IL NUOVO CIMENT0 VoL. 46 A, N. 4 21 ipr i le 1978

On M-Loop Amplitudes for Excited States in Dual-Resonance Models.

1~I. Qun~6s

In s t i~ to de Estructura de la Materia - Serrano, 119 Madr/d

(rieevuto il 14 Dicembre 1977)

Summary. - - A general formula for M-loop helicity amplitudes is given in the framework of the conventional dual-resonance model. Signaturized amplitudes are analysed in detail for orientable diagrams up to one loop. The pomeron and the renormalizability of the theory are discussed.

The N-reggeon ver tex allows us to compute helicity amplitudes for external excited states (1). General properties of scalar amplitudes as the Fubini-Veneziano factorizat ion and dual i ty must be satisfied automatical ly for helicity am-

plitudes due to the s t ructure of the vertex. However, other properties as factorizat ion of form factors and positive signature for the pomeron, pomeron singularity not violating uni tar i ty at critical dimensionality and renormal- izabili ty 5 la !~eveu-Seherk must be proved for helicity amplitudes. All these properties will be invest igated in the framework of the conventional dual-res- onance model (DRM) and the s t ructure of signaturized amplitudes analysed.

Our s tar t ing point will be the Y-reggeon M-loop integrand (1) which can be wri t ten as

- r

where a (')t. are creation operators in the space of excitations of the reggeon i.

(i) (with convention %/6 ~ 1), Zero modes are included as usual, a o = (2s189

(1) B. l~IOl~WT, and M. QumSs: Nucl. Phys., 55 B, 266 (1973).

579

580 ~ . q v m 6 s

~' being the slope of input Regge trajectories. The coefficients A(~ ''") are given by

~ m + n

(~) ~,~ - (~ ' , (~) , T , ( ~ ' ) ) , ~ : r

where N~(z~, z~) is the Neumann function of the original duality diagram (the real part of the third Abeli~n integral ~Q~,,0(zj) of a l~iemann sphere with M handles). T~(~) is a projective transformation which can be written in Love-

lace's notation (") as[~ z~ g~ ' where z; and g~ are lower and upper integration

limits of the Koba-Nielsen (KN) variable z~; They depend on the particular configuration or disposition Of external particles over the boundaries of the duality diagram. The coefficients ~0~ = (T~)0, + (TT~)0, and the matrix (T~)~ is the - - s - + 0 infinite-dimensional representation of the projective transformation T~(8). l~inally the symbol ~ ' means that the regular part of -ArM must be ex- tracted whenever i = j.

The integrand of the tensor amplitude for N spinning paxticles is given by the following matrix element:

r (i) N

(a) - / ~ ' = I I I I (~, II,,lO,...,,,)

with the shorthand notation (tz) = t~ .../~j, J = ~ J , and J, = ~ l~ ~ the spin n==l

of i-th particle. By contracting the Lorentz indices in (3) with traceless, cova- riant, wave functions (3), one can get the helicity amplitudes

(4) - / ~ = I I ~ ~ j( , , ' ~(/*D XM'i] M t

where wave functions are described by the Clcbsch-Oordan series

(5) (A) . . .

with A - ~ ~ , the helicity of the particle, and the subsidiary conditions

p~,e~(p) = 0. In the following we shall restrict ourselves to the production of particles on the leading l~egge trajectory and the mass shell condition will read p~ ---- + J~. I t amounts to keeping only first modes of excitation in (1),

(i) a 1 ~_ a s Using the standard properties of Neumaml functions and definition (2)i

(~) C. Lo~,m,~cv.: Phys. Lett., 32 B, 490 (1970). (a) hi. D. SCADI~O~: Phys. Rev., 165, 1640 (1968).

ON ~ - L O O P AMPLITUD]BS F O R ]~XCIT~ED S T A T E S I N D U A L - I 4 E S O . ~ A N C E MOD]BL8 5 8 1

we can write eq. (1) as

(6) I~ Io] - I ) .~ ' exp[ (2:r ~ ~ ~ ~ z~,~N(z,,zj)a:] z~ ~ = - - ~ p ~ o ~ ( z ~ , z ~ ) - - , . . . . .

where lo-~I~exp[s163 z~)] is the scalar integrand, i i , i L J

~--~ In 2~ and ~ means ~/~z~. We have suppressed diagonal terms from (6), because they will not contribute to helicity amplitudes.

