the d function in a phenomenological nd model

Volume 121B, number 4 PHYSICS LETTERS 3 February 1983

THE D FUNCTION IN A PHENOMENOLOGICAL N/D MODEL

Ramesh BHANDARI Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA and Physics Department, New Mexico State University, Las Cruces, NM 88003, USA 1

Received 26 July 1982 Revised manuscript received 18 October 1982

Analytic expressions for the.D function in a phenomenologicalN/D model are given. These expressions can be of con- siderable use in'phenomenologi~al studies of partial-wave scattering amplitudes pertaining to systems such as *rN, KN, NN, etc. Extension to the case in which one of the two particles in a channel is an isobar is also considered.

1. Introduction. The N/D formalism has long been recognized [ 1 ] as a useful tool to study the dynamical con- tent of partial-wave scattering amplitudes for systems such as nN, KN, NN, etc. Its practical use in general has, however, been limited by the massive amount of numerical work that is usually involved in its application. In hadronic phenomenology, the attempt to circumvent this difficulty has consisted in replacing the left-hand singularities of the amplitude (which are mainly of the branch point type) by an array of poles on the left-hand part of the energy axis. However, in spite of this approximation to the left-hand singularities, it turns out that the D-matrix exists as an integral, which more often than not, requires numerical calculation.

Motivated by a need for a tractable N/D model we present in this paper analytic expression for the aforemen- tioned integral in a phenomenological formulation of the N/D model. We consider the general case of scattering involving two particles with relative orbital angular momentum 1. Also near the end, we discuss the consequences of mass averaging which is required when one of the particles involved is an isobar.

2. Writing the S-matrix element for the single-channel case as

&(s) = 1 + 2 im( s )A t ( s ) , (1)

the reduced scattering amplitude A l in terms of the familiar functions, Nl and D1, is

A l (s) = g l (s)/O l (s) . (2)

The subscript l denotes the relative orbital angular momentum of the particles in the two-body channel under con- sideration. Pt in eq. (1) is the corresponding two-body phase-space factor. The Mandelstam variable s equals the square of the center-of-mass energy. The product of Pt and At is the conventional elastic partial-wave amplitude. Each of the functions Nl and Dl is real-analytic with Nl possessing the left-hand dynamical cut and DI the right- hand unitarity cut.

It is customary in a phenomenological treatment to approximate Im Al on the left-hand cut by a series of 8- functions, i.e.,

n Im A1 = -- ~ zrXfP)6 (s - s(P)). (3)

p=l

1 Address from August 26, 1982.

0 031-9163/83/0000-0000/$ 03.00 © 1983 North-HoUand 279


This then leads [1] to

x(P)DI(s (p)) 1 n / N l ( s ) = _ sfp) , O l ( s ) = 1 - - - - ~ X(P)DI(s(P)) #I(S') d$'

p=l s rr p=l (s' - s(P))(s ' - s) ' (4a,b) sl

where Sl is the threshold energy squared, a n d D l ( s ) ~ I as s ~ ~. Under the approximation o feq . ( 3 ) , N l ( S ) is represented by a series of poles simulating the effect of the left-hand cut. In phenomenological studies, the pole positions s (p) and the force parameters X (p) are determined by fitting the resulting expression for the partial-wave scattering amplitude to the given set of predetermined phase-shifts. Our primary aim here is to obtain an analytic expression for Dl( s ) given in eq. (4b). Clearly, it is the integral therein which we must cast into an analytic form. This we do below.

For a two-body channel, it is well-known that pl(S) near threshold behaves as q2/+l where q is the center-of- mass momentum. This behavior gives rise to a square-root unitarity cut at s = Sl in the complex s-plane. Conse- quently, for phenomenological purposes, one may define

p l ( s ) = [(s - S l ) / ( s - a ) l t + l l 2 . ( 9

The factor in the denominator of eq. (5), which incidentally gives rise to a left-hand branch point at a, is necessary to ensure the convergence of p/(s), i.e., as s ~ 0% pl(S) ~ 1. This parameter a can be fixed at a convenient value or simply set to zero. The form, eq. (5), for p t ( s ) has been effectively used before [2]. Using it in eq. (4b), one easily sees that the calculation of the Dl function involves integrals of the type

? ( s ' , - S l 1 I+1/2 ds' I I ( s ) = \ s - a ] (s' - sfP))(s ' - s) ' Sl whose analytic expression we are interested in.

Via a series of substitutions, we can easily cast the integral I l into the following form:

I l = (e - b ) - l ( c I (t) - b i l l ) ) ,

where

(6)

(7)

6 o= fl (1 a - c z - z)l+l12 ' Ip = fl (1 z)l+112 dz

0 0

and

a = s l - a > O , b = s(P) - ot , c = s - o~ .

We can further express the integral I~ t) as

' c (' = - - ~ ( ~ ) (~-)lI(cO ) 1 n - l j l _ n + ) , c n=l

(8a,b)

(9)

where

1 f = 2 J m = ( l - z ) m+l /2 dz 2 m + 3 ' 0

_ a)1/21 ) i(0) = 2 + (c ~ t In 1 - [(c - a)/c] 1/2 c c 1 + [(c - a)/c] 1/2 + iTr .

