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The structure of dual green functions
Volume 65B, number 5
PHYSICS LETTERS
THE STRUCTURE
20 December 1976
OF DUAL GREEN FUNCTIONS M.B. GREEN
Cavendish Laboratory, Cambridge, UK
Received 14 October 1976 We exhibit a s~mplerelationship between the position space and momentum space singularity structure of offshell dual amplitudes. This scheme is naturally described in terms of amplitudes which couple to the external sources via pairs of "confined" states.
The dual string model provides an interesting framework for investigating possible properties of extended quantum mechanical systems. The model is so constrained that it has proved very hard to obtain reasonable off-shell behaviour for dual amplitudes coupling to external sources of definite angular momenta (and hence, for example, reasonable form factors). However, recent developments have suggested [ 1 - 4 ] a physically plausible scheme of constructing such Green functions (although for presently existing open string models they are not completely satisfactory). We have investigated the space-time structure of some simple dual off-shell amplitudes and have noted a systematic relationship between their singularity structure in momentum space and position space [5]. This scheme has a simple diagramatic representation that involves "confined" states coupling in pairs to the external sources. The natural appearance of states which do not propagate in space-time but which carry quark flavour quantum numbers (when quantum numbers are incorporated in the most naive Chan-Paton fashion) is a particularly intriguing feature of this scheme. This note provides a description of our main results many of which are described in detail elsewhere (ref. [5] ). We will begin with a summary of known results on the construction of amplitudes with a single off-shell open string state [ 1 - 3 ] in order to introduce the basic ingredients of the scheme. Throughout the paper we shall, for simplicity, concentrate on scalar off-shell states although the construction of states of arbitrary angular momenta presents no essential difficulty. As usual there is a choice between two formalisms: (a) The manifestly unitary formalism that has a simple string interpretation. (b) The manifestly covariant formalism [1, 2]. Unfortunately these two methods 432
are not entirely equivalent for the ordinary Veneziano model (OVM) although they give identical amplitudes (at least in the case of processes with one or two offshell states). The covariant method only works for D = 16 (where D is the space-time dimensionality) in which case the OVM is known not to be perturbatively unitary. The unitary string method only exists for D = 26, a dimension for which the off-shell states of ref. [3] are not pure Lorentz scalars but carry zero helicity. However, for closed strings both formalisms turn out to be equivalent [4] and so we feel that this mismatch, although an obvious fault of the model, will not crucially affect our general observations. fa) The unitary string method. In this method the dual string is formulated m the GGRT gauge [6] and states are restricted to the 24 dimensional subspace. An off-shell state couples at light-cone time r 0 (r = (x 0 + x D - 1/X/~-)) to a section of string that has collapsed to a point (fig. la). A crucial feature that ensures finite form factors is that the collapsing section of string is permitted to break away from the rest of the string at time r 1 . In practice the off-shell state, Ic), can only be constrained to be point like in the transverse directions, i.e.: Xi(o)lc)=xilc),
i= 1 . . . . . (D - 2).
(1)
The longitudinal direction is not an independent degree of freedom and it turns out that quantum mechanical normal ordering terms prevent the state being a Lorentz scalar. These unwanted terms are, however, very sensitive to the precise model being considered for instance they do not arise at all in the case of the off-shell closed string states considered in ref. [4]. For convenience we shall refer to the state Ic) as a scalar state, absorbing non-invariant factors in the
Volume 65B, number 5
PHYSICS LETTERS
X'(O)=O
c
(a)
modes
(b)
. ,=t~v'I ~const. X ~ (c) Fig. 1. The single off-shell amplitude. (a) The string picture. (b) The covariant operator picture. (c) The world sheet displaying one abnormal boundary.
Green functions into overall multiplicative constants. Our hope is that in a fully consistent theory the localized states will truly be Lorentz scalars (as in the closed string model [4] ). (b) The covariant method [1, 2]. In this case the off-shell state, Id), is constructed to satisfy the same gauge conditions as the tree diagrams of the theory [I ]. However, for the OVM these conditions can only be satisfied in D = 16 dimensions. This dimensionality is of course model dependent [2]. A particularly elegant method of constructing amplitudes with one offshell state was introduced in ref. [2]. This made use of a Fock space of c-moded states, where the c-mode creation and annihilation operators satisfy: [Cn", Cmv + ] = -gZ~6nr n,
n, m =~-,~ 1 3 .....
