IL NUOVO C I ~ E N T O VOL. 7 A, N. 3 1 Febbraio 1972

I n c l u s i v e C r o s s - S e c t i o n s a n d D i s c o n t i n u i t i e s .

J . C. POLKINGHORNE

Department o/ Applied Mathematics and Theoretical Physics University o] Cambridge - Cambridge

(ricevuto il 14 Set tembre 1971)

Summary. - - The effect of physical-reglon Landau curves on the inclusive differential cross-section is invest igated for a typical class of curves. I t is concluded tha t the differential cross-section is a to ta l discontinuity in the missing mass variable M S of an ampli tude evaluated on an un- physical sheet. There is a point of nonanalyt ic i ty at each new singularity.

1. - Introduction.

T h e r e has r e c e n t l y b e e n c o n s i d e r a b l e i n t e r e s t in i n c l u s i v e p roces ses of t h e f o r m

(1.1) a ~ b - - > c ~ X ,

w h e r e X is a n u n o b s e r v e d h a d r o n i c s y s t e m a n d one is c o n c e r n e d w i t h t h e

i n c l u s i v e d i f f e r e n t i a l c ross - sec t ion c o r r e s p o n d i n g to s u m m i n g ove r a l l p o s s i b l e

s y s t e m s X for a g i v e n v a l u e of t h e a s s o c i a t e d m i s s i n g m a s s

(1.2) M2 = (p~ + p b - - p o ) 2 .

T h e d i f f e r e n t i a l c r o s s - s e c t i o n is a s u m of t e r m s of t h e f o r m shown in F ig . 1,

c o r r e s p o n d i n g to a l l t h e i n t e r m e d i a t e s t a t e s w h i c h a r e p h y s i c a l l y access ib le

for t h e g i v e n v a l u e of M ~, t h e + l a b e l d e n o t i n g p h y s i c a l a m p l i t u d e s a n d - -

t h e i r c o m p l e x c o n j u g a t e s .

555

556 5. c. POLK~NG~O~N~-

I~U~ELLER (1) has suggested t h a t th is cross-section corresponds to a discon- t i nu i t y of the s ix-point ampl i tude for

(1.3) a ~ b ~ c - - > a ~ b ~ e .

A consequence of th is would be t h a t Regge l imits for (1.3) could be used to find the f o r m of the different ia l cross-sect ion for (1.1) in a p p r o p r i a t e a s y m p t o t i c

o.

Fig. 1. - A term contributing to the inclusive differential cross-section.

regimes. This idea has p roved to be the s t a r t i ng po in t of a growing and impor - t a n t phenomenolog ica l discussion (2). I t is the re fo re of some in te res t to enquire

if the inclusive cross-sect ion is indeed a d i scont inu i ty and, if so, of wha t am- p l i tude in which var iable . There has a l r eady been some discussion of this ques- t ion (3.4). However , STAPP gives no deta i led account of the effect of phys ica l region L a n d a u curves (*), whilst TA~ relies on the ana ly t i c p roper t ies of a

s imple F e y n m a n d iag ram and i t is known (5) t h a t the s ingular i ty s t ruc tu re of ind iv idua l u n i t a r i t y integrals~ such as Fig. 1, is quite different.

The a im of this p a p e r is to m a k e i t p lausible t h a t the inclusive differential cross-section for (1.1) is a t o t a l d i scont inu i ty in M 2 of an unphys ica l ampl i - tude for the process (1.3). B y t o t a l d i scon t inu i ty is m e a n t the change in the va lue of t he func t ion eva lua t ed above e v e r y cut in M ~ to i ts va lue u n d e r n e a t h eve ry cut in M ~. The set of cuts to be encircled in this way includes not only the n o r m a l th resholds in M e b u t also higher physical-region L a n d a u curves (5.,).