IL l) has the same series expansion as I(c l) in eq. (9), except that c is replaced by b. The expression for I~ O) is

(10,11)

280


2 - b) 1/2 ( 1 - i [ (a - b)/b] 1 / 2 . . \ I~O)=-b + i (a b312 . l n l + i [ ( a b) /b] l lE * lTr)

(b > 0 is assumed here and hereaf ter) .

(12)

3. Analy t ic structure o f l l ( s ) . The integral I t(s) in eq. (6) is given by eqs. ( 7 ) - ( 1 2 ) . However, to unders tand its general analyt ic proper t ies , one need only consider the case corresponding to l = 0. Examina t ion o f /~ 0) in eq. (11) shows tha t a r ight-hand branch cut exists at s = Sl (see fig. 1) wi th the d iscont inu i ty across i t given by

l im [Ic(0)(s + ie) - I~°)(s - ie)] = 2rr[(s - Sl)/(s - t0] 1/2(s - a ) - 1 • (13) e-*0

When s < Sl, [(c - a)/c] 1/2 _+ i [(a - c)/c] 1/2 and 160) becomes real. Since 1~ 0) has no energy dependence and is

always real, Io(s ) is real-analyt ic . (i) A t s = s (p).

~ ( P ) ~-+~ "" a ~ b a < (a) On I (principal) sheet o f the S = Sl CUt. W h e n s ~ ,~ o a n a ( c - ) > 0 ( - ) 0. As a consequence, (e - a) 1/2-+ i (a - b) 1/2, implying/(c O) = I~ 0). We have here a s i tua t ion involving 0/0 for the value of I0. Expans ion o f / ~ 0) a round e = b shows tha t

rra ia 1 - i [ (a - b)/b] 1/2 1 l im lo(s ) = _ b ) l / 2 _ b ) l / 2 In (14) s--,s(P) 2b3/2(a - 2b3/2(a 1 + i [ (a - b)/b] 1/2 - b - ,

which is a finite quant i ty . (b) On H sheet o f the s = Sl cut. On this sheet of the cut reached by burrowing through it f rom the top or the

b o t t o m (see fig. 1), (c - a) 1/2 -~ - i ( a - c) 1/2 for s < Sl in the express ion for/~0). When c --- b, only the log terms cancel. Thus

27r [ Sl - s (p) ~1/2 Io(s = s(P))[ l /= l im -- ~ - ) ) ~ s - - - ~ 7 ~ _ a ] + the finite result o f eq. ( 1 4 ) , (15)

s-,sq') (s implying there is a pole on the II sheet wi th a residue = 27r[(sl - s(P~[(s (p) - a)] 1/2. This fact is also verif ied f rom cons idera t ion of pinching singulari ty tha t appears at s = s (p) in the analyt ic con t inua t ion of the integral represen- ta t ion of I0(s) , pe r fo rmed by deforming the con tou r o f integrat ion.

As a result o f the foregoing analyt ic s t ructure of I l ( s ) , Dl(s) has a pole at s = s (p) only on the II sheet of its cut . Fur the rmore , since Nl(s) h a s p o l e s on all the sheets o f the cut associated with Dl(S), the ampl i tude Al ( s ) = Nl(s ) /

(P) Dl(s) possesses a pole at s = s only on the I (physical) sheet, as one expects . In a similar way, one can show tha t the branch po in t at s' = a in the in tegrand o f eq. (6) is genera ted in complex s-plane at s = a bu t only on the unphysical sheets associa ted with the cut at s = Sl.

For completeness , we men t ion tha t near c = a, i.e., s = Sl,

Im Io(s ) = [rr[(c - b)] [(c - a)/c] 1/2, (16a)

for c - a > 0, while

X~.%^^~.^^:.^.^.^:^~^v^^v^.^.. s(~) s t

Fig. 1. The complex s-plane showing the right-hand unitarity cut at sl and the position s(P) of one of the input poles repre- senting the function N l (s).

281


Real/0(s) = bl/2(a l b),/2 ( i t - i l n 1 - i [ ( a - b)/b] 1/2 ) - 1 + i[(a b)/b] 1/2 + O((c - a)/c). (16b)

Furthermore, for the case s -+ o% one notes that c -+ oo and (c - a)/c ~ 1. Consequently, the argument of the log function in eq. (11) goes to zero. But appropriate expansion shows that in this limit, s ~ oo,

Im/ t ( s ) ~ l / s , Re lt(s) ~ ln(s)/s , (17a, b)

verifying that Dl(s) ~ 1 as s -+ oo. It may be noted here that the foregoing results, although derived for a one-channel case, are also applicable to

D functions which occur as elements of a D-matrix in a coupled-channel treatment. If, however, one of these D functions pertains to a quasi-two-body channel, such as zrA in ~rN scattering, NA i n N N scattering, etc., an average over the variable mass m of the isobar must be performed. This averaging can be carried out as follows:

S 1 O ( s ) = g

s l=(ml+m2+m3)2 (s' a ) t+ l /2 ( s ' -- s(P))(S' -- s)