(2)
The external source couples to a tree diagram via a pair of c-moded ground states as in fig. 1b. Since the c-modes are y-integral there Is no zeroth mode and so the "propagator": 1
( L 0 c - 1) - 1 ,
20 December 1976
correspond to the "'propagation" of c-moded states. The simplest example, fig. 1 c, is the world sheet for a single off-shell amplitude which is just a distortion of an analogue model diagram introduced in ref. [2] which was the original motivation for the c-moded formalism. The thick line corresponds to an "abnormal" boundary on which the analogue model potential, XU(z), is constant. On the other boundaries the normal derivative of XU(z) vanishes as usual. A strip with one normal and one abnormal boundary then corresponds to a c-moded state and has no m o m e n t u m space singularities. The current propagator is illustrated in fig. 2a (in the string picture) and fig. 2b (in the covariant picture). Notice that in the covariant approach the propagator may be viewed as an amplitude with four external ground state c-moded lines. We would thus expect some sort of duality-like equivalence between the sum over q2 resonance poles and a sum over "crossed channel" singularities (indicated by the dashed line in fig. 2b). There is clearly no m o m e n t u m space variable associated with this crossed channel - we shall see that the dashed line has a simple interpretation in terms of position space singularities. We find that the current propagator in either the unitary string picture of the covariant picture is given by [3] :
?
Cn -c exp (i(q 2 + 1)y} n=no q2 _ 2n 0
F(q 2) = ~
(4)
oo
X l-I (1 - e x p { - 2 n i y ) ) -12 dy, n=l
(which has a bad divergence that we shall ignore for the moment). Upon Fourier transforming eq. (4) we obtain:
.
(3)
has no m o m e n t u m space singularity (Loc = - Z n = 1/2 nC~n+Cnv)• Fig. l b therefore has no momentum space normal thresholds and must not be viewed as a normal Feynman diagram. Our main results will concern amplitudes with two or more external off-shell states. In order to illustrate the structure of these amplitudes we shall introduce some rather novel world sheet configurations which
(x2) = 1
f e _ i q x F ( q 2 ) dOq,
(5)
(where the conventional ie prescription is to be assumed in defining this integral). Since the dimension D enters explicitly in eq. (5) we will obtain different results from the two formalisms. We believe the most consistent picture is to use D = 16 (the covariant formalism) since only then is eq. (4) truly Lorentz invariant (recall that in the D = 26 transverse string picture the off-shell states only have zero helicity). In either case we have: 433
Volume 65B, number 5
PHYSICS LE'ITERS
x,(o,.ol
20 December 1976
x,,o,=o
I
T2
"t 1
(a)
"q
Z3x-,--
=
(b)
Fig. 2. The propagator. (a) The string picture. (b) The relationship between momentum space and position space singularities in the covariant picture. The dashed line is seen to correspond to the strip of world sheet in (c) with two abnormal boundaries.
F~ ( x 2 ) -_ ~ C D j? e x p ( - i x 2 / g y + iy) 0 ~. x I-I ( 1 - e x p { - 2 n i y ) ) - 1 2 ~ - l ---D~ /Y2}" - " n=l \n/
=
c
/ 0
dy.