(1) _A_. H. ~IUELLER: Phys. _t~,ev. D, 2, 2963 (1970). (2) H.M. CHAN, C. S. HSUE, C. QUIGG and J. M. WANG: Phys. Rev. Lett., 26, 672 (1971); C. D. DE TAR, C. E. JONES, F. E. Low, J. H. WEIS and J. E. YOUNG: Phys. l~ev. Lett., 26, 675 (1971); J. ELLIS, J. FINKELSTEIN, l ~. A. •RAMPTON and M. JAeos: Phys. Zett., 35B, 227 (1971); H. M. CHAN and P. HOYER: Phys. ZeSt., 36B, 79 (1971). (3) H . P . STAPP: Berkeley preprint, UCRL-20623. (4) C. I. TAN: Brown University preprint. (*) Note added in 10roo/s. - A more detailed discussion is given by CAHILL and STAPP; Berkeley preprint. (s) M . J . W . BLOXHAM, D. I. OLIV~ and J. C. POLKINGHORNE: Journ. Math. Phys., 10, 494, 545, 553 (1969). We also avail ourselves of some unpublished work of these authors. However, the attempt is made to make this paper reasonably self-contained. (6) R . J . EDEN, P. V. LANDSHOFF, D. I. 0LIV~ and J. C. POLKINGHORNE: The Analytic S-Matrix (London, 1966).

INCLUSIVE CROSS-SECTIONS AND DISCONTINUITIES 557

I n general , these l a t t e r also involve the va r i ab le

(1.4) S = ( p ~ ÷ p b ÷ p ~ ) ~,

and i t is unders tood t h a t the to ta l d i scont inui ty in M 2 is to be eva lua ted a t cons tan t S so t h a t all such curves count towards the M2-discontinuity. The

effect of these higher L a n d a u curves is the ma in difference be tween the inclu- sive differential cross-section and the t o t a l cross-section. The opt ical t heo rem relates the l a t t e r to a t o t a l d i scont inu i ty of the elastic two-par t ic le ampl i tude , bu t the only physical-region singulari t ies of this ampl i tude are no rma l thresholds.

The to t a l d iscont inui ty in M s defined in this way is not ana ly t ic bu t is only piecewise ana ly t i c wi th a po in t of nonana ly t i c i ty a t each new thresho ld

or L a n d a u curve. This nonana ly t i c i ty is essent ia l to p rese rve the r ea l i ty of the differential cross-section. The to t a l cross-section is s imilar ly nonanaly t ic

a t each n o r m a l threshold . The inclusive differential cross-section will be shown not to have no rma l

thresholds in S. In this sense i t satisfies the S te inmann- l ike re la t ion

(1.5) disc s [discM. A] ~- 0 ,

wi thin each region, bounded b y M 2 n v r m a l thresholds and physical-region L a n d a u curves, in which i t is analyt ic .

We only claim plausibi l i ty for the resul t for two reasons. One is t h a t we

do not a t t e m p t to discuss in deta i l cer ta in proper t ies of cross-energy no rma l threshold discontinuit ies which have only been inves t iga ted thoroughly by the methods of this p a p e r in cer ta in s imple cases (5.7.s). We assume in Sect. 2

t h a t the na tu ra l general izat ions of these resul ts hold for thresholds of high mult ipl ic i ty . Not only is this in tu i t ive ly l ikely bu t i t is also suppor ted by the inves t igat ions of COSTER and ST~,~P (9) who use somewhat different me thods and in i t ia l assumpt ions . The second l imi ta t ion is t h a t we shal l no t a t t e m p t an exhaus t ive discussion of all physical-region L a n d a u curves bu t will con ten t ourselves in Sect. 3 and 4 wi th the discussion of some simple, bu t we bel ieve typical , examples . Our conclusions are summar i s ed in Sect. 5 whils t an Ap- pendix s ta tes sufficient resul ts of previous analyses (s) of the s ingular i ty struc- ture of uu i t a r i t y in tegrals to make this p a p e r self-contained.

2. - G e n e r a l c o n s i d e r a t i o n s .

I n addi t ion to the var iab les M s and S defined b y (1.2) and (1.4), we are

also concerned wi th the square of the cent re-of -mass energy for the pro-

(7) D. I. OLIVE: .STuovo Cimento, 37, 1422 (1965). (8) P. V. LANDSHOFF, D. I. OLIVE and J. C. POLKINGHORNE: Journ. Math. Phys., 7, 1593 (1966); A. R. WHITE: Nuovo Cimento, 59A, 545 (1969). (9) J. COSTER and H. P. STAFF: Journ. Math. Phys., 11, 1441, 2745 (1971).

