Leptonic and semileptonic decays of pseudoscalar mesons

Ž .

Physics Letters B 456 1999 248–255

Leptonic and semileptonic decays of pseudoscalar mesons

M.A. Ivanov

a

_{, P. Santorelli}

b,c a

BogoliuboÕ Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia b

Dipartimento di Scienze Fisiche, UniÕersita ‘‘Federico II’’ di Napoli, Napoli, Italy`

c

INFN Sezione di Napoli, Napoli, Italy Received 19 March 1999

Editor: R. Gatto

Abstract

We employ the relativistic constituent quark model to give a unified description of the leptonic and semileptonic decays

Ž .

of pseudoscalar mesons p , K, D, D , B, B . The calculated leptonic decay constants and form factors are found to be ins s good agreement with available experimental data and other approaches. We reproduce the results of spin-flavor symmetry in the heavy quark limit. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 13.20.-v; 13.20.He; 12.38.Lg; 24.85.q p

Keywords: Pseudoscalar mesons; Leptonic and semileptonic decays; Quark models; Spin-flavor symmetry

1. Introduction

Semileptonic decays of pseudoscalar mesons al-low to evaluate the elements of the Cabibbo–

Ž .

Kobayashi–Maskawa CKM matrix, which are fun-damental parameters of the Standard Model. The decay K ™ p en provides the most accurate determi-nation of V , the semileptonic decays of D and Bus

Ž )

. Ž )

.

mesons, D ™ K K ln , B ™ D D ln and B ™

Ž . < < < <

p r ln , can be used to determine V , V_{cs cb} and

<_{V , respectively. The effects of strong interactions}<

ub

in these processes can be expressed in terms of form factors, which depend on q2_{, the squared momentum}

transferred to the leptonic pair. Information on the form factors are obtained by measuring the distribu-tions of q2 _{and decay angles.}

The decays of heavy D and B mesons are of particular interest due to the spin-flavor symmetry

w x

observed for infinite quark masses 1 . This symme-try allows to reduce the number of form factors and express them in terms of the universal Isgur–Wise

w x

function 2 . Also the scaling laws derived for some physical observables can be, in principle, tested ex-perimentally. Since the Isgur–Wise function cannot be calculated from first principles, many models and nonperturbative approaches, which exhibit the heavy quark symmetry, have been employed to describe relevant phenomena. However, it was found out, that the finite mass corrections are very important, espe-cially, in the charm sector. It appears that in some sense a step back should be done from using the heavy quark symmetry as a guide under model build-ing to the straightforward calculations with full quark propagators. Then one has to check the consistency of the results with the spin-flavor symmetry in the heavy quark limit.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.

Ž .

M.A. IÕanoÕ, P. Santorelli r Physics Letters B 456 1999 248–255 249

In this paper we employ the relativistic

con-Ž . w x

stituent quark model RCQM 3 for the simultane-ous description of both light and heavy flavored meson leptonic and semileptonic decays. This model is based on the effective Lagrangian describing the coupling of mesons with their quark constituents, and the compositeness condition. The physical pro-cesses are described by the one-loop quark diagrams with free constituent propagators and meson-quark vertices related to the Bethe–Salpeter amplitudes.

Ž .

The masses of lower-lying pseudoscalar PS mesons should be less than the sum of quark constituent masses to provide the absence of imaginary parts corresponding to quark production. The adjustable parameters, the widths of Bethe–Salpeter amplitudes in momentum space and constituent quark masses, are determined from the best fit of available experi-mental data and some lattice determinations. We found that our results are in good agreement with experimental data and other approaches. Also we reproduce the results of spin-flavor symmetry for leptonic decay constants and semileptonic form fac-tors in the heavy quark limit.

