# Decay Constants of Charged Pseudoscalar Mesons

###### Abstract

We review here the physics of purely leptonic decays of , , , , and pseudoscalar mesons. The measured decay rates are related to the product of the relevant weak interaction based CKM matrix element of the constituent quarks and a strong interaction parameter related to the overlap of the quark and anti-quark wave-functions in the meson, called the decay constant . The interplay between theory and experiment is different for each particle. Theoretical predictions that are necessary in the sector can be tested, for example, in the charm sector. One such measurement, that of , differs from the most precise unquenched lattice calculation and may indicate the presence of new intermediate particles, or the theoretical prediction could be misleading. The lighter and mesons provide stringent comparisons due to the accuracy of both the measurements and the theoretical predictions. This review was prepared for the Particle Data Group’s 2008 edition.

^{†}

^{†}preprint: EFI 08-03 SUHEP 08-03

Charged mesons formed from a quark and anti-quark can decay to a charged lepton pair when these objects annihilate via a virtual boson. Fig. 1 illustrates this process for the purely leptonic decay of a meson.

Similar quark-antiquark annihilations via a virtual () to the () final states occur for the , , , and mesons. Let be any of these pseudoscalar mesons. To lowest order, the decay width is

(1) |

Here is the mass, is the mass, is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element between the constituent quarks in , and is the Fermi coupling constant. The parameter is the decay constant, and is related to the wave-function overlap of the quark and antiquark.

The decay starts with a spin-0 meson, and ends up with a left-handed neutrino or right-handed antineutrino. By angular momentum conservation, the must then also be left-handed or right-handed, respectively. In the limit, the decay is forbidden, and can only occur as a result of the finite mass. This helicity suppression is the origin of the dependence of the decay width.

There is a complication in measuring purely leptonic decay rates. The process is not simply a radiative correction, although radiative corrections contribute. The can make a transition to a virtual , emitting a real photon, and the decays into , avoiding helicity suppression. The importance of this amplitude depends on the decaying particle and the detection technique. The rate for a heavy particle such as decaying into a light particle such as a muon can be larger than the width without photon emission Bradcor . On the other hand, for decays into a , the helicity suppression is mostly broken and these effects appear to be small.

Measurements of purely leptonic decay branching fractions and lifetimes allow an experimental determination of the product . If the CKM element is well known from other measurements, then can be well measured. If, on the other hand, the CKM element is less well or poorly measured, having theoretical input on can allow a determination of the CKM element. The importance of measuring depends on the particle being considered. For the system, is crucial for using measurements of - mixing to extract information on the fundamental CKM parameters. Knowledge of is also needed, but this parameter cannot be directly measured as the is neutral, so the violation of the SU(3) relation must be estimated theoretically. This difficulty does not occur for mesons as both the and are charged, allowing the direct measurement of SU(3) breaking and a direct comparison with theory. (In this note mention of specific particle charge also implies the use of the charge-conjugate partner.)

For and decays, the existence of a charged Higgs boson (or any other charged object beyond the Standard Model) would modify the decay rates; however, this would not necessarily be true for the Hou ; Akeroyd . More generally, the ratio of to decays can serve as one probe of lepton universality Hou ; Hewett .

As has been quite accurately measured in super-allowed decays Vud , with a value of 0.97418(26), measurements of yield a value for . Similarly, has been well measured in semileptonic kaon decays, so a value for from can be compared to theoretical calculations. Recently, however, lattice gauge theory calculations have been claimed to be very accurate in determining , and these have been used to predict Jutt .

Next we review current measurements, starting with the charm system. The CLEO collaboration has measured the branching fraction for and recently updated their published result fD . By using the well measured lifetime of 1.040(7) ps and assuming BM , they report

(2) |

This result includes a 1% correction for the radiative final state based on the estimate by Dobrescu and Kronfeld Kron .

Before we compare this result with theoretical predictions, we discuss the . Measurements of have been made by several groups and are listed in Table 1 CLEO-c ; CLEO-CSP ; Belle-munu ; CLEO ; BEAT ; ALEPH ; L3 ; OPAL ; Babar-munu . Early measurements actually determined the ratio of the leptonic decay to some hadronic decay, usually . This introduces a large additional source of error since the denominator is not well known. CLEO CLEO-c has published absolute branching fractions for and , , and in a separate paper CLEO-CSP for , ; there is also an as-yet-unpublished result from Belle Belle-munu for .

Exp. | Mode | (%) | (MeV) | |
---|---|---|---|---|

CLEO-c CLEO-c | ||||

CLEO-c CLEO-c | ||||

CLEO-c CLEO-CSP | ||||

CLEO-c | combined | – | ||

Belle Belle-munu | ||||

Average | ||||

Average with radiative correction | ||||

CLEO CLEO | 3.60.9 | |||

BEATRICE BEAT | 3.60.9 | |||

ALEPH ALEPH | 3.60.9 | |||

ALEPH ALEPH | ||||

L3 L3 | ||||

OPAL OPAL | ||||

BaBar Babar-munu | (6.740.830.260.66) | 4.710.46 |

Model | (MeV) | (MeV) | |
---|---|---|---|

Experiment (our averages) | |||

Lattice (HPQCD+UKQCD) Lat:Foll |
|||

Lattice (FNAL+MILC+HPQCD) Lat:Milc | |||

QL (QCDSF) QCDSF | 2206511 | 2066322 | |

QL (Taiwan) Lat:Taiwan | |||

QL (UKQCD) Lat:UKQCD | |||

QL Lat:Damir | |||

QCD Sum Rules Bordes | |||

QCD Sum Rules Chiral | |||

Field Correlators Field | |||

Isospin Splittings Isospin |

We extract the decay constant from the measured branching ratios using , and a lifetime of 0.50 ps. Our experimental average,

(3) |

uses only those results that are absolutely normalized reason . We note that the experiments do not correct explicitly for any that may have been included. We have included the radiative correction of 1% in the rates Kron (the rates need not be corrected). Other theoretical calculations show that this rate is a factor of 40–100 below the rate for charm theories-rad .