~ J. C. POLKINGHORNE

cess (1.1), given by

(2.1) s = (po + p~)~.

I t will be convenient to distinguish the s-variable associated with the top bubble in Fig. i f rom the s-variable associated wi th the lower bubble. Although they are numerical ly equal, the opposite signs on the bubbles imply t h a t these two variables are eva lua ted on opposite sides of the re levant cuts. The upper var iable will be denoted by sl, the lower by s~. The other invar iants associated wi th the bubbles

(2.2) t = ( p , - - ; o b V , u = ( p b - - p c ) * ,

m ay be t aken spacelike and so free f rom cuts. Because of the to ta l ly forward na ture of the react ion {1.3) there is a re la t ion

(2.3) S + M S = s l + s , + 2 m ~ .

Whilst in evaluat ing the inclusive differential cross-section we must respect (2.3)9 we are free in in te rmedia te steps to cont inue off this manifold.

We wish to consider an expression of the form of a sum of t e rms like Fig. 1 and see if i t can be identified wi th a d iscont inui ty in M S. I f this is to be done successfully th ree problems mus t be solved:

i) In the lower bubble s3 is below its cuts, t h a t is eva lua ted in the l imit f rom below the real axis, to correspond to the - - label. I t was soon recog- nized (~.4) t ha t no manipulat ions re la t ing to the MS-channel, and discon- t inuit ies thereof, could achieve this. I t is therefore necessary to consider discon- t inui t ies in M S of an ampl i tude which a l ready has s~ unde rnea th i ts cuts. T h a t is to say ins tead of the physical ampl i tude for the process (1.3) one considers

discontinuit ies of the ampl i tude

(2.4) A ( + , + , - - ) ,

where the signs re la te to the disposit ions of sl, M S and s~, respectively, with respect to the i r cuts. We consider in (2.4) t h a t S is also above its cut. Rela- t ions involving ampl i tudes with S below its cut may be obta ined by complex conjugation, if we recall the rea l i ty of the differential cross-section.

An ampl i tude of the form of (2.4) cannot be obta ined f rom the physical scat ter ing ampl i tude by analy t ic cont inuat ion within the to ta l ly forward sub- space for the process (1.3) (since necessari ly sl = s~ within tha t subspace), but, of course, i t can be obta ined via a pa th of analyt ic cont inuat ion going outside tha t subspace (and indeed outside the physical region).

INCLUSIVE CROSS-SECTIONS AND DISCONTINUITIES 5 5 9

ii) The resul t for the d iscont inui ty m u s t be in accord wi th the p roper t i es of no rma l thresholds in cross-energy var iables . The ana logy wi th the way in which the opt ical t heorem evalua tes a t o t a l d i scont inui ty in s across the n o r m a l thresholds in t h a t channel ce r ta in ly makes i t in tu i t ive ly plausible t h a t our expression does the same for no rm a l thresholds in the M~-ehannel. A more

detai led invest igat ion is technical ly quite complex. We are conten t to sketch

the considerat ions involved. Changing the bounda ry value on s2 will no t affect the discussion of no rma l

thresholds in M 2, which are therefore expected to exhibi t the same s t ruc ture as in the phys ica l ampl i tude (7-9). Eva lua t i ng the d iscont inui ty of this am- p l i tude across the threshold in the immed ia t e ne ighbourhood of the thres- hold, yields an express ion s imilar to Fig. 1 excep t t h a t the va r iab les as sociated wi th the M~-channel are not below all the i r thresholds (as a - label would require). I n s t ead , the t o t a l channe l energy M ~ is below only the par t icu la r threshold under considera t ion bu t above the lower-lying thresholds .

+----5_ X

(n-1)2~ 2

Fig. 2. - The /-limit associated with normal threshold discontinuities.