After a tedious but straightforward c~lculation we can pul; the integrand of the helicity amplitude in the following simple manner:

(7)

where stunmation over Clebsch-Gordan coefficients (5) has been omitted for simplicity. The summation in (7) is extended to all coefficients a~; == aj~, f l , j>0 satisfying the two set of conditions ~ (~j + fl~)-~ J~ and ~ a , - < J j . The

factor I~'t)(~, fl) is made of invariant quantities as

(8) = 1 - I (~ ,~ )~ , ,

and the combinatorial factor

Let us remark the cancellations of factors ),J' in (7). According to (7) the helicity amplitude can be decomposed as

(9) T (A) ~ I(m(a, fl) 7'~(a, fl) o:,,6

In the following we shall s tudy the contribution of planar and nonplanar orientable diagrams up to one-loop level.

1. - Tree d i a g r a m s .

The I~icmann surface corresponding to the zero-loop case is a sphere where particles a.re disposed along the equator. Thence the h~eumann function, or electrostatic energy, is simply No(Zl, zj)--~ In (z,--z~). The KN variables z~ can be complex numbers on the unit circle (cutting the sphere along the equator) or real numbers (cutting the sphere along a meridian). As measure of integra-

3 8 - I l Nuovo Cimento A.

582 .,~. qum6s

t ion we shall use Olive's one, as wri t ten by ALESSANDRINI and AMATI (4). In

this way the component To(a, ~) can be wri t ten as

(10)

with ~ =: 2~ ; + ~,~-+ fl~. Le t us no t e tha t a similar ampli tude has been obtained by I~OSENZWEIG et al. (5) using a dual ver tex operator in t roduced by CA~AONA el al. (s). Their method is only valuable for tree diagrams and can be shown to be a part icular case of ours.

Le t us particularize eq. (10) to the case of four external particles with in- var iants defined as s = (p~ + p2) 2, t = (pl + p3) -~, u = (p~ + p~)2. Taking the

~ s tandard frame ~> z, = z~ : - 0<z~<z~ == z3 ---- 1, zo = z~ = co, we can write the Born approximat ion for the helieity ampli tude as

(11) To(~, ~) = (-- 1)*Z<]"+a"B( - ~(s) + J , + J 2 - ~ , ~ , - ~(t) + dx + J 3 - ~,)

with vanishing coefficients fl~4 = fl~ = 0 due to the chosen frame. The dual i ty propert ies of (11) are evident by themselves. There are resonances in the s- channel a t a(s) ---- J~ + J ~ . - al~ -~ n (n>0 ) dual to Regge poles in the t-channel:

(12) ~, z c~,+~,,,, =(s))~,)~,+~_,,,)"

The extra power ~ 1 8 - - J 1 - J3 in (12) must be compensated in each case by I ~A) so as to reproduce the power a(t). Adding to (12) the corresponding

s-u crossed function, we get the signature ~ ( ~ , f l ) : (--1) J'+J'+~'. Thus two leading degenerate trajectories (with positive and negative signature) emerge. Only renormalizat ion effects could break this degeneracy.

2. - One-loop nonplanar orientable diagrams.

This class of diagrams is part icularly interesting because they are associated

to the propagat ion of the pomeron singularity in the crossed channel. I ts

analogue surface is the annulus q< [z l< l whose double is a torus sewn along

the equatorial Ks cycle. The K N variables are disposed over equatorial cycles (boundaries of the original annulus). One can get real K N variables cut t ing

(4) V. ALESSANDRINI and D. AMATI: Nuovo Cimento, 4 A, 793 (1971). (5) C. RosE~zw~m and U. P. SUKHAT~: Nuovo Ci~ento, 3A, 511 (1971). (s) P. CAmPAGnA, E. NAPOLX~ANO, S. SCIUTO and S. FUB~NI: Nuovo Cimento, 2A, 911 (1971).

ON M - L O O P A ~ P L I T U D E B F O R ~ X C I T ~ D S T A T E S I N DUA-L-It'E, SONANC]~ M O D E L S 5 8 3

the torus along the K~ cycle and straightening the resulting surface so as to get a new annulus ~o~< ]z I ~< 1. Boundary points of the analogue surface can be taken to be the segments of the real axis contained inside the new annulus. The Neumann functions corresponding to the real configuration have been studied by ALESSANDRINI (7)~ gett ing N , = in y~(x~/x~), if z~, z~ belong to the

same boundary and N~; = In ~fr(x~/x~), if they belong to different boundaries. y~ and ~ are the usual elliptic functions appearing in loop calculations (s). Le t us choose a configuration where all internal lines are twisted. The helicity ampli tude is

