The unitarity condition for the resonance-particle amplitude

1. B

[

Nuclear Physics A266 (1976) 163 -- 172; (~) North-Holland Pubhshin# Co, Amsterdam

i

Not to be reproduced by photoprmt or microfilm w~thout written permtsston from tile pubhsher

T H E UNITARITY C O N D I T I O N

FOR T H E RESONANCE-PARTICLE A M P L I T U D E Yu A. S I M O N O V t

Natuurkundig Laboratortum, Vr~ie Umversiteit Amsterdam, The Netherlands Received 17 October 1975 (Revised 8 March 1976) Al~traet: The unitarity conditions (UC) for the resonance-particle amplitude are explicitly deduced f r o m the exact three-body UC. In case where two pairs out o f three particles resonate an additional term changes the quasi two-body form of the UC. Its presence is shown to guarantee the real analyticity o f the resonance-particle amplitude. The U C obtained possess two classes o f solutions, singular or regular at the Peierls points. Comparison with the Fredholm denominator o f the Faddeev equation for the resonance-particle amplitude enables one to choose the unique solution having no Pelerls singularities. The linear integral equation for the denominator of the resonance-particle amplitude ~s explicitly written a n d discussed.

The concept of a resonance-particle amphtude and the unitarity conditions ( U C ) f o r ,t are widely used 1-3). Although the resonance-particle amplitude does not have a direct physical meaning (as a quantity wtuch can be measured in the experiment), it enters the unitarity condit~ons for the resonance production amplitude and therefore is essential in any dynamical calculations. Moreover, a pole in the resonance-particle amplitude due to the same UC appears also in the production amphtudes and can show up as a resonance or a bound-state (a virtual state) in the resonance-particle system with the ~mmediate experimental consequences. Therefore the correct treatment of UC for the resonance-particle amplitude is an important task for our understanding of the dynamics of the resonance particle system. There are two main difficulties which have so far prevented a clear understanding, even of the correct way of writing of the UC for the resonance-particle amplitude. One is connected with the fact that the driving force, essentially the discontinuity of the pole graph (fig. 1), lies on a part of the usual unitarity cut, extending from threshold to infinity in the complex energy plane. It was shown in refs. 6-8), and also will be shown here in another way, that the correct definition of the physical amplitude for the resonance-particle interaction requires both cuts to be exactly superimposed; as a result no Peierls singularities occur in the denominator of the resonance-particle amplitude. This is in contrast with the results of refs. 4, 5, 9). ÷ Permanent address: Institute of Theoretical and Experimental Physics (ITEP), Moscow. 163

Yu. A. SIMONOV

164

m~.--~(..(~ )

~'~ m$

Fig. 1. The pole graph, representing the driving force for the resonance-particle amphtude. Another point of importance in the resonance-particle UC appears in the situation, when two pairs out of three particles can resonate. With only one resonating pair the UC have the same form as for a particle-particle scattering amplitude and ha~e been written explicitly many times 2, 2, 4, 5). With two resonating pairs the UC were reported in refs. 7, 8), but no close investigation was made there. Here we give the exact derivation of the UC for the resonance-particle amplitude M in the situation when two pairs of particles may resonate. We discuss the real analiticity property of M and state that those are compatible with the unitarity conditions. We compare these results with the usual properties of the two-body scatterIng amplitude. Fmally we show how the ordinary N / D method should be modified to solve the UC for the resonance-particle amplitude and investigate the analytic properties of the denominator, especially in the neighbourhood of the Peierls "singular" points. As a result of our investigation we obtain the integral equation for the denominator D of the amplitude M and define its properties. With the usual definition of the T-matrix: S = 1 + i(2zr)4tS( Z P , - Z p~)T,f, the UC for T takes the form

1

Z

2i

(1)

3

where we have used the notation ]3

"

Z3 =

r-[

tt

ii p,

Z p, - Z p,) , 2e,(2

)3 •

(2)

From now on we confine ourselves to the three-particle initial, final and intermediate states and will show how the UC for the resonance-particle amplitude result. To this end we define as usual the sum of the connected 3 ~ 3 diagrams w:

T = g ti(ai)2E, 6~3)(P,- P;)(2rr) a + w(tr, a', s),

(3)

|

where t, is the two-body t-matrix for the ith pair (consisting of particles j, l # i) and tr, = (pj+pt) z.

UNITARITY CONDITIONS

165

The two-body t-matrix ~s normalized according to

l (t(a)-t(a*))

q(tr) (dO t(a) t(a*).