X
e x p t - i x 2 /4y)
[0 L
]-4 27r
(7)
J
The Jacobi transform [7] suggests the change of variable from y to:
y' = 7r2/y
(8)
which gives:
o X n=ll-I (1-exp{2niy'))-12(-~---~Y')(D/'2-8)dy' 434
(10)
(6)
This integral, which is an infinite sum of Feynman propagators can be re-expressed in terms of the Jacobi 01 function [7] as: F ( x 2) = ~ - ~ D
~ (n n=nl (x2/4rr2a ' -- 2n - ie)ao"
(9)
The result is thus a discrete set of simple power singularities in x 2, spaced by 81r2a ' where a ' is the Regge trajectory slope. The coefficients c n and Cn are all positive. Unfortunately there is no singularity on the lightcone and there is actually one singularity outside it (n 1 = - { ) which is a severe sickness of the model (this also causes the bad divergence at small y in the integral in eq. (4)). The nature of the singularities in x 2 depends on the value o f a 0. We see from eq. (9) that w h e n D = 16 (the only dimension with consistent Lorentz invariance) a 0 = 1 and we obtain pure poles in x 2. We have also examined the propagators for the Neveu-Schwarz offshell scalar and vector states proposed in ref. [2] (for which D = 10) and find once again the x 2 singularities are pure poles. The world sheets corresponding to the two configurations depicted in fig. 2b are shown in fig. 2c. Tlae dashed line is seen to correspond to a strip with two abnormal boundaries. This suggests the intriguing possibility of choosing a parametrization on this strip in which the analogue model potential has an expansion in position space in terms of integer modes*:
Volume 65B, number 5
PHYSICS LETTERS
0 ql"-~l
20 December 1976 "g
1:Z
pl--*~ (a)
I q2-,--> 'l p2--->
'~?,=q2%-P,/
xl=O 'l Xz /
(b)
(c) Fig. 3. The diagrams for an amplitude with two off-sheU and two on-shell statcs.They correspond to (a) the stringpicture and (b) the covariant picture.The world sheet configurations are shown in (c).
x"('7'O=x~-(x~-~'2) +~=_~ -nn s i n
n~ e -inn (11)
Eq. (10) would then be rewritten (reverting to the covariant case for whichD = 16)as: /~(x 2) = (tTl(L~ - 1) -1 ltt)
.(12)
where L~ = - ~ n® = l a+n a n - (Xl - x2) 2 and 1~7)is the state satisfying conditions analogous to the usual gauge conditions [1, 2]: (L x - L ~ + 1 - n ) I d )
= 0,
(13)
where L x is the same as the usual L n with qU replaced by (x 1 - x2)#. The relation expressed in fig. 2b now reads:
verse string diagram and the covariant c-moded diagrams for this process. We shall not discuss the calculation here apart from pointing out that once again the Jacobi transform plays a crucial role in simplifying the Fourier transform. Once again we find a simple connection between position space m o m e n t u m space singularities:
T(q2, q2, M 2 ' t) = f dD x eiq2 xT(x 2 , Pl x, p2 x, t),
(15) with
dklm(q 12,q2, M 2 2, t) T(q2,q2,M2,t) = L.I k,l,m =no (q2 _ k)(M 2 - / ) (q2 _ m) (16)
T(x2,PlX, P2X, t) = ~ ~ d-n(x2'plx'p2x't) (17) n=nl i (xE/4rrEot ' - 2n)ai ' and
x u = x~ - x~. 435
Volume 65B, number 5
PHYSICS LETTERS
Here n o and n I take the same values as for the propagator. In the calculation given in ref. 15] the index i takes three values with a 1 = D / 2 - 7, a 2 = 1 ) / 2 - 8 and a 3 = D / 2 - 9 . Again when D = 16 the leading x 2 singularities are pure poles. The diagram of fig. 3b involving c-moded states again serves to illustrate the singularity structure in the two representations (momentum space and position space). It seems probable that such diagrams will generalize to an arbitrary amplitude involving M onshell particles and N off-shell states. All diagrams have M + N - 1 independent external variables (taking momentum conservation or translation invariance into account). The labelling of space-time indices on the quark lines is once more motivated by the configurations of world sheets such as those shown in fig. 3c. By suitably distorting the world sheets singularities may be exhibited in various momentum space and position space channels in a manner very reminiscent of the usual duality connection between crossed momentum space channels (with the additional requirement of appropriate Fourier transformation). Strips with two normal boundaries have momentum space poles and those with two abnormal boundaries have position space singularities (when the appropriate Fourier transforms have been made). The instantaneous nature of the c-moded propagators of fig. 3b (which are strips in fig. 3c with one abnormal boundary) enables the unambiguous isolation of channels which carry definite values o f x . Notice that fig. 3b contains internal c-moded lines in the position space configuration which involve the "propagation" of excited c-moded states. We have not yet investigated how the x-space factorization that we described above for the propagator (eq. (I 2)) will generalize to these more complicated amplitudes. We end with several comments: (1) The Chan-Paton scheme for attaching flavour quantum numbers to dual amplitudes is automatic if the confined c-moded states are given quark quantum numbers. This suggests a possible identification of the c-moded states with confined quark states. (2) In the dual model calculations described above the lowest lying singularity in x 2 dominates the deep Euclidean limits - the correction from the next singularity (spaced by 8rt2ot' in x 2) giving rise to an exponentially damped contribution. We may envisage an 436
20 December 1976