1

0

where

( - )P</" H (D J , , I I , ( In e),' ~ ~<~

d#(y) : 1-I dyi(1 -- yi y~ t 1)~(~ -- w)a ~o-~(o)-1/(~o)-~+~(_ ln~o)-~ i

is the GNSS (~) measure of integrat ion modified b y addit ionnal factors coming from the elimination of spurious states propagat ing through the Riemann surface. The relation between the KN variables x~ and the Chin variables y~ ap- eparing in d/t(y) is given by y~ ~ x~/x~_~ and ~o ---- co ---- cxp [2~2/ln q]. The new

functions appearing in (13) are given by logarithmic derivatives of Jacobi 's theta-

functions (~), D~j-----0~/0~(~oj--~), if z~ and zj belong to the same boundary and D~j----0~/0,(~0;--~0~) otherwise. Angular variables are defined by ~0~

-- 2 In xJln ~. When one eliminates In ~ factors f rom N , by a suitable redefinition of the

functions, the factor (ln ~o) -J in (13) is cancelled thanks to the mass shell con- dition. On the other hand, using the series representat ion for D~j (9), one can see they depend ou q bu t not on In q. In this way, when the change of integrat ion (y~) -+ ( ~ , q) is performed in (13), by using the Hardy -Ramanu jan formula to t r ans form the par t i t ion function, there is a singularity at the endpoint q ~ 0

becoming a pole when only DD~ ~ states (lo) are propagating, and E ---- 2, G ~ 0,

D----26 (~). At this level the dual theory with external spinning particles does not show any iuconsistency.

(7) V. AL]~SSANDRI~'I: Nuovo Cimento, 2 A, 321 (1971). (~) D. GIr A. NEver, J. SC~RK and J. SCHWARZ: Phys. Rev. D, 4, 697 (1970). (9) Bateman Manuscript Project, Higher Transcendental _Functions, Vol. 2 (New York, N.Y., 1953). (~o) E. DEL GIUD~CE, P. D1 V~CCHIA and S. FuBIKI: Ann. o] Phys., 70, 378 (1972). (11) L. BRINK a.nd D. OTav~: Nucl. Phys., 58 B, 237 (1973); P. V. COLLINS and K. A. FRIEDMAN: NUOVO Cimento, 28A, 475 (1975).

5~4 3f, QUIRSS

Le t us s tudy in some detail the case N ---- 4. Using as variables of in tegra t ion

q, ~---- % - - ~a, ~ = % - - 9~ and a---- �89 + % - - 9~-- ~ ) , one can write

(l~t) T~(a, fl) = ( - 1),~<~ a'' g,f dq d~ d~ d~ #(q, q~, ~).

�9 exp [-- sV -- tV, + 2t z~ V + ~ J~ W,] 1-[ (D-) ~'' I - [ (D;y",

where the t r a n s f o r m e d measure of integrat ion /~ is given by (co) t~F(~176 �9 (1 - co)~176 -~('~(~176 (In q)~-,~(0)-~/~ {(1 - oH ~')(1 - co *'~') ( 1 - ~-~/~) - �9 (1--eox- ' t~')} ~(~ and a ( 0 ) - - - - ( D - - E ) / 1 2 is the in tercept of the pomeron

t ra jec tory . The functions V,, V~, V und W, are logari thms of Jacob i ' s thc ta-

functions. Their explicit form is easily deduced f rom N,-. We shall be con-

cerned now with the a sympto t i c behaviour of (14) when [s[ -~ oo, In this l imit

the integral is domina ted b y the contr ibut ion of the critical points q ~ 0 (~2) (the (~ pomeronic ~) contr ibut ion of the diagram, P) and 9 = 0, 2~; ~ - - 0, 27~

(the (( reggeonie ~) contribution).

The contr ibut ion of the critical point q = 0 is given b y

(~5) P ~ og, F ( - %(0 + o) ln% ( - ~(s))~.,-o(z + ( - 1)~,,}.

. {1 + (-- 1)"}/,3(t)[, ,(t) ,

where q is a l inear combinat ion of a~j, fli~, %(0 ---- a ( 0 ) + s is the pomeron t r a j ec to ry and 6 ~ G + 1 - - E/2. Ext r~ powers of ~(s) mus t again be compen-

sated b y I (a). The funct ions / ( t ) p lay the role of pomcron form factors desc r ib ing its coupling to initial and final states. They h~ve a simple integral represen- ta t ion, wi th poles a t %(0 ~- a(t) -- n, so t ha t the pomeron is coupled to external part icles th rough l~egge poles (fo dominance). The pomeron t ra jec to ry has positive signature and the ampl i tude ]actorizes as in the scalar case. J - o d d

s ta tes are decouplcd f rom the pomeron (x,~3). This is the way in which

Gribov-Morrison 's (~4) rule appears in the theory. The (~ rcggeonic ~) contr ibut ion

can be ar ranged to

R,-~ g~2( t )Ins / ' ( - a(t) + e ' ) s ~(')-~'.

