IC/68/97

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

CALCULATION OF THE K

FORM FACTORS

A.O. BARUT

and

K.C. TRIPATHY

1968MIRAMARE - TRIESTE

IC/68/97

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CALCULATION OF THE K FORM FACTORS

A.O. BARUT

International Centre for Theoretical Physics, Trieste, Italy

(on leave of absence from the University of Colorado, Boulder, Colo., USA)

and

K.C. TRIPATHY'

Department of Physics, Syracuse University, Syracuse, NY, USA.

MIRAMARE - TRIESTE

October 1968

* To be submitted for publication.

T Supported by the U.S. Atomic Energy Commission.

ABSTRACT

We have determined the K., form factors f (t) in a model

of strong interactions based on the rest frame group O(4, 2) ® SU(3).

Using the values of the parameters determined from the boson mass

spectrum and pion electromagnetic form factor, we obtain at2 m - "Tt

t = t = (m - m ) , f (t ) - ^—& __JL = 0. 27. We then test theK TT

validity of exact SU(3) symmet ry for the r e s t f rame states. The

condition f̂ ( t ) = | f^(t ) yields X+ = 0.0152, X_ = 0.199 and

5(0) = 0. 097, in agreement with the latest exper imental values .

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CALCULATION OF THE K FORM FACTORS

I. INTRODUCTION

The form factors occurring in the reaction K => w&v have

been the subject of a large number of investigations all using the

techniques of current algebra or field algebra. The results

obtained differ widely. It is therefore of interest to have a

calculation based on an entirely different approach of symmetry

breaking. This will be done here.

A number of hadron properties {mass spectra,, form factors,

decay rates, diffraction scattering) have been successfully described

by the relativistic O(4, 2) model of strong interactions. The

underlying general hypotheses can be applied to weak interactions

of hadrons and thereby further tested. In this model the hadrons

are described by the relativistic analogue of "wave functions" and

the information contained in these wave functions enters into

strong, electromagnetic and weak interactions in the same way.

3)In a previous paper the general hypotheses concerning the

weak currents (scalar, pseudoscalar, vector and axial vector)

acting on the "wave functions" of the hadrons have been discussed

and the form factors in non-leptonic decays have been calculated

using scalar and pseudoscalar couplings. In the preseit paper we

discuss the important case of vector currents.

Let up emphasize that the theory amounts essentially to a direct

generalization of the usual weak interaction currents: instead of

writing the currents between Dirac spinors (i. e. , simplest O(4, 2)

spinors) we write them between the relativistic hadron "wave functions11,

the more general O(4, 2) spinors. The rest frame hadron states are

labelled by | njm ±, 1,1,, YJ , where the range of (njm ±, n = principal

quantum number, (jm) = spin and spin component, ± parity) are given

by the O(4, 2) representation used, and the range of I,IqY by the rest

- 2 -

frame SU(3) representation, for example. The minimal conserved

current elements are the matrix elements of the following operator:

T [

where r , S, L are O(4, 2) generators, P = (p1 + p) , q = (p' - p) ,

TI the simple pseudoscalar operator in O(4, 2), taken between the

so-called "tilted" states of momentum p = (mchf;, § msh?) to be

defined in Sec. 3.1.

The coefficients a. in eq. (1.1) are tensor operators with respect

to an algebra of currents, SU(3) x SU(3), for example. Note the

difference between this algebra of currents and the SU(3) algebra2)

which labels the multiplets.

The conserved current (1.1) fixes a mass spectrum depending

on the coefficients a ,a and a (a and a do not contribute to the

mass spectrum).

The hypothesis was made that the electromagnetic current,

the weak vector current and even (perhaps) the strong vector current2) 3)

are conserved and proportional to the matter current (1.1).

In the next section we apply the current (1.1) to derive K »̂ decay form

factors f (t) and f (t). We derive general formulae for the

complete functional form of f, (t). We then evaluate these functions

numerically by using the estimated values of a. from the mass spectrum.