The shapes of vertex functions and quark propa-gators should be found from the Bethe–Salpeter and Dyson–Schwinger equations, respectively. This is

Ž .

provided by the Dyson–Schwinger Equation DSE

w x_{4 studies. A DSE-approach has been employed to}

provide a unified and uniformly accurate description

w x

of light- and heavy-meson observables 5,6 . A similar approach, based on the effective heavy

w x

meson Lagrangian, has been described in Ref. 7 in terms of a model based on meson-quark interactions, where mesonic transition amplitudes are represented by diagrams with heavy mesons attached to quark loops. The free propagator has been used for light quarks. However, the quark propagator obtained in the heavy quark limit has been employed for heavy quarks.

2. The model

w x

We employ an approach 3 based on the effective interaction Lagrangian which describes the transition of hadron into quarks. For example, the transition of

the meson H into its constituents q and q is given1 2

by the Lagrangian

L

Lint

Ž

x s g H x

.

H

Ž

.

_H

dx1

_H

dx F2 H

Ž

x ; x , x1 2

. Ž

q x1

.

=_{G l q xH H}

_Ž

₂

_.

_.

_{Ž .}

₁

Here, lH and GH are the Gell–Mann and Dirac matrices, respectively, which provide the flavor and spin numbers of the meson H. The function FH is related to the scalar part of Bethe–Salpeter

ampli-Ž .

tude. For instance, the separable form FH x; x , x1 2

Ž Ž . . ŽŽ .2.

s_{d x y x q x r2 f x y x has been used}

1 2 1 2

w x

in Ref. 3 for pions.

The coupling constants g_H is given by the so

w x

called compositeness condition proposed in Ref. 8

w x

and extensively used in Ref. 9 . That condition means that the renormalization constant of the meson field is equal to zero:

3 gH2 X ₂

˜

Z s 1 y_H P_H

_Ž

_mH

_.

s_{0 ,}

_{Ž .}

₂ 2 4p

˜

X

where PH is the derivative of the meson mass operator defined by d4k 2 2 2

˜

P_H

_Ž

_p

_.

s

_H

f_H

_Ž

y_k

_.

2 4p i =tr G S

_{Ž .}

k

u

_{G S k}

_Ž

u

q_p

u

_.

_.

_{Ž .}

₃ H 2 H 1

The invariant amplitudes describing the leptonic

Ž . Ž . XŽ X.

H p ™ ln and semileptonic H p ™ H p ln

de-cays are written down

GF m X A H p ™ en s

Ž

Ž

.

.

Vq q

Ž

eO n Mm

.

H

Ž

p

.

Ž .

4

'

2 A H p ™ H

Ž

Ž

.

X

Ž

pX

.

en

.

GF m X X X s _{Vq q}

_Ž

_{eO n Mm}

_.

_{H H}

_Ž

_{p, p ,}

_.

_{Ž .}

₅

'

2

where G is the Fermi weak-decay constant, V X is

F q q

the appropriate element of the CKM matrix. The matrix elements of the hadronic currents are given by 3 d4_k m 2 M

_Ž

p s

_.

g

_H

f

_Ž

y_k

_.

H _4p2 H _{4p i}2 H 5 m m = tr g S

_{Ž .}

k

u

_{O S k}

_Ž

u

q_p

u

_.

s_{f p} 2 1 H 6

Ž .

Mm X p, p X

Ž

.

H H 3 d4_k 2 2 X X s _{g g}

_H

f

_Ž

y_k

_.

f

_Ž

y_k

_.

H H H H 2 2 4p 4p i = 5 5 X m tr g S

_{Ž .}

k

u

_{g S k}

_Ž

u

q_p

u

_.

_{O S k}

_Ž

u

q_p

u

_.

_{Ž .}

₇ 3 2 1 m m X X 2 2 s_fq

_Ž

_q

_.

_Ž

_{p q p}

_.

q_fy

_Ž

_q

_.

_Ž

_{p y p}

_.

_{Ž .}

₈ Ž 2.

where f y_{k is related to the BS-amplitude in} H

momentum space, and 1

S k

_{Ž .}

u

s

_{Ž .}

₉

i

m y k

u

i

is the propagator of the constituent quark with mass

m . As discussed before, to avoid the appearance of_i

Ž . Ž .

imaginary parts in Eqs. 6 and 7 , we assume that

m - m q m_{H q q} which is a reliable approximation

1 2

for the lower-lying mesons considered here.