Table 2 compares the experimental with theoretical calculations Lat:Foll ; Lat:Milc ; QCDSF ; Lat:Taiwan ; Lat:UKQCD ; Lat:Damir ; Bordes ; Chiral ; Field ; Isospin . While most theories give values lower than the measurement, the errors are sufficiently large, in most cases, to declare success. A recent unquenched lattice calculation Lat:Foll , however, differs by more than three standard deviations Crit-follana . Remarkably it agrees with and consequently disagrees in the ratio .

Akeroyd and Chen AkeroydC first pointed out that leptonic decay widths are modified by new physics. Specifically, for the and , in the case of the two-Higgs doublet model (2HDM), Eq. (1) is modified by a factor multiplying the right-hand side:

(4) |

where is the charged Higgs mass, is the mass of the meson (containing the light quark ), is the charm quark mass, is the light-quark mass, and is the ratio of the vacuum expectation values of the two Higgs doublets. (Here we modified the original formula to take into account the charm quark coupling KronPC .) For the , , and the change due to the is very small. For the , however, the effect can be substantial. One major concern is that we need to know the value of in the Standard Model (SM). We can take that from a theoretical model. Our most aggressive choice is that of the unquenched lattice calculation Lat:Foll , because they claim the smallest error. Since the charged Higgs would lower the rate compared to the SM, in principle, experiment gives a lower limit on the charged Higgs mass. However, the value for the predicted decay constant using this model is more than 3 standard deviations below the measurement, implying that (a) either the model of Ref. Lat:Foll is not representative; or (b) no value of in the two-Higgs doublet model will satisfy the constraint at 99.9% confidence level; or (c) there is new physics, different from the 2HDM, that interferes constructively with the SM amplitude Rviolating .

Dobrescu and Kronfeld Kron emphasize that the discrepancy between the theoretical lattice calculation and the CLEO data are substantial and “is worth interpreting in terms of new physics.” They give three possible examples of new physics models that might be responsible. These include a specific two-Higgs doublet model and two leptoquark models.

The Belle Belle-taunu and BaBar Babar-taunu collaborations have found evidence for decays. The measurements are

The Belle and BaBar values have 3.5 and 2.6 standard-deviation significances. More data are needed, and the average can only be provisional. Here the effect of a charged Higgs is different as it can either increase or decrease the expected SM branching ratio. The factor is given in terms of the meson mass, , by Hou

(7) |

In principle, we can get a limit in the – plane even with this statistically limited set of data. Again, we need to know the SM prediction of this decay rate. We ascertain this value using Eq. (1). Here theory provides a value of MeV fBl . The subject of the value of is addressed elsewhere HFAG . Taking an average over inclusive and exclusive determinations, and enlarging the error using the PDG prescription because the results differ, we find , where the error is dominantly theoretical. We thus arrive at the SM prediction for the branching fraction of . Taking the ratio of the experimental value to the predicted branching ratio at its 90% c.l. upper limit and using Eq. (7), we find that we can limit GeV. The 90% c.l. lower limit also permits us to exclude the region 4.1 GeV GeV IsiPar .

We now discuss the determination of charged pion and kaon decay constants. The sum of branching fractions for and is 99.98770(4)%. The two modes are difficult to separate experimentally, so we use this sum, with Eq. (1) modified to include photon emission and radiative corrections Marciano-Sirlin . The branching fraction together with the lifetime 26.033(5) ns gives

(8) |

The first error is due to the error on , 0.97418(26) Vud ; the second is due to the higher-order corrections.

Similarly, the sum of branching fractions for and is 63.57(11)%, and the lifetime is 12.3840(193) ns Antonelli . We use a value for obtained from the average of semileptonic kaon decays of 0.21661(46). The must be determined theoretically. We follow Blucher and Marciano BM in using the Leutwyler-Roos calculation , that gives , yielding

(9) |

The first error is due to the error on ; the second is due to the CKM factor , and the third is due to the higher-order corrections. The largest source of error in these corrections depends on the QCD part, which is based on one calculation in the large framework. We have doubled the quoted error here; this would probably be unnecessary if other calculations were to come to similar conclusions. A large part of the additional uncertainty vanishes in the ratio of the and decay constants, which is

(10) |

The first error is due to the measured decay rates; the second is due to the uncertainties on the CKM factors; the third is due to the uncertainties in the radiative correction ratio.

These measurements have been used in conjunction with lattice calculations that predict in order to find a value for . Together with the precisely measured , this gives an independent measure of Jutt ; Antonelli .

This work was supported by the U. S. National Science Foundation. We thank Bostjan Golob, William Marciano, Soeren Prell and Charles Wohl for useful comments.

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