EXTRAPOLATION OF THE S-MATRIX FROM THE LEHMANN'S REPRESENTATION

Transcription

1 IC/65/9 INTERNATIONAL ATOMIC ENERGY AGENCY V *' INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS EXTRAPOLATION OF THE S-MATRIX FROM THE LEHMANN'S REPRESENTATION N. LIMIC v,,x * t - \; 1965 PIAZZA OBERDAN TRIESTE

2 10/65/9 ATOMIC EIJBRGY AGEUCY INTERNATIONAL CENTEE FOE TESOEETICAL PHYSICS EXTRAPOLATION OP THS S-MATBIX FROM THE LEHMAOPS ESPEESBNTATIOU U. Limi6 TEIESTE February On leave of absence from the Institute "Eudjer Boskovic", Zagret.

3 ABE The extrapolation of the S-matrix to the complex angular momenta is obtained using the Lehmann's representation of the amplitude for the elastic scattering and the uni^nrity condition. In the half plane R&2- > Q "the S- ~ctrix is analytic rojular and satisfies the extended unitarity condition.

4 OP THE S-MATRIX PROM THE LEHMAEKT' S REPRESENTATION 1_. Xntr oduc t lor. In recent tin:es there have been extended investigations concerning the amplitude o_" L.ia -tn partial wave for elastic scattering and concerning the possioi...ity of the extrapolation of the complex,^-plane from its physical valu^ for integer Jc, The reasonable extrapolation was found using Kandela'jc;:. ; i proposition at out the analytic i *y of the amplitude for elastic scattering in the variables s,t and u. On the other hand, the amplitude of the st,~la partial wave is the magnitude which exists without regard to the possession or nonpossession of the mentioned analytic properties of the whole amplitude. That is, the analyticity of the amplitude in the Lehmann's ellipse guarantees the existence of the partial wave amplitudes. How the question appears ; is not the analyticity of the amplitude in the Lehmann's ellipse sufficient to discover a reasonable extrapolation of the S-matrix to the complex values > 1 Our intention is to show here that really the extrapolation of the partial wave amplitude follows from the existence of the analyticity of the whole amplitude for the elastic scattering in the Lehraann's ellipse if we keep to the usual supposition on the polynomial increase of the amplitude for large complex energies. The uniqueness of the extrapolation can be shown also if unitarity is required - It seems that the unitarity condition is not any rough supplementary condition as it is built in the Lehmann's representation in a certain iray. For ;;inolicity we shall treat the case of two l) 2) scalar fields. Irenes we shall use '' A & - co/mcosot)

5 The ur.itarity condition will be used for the partial wave amplitudes. The expansion of the whole amplitude in the Legendre series is: ~. X. (& if The function pp"/ has to be {_($ ~j^' J/Sj in the case of the scattering 3) of two equal particles and. its value must be positive on the upper side of the cut S > -4^ The generalisation of the function for the scattering of the particles with nonequal masses can be found in reference. The unitarity condition for the partial wave ampli tude reads; ct(^s).= f(s) jclf^s)! 2 ; d-5) The equality (1.5) enables us to write the amplitude in the form 2_; Extrapolation of the S-matrix '",:: vrant to decompose the fractions under the integral sign in (l.l) and (1.2) into the sum - "*' -4- L^. where C± and J?± are the functions of og and al only. The range of the functions db will be outside the Lehmann's ellipse and so the Heine's formula for the denominators ir-.zj.cari be used in order to obtain the expansion of the fraction in the Legendre's series. We prefer to use a new variable X instead of the variable OJ where the new one is defined by O =s co&hx,, X>0 r After transforming the amplitudes (l.l) and (1.2) in the suggested way they can be rewritten in the form:

6 fa-it*), L y S( 2.2 where? ~ Our intention i^ to consider the partial wave amplitude defined ~oy (l.4) and to prov-~ zhzx, it can be represented in the form:. If the weight function! p in this expression were a bounded function the equality (2.3) would be the consequence of the analyticity in the Lehmann's ellipse. As generally f> is distribution, we must prove that (2.3) follows from (2.1). Let C (R)\>Q the linear space of the infinitely differentiable functions from 8. = (-*-> i <1 ) to the complex values. Let B(M.f) be the linear sp;=.ce of the continuous uniformly bounded functions fro-t; &i - (Xi / <* &) o <x y -< X o, "to the complex values. We define the linear space which is induced by zhe seminorms <0 = C**(R)X B(ft*) and introduce the topology ^ % l, s (2.4} < x P where j^feji is the collection of the continuous functions with compact support and which vanish for large enough i?. "We can define the dual space,0 of the space oo and introduce the usual topology in j >*; p*(a) - la( A 0~ A ~. In this way our representations (2.1) and (2.3) become -she elements AO^/ S f Z) and (Z/j^SJ of the space c0 if we suppose their linearity on the space 0. To prove that (l.4j implies (2,3) it is sufficient to proclaim our functionals to be continuous on the space J&. That is we can find the sequence of the elements -3-