-

F

32~2x/a,J

Now we define the resonance-particle amplitude C(a,, try,,s), assuming that two pairs out of three can resonate, producing a resonance with the same width and position [we always have in mind the situation 4, s) of three spinless particles with masses m, # and/z, the resonance being formed in the system of particles m and l~, and the interaction in the system g, # being switched off]:

w(a, a', s) = E

•

t(a,)CL(tr,, ¢~.; s)PL(cos O,~)t(tr'~).

(4)

L i,r=l,2

Inserting (4) and (3) into UC (1) we get the UC for C(a, a', s), which have the form 1

t

2i [Ct'(tr+' o'+,

s+)-C~.(tr_, tr_, s_)]

Jl +S2+Ja+J4+Ss,

(5)

where the subscripts denote the signs of the imaginary parts; Jl does not contain CU

J1

=

+

7"C

4k(s, a)k(s, a')

PL(u)O(t-lul),

(6)

but it represents the discontinuity of the pole graph fig. 1, where k(s, a) and k(s, a') are c.m. momenta of the initial and final resonances respectively, and u is the cosine of the scattering angle at the point where the denominator of the pole vanishes:

11,,2 - m2) - 4s[ma + k2(s, a) + k2(s, a')]

(7)

8sk(s, ,)k(s, The terms Jz and J3 contain CL linearly, and we shall not need them. We can now use the arguments given in ref. a), to go over to the UC for Ct. with fixed arguments O', 0 " :

1 [CL(a, a', s+)--CL(a, a', s_)] 2i

Js

--St +Jg(a, a')+Js(a, a').

(8)

+ ~ = - . ~ _ . . , x , .m3 .__j~ (a)

(b)

(c)

FJg. 2. The diagrammatic representation of the UC for tile resonance-particle amphtude, eq. (8).

166

Yu. A. SIMONOV

The d~scontmmty gwen by eq. (8) xs illustrated in fig. 2, where we have attributed to figs. 2a, b and c the terms J l , J~ and J5 respecUvely. (The crosses at the hnes in fig. 2 suppose the propagators to be taken on the mass shell.) The explicit expressions for J4 and J5 are [see also ref. 8)]:

1 L f("-~-"")' da"k(s, a")q(a") CL(a, a+, s+)Cz(c'-', a', s_)lt(o-")l z,

J4 = 64/l:3j(,,,,+,,2)z

J5 =

) ~,,

(9)

N' SO"

[ f d(~l d~3 6~-~aj 4s

CL((Y ,

at+"

S+)t(a, +)CL(~F 3 _ ~', S_)t(O-3_ )

x Pt.(u(a,, O'a))O(1 -I,,(o',, o~)1). (lO) We see from the exact form of the UC for CL, given m (9) and (10), that these conditions are integral equations. The crucial step usually done ~s to exploit the resonance character of the tmatrix 8rrFa

t(o)

(I l)

-

q(a)(aR--a ) ' where aR = (m+#+e-½iF) 2, and e and F are the positron and the width of the resonance. We can now take all factors except [t(O"')[ 2 o u t of the integral in (9) at a = Re aR -- if, and do the remaining integration with the assumptions e E ~F and x / s - r e x - m 2 - m s - e > > ½ F . The result is

J, -

~rk(s, ~) Ca(a, ~+, s+)CL(~-, - -

q(~)4s

a',s_)0(~/s--,,,a -- N/a).

(12)

We used the notation, ff -- Re aR, t~+ = ff+i0, if_ = if--10. The same procedure can be applied to J5 with the result

F2~ 2

J5 = ¼rr~

Cr(a,

~+,

s + ) C r ( # - , a', s-)PL(u(#, ff))z(s),

(13)

where z(s) is the integral of two resonance factors over the Dalitz plot:

z(s) = 1 ( f dal_ d a a0 [ 1 - u(a, ,_~3)] ~ j j (,,.-,,,)(,,*-~)

(14) -

The U C (8) with the definitions(6),(12)-(14)willbe our main point of discussion Let us note first that we can put ~, a ' equal to a~., a ± with any signs for the imaginary parts, obtaining in this manner four amplitudes C(~±, ~±, s). One equation separates, that for C(6_, ~+, s), while the others are connected with each other. One can easily see that the same amplitude C(~_, t~+, s)enters the U C for the amplitude describing the process where two stable particles produce a resonance in the final state.

UNITARITY CONDITIONS

167

Calling this amplitude B(M 2, a+, s), we get the following UC:

25

I_ q~-s

+¼re--q2s t

BL(M 2, e+, s+)CL(ff-, if+, s_) "3i- . . . .