T h a t is, i t is eva lua ted in the / - l imit of Fig. 2. Similar ly the subenergies as- sociated wi th the subsets of par t ic les in the M 2 channe l are eva lua ted in the i r co r respond ing / - l imi t s .

However , the d iscont inui ty of the ampl i t ude across the (ntu) 2 threshold i tself has all the higher (n '#) ~ thresholds , wi th n ' ~ n, and these m u s t be t a k e n into account in eva lua t ing the to t a l discont inui ty . A s imilar effect oc- curs in considering discontinuit ies of the two-par t ic le elastic ampl i tude across i ts s-channel normal thresholds . This l a t t e r case is comple te ly unders tood (6) and one knows t h a t the effect is to give a to t a l d i scont inui ty in accord wi th

uni ta r i ty , t h a t is to say a sum of phase-space integrals wi th one of the bubb les eva lua ted in the - - l imi t below all the s-channel cuts. An exac t ly s imi lar effect is to be expected wi th the M 2 discont inui ty , so t h a t one will indeed ge t

a sum of expressions of the fo rm of Fig. 1 for the to t a l d iscont inui ty . We do

not a t t e m p t a detai led proof (which was one of our reasons s ta ted in the In t ro - duction for only claiming plaus ibi l i ty for our results) bu t the resul t can hard ly

be in doubt .

However , there is a way in which cross-energy in tegra ls l ike Fig. 1 differ f rom the uu i t a r i ty in tegrals associated with a two-par t ic le ampl i tude and th is furnishes our th i rd requi rement .

5 6 0 J . C. POLKINGHORNE

iii) We have argued t ha t expressions of the desired form are re la ted to discontinuit ies in M 2 at least near the normal threshold in question. However , as the expressions are cont inued in M ~ away f rom the normal threshold, sin- gulari t ies in the bubbles of Fig. 1 may impinge on the in tegra t ion region and dis tor t the phase-space integrals. I t is essential for the in te rp re ta t ion as an inclusive differential cross-section t ha t this dis tor t ion should not in fact be allowed to occur. The possibil i ty of dis tor t ion corresponds to the encounter ing physical-region Landau curves (s.6), and the enforcement of the no-dis tor t ion prescr ipt ion determines how these singularities mus t be t rea ted . This problem has no coun te rpa r t for the to ta l crossJseetion where there are no possible sin- gulari t ies of the bubbles in the phase-space integrals capable of causing dis tor t ion.

Some simple examples of these effects have a l ready been s tudied (8). I t is the main purpose of this paper to de te rmine how behaviour in M s mus t be prescribed in order to main ta in undis tor ted phase-space integrals in Fig. 1 and to show tha t this is consistent with the in t e rp re t a t ion of the inclusive dif- ferent ia l cross-section as a to ta l d iscont inui ty in M 2, t aken across bo th the set of normal thresholds and also the L a n d a u curves in question.

This is invest igated in the following two Sections and the conclusions sum-

mar ised in Sect. $.

3 . - T r i a n g l e c u r v e s .

Triangle singularit ies like those associated wi th Fig. 3 correspond to the effect of normal threshold singularit ies within the % bubble of Fig. 1. The

61

Fig. 3. - A triangle diagram.

corresponding Landau curve is sketched in Fig 4, which also represents the

fact t h a t the physical region cer ta in ly satisfies sl > M ~. The are BC cor- responds to posi t ive ~'s and the arc AB to g's of mixed signs as indicated. B is the point of so-called effective intersect ion wi th the normal threshold h r

in M ~ corresponding to ~1----0. The Landau curve corresponds to the points a t which the normal threshold

in the bubble first impinges in the region of phase-space in tegra t ion of Fig. 1. We wish to cont inue across the curve in such a way t h a t this normal threshold

I N C L U S I V E C R O S S - S E C T I O N S A N D D I S C O N T I N U I T I E S 561

will not cause the integrat ion region to be dis tor ted f rom its real plane. This

is just what is called the na tura l continuation, discussed in the Appendix, where

simple rules are also given for determining how to make this continuation. These rules tell us t ha t we must continue across the arc BC in an M S --F i~

sense whilst on A B we must continue in an M S - i s sense (*).