ideal dual model in which the leading x 2 singularity is on the light-cone instead of outside it. In this case we would get scaling with a specific correction term related to the hadronic resonance separation (i.e. to the Regge slope a'). The scale of this correction term is set by x/-g-~-2~,the size of the spatial separation of the singularities, which is of the same order as the typical size of a hadron. Our diagrams serve to illustrate the relationship between Regge pole, resonance pole and light-cone dominated kinematic limits. The narrow resonance approximation that we have been discussing is just the Born term of the complete theory. We may expect that in the complete theory the discrete x 2 singularities will be shielded by cuts in much the same was as the narrow q2 poles are shielded by normal threshold cuts. (3) It would be satisfying to be sure that the simple structure of eqs. (4) and (10) does not depend crucially on having a singularity outside the light-cone. It is easy to construct mathematical examples of functions which have an infinite set of q2 poles with positive residues and whose Fourier transforms have singularities only inside or on the light-cone. (An example is given in ref. [5] - in this example the x 2 singularities cannot be poles). (4) Closed loops of c-moded states may be inserted into the usual dual propagator. This corresponds, in the string language, to a covariant reweighting of the modes oF the string which emphasizes those histories in which the string has points with infinite momentum density [4] - which may be a physically reasonable modification of the dual model. I am grateful to C.B. Thorn, J.A. Shapiro and A.H. Mueller for useful discussions and to the CERN theory group for its hospitality during the summer.
References
[1] [2] [3] [4]
J.H. Schwarz, Nucl. Phys. B65 (1973) 131. E. Corrigan and D.B. Fairlie, Nucl. Phys. B91 (1975) 527. M.B. Green, Nucl. Phys. B103 (1976) 333. M.B. Green and J.A. Shapiro, CERN preprint TH.2216 (Aug. 1976). [5] I~I.B.Green, CERN preprint TH.2219 (Aug. 1976). [6] P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109. [7] A. Erdelyi, ed., Bateman Manuscript Project, Higher transcendental functions, voi. II (McGraw-Hill,1953).
PHYSICS LETTERS
THE STRUCTURE
20 December 1976
OF DUAL GREEN FUNCTIONS M.B. GREEN
Cavendish Laboratory, Cambridge, UK
Received 14 October 1976 We exhibit a s~mplerelationship between the position space and momentum space singularity structure of offshell dual amplitudes. This scheme is naturally described in terms of amplitudes which couple to the external sources via pairs of "confined" states.
The dual string model provides an interesting framework for investigating possible properties of extended quantum mechanical systems. The model is so constrained that it has proved very hard to obtain reasonable off-shell behaviour for dual amplitudes coupling to external sources of definite angular momenta (and hence, for example, reasonable form factors). However, recent developments have suggested [ 1 - 4 ] a physically plausible scheme of constructing such Green functions (although for presently existing open string models they are not completely satisfactory). We have investigated the space-time structure of some simple dual off-shell amplitudes and have noted a systematic relationship between their singularity structure in momentum space and position space [5]. This scheme has a simple diagramatic representation that involves "confined" states coupling in pairs to the external sources. The natural appearance of states which do not propagate in space-time but which carry quark flavour quantum numbers (when quantum numbers are incorporated in the most naive Chan-Paton fashion) is a particularly intriguing feature of this scheme. This note provides a description of our main results many of which are described in detail elsewhere (ref. [5] ). We will begin with a summary of known results on the construction of amplitudes with a single off-shell open string state [ 1 - 3 ] in order to introduce the basic ingredients of the scheme. Throughout the paper we shall, for simplicity, concentrate on scalar off-shell states although the construction of states of arbitrary angular momenta presents no essential difficulty. As usual there is a choice between two formalisms: (a) The manifestly unitary formalism that has a simple string interpretation. (b) The manifestly covariant formalism [1, 2]. Unfortunately these two methods 432
are not entirely equivalent for the ordinary Veneziano model (OVM) although they give identical amplitudes (at least in the case of processes with one or two offshell states). The covariant method only works for D = 16 (where D is the space-time dimensionality) in which case the OVM is known not to be perturbatively unitary. The unitary string method only exists for D = 26, a dimension for which the off-shell states of ref. [3] are not pure Lorentz scalars but carry zero helicity. However, for closed strings both formalisms turn out to be equivalent [4] and so we feel that this mismatch, although an obvious fault of the model, will not crucially affect our general observations. fa) The unitary string method. In this method the dual string is formulated m the GGRT gauge [6] and states are restricted to the 24 dimensional subspace. An off-shell state couples at light-cone time r 0 (r = (x 0 + x D - 1/X/~-)) to a section of string that has collapsed to a point (fig. la). A crucial feature that ensures finite form factors is that the collapsing section of string is permitted to break away from the rest of the string at time r 1 . In practice the off-shell state, Ic), can only be constrained to be point like in the transverse directions, i.e.: Xi(o)lc)=xilc),
i= 1 . . . . . (D - 2).