N/8-m 3 ×(/ m T = m l + m 2

+ t

[s' - (m 3 + m)2] l 1/2(m -- m T ) l +1/2 dm )ds ' (18a)

oo (m - mT) 1'+1/2 ( S [ s ' - (m3 + m)2]/+1/2 ds' ) dm

=g mT=ml+m2f [(m 0 - m)2+F2/a]f(m)~(m+m3)2 (s' ---- a)/-~7-/2(sT-- s - ~ ( s ; - - s) , (18b)

where

g = (P/27r) f(mo)/(mo - mT) l'+1/2 (19)

and l / f (m) is some form factor in the m dependence, which also ensures the convergence of the integral in the m variable [3]. It may, for example, be of the form: f (m) = (m - 13)2/+/'+1 where t3 is a phenomenological parameter less than (ml + m2). ml and m2 are the masses of the particles into which the isobar decays, and l ' is their relative orbital angular momentum, m3 is the mass of the stable particle produced along with the isobar. This isobar cor- responds to a complex mass of m 0 - iF/2. One may note here that, in the limit P -+ 0, the foregoing double integral reduces to the integral of eq. (6) as expected. One also observes here that the integral over s' in eq. (18b) can be replaced by analytic expressions since it is of the type given in eq. (6). But obtaining an analytic expression for the subsequent integral in the m variable in terms of simple mathematical functions does not seem to be possible. Nevertheless, despite the lack of such expressions, one can still point out some interesting analytic properties of the function lt(s) from a study of its integral representations in eqs. (18a) and (18b). From eq. (18a), we see that

27rig x/s-m3 [s r... +...~21l+1/21 m m ~l'+I/2n~. E-~olim [I(s + ie) - I(s - ie)] = (s - a)l+l/2(s - s(P)) f - ~'m3[(m0m)-~ m--~I ~/4-]~" - f(~m)T) u,,, (20a)

ml+m2

(x/S-- X/~-l) 1+1'+2 near s = Sl . (20b)

where Sl = (ml + m2 + m3) 2 [3]. The integral nature of the exponent implies that the right-hand unitarity cut at s = Sl in fig. 1 is no longer a square-root cut. Rather, it is of the logarithmic type with an infinite number of sheets, as discussed in ref. [3]. Eq. (20a) further suggests that the analytic continuation ofll(s) to the unphysical sheet (say II sheets) reached by burrowing from the top of the cut at s = Sl can be achieved in the following way:

282


/ /(s)l II = / / ( s ) + right-hand-side of eq. (20a) , (21)

In the lower-half of this unphysical sheet (on which the resonance poles of a scattering amplitude can exist) is present a square-root branch cut at s = Sc = (m0 + m3 - iF/2) 2. This square-root cut appears due to the pole at m 0 - i I ' / 2 in the integrand of the integral of eq. (20a). It can be taken to run to the right and paraUel to the Re s axis in fig. 1. By the procedure of analytic continuation described in detail in ref. [3], one can show that the discontinuity

across this cut (i.e., for Re s > Re Sc) is given by

lim [//(Re s + i Im Sc+ ie)l Ii - / / ( R e s + i Im S c - ie)[ i i ] e-*0

= - 2 zri X residue of the pole at m0 - iP/2 in the integrand of eq. (20a)

X the factors multiplying the same integral (22a)

47r2ig [s - (m 3 + m 0 - i i ' / 2 )211+1 /2 (m0 - m l - m 2 - i P / 2 ) 1'+1/2 (22b)

= (s -- 001+1/2(s -- s (p)) Pf(m0 - iP/2)

In a manner analogous to the analytic continuation o f I t ( s ) into the II sheet associated with the cut at s = Sl = (ml + m2 + m3) 2 [see eqs. (20a) and (21)], the knowledge of the above discontinuity provides analytic continuation o f l l ( s ) l II into the unphysical sheet associated with the above square-root cut at s = Sc. Analytic continuation into this region of the complex s-plane is necessary to uncover any influential nearby poles of the scattering T- matrix that may be present.

In conclusion, we have presented analytic expressions for the integral I t ( s ) in a phenomenological N / D formulation for two-body scattering states. When one of the two particles is an isobar, an average over the variable mass of the isobar is required. This averaging leads to different analytic properties of the function in I t ( s ) . Although its sheet structure is more complicated in this case, one finds one can still provide a prescription for analytic continuation o f l l ( s ) into the unphysical regions of interest.

R e f e r e n c e s

[1] See, e.g.H. Burkhardt, Dispersion relation dynamics (North-Holland, Amsterdam, 1969). [2] B.J. Edwards and G.H. Thomas, Phys. Rev. D22 (1980) 2772;

B.J. Edwards, Phys. Rev. D23 (1981) 1978; R. Bhandari et al., Phys. Rev. Lett. 46 (1981) 1111; I. Duck, Rice University preprint (1981); R. Bhandari, Virginia Tech preprint (1982).

[3] R. Bhandari, Phys. Rev. D25 (1982) 1262.

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the d function in a phenomenological nd model

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