(15)

We have omitted above the contributions from all channels other than resonanceparticle channel. We call in the following the amplitude CL(~-, 6÷, s) the "physical" resonanceparticle amplitude - the name refers to the physical process where an incoming wave ~s present m the initial state and outgoing wave in the final state. An immediate consequence of the above choice is that the dynamical cut (6) occupies a part of the real S-axis. Indeed, we have used 6 +_ everywhere in the definition of the resonance-particle amplitude C(tY_, 0+, s), that is tr, a' with negligtbly small imaginary parts which produce a real dynamical cut in a trivial way. If instead we used a resonance-particle amphtude C(tr, tr', s) with a and a' taken at the corresponding pole positions.

tr = tr_ = m+l~+e-½iF,

a' = tr+ = m+It+~+½iF,

(16)

then the dynamical cut of the newly defined amplitude would again occupy a part of the real S-axis. The situation is different however for the amplitudes C(tr+, tr'+, s) and C(a_, ¢'_, s). Let us now introduce a new dimensionless energetic variably y instead of s and a new function M(y) instead of C(~_, 6+, s), the connection between old and new variables being s - ( x / a + ~t)2

ML(y) =

(17)

"s ,

(18)

where 7(s) = {[s - (x/~ - #)212~(x '~ + tO}~/2s

(19)

and So = (x/~ +#)z. With those definitions the UC (8) take the form

t A M = y(s°) ?(s) M ( y ) M ( y * ) - ~(y)O 1 ?(s) 2~ y(s) ¢(y)O 1 + x/),O(y) ~)So) ~ ytL(y)M(y)MO,*), (20) where the discontinuity ~(y) across the dynamical cut is

¢(y) = -¼r~ PL(u(ff, O)) F .~.,

(21)

168

Yu. A. SIMONOV

while the positive-valued function ~ Js defined as

( = (m+#+e)3[21a(m+lt+e)+zY(m+2#+e)]-~' [(2m + 21t + e)(m + ½c)(/a+ ½~)(2ju+ m + ~)]'~

(22)

The symbol 01 in the first and third terms on the r.h.s, of (20) means O(y-yl)O(y2-y), and 71, Y2 are the so-called Peierls singular points lo): Yi -

2m + e 2(2/a + m + c ) '

72 =

(m + 2# + e)(Zm + e) 2m 2

(23)

The definitions (17), (21)-(23) coincide with those in refs. 7, s). Note that M z (y) is the same Ct. up to a constant and therefore Mz (y) does not contain new singular~ties. In addition to the singularities present in UC (20) the function M= (y) of course contains the left-hand cuts due to the t-channel exchanges of any mass and due to u-channel exchanges with the mass M > x / ~ - # It appears that the pole diagram fig. 1 produces not only the disconutinity J l (eq. (6)), but also a discontinuity - J l on the cut ( - oo, 0) in the s-plane. The latter should be combined with other t- and u-channel exchange contributions, the sum of all discontinuities on the left-hand cut we denote by ~(y), assuming it to be non-zero in the interval [Ys, Y4] in the y-plane. It is convenient now to introduce the analytic function

PO')

' 'z (Y)~(Y ' ' )(Y(Y' )/7(Yo))0 i = y rJ o ° x / -Y; (Y(Y)/(Y(Yo))--Y

~

Y'(Y'--Y)

, dy,

(24)

7(Yo) -- 7o, so that the UC (20) take the form 1

2i

AM = ~(y)O(y-y3)O(y4-y)+~(y) 7(s°) 01 + P(Y)-P(Y*) M(y)M(y*). y(s) 2i

(25)

We wdl solve the UC (25) by the use of the well-known N/D method, namely

= yL U(y)

(26)

D(y) " Then as is shown in refs. 6, 7), associating the singularities of ~ and ~ with N(y) and those of p(y) with D(y), we get in general two kinds of equations for N, D:

A 2ii D I(Y) =

I

P(Y)--P(Y*) ytNl(y-i&) 2i (2v)

2i

UNITARITY CONDITIONS

169

or

A D , (y) = • -

I

AN2(y) -

P(Y) - P(Y*) N , (y + i6)y L 2i

(28)

@ '° +~) Oz(y+i6)y-L.