7 which tends to the element ^fif/x/x?) which is represented in the brackets of the expression (2.l) in the topology defined by the seminorms (2.4). This can be verified using the second Christoffel's formula (3.8 (21) of the reference^). Hence A C^t,, S, Z) -^ A {/, s, On the other hand Aft, >*,*)- Z.^e (Mf f)%(z] ct(s ei S ) where j >(*t,x} are the elements which are represented by the brackets in the expression (2.3). This proves our assertion. Before the analytic extrapolation of the amplitude defined by (2.3) let us make some comments about the existence of the representations (2.l) - (2.3). It was pointed out by JOST and L'SKViAXN ' that the integrals of the form (l.l) to (2.3) may not exist if the weight functions do not decrease strongly enough for large (O or coshx But if Lehmann's representation exists, then an integer^ can be found such that the quotient 9 } f s t c oso<[ / c^)/ccoi-m) possesses all desired properties. ' In this case we have for the scattering amplitude ' } Wow it is not difficult to calculate the partial wave amplitude defined by (1.4). Particularly if * "^.Zti-ithe corresponding amplitude Ct(^5) has the same form as if the additional difficulties were absent. That is, the amplitudeflf/vvconsists ^ ^^e integrals over the functions i P{s,e6Se( 1 io) 0 e f<ast](xt>i*c)) which decrease strongly enough if ^Zn-i and the limiting value M~> 00 can be performed in order to obtain the expression (2.3). Therefore we shall consider the amplitude (2.3) for ^ larger than some number < o in- the following. How the form of the amplitude df^s)appears to us to be the extra polation of this amplitude to the complex ^-plane. Using linearity and continuity of the functional Ct(*iSj and the fact that the derivative jkq^fcoij}(k±jec)) is again an element of the space }, we can prove the analyticity in Jir of our functional. Hence the amplitude represented by (2.3) is analytic regular function in the half plane In the same way for &e# > o it holds : Ah*)' Z ~ (i- -4-

8 (2.5) Let us now establish extended unitarity condition for the S-raatrix S(#,s) a j+ a.(p f s). Because of the analyticity and unitarity for the integers j h of the S-matrix, there must exist domain for any^)/^,>^, such that the function af^sj^ -4--&n<S({' l 5\ is regular and analytic in ocl. Even more there exists an interval I on the real - n axis in the domain oc+,, such that the function 6 is.real in : the interval I. That is, cos2s=?-2sjn 2 is real for real y ^± J^. Then the continuity of the function cosza the existence of the mentioned interval I. unitarity condition *&,$)- (?*$) or S/%s)S*/S*s)~ and reality of d/^, / sy ensure Therefore we have the / at least in Jt^. The unitarity condition for the S-matrix can be easily extended to the whole half plane X > <?^'>^ because of the analyticity of the function there.,5 /e, As the phase shift is a-regular function in the half plane Ke >-e a we conclude from the relation (2.6) that there is no pole or zero of the 5-matrix in the half plane / # > o > The representations (2.3) and (2«5) seem to have appropriate forms for the investigation of their asymptotic "behaviour for large variable & with fiz&> Je o. Really if the weight functions were continuous functions in the range (o f Jt ) the asymptotic forms can be easily guessed. In any case ne can establish an upper bound on the S-matrix in the variable using the uniform asymptotic expansions and the corresponding bounds of the Legendre's functions of the second kind. As we required the functional Afys) to be the linear continuous functional on the space au there must be a seminorm of type (2 4J such that the functional is bounded in the way I*Afys)\< Cp{i/J'. Hence we have the estimate

9 z The D-$ derivative of the function ak&yis equal to the function Q( W )^Q asymptotic form of the function Q^(w)fOT large This asymptotic form is uniform with respect to IV outside the ellipse with foci at i-/. We have to be careful only when w approaches the cut (\N -d. +*/) as w has different arguments coming from different sides of the cut. Hence a rough ("but for us sufficient) estimate is : (2. 7) Essentially the same bound is obtained for the increase of the function Sin^ (ff-f t s) along the straight lines in the plane & #>/ o which are parallel with the imaginary axis. In order to prove this the integral representation 3.7(5) of the reference of the function Q# fw) can be used. After rough estimation the following bound appears: In both inequalities (2.7) and (2.8) (& is a non-negative constant which can be arbitrarily small. can be represented in the form If we suppose that our functionals where Mp are measures on the set PocR, then the constant & appearing in estimates (2.7), (2.8) will be equal to zero. At the end of this section we shall prove that there exists only one analytic S-matrix which is regular for $j? ^ -e^olf; 1; it is bounded by the exponential function of * in Mee^ e c / 2) if the extended unitarity condition holds;and 3} if S-matrix tends to unity along the real axis. That is v because of the regularity and the unitarity, the S-matrix has no oole and no zero in Re#^ &. Hence the function S" f, S ) is o

10 uniquely defined if we prescribe that it tends to unity along a real axis. How if there are two extrapolated S-matrices, say S^ and $ 2 ({, s), then fl can be found such that -^ Ytfs) and -^ ^ffa) are hounded by <?*/>(cwl) t C<7 in & > c. As the difference 5f"(O)~^ * *, has zeros for integers, -c> $ this difference must vanish identically by the Carlson theorem. Hence our extrapolated amplitude CLf^s) is unique in its class. Acknowledgement The author is grateful to Professor Ahdus Salam and the IAEA for the hospitality extended to him at the International Centre for Theoretical Physics, Trieste. The author also wishes to thank Professor A.Martin, Professor K. Proissart and Dr. R. Raczka for helpful discussions.

11 H3PEREKTC3S 1) H. LEHMNU : Nuovo Cimento, K>» 579 (1958). 2) H. LEHMAOT : ffuovo Cimento, Supply L4_, 153 (1959). 3) R. G. MOORHOUSE : Huovo Cimento, 23., 233 (1962). 4) R. OEHME : Phys. Rev., 121, 1840 (l96l). 5) Bateman Manuscript Project : Higher Transcendental Functions, Vol. I, McGraw-Hill, New York (1953). 6) R. JOST and H. L3HMAMT : Huovo Cimento, 1, 1598 (1957) -8-