�9 {1 + ( - 1) ''~ exp [ - iz~(t)]} {1 + ( - 1)J},

where ~' is again a linear combinat ion of cqj and f t , . When ,1/2 is even (odd)

~7(t) m u s t be in terpre ted as a renormalizat ion effect, to order g2, of the leading

(12) V. AlmSSAXD~I~I, D. AMAT[ and B. MOm~L: Nuovo Cimento, 7A, 797 (1972). (is) D. EBERT and H. J. OTTO: Ann. tier Phys., 32, 47 (1975). (14) D. R. O. MOlmlSON: Phys. Rev., 165, 1699 (1968).

ON ~/]//-LOOP AMPLITUDES FOR :EXCITED STAT:ES IN DU2~L-I:~:ESOKANC:E MOD:ELS 5 8 ~

t rajectory with positive (negative) signature. When J is odd 1~ vanishes iden- tically.

The infra-red behaviour of the amplitude is governed by the critical point q ---- 1. To estimate the behaviour of the intcgrand in the limit ~o --> O, it is convenient to handle elliptic functions before performing Jacobgs imaginary transformation, thus getting

(17) { x~.~jiV~, = ~ - ~ § 1 6 2 1 6 2

x~x~ ~,~ ~V,~ = - 1/ln ~ • ~ / (o~r T 1)~ § 0(o~),

where ~----(~0j--~)/2~ and the first (second) sign corresponds to untwisted (twisted) functions. Using eq. (17) and s ~- t -}- u ---- 4#" ~- J, we get for the amplitude (14) branching points corresponding to normal thresholds and coming from the integration corners (~, ~, ~) = (0, ~, =), (7:, 0, 0) and (=, =, 7:).

3. - One-loop planar diagrams.

The corresponding amplitude is given by (13) with untwisted functions and the corresponding GNSS (s) measure. By expanding N~j and D~ around the critical surface (~5) the asymptotic behaviour is found as in (16), bu t ~(t) is replaced by a divergent function 2:(t). I t can be shown tha t the divergent par t of 27(t) is dependent neither on a(t) nor on J~, so tha t it can be regularized as in the scalar case (~5).

~qonorientuble diagrams could be ~nalysed in the s~me way. Our procedure seems the most general one to include excited states in the DI~]~L For states belonging to daughters, care must be taken of ghosts. The formalism is general- izable to other DI-CM, once one knows the automorphie structure of amplitudes. For the Shapiro-Virasoro model only the region of integration is changed (~e). For the Neveu-Schwarz model the automorphic structure has been analysed by FArI~LIE~, NIAlVrIlV (~7) and ~IO~TO~E~ (~s).

I t is a pleasure to thank the Theoretical Physics Depar tment of the Uni- versi ty of Geneva and the CEI~N Theory Division for the hospitali ty offered to the author.

(15) A. N~'vEu and J. ScH~a~i~: Phys. Rev. D, 1, 2355 (1970). (16) j . A. Sl~AemO: Phys. ~ett., 33 B, 361 (1970). (17) D. B. FAIRT.I]~ and D. ~[A.RTIN: l~'rCtOVO Cimento, 18 A, 373 (1973). (is) C. MONTO~]~: NUOVO Cimento, 19 A, 69 (1974).

586 M. QuII~bS

�9 R I A S S U N T O (*)

Si d's una formula generale per le ampiezze dell'elicits a Ma nse nel eontesto del sistema del modello della risonanza duale convenzionale. Si analizzano in maniera det~agliata le ampiezzo con scgno per cUagrammi orientabili fino ad un'ansa. Si discutono il po- merone e la rinormalizzabilits della teoria.

(*) Traduzione a cura della Redazio~e.

O M-neTe.,mm~x aMnYmTy~ax ~ q Bo36y~'[C,~OHHblX COCTOIIHHfi B ~ya~bm,~x pe3onaHeHb~X

MO~OJLqX.

PeamMe (*). - - B paMrax o6~eupaHaToi~ ~ya~1,HO~ peaonaHCHO~ Mo~e/IH npe~aaraerca o6taa~t qbopMyJm ~ t~ M-nere~bHSlX cnapa~brmlx aMn3trlTy~I. I Io~po6ao anaJm31~- pymTC~t aMrmnTy~Ir, r ~n~t opi~eHTnpyeMl, iX ~rmrpaMM c o~noti neT~e~. O6cyx~aeTc~ noMepoH H nepeHOpMHpyeMOCTra TeopnH.

(*) 1-IepeaeOeno peOatatue?t.