S~ ~ N , 4 physical / y region

Fig. 4. - A triangle Landau curve in the physical region corresponding to Fig. 3. The signs correspond to the signs of the £s.

However, this is not the whole story, for the tr iangle diagram of Fig. 5

gives an identical Landau curve to t ha t of Fig. 4 except t h a t sl is replaced by

(the numerically equal) s~. Because the normal threshold which generates Fig. 5

is contained within t h e - bubble of Fig. 1 the na tura l continuat ions are just

opposite to those of Fig. 3, t ha t is M S - - i s on BC and M 2 + is on A B .

6 3

Fig. 5. - A second triangle diagram associated with Fig. 3.

The clash of prescriptions for Fig. 3 and 5 is an example of the nonanaly-

t ici ty which is inevitable if one is to preserve the real i ty of the inclusive dif-

ferential cross-section. I t is clearly a general feature, for all physical-region

Landau curves will occur in clashing pairs in this way for the total ly forward

si tuation (1.3).

(*) This is because the normal threshold associated with c¢ 1 has a + ie displacement since it comes from a ~- bubble. A normal threshold from a -- bubble has an associated -- it diplacement.

3 7 - I I Nuovo Cimento A.

5 ~ J. C. POLKII~GHORlq]E:

We now wish to see t ha t these clashing prescr ipt ions are also consis tent wi th the not ion t h a t we are evalua t ing a t o t a l d i scont inui ty in M ~. In the no- t a t ion of (2.4) this means t h a t we are considering

(3.1) A(-}-, ~-,--)--A(+,--,--).

Any Lan dau curve t h a t we encounte r which is a s ingular i ty of the inclusiv~ differential cross-section t hen has to be a s ingular i ty of one or bo th of t h e te rms in (3.1). ]Koreover, to preserve the to ta l d iscont inui ty in te rpre ta t ion i t is necessary t h a t curves which we cont inue pas t in an M 2-{- i t sense are sin- gulari t ies only of A ( ~ , ~- , - - ) and curves which we cont inue pas t in a n M~--i~ sense are singulari t ies only of A ( - ~ , - - , - - ) . In this way we shall be cont inuing to evaluate the difference between being above all singularit ies in M S (including the Landau curves as well as normal thresholds) and being be low all singularit ies in M ~.

I t is known t h a t the singulari t ies of A(~-, ~ , -/-) are the same as those, of pe r tu rba t i on t heo ry (5) and we shall assume t h a t the same is t rue for t h e ampl i tudes in (3.1). Then (s) A( ~-, ~-, --) is s ingular on the arc ~ C for Fig. 3 and on the arc AB for Fig. 5, whils t A ( - ~ , - - , --) is s ingular for Fig. 5 on Bf f a n d for Fig. 3 on AB. These are jus t the proper t ies we require to be consistent~ with the differential cross-section being a t o t a l discontinui ty. A(~-, -~,--)~ always requires an M ~ -~ is cont inua t ion and A(-~, - - , - - ) an M2--ie continua~

t ion.

4. - H i g h e r L a n d a u eurves .

I~ormal threshold singularit ies in bo th the bubbles of Fig. 1 can generate, singularities corresponding to box diagrams of the t y p e of Fig. 6. The rules of:

a2 %

Fig. 6. - A square diagram.

the Appendix tel l us t ha t this would not be singular on posi t ive g arcs, because of the opposite /e-prescriptions associated wi th ~- and -- . This is a s imple i l lustrat ion of the difference between un i t a r i t y integrals and physical scat- te r ing ampli tudes . However , an arc wi th a 's having signs ( 4 ~ - - -~ - ) c a a enter the physical region and the na tu ra l cont inuat ion across i t would b e

INCLUSIVE CROSS-SECTIONS AND DISCONTINUITIES 5 6 ~

M 2 ~ is. There will necessar i ly be a coincident L a n d a u curve wi th signs (-- ~- ~- ~ ) requir ing an (M s - - ie)-continuation, so ano the r nonana ly t i c i t y is encountered, as expected. Again we can i n t e r p r e t this as due to cont inuing A ( ~ , ~ - , - - ) above i ts MS-cut and A( ~-, - - , - - ) below i ts MS-cut, so t h a t the to t a l d iscont inui ty i n t e rp re t a t i on is ma in ta ined . The curve associa ted wi th Fig. 6 also depends upon S bu t we are regard ing i t as giving a s ingular i ty in M s a t fixed S.