We then test the hypothesis that the (unboosted) rest frame states

satisfy the SU(3) symmetry. The theory, as we shall see, contains

definite procedures for evaluating the effect of symmetry breaking.

-3 -

' ¥ ¥ : * . i , ^ * •* '•••- • *

II. K DECAY FORM FACTORS

The basic vector vertex functions in the processes like

K -> ff + {iv) is given by

Here i, j , k a r e the SU(3) indices , If . . . J the re levant SU(3) reducedU U

matrix elements; prj., p the four momenta of K and it, and f , (t) theK. IT x

(invariant) scalar amplitudes.

We now have a dynamical theory to evaluate the left-hand side

of (2,1). Thus, by comparing it with the right-hand side we can

evaluate the complete functional form of the two form factors f and

f .

1. The states

The states {K,p "> , | TT , p >̂ , . . . , are the tilted, rotated

and boosted O(4, 2) ® SU(3) states. This means we take the basis

vectors | njm;II~Y) and define the new states

|njm ; IlgY ; p \ = jj e ~ ~- e e I njm , Ug

(2.2)

The reason one defines these new "physical" states is that the current

operators have simple transformation properties as elements of the

Lie algebra with respect to these new states j \ / and not with

respect to the old basis vectors j \ . In eq. (2. 2) , on the right, the-2i^X 4)

first operation e ' i s simply the Cabibbo rotation; the2)second operation is the non-compact tilting in O(4, 2), and the third

is a pure Lorentz transformation; N is a normalization factor. The

- 4 -

generators of the pure Lorentz transformations a re the L. ^ = M.

elements of the 0(4, 2) Lie algebra.

2. The current

With respect to the new states (2. 2) the current operator (1.1),

in the case of bosons, has the simple form

XX® OLS F^ . (2.3)

The terms a and a do not contribute in the case of bosons. Here X1

are the usual SU(3) generators. Thus the current operator factorizes,

The situation is more complicated for baryons where aA and a terms

are also present. The current (2, 3) satisfies the charged current

algebra at infinite momentum.

3. The vector vertex amplitudes

In our model the left-hand side of (2.1) is given by

with the states j >̂ defined in (2.2) and j defined in (2.3). Because of

the direct product form assumed in (2. 3) we can evaluate the SU(3)

part of (2. 3) in a standard manner, If the octet of mesons a re assigned

to the SU(3) tensor T.1, the SU(3) part of (2. 3) isJ

g I [ 8] ; 1, 0, 0> (2.5)a = Asin0

t = (pj. - p ) = (m K + rn - 2m E ) (2. sj)

and for the p a r a m e t e r £ of Lorentz t rans format ions in (2.2) we have

2 . .2

7r K 7T L (m +m ) - tis ir

(2.7)

5. Spin part of the vertex amplitudes

In the rest frame of K, we then have from {2. 3), (2. 4) and

(2.5)

| / | ° ^ = a T which are

- 6 -

common to both the ir tower and the K tower of states (that is statesp - 4- +

with j = 0 ,1 ,2 , .. . , with the same internal quantum numbers as

the IT or the K). Now the condition of current conservation expressesfi \

the mass as a function of a , a a and two new parameters J3 and y.{ Z o

This can also be seen by noting that the current conservation is

equivalent to writing down an equation of the form

( j ' p " + /3]S + yl) 0(p) = 0 (2.10)

with j given by (2, 3). Here S = L is the Lorentz scalar element

of the Lie algebra of O(4, 2). The parameters /3 and y are, in

contrast to a , a a , different for t and K tower because of the

K-7r mass differences which cannot be neglected. Our formalismi

takes ca re of these mass differences via the values of £ and via the

tilting angles Q and 8 in eq. (2.9). These angles a r e , however,

not new p a r a m e t e r s ; they a r e related to a. and to j3 and y by

j3-a M2

s inh^ n = n ——• (2.11)y-a M

2 n

where n is the principal quantum number, n = 1 for both ir and K.

