Kaon semileptonic vector form factor with $N_f=2+1+1$ Twisted Mass fermions

We investigate the vector form factor relevant for the $K_{\ell 3}$ semileptonic decay using maximally twisted-mass fermions with 4 dynamical flavours ($N_f=2+1+1$). Our simulations feature pion masses ranging from $210$ MeV to approximately $450$ MeV and lattice spacing values as small as $0.06$fm. Our main result for the vector form factor at zero 4-momentum transfer is $f_+(0)=0.9683(65)$ where the uncertainty is both statistical and systematic. By combining our result with the experimental value of $f_+(0)|V_{us}|$ we obtain $|V_{us}|=0.2234(16)$, which satisfies the unitarity constraint of the Standard Model at the permille level.

e quark underlying process is q → q lν l and the quark q participate (see fig. 1.4).
mileptonic decay prototype -Feynman diagram which contribute to decay process M → M lν l . The hadronic part (bottom) must be evaluated perturbative methods.
he situation is much more complicated, with respect to the leptonic the composition of the final state (leptons plus an hadron); writing Extracting f+(0) from Lattice QCD allows us to estimate |V us | This section summarises state-of-the art lattice calculations of th decay constants and kaon the semileptonic decay form factor and pr of the Standard Model. With respect to the previous edition of F section has been updated, correlations of lattice data are now tak analysis and a sub-section on the individual decay constants f K the ratio) has been included. Furthermore, when combining lattic results we now take into account the strong SU(2) isospin correct theory for the ratio of leptonic decay constants f K /f π .

Experimental information concerning |V ud |, |V us |,
The following review relies on the fact that precision experimental accurately determine the product |V us |f + (0) and the ratio |V us /V ud |V us |f + (0) = 0.2163(5) , V us V ud f K ± f π ± = 0.27 Here and in the following f K ± and f π ± are the isospin broken decay QCD (the electromagnetic effects have already been subtracted in using chiral perturbation theory). We will refer to the decay const symmetric limit as f K and f π . |V ud | and |V us | are elements of Maskawa matrix and f + (t) represents one of the form factors rele decay K 0 → π − ν, which depends on the momentum transfer t   Range of the simulated pion masses f 0 (MeV) 120.8(1) 120.9(1) 120.9(1) 121.1(2)      Range of the simulated pion masses f 0 (MeV) 120.8(1) 120.9(1) 120.9(1) 121.1(2)      Range of the simulated pion masses f 0 (MeV) 120.8(1) 120.9(1) 120.9(1) 121.1(2)

Simulation Details
The valence light quark mass is put equal to the sea quark mass Details of the ensembles used in this N f =2+1+1 analysis   General strategy ✦ we used the ratio extract the matrix element from appropriate ratio of three-points correlation function to build f 0 (q 2 ) and f + (q 2 ) ✦ z expansion ✦ Polynomial fit Fit simultaneously f 0 (q 2 ) and f + (q 2 ) to get f 0 (0)=f + (0) Perform the Chiral and continuum extrapolation of f(0) We define the ratio: The matrix element of the vector current between two PS mesons decomposes into two form factors depending on the momentum transfer The matrix element can be derived in lattice QCD from a combination of Euclidean three-point functions in which the renormalization Z V and Z K and Z π cancels giovedì 26 giugno 2014 General strategy We can define V 0 and V 1 related to the form factors by the relations The matrix element of the vector current between two PS mesons decomposes into two form factors   Table 3. Input values for the renormalization constant Z MS P (2 GeV), corresponding to the methods M1 and the chirally extrapolated values of r 0 /a for each value of β (see text).
Since the renormalization constants Z P and the values of r 0 /a have been evalua of gauge configurations, their uncertainties have been taken into account in the fitting we generated randomly a set of values of (r 0 /a) i and (Z P ) i for the bootstrap event i as corresponding to the central values and the standard deviations given in Table 3. Then the χ 2 the following contribution where (r 0 /a) f it i and (Z P ) f it i are free parameters of the fitting procedure for the bootstra allows the quantities r 0 /a and Z P to slightly change from their central values (in the weight in the χ 2 given by their uncertainties. This procedure corresponds to impose a g Before closing this section we have collected in Table 4 the time intervals (conserva tion of the PS meson masses (and of the pion decay constant) from the 2-point correlato in the light, strange and charm sectors.