Similar considerat ions a p p l y to more compl ica ted L a n d a u curves, such as

the square wi th d iagonal of Fig. 7. Singular curves a lways come in pa i rs wi th

Fig. 7. - The square with diagonal.

clashing MS-prescript ions for na tu ra l cont inuat ion . The in t e rp re t a t i on as a t o t a l M2-discontinuity is more compl ica ted to ver i fy in general , though i t can be checked for Fig. 7. There seems no reason to doub t i ts genera l t r u t h bu t

the absence of a detai led a r g u m e n t to this end is the second reason for c la iming only plaus ibi l i ty for the general conclusion of th is paper .

The differential cross-section will be piecewise ana ly t i c in regions bounded by M 2 no rma l thresholds and physical-region L a n d a u curves. Wi th one such region the only possible singulari t ies are no rma l thresholds in S. However , the rules of the Append ix immed ia t e ly show t h a t these are not s ingulari t ies of Fig. 1 so t h a t eq. (1.5) follows, in the l imi ted sense exp la ined in the In t roduc t ion .

5 . - C o n c l u s i o n .

The inclusive differential cross-section is a t o t a l d i scont inu i ty in M S of the unphys ica l amp l i t ude (3.5). This to ta l d i scont inui ty is to be calculated a t fixed S and all physical - region L a n d a u curves which depend upon S and M 2

are to be t r e a t e d as cont r ibu t ing to the d iscont inui ty . We have only expl i -

c i t ly considered the curves of Fig. 3, 5, 6 and 7 bu t there is every reason to believe t h a t the resul t is general .

This to ta l d iscont inui ty is piecewise ana ly t i c wi th a po in t of nonana ly-

t i c i ty a t each new threshold or higher s ingular i ty . This nonana ly t i c behav iour

enables the differential cross-section to r ema in rea l for all M 2.

This t o t a l d iscont inui ty resul t provides a just i f icat ion of the phenomeno- logical discussions (3) p rov ided t h a t Regge theo ry appl ies on the unphys ica l sheet corresponding to (2.4). Of course, i t is r e m a r k a b l e t h a t the Regge fo rm

has an ana ly t i c s t ruc tu re which shows no sign of the L a n d a u curves we have

~ 4 J . C. POLKINGHORNE

been discussing. The re la t ion of physical-region Landau curves to the Regge regime has been discussed by GODDARD and WHITE (~0), who connect un i fo rmi ty of the Regge l imit wi th the fixed-angle behaviour of subampli tudes.

F inal ly we note t ha t these conclusions appear to be confirmed by the s tudy of models (n).

I wish to t h a n k Drs. P. V. LAI~DSHOFF and A. R. WI~TE for in teres t ing discussions. I also wish to t h a n k Prof. B. Zln~i~¢o for the hospi ta l i ty of the CERN Theory group where most of this work was done.

APPENDIX

In this A p p e n d i x we summarize known results (5) on the singularities of integrals over real ranges of the variables of integrat ion which we use in this paper. We consider integrals of the form

(A.1)

where ] is supposed to be analyt ic for real values of p~ and kj satisfying

(A.2) Dz ~ 0 , l = 1, 2, ...

except a t singularities given by

(A.3) S,,,(p~, k3) = O, m ---- 1, 2, . . . .

The D's and S's are supposed to be real analyt ic functions. The contours mus t be infinitesimally dis tor ted to avoid the singularities (A.3), or equivalently those singularities must be infinitesimally displaced from the real plane. We choose the la t ter description and prescribe the sense of the displacement b y replacing each by

(A.4) S~ ~- ie~,

where em is a small quant i ty whose sign determines the sense of the displacement. Different Landau curves correspond to subsets of the D~ and S ~ and their

(lo) p. GODDARD and A. R. WHITE: £Vuovo Cimento, 3A, 25 (1971). (n) j . D. DORREN: Weizman Institute preprint; I. G. HALLIDAY and G. W. PARRY: Imperial College preprint; P. V. LANDSHOFF: private communication.