Ž .

To evaluate the integral in Eq. 7

d4_k X ₂ X IH H

Ž

p, p s

.

_H

_{4p i}2 FF yk

Ž

.

=tr g5_{S k}

_u

_g5_{S k}

_u

_qp

_u

X

Ž .



3 2

Ž

.

=gm S k

_Ž

u

q_p

u

_.

₄

_,

_Ž

₁₀

_.

1 Ž 2. Ž 2. XŽ 2. where FF yk sf y_k Pf y_{k , we need to} H H

calculate the following integrals:

JŽ0 , m , mn , mn d . 4 Ž m m n m n d. Ž 2. d k 1, k , k k , k k k FF y k s

_H

_{p i}2 2 2 2 X2 2 2 . Ž . Ž . w x m y k q p m y k q p m y k w ₁ x w ₂ x ₃ 11

Ž

.

Using the Cauchy representation for the function

Ž 2. F

F yk and then the standard techniques of the

Ž XŽ .

Feynman a y parametrization one finds FF z s

Ž . . dFF z rdz 2 3 ` _t X 0 3 J s

_H

dt

_H

d a d 1 y

_Ý

a_i

_Ž

_yFF

_Ž

_zI

_.

_.

ž

1 q t

/

_ž

_/

0 _is1 12

Ž

.

3 3 ` t m 3 J s y

_H

dt

_ž

_/

_H

d a d 1 y

_Ý

a_i

ž

/

1 q t 0 _is1 =Pam

Ž

yFFX

Ž

zI

.

.

Ž

13

.

2 3 ` t mn 3 J s

_H

_dt

_ž

_/

_H

_{d a d 1 y}

_Ý

a_i

ž

/

1 q t 0 _is1 = 1 mn 1 y _{g FF z}

_Ž

_I

_.

2

½

1 q t 2 t X m n y_{P Pa a FF}

_Ž

_zI

_.

_Ž

₁₄

_.

5

ž

1 q t

/

2 3 ` t mn d 3 J s

_H

_dt

_ž

_/

_H

_{d a d 1 y}

_Ý

a_i

ž

/

1 q t 0 _is1 = 1 mn d m d n n d m g P q ga P q ga Pa 2

½

= t _{FF z}

_Ž

_I

_.

2 1 q t

Ž

.

3 t X m n d q_{P P P FF}

_Ž

_z

_.

_Ž

₁₅

_.

a a a

ž

/

I

5

1 q t where q s p y pX, P s a p q a pX, D s a a p2 a 1 2 3 1 3 X_{2 2} w 3 2 _x q_{a a p q a a q , and z s t Ý a m y D} 2 3 1 2 I is1 i i 3 2 _{Ž .} y_{P tr 1 q t .a} Ž .

Finally, Eq. 10 becomes

m X X X m X 2 2 2 IH H

Ž

p, p s p q p

.

Ž

.

Iq

Ž

p , p , q

.

m X ₂ X_{2 2} q

_Ž

_{p y p}

_.

_Iy

_Ž

_{p , p , q}

_.

with I p2, pX₂ , q2

Ž

.

q 2 3 ` t 1 3 s₂

_H

_dt

_ž

_/

_H

_{d a d 1 y}

_Ý

a_i

ž

/

1 q t 0 _is1 1 t = _{FF z}

_Ž

_.

_{4 y 3 a q a}

_Ž

_.

I 1 2

½

1 q t 1 q t t X yFF

Ž

zI

. Ž

m q m1 2

.

m q3 1 q t = y

Ž

Ž

a q a1 2

. Ž

m m q m m y m m1 3 2 3 1 2

.

qa p2_{qa p}X₂

.

1 2 2 t t 2 y_{P 2 y a q a}

_Ž

_.

_. a

ž

/ ž

1 2

/

5

1 q t 1 q t 16

Ž

.

M.A. IÕanoÕ, P. Santorelli r Physics Letters B 456 1999 248–255 251

The normalization condition is written in the form 3 gH2 _{2 2}

Iq

Ž

p , p ,0 s 1

.