Extracting f(0)
We fitted simultaneously f + (q 2 ) and f 0 (q 2 ) to extract f(0) using the z expansion * to interpolate at q 2 =0 we neglect the points corresponding to large negative q 2

Extracting f(0)
We fitted simultaneously f + (q 2 ) and f 0 (q 2 ) to extract f(0) using the z expansion

Extracting f(0)
We fitted simultaneously f + (q 2 ) and f 0 (q 2 ) to extract f(0) using the z expansion The results obtained with polynomial fit formulae are also compatible Ensemble A.60.24 aµ s =0.0225 f(0)=0.9880(7) polar fit in q 2 f(0)=0.9877(7) polynomial fit in q 2 f 0 (q 2 ) = f (0) 1 + P 3 q 2 + P 4 q 4 (16) f (0) con la formula SU (2) to interpolate at q 2 =0 we neglect the points corresponding to large negative q 2 giovedì 26  Two different approaches for the chiral extrapolation l e a 2 di f (0) con la formula "SU(3)" Fit in m l e a 2 di f (0) con la formula SU(2) Fit in m l e a 2 di f (0) con la formula "SU(3)" m l e a 2 di f (0) con la formula SU(2) m l e a 2 di f (0) con la formula "SU(3)" Fit in m l e a 2 di f (0) con la formula SU(2) Fit in m l e a 2 di f (0) con la formula "SU(3)"  Fig. 3.2, but for the decay constant r 0 f π . Scale setting done with f π + = 130.41 MeV  we performed a small interpolation in the data to arrive at m s phys f + (q ) = f (0) 1 + P 1 q + P 2 q f 0 (q 2 ) = f (0) 1 + P 3 q 2 + P 4 q 4 Fit in m l e a 2 di f (0) con la formula SU(2) Fit in m l e a 2 di f (0) con la formula "SU(3)" tabelle 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 r 0 m l R 10 11 Figure 3.2: Chiral and continuum extrapolation of r 0 M 2 π /m l based on the NLO ChPT fit given by Eq. (3.12). Lattice data have been corrected for FSE using the CWW approach. The dependence of our lattice data for r 0 M 2 π /m l and r 0 f π on the renormalized quark mass r 0 m l is shown at each lattice spacing in Figs. 3.2 and 3.3, respectively. The behavior of the chiral extrapolation in the continuum limit is also reported. From Figs. 3.2 and 3.3 it can be seen that the impact of discretization effects using the values of r 0 /a is almost completely negligible in the case of r 0 f π , while it is at the level of 10% in the case of r 0 M 2 π /m l (if estimated from the difference between the 0,03 0,06 0,09 0,12 m l r 0 6 "=2.10 V=48 x96 Continuum limit 0,04 0,08 0,12 m l r 0 0,25 "=2.10 V=48 x96

