) - 2  a_CP^ind
       = -0.002± 0.002 ,

where we do not take into account correlations, which may be sizable.

  * The phases δ_KK and δ_ππ have not yet been measured, and we cannot get
any indirect information about them.


From Eqs. (<ref>)–(<ref>) it follows that 

    Re( t̃_1 )    = 0.109 ± 0.011 , 
    
    Im( t̃_1 )    = -0.15^+0.16_-0.17 , 
    
    s̃_1    = -0.2668 ± 0.0045 ,  
    
    -1/4(Imt̃_1)^2 + Re(t̃_2)     = 
    			0.075±  0.018	  .

Employing <cit.> 

    Im(λ_b/Σ) = (-6.3± 0.3)· 10^-4 ,

and inserting the measurement of Δ a_CP^dir into
Eq. (<ref>), we obtain

    Im p̃_0    = 0.65 ± 0.12  .

Using Σ a_CP^dir we get

    2 Im(p̃_0 ) s̃_1 + Im(p̃_1)    =  1.7± 1.6 .


Few remarks are in order regarding the numerical values we obtained.


  * Among the five parameters defined in Eq. (<ref>),
p̃_1 is the least constrained parameter as we have basically no information
about it.
In order to learn more about it we need measurements of Σ a_CP^dir 
as well as of the phases δ_KK and δ_ππ. 

  * The higher order U-spin breaking parameters are
consistently smaller than the first order ones, and the second order ones
are even smaller. This is what we expect assuming the U-spin expansion.

  * Eqs. (<ref>)–(<ref>)
suggest that the SU(3)_F breaking of the tree amplitude t̃_1 is smaller than the broken penguin contained in 
s̃_1. 

  * Using Eqs. (<ref>)–(<ref>) 
we can get a rough estimate for the 𝒪(ε^2) corrections
that enter the expression for Δ a_CP^dir in Eq. (<ref>).
The results on the broken penguin suggest that these corrections do not exceed a level of ∼ 10%. 
We cannot, however, determine these corrections completely without further knowledge on p̃_1. 





§ THE Δ U=0 RULE  


We now turn to discuss the implications of Eq. (<ref>).
We rewrite
Eq. (<ref>) as

    Δ a_CP^dir   = 
    4 Im(λ_b/Σ) |p̃_0 | sin( δ_strong) ,

with the unknown strong phase 

    δ_strong   = arg(p̃_0) .

Then the numerical result in Eq. (<ref>) reads 

    |p̃_0| sin( δ_strong)    = 0.65 ± 0.12 .

Recall that in the group theoretical language the parameters t_0 and p_0 are 
the matrix elements of the Δ U=1 and Δ U=0 operators, respectively <cit.>.
For the ratio of the matrix elements of these operators we employ now the following parametrization

    p̃_0 = B + C e^i δ ,

such that B is the short-distance (SD) ratio and the second term arises
from long-distance (LD) effects. While the separation between SD and LD
is not well-defined, what we
have in mind here is that diagrams with a b quark in the loop are perturbative
and those with quarks lighter than the charm are not.

In Eq. (<ref>) of Sec. <ref> below we apply the same decomposition into a no QCD part and corrections to that also to the Δ I=1/2 rules in K, D and B decays to pions. It is instructive to compare all of these systems in the same language.

We first argue that in Eq. (<ref>) to a very good approximation B=1. This is
basically the statement that perturbatively, the diagrams with
intermediate b are tiny. More explicitly, in that case, that is when
we neglect the SD b penguins, we have 

    Q^Δ U=1≡Q^s̅s - Q^d̅d/2 ,     
    Q^Δ U=0≡Q^s̅s + Q^d̅d/2 .

Setting C=0 then corresponds to the statement that
only Q^s̅s can produce K^+K^- and only Q^d̅d 
can produce π^+π^-. This implies that for C=0

    ⟨K^+K^-| Q^d̅d|D^0⟩   =  
    ⟨π^+π^-| Q^s̅s|D^0⟩ = 0 ,

and 

    ⟨K^+K^-| Q^s̅s|D^0⟩≠ 0 ,     ⟨π^+π^-| Q^d̅d|D^0⟩≠ 0 .

We then see that B=1 since

    ⟨K^+K^-| Q^Δ U=0|D^0⟩/⟨K^+K^-| Q^Δ U=1|D^0⟩   = 1 ,      
    ⟨π^+π^-| Q^Δ U=0|D^0⟩/⟨π^+π^-| Q^Δ U=1|D^0⟩  = -1 .