2i

One can prove that the linear combination of the type (~Nx +flN2)/(~DI +flDz) with ~ and fl as arbitrary constants is also a solution of (25) if NI, D1 and Nz, Dz satisfy eqs. (27) and (28)*. Assuming that D and N do not possess poles (which essentially takes away the socalled C D D poles), we can eqmvalently write (27) as (L = 0, 1)

o1=1-

Y f ~ P(Y')-P*(Y') y'LN~(y'--i~)dy', o y'(y'-y)

f,~'4

1 dy'~(y') D,(y'-i6)+ l i yz dy'¢(y') 7o Dl(y'-i6). N l = ~ , ~ ( y ' - y)3''L ~-'r, ( y ' - . r ) y 'L 7

(29)

(3O)

Now we can use the procedure described m ref. 7) to get the linear equation for

D,(y): 1 fr, dy,,~(y,,)Dz(y,, ) Dl(y ) = 1--~,Jr3 y,,L+X(y,,_y) [yLp(y)y,,_yy,,Zp(y,,)] ~Y2

--11 - =| ~ yt

P!

tt

t!

t!

•

dy ¢(y )(~o/~(Y ))Ot(Y -,6)~ L . . . . , ,,L . . . . ~ LY pl,y)y -3Y PlY +i6)]. y

[y - y)

(31)

The equation for Dz(y) obtains by putting 6 --, - 6 , in the arguments of D and p under the integral in (31). Note that the physical boundary consists of the points of y with a positive small imaginary part• Therefore the last integral in (31) is regular at y = y~, Y2 on the physical boundary and hence Dx(y) is not singular there, whereas D2(y) is singular at y = y~, Y2. In other words D2(y) possesses Peierls singularities, while DI does not. To choose the appropriate solution of the UC (25), one can use the same procedure, as given in ref. 6), the only distinction is that in ref. 6) the special case Js = 0 was considered. In that case in refs. 6, 7) [assuming also r(s)/7(so) = I ] the exact solution of the UC was found for Da and D2. The same procedure applies in our case. Indeed, let us consider for simplicity the case when e << m, #, and m = #. The first integral (31) is irrelevant for our purposes and we shall omit it for a moment. It is an easy matter to get the expressions for D~ and Dz keeping only terms of zeroth and first power in ~, that is in F/~. ? This has been shown by E. S. Nikolaevski and M. I. Pohkarpov.

170

Yu. A. SIMONOV

The answer is (for L = 0)

Dl(y )

2iF 3(x/y +x/~) = 1-- __e~yln \ / ~ + x / ~ ,

D2(y)

= 1- ~

2 i r . 3 ( x / ~ - x/y ) In ex/3Y x/3-x/Y

(32)

(33)

Now we can obtain instead the expression for the Fredholm denominator of the resonance-particle amphtude in the same order of Fie, using the Faddeev equation for the resonance-amplitude scattering. The result is given in ref. 6) and we quote it here for the convenience of the reader: Diet'(y) = 1 -

2iF_ In ex/3y

x/3(x/Y+X/~ x/~+ x/¼(1 + y )

(34)

The expression (34) is analytic at the Pelerls singularities and it should be the same (up to the off-sheU corrections) as D (y), so the only permissible solution is Dl(y ). and not D2(y) or any hnear combination of Dx and Dz. In this way we have proved, that the denominator of the resonance-parttcle amplitude does not possess Pelerls ~mgulant~es on the physical boundary, contrary to the claim made in ref. 9). Therefore the Peierls mechamsm for the generation of a resonance in the system cons,sting of a resonance and a particle does not work. All results obtained m refs. 4, 5) seem to refer to the amplitude M2 = yr. N2/D2 and not to the physical amphtude Mx =

yr. NI/D1

The integral equation (31) for the denominator of the physical amplitude of the resonance-particle interaction, as it is written, should be solved not on the physical boundary Im y ~ 0, Im y > 0, but rather in the region Im y < 0, because this is just where D~(y') enters in the mtegrand Then the kernel of the first integral on the r h s of (31) is not singular, whereas the kernel of the second integral is singular; the singularity being of 1/(x-y) type. When the values of D~ (y-i6) are found numerically, we can analytically continue the known solution onto the physical boundary, y > 0, Im y > 0 In the last part of this paper we d~scuss the general features of the UC (20). Let us define the "resonance-particle S-matrix":

SR(y)

= l + 2i(v/;--

~yzr(y)) y(y) M(y), (yo)

(35)

so that the UC (20) can be written as

Sa(y) S~(y*)

= l - 4~(x/~-

~yxL(y)).