Ž

17

.

2

4p

with m s m ' m.1 2

The integrals corresponding to the matrix element

Ž .

of the leptonic decay H p ™ ln and radiative decay

Ž . Ž . Ž .

of neutral meson H p ™ g q q g q1 2 are calcu-lated following the same procedure. We have

d4k m 2 Y

Ž

p s

.

_H

₂ f yk

Ž

.

4p i =tr g5_{S k}

_u

_gm I y g5 _{S k}

_u

_qp

_u

Ž .



2

Ž

.

1

Ž

.

4

s_pm_{Y p}

_Ž

2

_.

` t 2 Y p

_Ž

_.

s

_H

_dt 2 0

_Ž

_{1 q t}

_.

=

_H

1_{da m q m y m}

_Ž

_.

a t _{f z}

_Ž

_.

2 1 2 Y 1 q t 0 18

Ž

.

d4k mn 2 K

Ž

q , q1 2

.

s

_H

2 f yk

Ž

.

4p i =tr g5 S k

u

y_q

u

gm S k

_{Ž .}

u



Ž

2

.

=gnS k

_Ž

u

q_q

u

_.

₄

1 s_i´mn a b_qa_qb_{K p}

_Ž

2

_.

1 2 2 ` _t 2 K p

_Ž

_.

s_m

_H

_dt

ž

1 q t

/

0 =

_H

1_da1

_H

1ya1_{da yf2}

_Ž

X

_Ž

_zK

_.

_.

_Ž

₁₉

_.

0 0 w 2 _{Ž .} 2 2 2 2 where z s t a m q 1 y a m y a p q a p tr_{Y 1 2} Ž1 q t.x and z s t m y a a p_K w 2_{1 1 2} 2xq_{a a p tr1 2} 2 Ž1 q t ..

The physical observables are expressed in terms

Ž . Ž .

of the structural integrals written in Eqs. 16 , 18

Ž . and 19 : gP ₂ g s _{K m}

_Ž

_.

_, Pgg _{2 2 p}

_'

2 P p 2 3 2 G

Ž

_{P ™ gg s}

.

a m gP Pgg,

Ž

20

.

4 3 2 f sP 2 g Y mP

Ž

P

.

, 4p 2 2 2 2 G fF P ml 2 2 < X< G

_Ž

_{P ™ ln s V}

_.

_{q q m m 1 yP l ,}

_Ž

₂₁

_.

2 8p mP 3 2 X 2 2X 2 f_q

_Ž

q

_.

s _{g g I}

_Ž

_{m , m , q}

_.

_, P P q _{P P} 2 4p G2 t F y 2 2 X < X< < < G

_Ž

_{P ™ P ln s V}

_.

_{q q 3 3}

_H

_{dt f}q

_{Ž .}

_t 192p m_P 0 =

_Ž

_{t y tq}

_{. Ž}

_{t y ty}

_.

3r2_,

_Ž

₂₂

_.

Ž X.2 Ž

with t s m " m" P P the extra factor 1r2 appears

0 _.

for p in the final state .

2.1. HeaÕy quark limit

The leptonic heavy decay constants and semilep-tonic heavy to heavy form factors acquire a simple form in the heavy quark limit, i.e. when m ' M ™ `,1

X ₂

Ž .2 X2 Ž X .2

m ' M ™ ` and p s M q E , p s M q E2

Ž .

with E being a constant value. From Eq. 16 by replacing the variables a ™ a rM and a ™1 1 2

a rM2 X, one obtains X ₂ ` M q M t I ™_{q X} P

_H

_dt

ž

/

2 MM 0 1 q t =

_H

1_daa

_H

1_{dt yF}

_Ž

_FX

_{Ž .}

_z

_.

_{m q} a t 1 q t 0 0 X

'

` M q M 1dt m q u 1 s _X P

_H

_H

_{duFF z}

_{Ž .}

_˜

_Ž

₂₃

_.