We note that in the SU(3)_F limit we also have

    ⟨K^+K^-| Q^Δ U=1|D^0⟩   = - ⟨π^+π^-| Q^Δ U=1|D^0⟩ , 
    ⟨K^+K^-| Q^Δ U=0|D^0⟩   = ⟨π^+π^-| Q^Δ U=0|D^0⟩ ,

but this is not used to argue that B=1.

We then argue that δ∼𝒪(1). The reason is that
non-perturbative effects involve on-shell particles, or in other words,
rescattering, and such effects give rise to large strong phases to the
LD effects
independent of the magnitude of the LD amplitude.

In the case that B=1, δ∼𝒪(1) and using the fact that the CKM ratios are
small we conclude that the CP asymmetry is roughly given by the CKM
factor times C

    Δ a_CP^dir = 4 Im(λ_b/Σ) × C ×sinδ .

Now the question is: what is C? 
As at this time no method is available in order to calculate C with a well-defined theoretical uncertainty, 
we do not employ here a dynamical calculation in order to provide a SM prediction for C and Δ a_CP^dir.
We rather show the different principal possibilities and how to interpret them in view of the current data.
In order to do so we measure the order of magnitude of the QCD correction term C relative to the no QCD limit 
p̃_0=1. Relative to that limit, we differentiate between three cases


  * C = 𝒪(α_s/π): Perturbative corrections  to
  p̃_0.

  * C = 𝒪(1): Non-perturbative corrections that produce
  strong phases from rescattering but do not significantly change the
  magnitude of p̃_0.

  * C ≫𝒪(1): Large non-perturbative effects with
  significant magnitude changes and strong phases from rescattering to
  p̃_0.

Note that category (2) and (3) are in principle not different, as they both include non-perturbative effects, which 
differ only in their size.

Some perturbative results concluded that C=𝒪(α_s/π), leading to 
Δ a_CP^dir∼ 10^-4 <cit.>.
Note that the value  Δ a_CP^dir = 1× 10^-4,
assuming O(1) strong phase, 
would correspond numerically to
C ∼ 0.04. We conclude that if there is a good argument that C is of category (1), the measurement of Δ a_CP^dir would be a sign of beyond the SM (BSM) physics, because it would indicate a relative 𝒪(10) enhancement. 

If the value of Δ a_CP^dir would have turned out as large as 
suggested by the central value of some (statistically unsignificant) earlier measurements <cit.>,
we would clearly need category (3) in order to explain that, i.e. penguin diagrams that are 
enhanced in magnitude, see e.g. Refs. <cit.>. Another example for category (3) is the Δ I=1/2 rule in the kaon sector 
which is further discussed in sections <ref> and <ref>. 

The current data, Eq. (<ref>), is 
consistent with category (2).
In the SM picture, the measurement of Δ a_CP^dir proves the non-perturbative nature of the Δ U=0 matrix 
elements with a mild enhancement from 𝒪(1) rescattering effects. This is the Δ U=0 rule for charm. 

Note that the predictions for Δ a_CP^dir of category (i) and (ii) differ by 𝒪(10), although 
category (ii) contains only an 𝒪(1) nonperturbative enhancement with respect to the no QCD limit 
p̃_0=1. We emphasize that a measure for a QCD enhancement is not necessarily its impact on an observable, but the amplitude level comparison with the absence of QCD effects. 

We also mention that we do not need SU(3)_F breaking effects to
explain the data.
Yet, the observation of 
|s̃_1| > |t̃_1| in 
Eqs. (<ref>)–(<ref>)
provide additional supporting evidence that rescattering is
significant. 
Though no proof of the Δ U=0 rule on its own, 
this matches its upshot and is indicative of the importance of rescattering effects also 
in the broken penguin which is contained in s̃_1.

With future data on the phases δ_KK and δ_ππ we will be 
able to determine the strong phase δ of Eq. (<ref>). In that way it will be possible to completely determine 
the characteristics of the emerging Δ U=0 rule.




§ Δ I=1/2 RULES IN K, D AND B DECAYS 


It is instructive to compare the Δ U=0 rule in charm with the
Δ I=1/2 rule in kaon physics, and furthermore also to the corresponding ratios of isospin matrix elements 
of D and B decays.
For a review of the Δ I=1/2 rule see e.g. Ref. <cit.>.