I N C L U S I V E CROSS-SECTIONS AND D I S C O N T I N U I T I E S ~ 6 ~

equations are given by the set of p ' s which solve

Sm~--O, DL=O,

(A.5) ~ ~., ~Sm ~ ~D~

where m and l run over the par t ic ipa t ing subsets, and the ~'s are defined by (A.5). The basic t heo rem is t h a t I is only singular on Landau curves (A.5) for which ~m~,, all have the same sign en~t for all par t ic ipat ing Sin. l~ote t h a t there is no corresponding condition for the Dz and if a Landau curve is gen- e ra ted b y D ' s alone it is necessari ly singular.

In our applicat ions the p ' s will be externa l m o m e n t a and the k's loop m o m e n t a in un i ta r i ty integrals. The D ' s will correspond to mass-shell 5-func- tions and the S 's to singularit ies of the bubbles represent ing subampli tudes in the un i t a r i ty integrals. The -k signs on these bubbles will de termine the corresponding em's. The a ' s of (A.5) are then just the usual ~'s associated wi th Landau curves (~).

When (A.5) is satisfied together with the a~e~ condition there is a pinch in the in tegral (A.1) giving a singulari ty. When one continues the p~s pas t th is s ingular i ty ei ther the par t ic ipat ing singularities S~ just slide over the contour of in tegrat ion, leaving i t undis tor ted , or they m a y impinge on the contour in such a way as to make it d is tor ted (see Fig. 8). The fo rmer case is called the

1 s' S21 X ~ X

X

s2 ~o

#inch

Fig. 8. - Natural and unnatural continuations.

Ist X

X T .s 2

× ×

s 2

natural continuation and is par t icular ly i m p o r t a n t for us since we wish to con- t inue integrals like those of Fig. 1 in such a way tha t they remain simple un- dis tor ted phase-space integlals . The second impor t an t theorem tells us how this can be done.

I f (A.5) is solved for the k's and the ~'s in t e rms of the p 's the Landau curve has the equat ion

(A.6) L(p) : ~ ~,~S,. + ~,~zDt. n

566 J. C. POLKINGHORNE

One can define a var iab le ~/ a long t h e n o r m a l to t he curve by

(A.7) d~7 = ~ ~ dp~,

and t h e n t h e na t u r a l con t i nua t i on is a lways given by a con t inua t i on fol lowing a ~7 q - i s~ t p a t h a r o u n d t h e s ingular i ty .

Iffo~ice t h a t th is na t u r a l con t inua t ion is on ly defined for s ingular i t ies wi th a t least one pa r t i c ipa t ing S. A s ingu la r i ty g iven b y D ' s a lone does no t possess a n a t u r a l con t inua t ion . I n fact , in t h a t case t h e va lue of I on t h e two sides of t h e curve are no t ana ly t i ca l ly r e l a t ed to each o ther . This is v e r y famil iar . The u n i t a r i t y in teg ra l be low a n o r m a l t h r e sho ld is zero, nonzero above , a n d t h e n o r m a l t h r e s h o l d is jus t such a s ingu la r i t y g iven b y D ' s alone. H o w e v e r our in t e res t s will be m a i n l y in h igher L a n d a u s ingular i t ies and for these t he p a r t i c i p a t i o n of some S ' s is necessary .

• R I A S S U N T O (*)

8i investiga sull'effetto delle curve di Landau della regione fisica sulla sezione d'urto differenziale completa, per una classe tipica di curve. Si conclude ehe In sezione d 'ur to differenziale ~ una diseontinuit~ totale nel difetto di massa variabile M 2 di una am- piezza valutata su di un piano non fisico. C'~ un punto di non analitieit~ in eiascuna nuova singolarit~t.

(*) Traduzione a eura della Redazione.

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