F r o m the mass spect rum the normalizat ion condition and pion7)

e lec t romagnet ic form factor we determine

% S 0, ^ = y^ = 0 (2.12)

and consequently

" I 2 " 1 • * 2 a = - • " 3 . " ° ( 2 - 1 3 )7T

With these values, eqs , (A. 11) and (A. 12) become

Q r 9 &

9 6^ c h " 2 | ( l -2 th 2 -^ th 2 | ) J (2.14)

- 7 -

8,(0 • 10-th*

Eq. (2.14) inserted into (2.8) together with (2. 5) and (2. 1) completes

the evaluation of the vector vertex amplitudes.

7. Determination of the form factors

Having determined the left-hand side of (2.1) we can now

determine f, (t) occurring on the right-hand side. In the rest frame

of K, and taking p = (E , 0, 0, p ), we haveIt It IT

! »

go( 9 - 1.

. | (m +E ) f ( t ) + (m - E ) f ( t )' i s . Tr + IS. Tf

(2.15)

the components V and V vanish.1 Ci

(2.16)

a) *TC"p form factors

Current conservation implies that f { t ) = 0, and we

immediately obtain from (2. 14) and either (2.18) or (2.19)

(4m E ) ' J t/r

tah -r~~th T9 n P 9 7Tj-cosh"^ |(l-2ih -,-

(2.17)

- 8 -

or , with our value

f + ( t , .

IT

mch -

(2.18,

Hence

f+(? = 0) = >/2 (2.19)

(f = 0 is t = 0 for 7r£3).

The form factor f+(t) falls off much faster with t than doer, the

electromagnetic form factor. l)

b) K. form factors

The two eqs . (2.15) and (2.16) a r e now

f

and

ch- ^ ch" I

a- ( m +EJsh-jj

4 ?ch - ch

ch

- 2 { >

' 2 c h T c h

(2. 20)

ich — ch >-

0

(2.21)

- 9 -

From these two equations we obtain, using the parameters (2.12) and

(2.13), 0 ^ - 1 , 0 ^ 1 / 1 1 ^ ,

The right-hand side of (2. 25) has been determined above in eq, (2. ]9).

Consequently we obtain

/ m m

Now we have the complete values of the form factors

f+(t) « 0.7(1+0.015 t/m2) (2.27)

f (t) = 0.068 (1 + 0.199 t/m2) . (2.28)

We predict therefore, subject to small uncertainties in the parameters

a ,ot ,a that we have chosen1 Ct O

X+ = 0.0152 , X"_ = 0.J99

f (t = 0)g(0) = f " ( t = Q) = 0.097. (2.29)

The form factor f (t) is a very slowly varying function in the range2

t - 0 to t = (tn-j, - m ) . We could equally well write the symmetry

condition (2. 25) at t = 0. The other form factor.f_ .which is the result

of the symmetry breaking, is, although more than an order of

magnitude smaller, a rapidly varying function. These results are in

agreement with the present experimental values within the errors

\JK+) = 0. 023 ± 0. 008 8^

?(t = 4mir) = -0.06 ± 0. 2 9 '

| ( t = 0) = -0.13 ± 0.25 ' (assuming X =0.1)

Whether the theoretical requirement (2. 25) is true is not clear. In

fact, more accurate experiments would test this symmetry hypothesis.

Another possibility of determining the parameter £?„, instead of (2. 26),

would be to use the mass spectrum and form factor of the K-tower as

we did for the r-tower, eqs. (2.12) and (2.13).

ACKNOWLEDGMENTS

One of the authors (AOB) would like to thank Professors Abdus Salam and P. Budini and the IAEAfor hospitality at the International Centre for Theoretical Physics, Trieste.