In kaon physics we consider K →ππ decays. Employing an isospin parametrization we have <cit.>

    (K^+→π^+π^0)    = 3/2 A_2^K e^iδ_2^K , 
    
    (K^0→π^+π^-)    = A_0^K e^iδ_0^K + √(1/2) A_2^K e^iδ_2^K , 
    
    (K^0→π^0π^0)    = A_0^K e^iδ_0^K - √(2) A_2^K e^iδ_2^K .

Note that the strong phases of A_0^K and A_2^K are factored out, so that A_0,2^K contain weak phases only.
The data give

    |A_0^K/A_2^K| ≈ 22.35 ,    δ_0^K - δ_2^K = (47.5± 0.9)^∘ ,

see Ref. <cit.> and references therein for more
details. A_0,2^K have a small imaginary part stemming from the CKM matrix elements only. 
To a very good approximation the real parts Re(A_0^K) and Re(A_2^K) in the Δ I=1/2 rule 
depend only on the tree operators <cit.>

    Q_1    = (s̅_α u_β)_V-A (u̅_β d_α)_V-A ,      
    Q_2  = (s̅ u)_V-A (u̅ d)_V-A .

The lattice results Refs. <cit.> show an emerging physical interpretation of the Δ I=1/2 rule, that is an approximate cancellation of two contributions in Re(A_2^K), which does not
take place in Re(A_0^K). These two contributions are different color contractions of the same operator.

The isospin decompositions of D→ππ and B→ππ are completely analog to Eq. (<ref>).
To differentiate the charm and beauty isospin decompositions from the kaon one, we put the corresponding superscripts
to the respective analog matrix elements. Leaving away the superscripts indicates generic formulas that are valid for all three 
meson systems. 

In order to understand better the anatomy of the Δ I=1/2 rule we use again the form

    A_0/A_2   = B + C e^iδ ,

analogously to Eq. (<ref>) in Sec. <ref> for the Δ U=0 rule. 
Here, B is again the contribution in the limit of no QCD, and C e^iδ contains the 
corrections to that limit.
Now, as discussed in Refs. <cit.>, in the limit of no strong interactions
only the Q_2 operator contributes in Eq. (<ref>). 
Note that the operator Q_1 is only generated from QCD corrections.
When we switch off QCD, the amplitude into neutral pions vanishes and we have for K,D,B→ππ 
equally <cit.>

    B    = √(2) .

This corresponds to the limit p̃_0 = 1 that we considered in Sec. <ref> for the Δ U=0 rule.
The exact numerical value in Eq. (<ref>) of course depends on the convention used for 
the normalization of A_0,2 in the isospin decomposition Eq. (<ref>), where 
we use the one present in the literature.

For the isospin decomposition of D^+→π^+π^0, D^0→π^+π^- and 
D^0→π^0π^0, we simply combine the fit of Ref. <cit.> to get 

    |A_0^D/A_2^D|    = 2.47± 0.07 ,     δ_0^D - δ_2^D = (± 93 ± 3)^∘ .

Reproducing the Δ I=1/2 rule for charm Eq. (<ref>) is an optimal future testing ground for 
emerging new interesting non-perturbative methods <cit.>. 
Very promising steps on a conceptual level are also taken by lattice QCD <cit.>.

In K and D decays the contributions of penguin operators to A_0 is CKM-suppressed, i.e. to a good 
approximation A_0 is generated from tree operators only.
In B decays the situation is more involved because there is no relative hierarchy between the relevant CKM matrix elements.
However, one can separate tree and penguin contributions by including the measurements of CP asymmetries within a global fit,
as done in Ref. <cit.>. From Fig. 3 therein we find for the ratio of matrix elements of tree operators that 

    | A_0^B/A_2^B| ∼√(2)
 
is well compatible with the data, the best fit point having | A_0^B / A_2^B| = 1.5.
The fit result for the phase difference δ_0^B - δ_2^B is not given in Ref. <cit.>.

The emerging picture is: 
The Δ I=1/2 rule in B decays is compatible or close to the no QCD limit.
The Δ I=1/2 rule in kaon physics clearly belongs to category (3) of Sec. <ref>.
Here, the non-perturbative rescattering affects not only the phases but also the magnitudes of the corresponding matrix elements.
Finally, the Δ I=1/2 rule in charm decays is intermediate and shows an 𝒪(1) enhancement, 
similar to the Δ U=0 rule that we found in Sec. <ref>.

We can understand these differences from the different mass scales that govern K, D and B decays. 
Rescattering effects are most important in K decays, less important but still significant in D decays, 
and small in B decays.