(1)
The longitudinal direction is not an independent degree of freedom and it turns out that quantum mechanical normal ordering terms prevent the state being a Lorentz scalar. These unwanted terms are, however, very sensitive to the precise model being considered for instance they do not arise at all in the case of the off-shell closed string states considered in ref. [4]. For convenience we shall refer to the state Ic) as a scalar state, absorbing non-invariant factors in the
Volume 65B, number 5
PHYSICS LETTERS
X'(O)=O
c
(a)
modes
(b)
. ,=t~v'I ~const. X ~ (c) Fig. 1. The single off-shell amplitude. (a) The string picture. (b) The covariant operator picture. (c) The world sheet displaying one abnormal boundary.
Green functions into overall multiplicative constants. Our hope is that in a fully consistent theory the localized states will truly be Lorentz scalars (as in the closed string model [4] ). (b) The covariant method [1, 2]. In this case the off-shell state, Id), is constructed to satisfy the same gauge conditions as the tree diagrams of the theory [I ]. However, for the OVM these conditions can only be satisfied in D = 16 dimensions. This dimensionality is of course model dependent [2]. A particularly elegant method of constructing amplitudes with one offshell state was introduced in ref. [2]. This made use of a Fock space of c-moded states, where the c-mode creation and annihilation operators satisfy: [Cn", Cmv + ] = -gZ~6nr n,
n, m =~-,~ 1 3 .....
(2)
The external source couples to a tree diagram via a pair of c-moded ground states as in fig. 1b. Since the c-modes are y-integral there Is no zeroth mode and so the "propagator": 1
( L 0 c - 1) - 1 ,
20 December 1976
correspond to the "'propagation" of c-moded states. The simplest example, fig. 1 c, is the world sheet for a single off-shell amplitude which is just a distortion of an analogue model diagram introduced in ref. [2] which was the original motivation for the c-moded formalism. The thick line corresponds to an "abnormal" boundary on which the analogue model potential, XU(z), is constant. On the other boundaries the normal derivative of XU(z) vanishes as usual. A strip with one normal and one abnormal boundary then corresponds to a c-moded state and has no m o m e n t u m space singularities. The current propagator is illustrated in fig. 2a (in the string picture) and fig. 2b (in the covariant picture). Notice that in the covariant approach the propagator may be viewed as an amplitude with four external ground state c-moded lines. We would thus expect some sort of duality-like equivalence between the sum over q2 resonance poles and a sum over "crossed channel" singularities (indicated by the dashed line in fig. 2b). There is clearly no m o m e n t u m space variable associated with this crossed channel - we shall see that the dashed line has a simple interpretation in terms of position space singularities. We find that the current propagator in either the unitary string picture of the covariant picture is given by [3] :
?
Cn -c exp (i(q 2 + 1)y} n=no q2 _ 2n 0
F(q 2) = ~
(4)
oo
X l-I (1 - e x p { - 2 n i y ) ) -12 dy, n=l
(which has a bad divergence that we shall ignore for the moment). Upon Fourier transforming eq. (4) we obtain:
.