2 ₂ 2 MM 0 W 0 m q z

˜

Ž .Ž

'

where z s u y 2 E urW , W s 1 q 2t 1 y t

˜

w y . Ž 2 X2 X 2. Ž X. 1 and w s M q M y 2 MM q r 2 MM . The normalization condition can be obtained from

Ž . X

Eq. 23 by putting w s 1 and M s M. We have 3 gH2 _{Ž0. Ž0.} 1 P I_q s_{1, Iq}s _{I ,N} 2 2 M 4p

'

` _{m q u} 2 I sN

_H

dufH

Ž

z

˜

0

.

₂

Ž

24

.

m q z 0

˜

₀

Table 1

Ž y1 .

Calculated values of a range of observables gpgg in GeV , leptonic decay constants in GeV, form factors and ratios are dimensionless .

w x

The ‘‘Observed’’ are extracted from Refs. 10–16 . The quantities used in fitting our parameters are marked by ‘‘)’’

Observed Calculated Observed Calculated

Kp Ž . ) gpgg 0.274 0.242 fq 0 0.98 0.98 D K Ž . ) f_p 0.131 0.131 ) fq 0 0.74 " 0.03 0.74 B D Ž . ) f_K 0.160 0.160 fq 0 0.73 q19 Bp Ž . ) f_D 0.191y28 0.191 fq 0 0.27 " 0.11 0.51 fDs _{Ž . Ž . Ž .} y2 y2 ) 1.08 8 1.08 Br K ™ p ln 4.82 " 0.06 P 10 4.4 P 10 fD q1 8 y2 y2 Ž . Ž . fD_s 0.206y28 0.206 Br D ™ Kln 6.8 " 0.8 P 10 8.1 P 10 q2 7 y2 y2 Ž . Ž . ) fB 0.172y31 0.172 Br B ™ Dln 2.00 " 0.25 P 10 2.3 P 10 fBs _{Ž . Ž . Ž .} y4 y4 ) 1.14 8 1.14 Br B ™ p ln 1.8 " 0.6 P 10 2.1 P 10 fB fB_s 0.196

'

where z s u y 2 E u . Then the leptonic decay con-

˜

0

stant and semileptonic form factors are written as

` 1 3

'

f ™ P

_H

_{du u y E f}

_Ž

_z

_˜

_.

P

_'

(

2 H 0 2p I M N 0 =m q u r2

'

_Ž

₂₅

_.

2 m q z

˜

0 MX" M f ™" X j w ,

Ž

.

'

2 MM

'

` 1 1dt ₂ m q u j w s

Ž

.

_H

_H

dufH

Ž .

z

˜

₂ .

Ž

26

.

IN 0 W 0 m q z

˜

It is readily seen that we reproduce the scaling law for both leptonic decay constants and form factors, and obtain the explicit expression for the Isgur–Wise

w x

function 1,2 .

3. Results and discussion

The expressions obtained in the previous section for physical observables are valid for any vertex

Table 2 Values in GeV Lp 1.16 LK 1.82 LD 1.87 LD_s 1.95 LB 2.16 LB_s 2.27 Ž 2.

function f y_{k . Here, we choose a Gaussian} H

Ž 2.  2 24

form f yk s_{exp k rL in Minkowski space.} H

The magnitude of LH characterizes the size of the BS-amplitude and is an adjustable parameter in our approach. Thus, we have six L-parameters plus the four quark masses, all of which are fixed via the least-squares fit to the observables measured

experi-Ž

mentally or taken from lattice simulations see Table

.

1 .

The fit yields the values of L-parameters and the constituent quark masses which are listed in Tables 2 and 3.

The values of L are such that L - Lmi mj if

m - m . This corresponds to the ordering law for_{i j}

sizes of bound states. The values of L s 1.87 GeV_D and L s 2.16 GeV are larger than those obtained in_B

w x

Ref. 6 : L s 1.41 GeV and L s 1.65 GeV. The_{D B} mass of u-quark and the parameter Lp are almost fixed from the decays p ™ mn and p0_{™ gg with}

an accuracy of a few percent. The obtained value of the u-quark mass m s 0.235 GeV is less than theu

constituent-light-quark mass typically employed in

Ž

quark models for baryon physics m ) m r3 su N .