(36)

Note, that UC for the scattering amphtude S(y) of two stable particles with a given inelasticity q2 = 1 - 4 ~ x/Y, have the same form as in (36) with z L = 0. Of course, the

UNITARITY CON DITIONS

171

two-body scattering amplitude is real analytic: S0,": ) = S*(y), which is compatible with the UC, assuming ~ < l/4x/y. For the resonance-particle scattering amplitude, on the other hand, ~ may be large due to the mass ratio in ~ eq. (22). Moreover, for particles with spin and ~sospm the sign of ~ may be reversed. The Investigation of the r.h.s, of eq. (36) shows directly that its sign is positive for all real ~ if rL(y) > 1. Now it remains only to show that those functions are indeed real analytic. We can use the following expressions, valid for or, a' below the three- and many-particle thresholds:

1 , 2i [CL(a+, a , s + ) - C z ( a - ,

, ( d a " t(cT+)C(a~ a' s+), a , s+] = , , 32fowls

1 f - -da" -' s+)-C,~(a, ,~_, ' s+)] = a32~,, t ( a ~s + ) C ( ,,a , 2i [CL(cr, a+,

a +,, , s+),

(37)

(38)

where the integration in the integrals is to be performed over the region, defined by the inequalities lu(a, a")[ < 1 and [u (a', a")l < 1 for eqs. (37) and (38) respectively. Now, assuming a' = a, and using (37), (38) one can easily prove C(a_, a+, ~) = C(a+, a_, ~).

(39)

But the generalized Schwarz reflection principle tells us that

c*(~, ,', s) = c(~*, ~'*, s*).

(40)

Combining (40) and (39) we conclude that M*O, ) = M(y*), (41) and hence SR(y*) = SR*(y). It is important to note, however, that both N and D for the resonance-particle case are not real analytic, in contrast to the two-particle scattering amplitude. From (41) it follows, that ( l / 2 i ) d M ( y ) = ImM(y), and with the help of the trivial inequality [lm M(y)[ =< [M(y)[ we easily get the following bounds for the modulus of the resonance-particle amplitude:

12~o/71 -< IM(y)I < 21~170/~ I+\/1-4~(x/Y-yT~ ) = II-\/1-4~(~,/y-yr~)l

(42)

Note, that the UC do not give any restriction for M ( y ) in the case ~ = l/rx/y, because the upper bound in (42) then tends to infimty. However, this infinity is of no consequence for a measurable quantity, such as BL [eq. (15)], since the UC (15) contain exactly the factor x/Y-vYq, multiplying M(y).

172

Yu. A. SIMONOV

Our last remark concerns the explicit expression of M(y) in terms o f D(y). Indeed, using (27) after simple algebraic manipulations one gets:

M(y) = D ( y _ ) [ 1 - 4 ¢ ( ~ / y - y ~ ¢ ) ] - D ( y + ) D(y +)2i[r(y)/~(yo)] [~/3' -- yz¢] "

(43)

which is valid for y on the cut (0, ~ ) . This paper was completed while the a u t h o r was staying at the Vrije Universiteit in A m s t e r d a m in the f r a m e w o r k o f the exchange p r o g r a m between R e a c t o r Center of Nederland and A t o m i c Energy Commission of USSR. It is a pleasure for the a u t h o r to t h a n k all the authorities involved in arranging this visit and the theoretical groups of the Vrije Universlteit and the Rijksuniversiteit Utrecht for kind hospitality. H e acknowledges also the discussions with B. Bakker, H.Boersma, L . K o n d r a t y u k . T. Ruigrok, J. Tjon and M. van der Velde.

Note added in proof." The present formulation is different from that o f Gustafson (Nucl. Phys. B63 ( 1 9 7 5 ) 3 2 5 ) w h e r e the resonance-particle amplitude is defined as containing at least one direct (non-rescattering) interaction. Our resonance-particle amplitude contains in addition all rescattering terms. References I) 2) 3) 4) 5) 6) 7)

J. S. Ball, W. R. Frazer and M. Nauenberg, Phys. Rev. 128 (1962) 478 S. Mandelstam, J. E. Paton, R. F. Peierls and A. Q. Sarker, Ann. of Phys. 18 (1962) 198 G. F. Chew, The analytic S-matrix (Benjamin, New York, 1966) I. J. R. Aitchison and D. Krupa, Nucl. Phys. A182 (1972) 449 M. Kac, Ann. of Phys. 81 (1973) 113 A. M. Badalyan, M. I. Polikarpov and Ya. A. Slmo~ov, preprint ITEP-38 (1975) (m English) A. M. Badalyan and Yu. A. Slmonov, Yad. Fiz. 21 (1975) 890; Particles and Fields B6 (1975) 300 8) Yu. A. Slmonov, Yad. Flz. 22 (1975) 845 9) D. D. Brayshaw and R. F. Peierls, Phys. Rev. 177 (1968) 2539 10) R. F. Peierls, Phys. Rev. Lett. 6 (1961) 641