§ PHENOMENOLOGY OF THE Δ U=0 VS. Δ I=1/2 RULE 


An interesting difference between the Δ I=1/2 rule in kaon decays and the 
Δ U=0 rule in charm decays is their effect on the phenomenology. 
Large rescattering enhances the CP violation
effects in D decays, but it reduces the effect in kaon decays. 

The reason for the difference lies in the fact that in kaon decays the
SD decay generates only a u u̅ final state, while in charm decays
it generates to a very good approximation the same amount of d d̅
and s s̅ states.

We write the amplitudes very generally and up to a normalization factor as

    A = 1 + r a e^i(ϕ+δ) ,

such that r is real and depends on CKM matrix elements, a is real
and corresponds to the ratio of the respective
hadronic matrix elements, ϕ is a weak phase and
δ is a strong phase. For kaons a is the ratio of matrix elements of the operators 
Q^Δ I=1/2 over Q^Δ I=3/2, while for charm it is the ratio of matrix elements of the operators  
Q^Δ U=0 over Q^Δ U=1. 

We first consider the case where we neglect the third
generation. In that limit
for kaons we have the decomposition

    A_K = V_usV_ud^* (A_1/2 + r_CG A_3/2) ,

where r_CG is the CG coefficient that can be read from Eq. (<ref>).
For charm we have

    A_D = V_csV_us^* A_1.

That means that in the two-generational limit for kaons we have r=1 
and in charm r=0. If we switch on the third generation we get small corrections to these
values in each case: r≪ 1 for charm and |r-1|≪ 1 for
kaons. These effects come from the non-unitarity of the 2 × 2
CKM. For the kaon case   
there is an extra effect that stems from SD penguins that
come with V_tsV_td^*. In both cases we have δ∼𝒪(1) from non-perturbative rescattering, as
well as ϕ∼𝒪(1).

The general formula for direct CP asymmetry is given as <cit.>

    A_CP   = -2 r a  sin(δ) sin(ϕ)
    / 
    1 + (ra)^2+ 2 ra cos(δ) cos(ϕ)
    ≈
    2 r a  sin(δ) sin(ϕ)   ,
    
    2 (r a)^-1sin(δ) sin(ϕ)   .

Non-perturbative effects enhance a in both kaon and charm decays.
This means the effect which is visible in the CP asymmetry is different depending on the value of r.
For r a≪ 1 increasing a results in enhancement of the CP
asymmetry, while for r a ≫ 1 it is suppressed. 
These two cases correspond to the charm and kaon cases,
respectively. It follows that the Δ I=1/2 rule in kaons reduces
CP violating effects, while the Δ U=0 rule in charm enhances them.




§ CONCLUSIONS 


From the recent determination of Δ a_CP^dir we derive the ratio of Δ U=0 over Δ U=1 
amplitudes as  

    |p̃_0|sin(δ_strong)    = 0.65 ± 0.12 .

In principle two options are possible in order to explain this result: In the perturbative 
picture beyond the SM (BSM) physics is necessary to explain Eq. (<ref>). 
On the other hand, in the SM picture, we find that all that is
required in order to explain the result is a 
mild non-perturbative enhancement due to rescattering
effects. Therefore, it is hard to argue that BSM physics is required. 

Our interpretation of the result is that the measurement of Δ a_CP^dir 
provides a proof for the Δ U=0 rule in charm.
The enhancement of the  Δ U=0 amplitude is not as significant
as the one present in the Δ I=1/2 rule for kaons. 
In the future, with more information on the strong phase of p̃_0 from time-dependent measurements 
or measurements of correlated D^0D^0 decays, we will be able to completely
determine the extent of the Δ U=0 rule. 


Interpreting the result within the SM implies that we expect a moderate
non-perturbative effect and nominal SU(3)_F breaking. The former fact
implies that we expect U-spin invariant strong phases to be
𝒪(1). The latter implies that we anticipate the yet to be
determined  SU(3)_F breaking effects not to be large. Thus, there are
two qualitative predictions we can make

    δ_strong∼𝒪(1),     
    a_CP^dir (D^0→ K^+K^-) ≈ -a_CP^dir(D^0→π^+π^-) .

Verifying these predictions will make the SM interpretation of the
data more solid.



We thank Alex Kagan, Yossi Nir, Giovanni Punzi and Alan Schwartz for useful discussions.
The work of YG is supported in part by the NSF grant PHY1316222.
SS is supported by a DFG Forschungs­stipendium under contract no. SCHA 2125/1-1.




apsrev4-1