0.313 GeV . For instance, the value of m s 0.420u

GeV was extracted from fitting nucleon observables

Table 3 Values in GeV mu 0.235 ms 0.333 mc 1.67 mb 5.06

M.A. IÕanoÕ, P. Santorelli r Physics Letters B 456 1999 248–255 253 Table 4 K ™p D™ K B ™ D B ™p b0 0.28 0.64 0.77 0.52 b1 0.057 0.20 0.19 0.38 w x

within our approach 3 . The different choice of constituent quark masses is a common feature of

quark models with free propagators due to the lack of confinement. However, we consider here the low-lying mesons that allows us to fix the con-stituent quark masses in a self-consistent manner. As mentioned above, the meson masses must be less than the sum of masses of their constituents. This gives the restrictions on the choice of the meson binding energies: E s m y m - m , E s m y_{K K s u D D}

Fig. 1. The semileptonic K ™ p , D ™ K, B ™ D and B ™ p form factors with, for comparison, a vector dominance, monopole model Eq.

m - m and E s m y m - m , which means thatc u B B b u

the binding energy cannot be relatively large as

w x

compared with those obtained in Ref. 6 : E s 0.58D

GeV and E s 0.74 GeV._B

Let us now consider the q2_{-behaviour of the form}

factors. We use the three-parameter function for the four fq form factors

f 0

Ž .

X H H 2 f_q

_Ž

q

_.

s

_Ž

₂₇

_.

2 2 2 2 2 1 y b₀

_Ž

q rm_H

_.

y_{b q rm1}

_Ž

_H

_.

Ž .

here b , b0 1 and f 0 are parameters to be fitted. We collect the fitted values in Table 4 and report the

q2-dependence in Fig. 1.

For comparison, we plot, together with our re-sults, the predictions of a vector dominance monopole model: fq ™ qX ₀

Ž .

X q q ™ q 2 f_q

_Ž

q

_.

s

_Ž

₂₈

_.

2 2 1 y q rm_V X q q X 2

with m_V X being a mass of lower-lying qq -vector

q q

meson. We choose m )s2.11 GeV for c ™ s, m )

Ds B

w x

)

s_{5.325 GeV for b ™ u, m} f_{m s 6.4 GeV 17} Bc Bc

q qX_{Ž .}

for b ™ c transitions. The values of fq 0 are taken from the Table 1. Also we calculate the branching ratios of semileptonic decays by using widely

ac-w x

cepted values of the CKM matrix elements 10 . Our result for the slope of the K_l3 form factor

fKp X 0

Ž .

q 2 l s m_{q p} s_{0.023 ,}

_Ž

₂₉

_.

Kp fq

Ž .

0

is in good agreement with experiment: lexpt_q s

w x VD M

0.0286 " 0.0022 10 and VDM prediction: l s

q

m2r_m2)s0.025. This value is also consistent with

p K

w x

Refs. 18

One can see that the agreement with experimental data and lattice results is very good, with the

excep-bu_{Ž .}

tion of the value of fq 0 which is found to be larger than the monopole extrapolation of a lattice

Ž w _x.

simulation, QCD Sum Rules cf. 19 and some

Ž w x.

other quark models see, for example, 20,21 . How-ever, this result is consistent with the value

calcu-w x

lated from Refs. 6,22 and allows us to reproduce the experimental data for B ™ p ln with quite good accuracy.

Acknowledgements

We appreciate F. Buccella, V.E. Lyubovitskij, G. Nardulli and C.D. Roberts for many interesting dis-cussions and critical remarks. We thank G. Esposito for reading the manuscript. M.A.I. gratefully ac-knowledges the hospitality and support of the Theory Group at Naples University where this work was conducted. This work was supported in part by the Russian Fund for Fundamental Research, under con-tract number 99-02-17731-a.

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Ž .

B 343 1990 1.

w x3 M.A. Ivanov, V.E. Lyubovitskij, Phys. Lett. B 408 1997Ž .

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