Continuum limit
Scale setting done with f π + = 130.41 MeV (M π r 0 ) 2 = 2(B 0 r 0 )(m l r 0 ) 1 + ξ l log(ξ l ) + P 3 ξ l + P 4  1 − 3 4 ξ log ξ + P 2 ξ + P 3 a 2 (17 Fit in m l e a 2 di f (0) con la formula "SU(3)" tabelle M 2 K , M 2 π and f π appearing in f 2 are expressed at LO We also tried the same fit using f π instead of f 0 in the definition of f 2 obtaining cosistent results f + (0)=0.9734 (40) giovedì 26 giugno 2014 f + (0) -results and sistematics The results from our analysis: in particular: stat+fit is referred to both the statistical uncertainties (including the total error on the light and strange quark mass determination) and the uncertainties due to the fitting procedure Chiral extrapolation systematic uncertainties have been evaluated comparing the results obtained from two different fit formulae i.e. SU(2) ChPT and SU (3) om |V ub | is known much better than needed in the present context: 88]. In the following, we first discuss the evidence for the validity of y then use it to analyze the lattice data within the Standard Model. lation between |V ud | and |V us | imposed by the unitarity of the CKM dotted arc (more precisely, in view of the uncertainty in |V ub |, the to a band of finite width, but the effect is too small to be seen here).
data for N f = 2 + 1 are in good agreement with this constraint. e for the sum of the squares of the first row of the CKM matrix reads tandard Model thus passes a nontrivial test that exclusively involves ablished kaon decay branching ratios. Combining the lattice results in (38) and (40) with the β-decay value of |V ud | quoted in (33), the ly: the lattice result for f + (0) leads to |V u | 2 = 0.9992(6), while the |V u | 2 = 1.0000(6), thus confirming CKM unitarity at the permille is for N f = 2, we find |V u | 2 = 1.029(35) with the lattice data alone. t larger than 1, in accordance with the fact that the dotted curve This matrix connects the interaction eigenstates (d , s , b (d, s, b): The mass eigenstates are different from the interaction o interactions mix different flavours with weight V ij in th is worthwhile to underline that within the Standard Mo mechanism is represented by this matrix and that there a where s ij = sin θ ij , c ij = cos θ ij (and θ 12 is t What is important to emphasize is that if δ has no CP violation in the quark sector to Another interesting feature of the matr in the Standard Model the number of qu the old Cabibbo version of the theory, wh pect to the previous edition of FLAG [1] the data in this tions of lattice data are now taken into account in all the individual decay constants f K and f π (rather than only thermore, when combining lattice data with experimental the strong SU(2) isospin correction in chiral perturbation cay constants f K /f π .
(1) f π ± are the isospin broken decay constants, respectively, in have already been subtracted in the experimental analysis We will refer to the decay constants in the SU(2)-isospin |V ud | and |V us | are elements of the Cabibbo-Kobayashients one of the form factors relevant for the semileptonic ds on the momentum transfer t between the two mesons.
Experimental input (1) from β-decay (3) y constants and kaon the semileptonic decay form factor and provides an analysis in vi e Standard Model. With respect to the previous edition of FLAG [1] the data in t on has been updated, correlations of lattice data are now taken into account in all t ysis and a sub-section on the individual decay constants f K and f π (rather than o atio) has been included. Furthermore, when combining lattice data with experimen ts we now take into account the strong SU(2) isospin correction in chiral perturbat ry for the ratio of leptonic decay constants f K /f π .
Experimental information concerning |V ud |, |V us |, f + (0) and f K ± /f π ± following review relies on the fact that precision experimental data on kaon decays v rately determine the product |V us |f + (0) and the ratio |V us /V ud |f K ± /f π ± [2]: and in the following f K ± and f π ± are the isospin broken decay constants, respectively, (the electromagnetic effects have already been subtracted in the experimental analy chiral perturbation theory). We will refer to the decay constants in the SU(2)-isosp etric limit as f K and f π . |V ud | and |V us | are elements of the Cabibbo-Kobayas kawa matrix and f + (t) represents one of the form factors relevant for the semilepto y K 0 → π − ν, which depends on the momentum transfer t between the two meso t matters here is the value at t = 0: f + (0) ≡ f K 0 π − + (t) t→0 . The pion and kaon dec tants are defined by 1  ✦ compare with an estimate of f + (0) coming from the scalar density ✦ A more detailed analysis of the q 2 dependence of the form factor and a comparison with the experimental data 4.83(9) 4.77(9) 4.76(9) 4.69(10) B 0 (MeV) 2548 (99) 2497 (97) 2500(93) 2515(90) f 0 (MeV) 120.8(1) 120.9(1) 120.9(1) 121.1(2)

Ademollo Gatto Theorem
The AG theorem states that in SU(3) limits f + (0)=1 and deviation from unity are of the order of (m s -m l ) 2 We can test AG theorem plotting Δf=f + (0)-1-f 2 divided by (m s -m l ) 2 as a function m l notice the the data as been extrapolated at m s phys