(3)
has no m o m e n t u m space singularity (Loc = - Z n = 1/2 nC~n+Cnv)• Fig. l b therefore has no momentum space normal thresholds and must not be viewed as a normal Feynman diagram. Our main results will concern amplitudes with two or more external off-shell states. In order to illustrate the structure of these amplitudes we shall introduce some rather novel world sheet configurations which
(x2) = 1
f e _ i q x F ( q 2 ) dOq,
(5)
(where the conventional ie prescription is to be assumed in defining this integral). Since the dimension D enters explicitly in eq. (5) we will obtain different results from the two formalisms. We believe the most consistent picture is to use D = 16 (the covariant formalism) since only then is eq. (4) truly Lorentz invariant (recall that in the D = 26 transverse string picture the off-shell states only have zero helicity). In either case we have: 433
Volume 65B, number 5
PHYSICS LE'ITERS
x,(o,.ol
20 December 1976
x,,o,=o
I
T2
"t 1
(a)
"q
Z3x-,--
=
(b)
Fig. 2. The propagator. (a) The string picture. (b) The relationship between momentum space and position space singularities in the covariant picture. The dashed line is seen to correspond to the strip of world sheet in (c) with two abnormal boundaries.
F~ ( x 2 ) -_ ~ C D j? e x p ( - i x 2 / g y + iy) 0 ~. x I-I ( 1 - e x p { - 2 n i y ) ) - 1 2 ~ - l ---D~ /Y2}" - " n=l \n/
=
c
/ 0
dy.
X
e x p t - i x 2 /4y)
[0 L
]-4 27r
(7)
J
The Jacobi transform [7] suggests the change of variable from y to:
y' = 7r2/y
(8)
which gives:
o X n=ll-I (1-exp{2niy'))-12(-~---~Y')(D/'2-8)dy' 434
(10)
(6)
This integral, which is an infinite sum of Feynman propagators can be re-expressed in terms of the Jacobi 01 function [7] as: F ( x 2) = ~ - ~ D
~ (n n=nl (x2/4rr2a ' -- 2n - ie)ao"
(9)
The result is thus a discrete set of simple power singularities in x 2, spaced by 81r2a ' where a ' is the Regge trajectory slope. The coefficients c n and Cn are all positive. Unfortunately there is no singularity on the lightcone and there is actually one singularity outside it (n 1 = - { ) which is a severe sickness of the model (this also causes the bad divergence at small y in the integral in eq. (4)). The nature of the singularities in x 2 depends on the value o f a 0. We see from eq. (9) that w h e n D = 16 (the only dimension with consistent Lorentz invariance) a 0 = 1 and we obtain pure poles in x 2. We have also examined the propagators for the Neveu-Schwarz offshell scalar and vector states proposed in ref. [2] (for which D = 10) and find once again the x 2 singularities are pure poles. The world sheets corresponding to the two configurations depicted in fig. 2b are shown in fig. 2c. Tlae dashed line is seen to correspond to a strip with two abnormal boundaries. This suggests the intriguing possibility of choosing a parametrization on this strip in which the analogue model potential has an expansion in position space in terms of integer modes*:
Volume 65B, number 5
PHYSICS LETTERS
0 ql"-~l
20 December 1976 "g
1:Z
pl--*~ (a)
I q2-,--> 'l p2--->
'~?,=q2%-P,/
xl=O 'l Xz /
(b)
(c) Fig. 3. The diagrams for an amplitude with two off-sheU and two on-shell statcs.They correspond to (a) the stringpicture and (b) the covariant picture.The world sheet configurations are shown in (c).