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APPENDIX A

Evaluation of g (jf)

From (2. 9),

g (?) = ( l00 | e e

with

sinh £ = cosh 8 sinh — ; sin«2 it 2

- _ coshOT/2)cosh(/3/2) , 7 = a + it

Thus we need the matrix elements of the form

I (mon

n00> (noo

) n (100 [ e

nand

(100 4 5

|n00)

|n00) (noo|Xf(?) | lOo)

(A. 4)

n= L46 a n d L56

These matrix elements can easily be evaluated in the parabolic basis

n n m) and transformed back to j nimj by means of the relation

n-1 n-1i /

nim1)

S. \

m-n +n -n- m

{A. 5)

We also have

L46

~

L = N 2 - N2 =36 2 1 Z K 2 2

N+ + N1 1

(A. 6)

with

+N, - [

n n m)

N

N"

n r n2+l, m)

[n (n +m)]2 n n - l . ra )6 Ci 1 £,

(A. 7)

-13-

Is I";*.' i, :J :* -.."

In the basis |n n m\ we have, because L = N + N

(•I I t I

n,n m | e 45

= V

f n

)m'm

1 2 ' 1 2 '2 2 ' 2 2

(A. 8)

where V (a) are the matrix elements of O(2,1) used extensively before .mn

of K

Thus, collecting terms together, we find;finally, in the rest frame

\ /

n nl"2 y\

T

fn2 n r n 2 /_ _ _ _ 0

sinh7T

-sinh

fl I 0 -i«(n -n )

(sinh -f~) Vl+X x (shf) e

V

(n2+l) -iordij-ng-l) i

and similarly

S,1 L Lo ^ r o

n 'Vn2 nrn 2

n-12

n-12

ni+n2 n r n 2

0

0 /

2~) V in 2 +>^i n h

2

K

y\.

Tl'2

( 2

>7 -i '

^ (n2+i) e'±

-shiL )

{A.10)

-15-

g3(g)The current conservation condition

sinh I

c o s h |would be

satisfied if the K-n mass difference could be neglected. It follows

from (A. 9) that the contribution of the intermediate state ) nQO )

goes like

| - Ch tiy. ^—ti rc ^ ,...

so that for small ?, i. e. , small [ (nru. - m_) - t] , higher n-values

can be neglected in the numerical evaluation. Thus to a good

approximation we find

- 1 6 -

41/2

OK

4 l i ^ A ( I -f Z O0S2 2 Coc -fo( 'j }

-17-

L2

4--r

(A.I?)-18-

REFERENCES AND FOOTNOTES

J) The recent l i te ra ture on K. form factors is in fact so3

extensive that we do not attempt to list all the papers

published on this subject. We refer to reviews in the

Proceedings of the Heidelberg Conference on Elementary

Par t i c l e s , H. Filthuth, ed. (North-Holland, 1968) and

Proceedings of the XIV. International Conference on High-

Energy Physics (Vienna, 1968).

2) References can be found in two recent reviews, one in

Lectures in Theoretical Physics , Vol. XB (Gordon and

Breach, 1968) the other in the Proceedings of the Symposium

on Hadron Spectroscopy (Aeta Physica Hungarioae. to he

published) (ICTP, Tr ies te ,prepr in t IC/68/88).

3) A .O. Barut and S. Malin, Uucl. Phys. (in p r e s s ) .

4) N. Cabibbo, Phys. Rev. Let ters .10, 531(1963).

5) B. Hamprecht and H. Kleinert, Univ. of Colorado preprint 1968.

6) A .O. Barut, D. Corrigan and H. Kleinert, Phys. Rev.

Let ters J20, 167 (1968) and Phys. Rev. 167,, 1527 (1968).

7) A .O. Barut, Nucl. Phys. B4_, 455 (1968).

8) W. J. Willis, Proceedings of the Heidelberg Conference on

Elementary Par t i c les , H. Filthuth, ed. (North-Holland,

1968) p. 278.

9) T. Eichen, et a l . Phys. Let ters 27B , 586 (1968).

10) D.R. Botterill et a l . Phys . Rev. Le t te rs 21_, 766 (1968).

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