x"('7'O=x~-(x~-~'2) +~=_~ -nn s i n
n~ e -inn (11)
Eq. (10) would then be rewritten (reverting to the covariant case for whichD = 16)as: /~(x 2) = (tTl(L~ - 1) -1 ltt)
.(12)
where L~ = - ~ n® = l a+n a n - (Xl - x2) 2 and 1~7)is the state satisfying conditions analogous to the usual gauge conditions [1, 2]: (L x - L ~ + 1 - n ) I d )
= 0,
(13)
where L x is the same as the usual L n with qU replaced by (x 1 - x2)#. The relation expressed in fig. 2b now reads:
verse string diagram and the covariant c-moded diagrams for this process. We shall not discuss the calculation here apart from pointing out that once again the Jacobi transform plays a crucial role in simplifying the Fourier transform. Once again we find a simple connection between position space m o m e n t u m space singularities:
T(q2, q2, M 2 ' t) = f dD x eiq2 xT(x 2 , Pl x, p2 x, t),
(15) with
dklm(q 12,q2, M 2 2, t) T(q2,q2,M2,t) = L.I k,l,m =no (q2 _ k)(M 2 - / ) (q2 _ m) (16)
T(x2,PlX, P2X, t) = ~ ~ d-n(x2'plx'p2x't) (17) n=nl i (xE/4rrEot ' - 2n)ai ' and
x u = x~ - x~. 435
Volume 65B, number 5
PHYSICS LETTERS
Here n o and n I take the same values as for the propagator. In the calculation given in ref. 15] the index i takes three values with a 1 = D / 2 - 7, a 2 = 1 ) / 2 - 8 and a 3 = D / 2 - 9 . Again when D = 16 the leading x 2 singularities are pure poles. The diagram of fig. 3b involving c-moded states again serves to illustrate the singularity structure in the two representations (momentum space and position space). It seems probable that such diagrams will generalize to an arbitrary amplitude involving M onshell particles and N off-shell states. All diagrams have M + N - 1 independent external variables (taking momentum conservation or translation invariance into account). The labelling of space-time indices on the quark lines is once more motivated by the configurations of world sheets such as those shown in fig. 3c. By suitably distorting the world sheets singularities may be exhibited in various momentum space and position space channels in a manner very reminiscent of the usual duality connection between crossed momentum space channels (with the additional requirement of appropriate Fourier transformation). Strips with two normal boundaries have momentum space poles and those with two abnormal boundaries have position space singularities (when the appropriate Fourier transforms have been made). The instantaneous nature of the c-moded propagators of fig. 3b (which are strips in fig. 3c with one abnormal boundary) enables the unambiguous isolation of channels which carry definite values o f x . Notice that fig. 3b contains internal c-moded lines in the position space configuration which involve the "propagation" of excited c-moded states. We have not yet investigated how the x-space factorization that we described above for the propagator (eq. (I 2)) will generalize to these more complicated amplitudes. We end with several comments: (1) The Chan-Paton scheme for attaching flavour quantum numbers to dual amplitudes is automatic if the confined c-moded states are given quark quantum numbers. This suggests a possible identification of the c-moded states with confined quark states. (2) In the dual model calculations described above the lowest lying singularity in x 2 dominates the deep Euclidean limits - the correction from the next singularity (spaced by 8rt2ot' in x 2) giving rise to an exponentially damped contribution. We may envisage an 436
20 December 1976
ideal dual model in which the leading x 2 singularity is on the light-cone instead of outside it. In this case we would get scaling with a specific correction term related to the hadronic resonance separation (i.e. to the Regge slope a'). The scale of this correction term is set by x/-g-~-2~,the size of the spatial separation of the singularities, which is of the same order as the typical size of a hadron. Our diagrams serve to illustrate the relationship between Regge pole, resonance pole and light-cone dominated kinematic limits. The narrow resonance approximation that we have been discussing is just the Born term of the complete theory. We may expect that in the complete theory the discrete x 2 singularities will be shielded by cuts in much the same was as the narrow q2 poles are shielded by normal threshold cuts. (3) It would be satisfying to be sure that the simple structure of eqs. (4) and (10) does not depend crucially on having a singularity outside the light-cone. It is easy to construct mathematical examples of functions which have an infinite set of q2 poles with positive residues and whose Fourier transforms have singularities only inside or on the light-cone. (An example is given in ref. [5] - in this example the x 2 singularities cannot be poles). (4) Closed loops of c-moded states may be inserted into the usual dual propagator. This corresponds, in the string language, to a covariant reweighting of the modes oF the string which emphasizes those histories in which the string has points with infinite momentum density [4] - which may be a physically reasonable modification of the dual model. I am grateful to C.B. Thorn, J.A. Shapiro and A.H. Mueller for useful discussions and to the CERN theory group for its hospitality during the summer.
References
[1] [2] [3] [4]
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