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Measuring the size and shape of strings
Nuclear Physics B349 (1991~ 159-167 No.h-Hal|and
MEASURING THE SIZE A.ND SHAPE OF SWRJNGS David MITCHELL and i ~ S U N D B O R ~ *
Primers, N~ (~544, USA Receded 4~~ul~~~
We define s~e and shape in terms of ~ r ~ a b ~ e s . ioe° scattering a m # i t u ~ . E x ~ s~n~gs can be measured b~, scauefing ground-stale string p ~ e s ~ fi~temo We ~ k ~ e ~ e ~#~e~e, classically, the ends of the string trace out a circle. Our ~ l a t ~ s ~ s that t ~ r ~ o~ t ~ circle is proportional to the mass of li~e siting. M, m agreemen~ ~ i ~ ¢ l a s s ~ e ~ ~ , ~ . However, on distance scales less lhan ~ M . quamum f l u c t u a t ~ are ~ m ~ r ~ ' ~ ~ a ~ ~ space-time #c|ure becomes questkmab|e.
1. l n t r o d ~ n
The fundamental difference between string theory and point partic|e theory lies in its spectrum of massive states. Duality, modular invariance and u i t r a ~ e t finiteness are all consequences of the properties of this spectrum. In some respects strings behave as "soliton-like'" objects of the low-energy field theory [t~ but they are probably more important than sol|tons as they already appear at lowest order in the perturbative expansion of the full theory. It is clear from classical comiderations [2] that strings are extended, their size increasing with mass. Strings therefore have the potential of modifying not only ultraviolet properties, but also infrared physics [3]. As a first attempt to understand massive strings one may want to take the classical limit by studying solutions to the classical string equations. ~ e r e are, however, certain limitations to this approach. Firstly, splitting and joining interactions cannot be properly taken into account. Secondly, it may be objected that violent vacuum fluctuations ruin attempts to define classical trajectories (string coordinates X~(tr, ¢) are massless quantum fields in two dimensions) [4,5]. In this paper we work directly with quantum amplitudes in order to find a space-time picture from first principles. The string coordinates X-~((r, ~') are not themselves observables, but appear in vertex operators used in calculating string amplitudes. * Present address: Institute of Theoretical Physics, University of Stockholm, Vanadi~'~igen 9. S-113 46, Stockholm, Sweden. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
D. Mitchell, B. Sundborg / Size and shape of strings
160
Therefore we can get information about the string's embedding in spacetime indirectly by studying the true observables of the theory, i.e. scattering amplitudes. On top of this we find to what extent the vacuum fluctuations are regularized automatically in this measurement process. The paper is organized as follows. In sect. 2 we calculate a diagram for the scattering of a tachyon off an excited open string, which in the classical limit corresponds to a rotating rod. In sect. 3 we pick a convenient kinematic configuration so that we can easily extract a form factor and determine the shape of the string. In sect. 4 we discuss additional diagrams that contribute to the full amplitude for the case of a U(1) gauge group. Strings are then dipoles and we verify that the Coulomb field vanishes. All calculations are very basic and just slight extensions of what can be found in ref. [6]. In the conclusion we summarize our results and what remains to be done.
2. The elastic string-tachyon amplitude This section contains some well-known physics but we include it here in order to make our discussion self-contained and to introduce our notation. We will use units where the string tension T is set equal to rr-~, and our convention is such that all incoming momenta are positive and the metric is of the form d i a g ( - , + , + , + ). Also we will choose our strings to be described by a U(1) gauge group, so we may think of the ends of the string as carrying opposite charges. Thus in terms of the standard Mandelstam invariants, S=
--(Pl
+/02) 2 ,
t = - ( p 2 + p 3 ) 2,
u= -(p,
+ p 3 ) 2,
(1)
one can define the following trajectories: !
a ( s ) = 5s + 1, and similarly for t and u. They then satisfy the identity 1
4
a(s) +a(t) +a(u)=-~ E M 2 + 3 .
(2)
i=1
For the open bosonic string we have M 2 = 2 ( N - 1), where N is the eigenvalue of the level number operator and M is the mass. The process we will be calculating is shown diagrammatically below (fig. 1). Here states 1 and 4 are tachyonic and states 2 and 3 describe equally massive strings. This is not the only diagram that contributes to the overall amplitude; we shall discuss the full amplitude in sect. 4. For ease of calculation we shall take the massive states to lie on the leading Regge trajectory. Classically they correspond to
D. Mitchell, B. Sundborg / Size and shape of strings
t6i
4
|
Fig. 1. The four-point string amplitude where states 1 and 4 are tachyonk: and states 2 ~ massive.
3
rigidly rotating rods. The normalized oscillator representation of one of these states at level N is 1
N indices
(3)
v~N! _~,.p... a~_,a~_,aP_,....__ -..-_. [0, p ) . N oscillators
Here ~: is a totally symmetric polarisation tensor. In order that (3) be physical ~: must be transverse and traceless: =
=0,
= 0.
(4)
Massive physical states are then created by the following v e r t e x operators: V ( p ) =~rr..." P~P~"" e i P x :
(5)
N factors
the • - denote normal ordering and the X ~ and P~ are the usual mode expansions for the string position and momentum operator respectively. By introducing the dummy variable A, we can rewrite eq. (5) as 0
V(p)
0
= ~:,~... 0A,~ 0Ar " " ' e i p X + a P : "
(6)
The amplitude we are interested in is then formally given by A = g2(0, - P 4 I :I/3(P3) :A( Pm + P2):V2( P2)" 10, P l ) ,
(7)
where A is the usual string propagator and g is the string coupling constant. The algebra involved in calculating (7) is greatly facilitated by the use of (6) - indeed, using only a slight generalisation of the coherent state techniques employed in
162
D. Mitchell, B. &mdborg / Size and shape of strings
calculating the Veneziano amplitude, one finds A = f01 d y y - " - ' / ' ( 1
N
x E
_y)-2-,/2
N!
!(
r!)
,y x_,~(l _y)2~Q({p}, y, r,{_~}, N ) ,
[
Q-- - p 4 - ( p , + P 4 ) ~ _ y
Nr I
r'~'* " ~¢2"r P I + ( P ~ + P 4 )
y
®r
,8,
The notation used in the definition of Q in eq. (8) is somewhat condensed and needs a little explanation. There are r contractions between each of the polarization tensors and the respective term in square brackets. The remaining ( N - r) indices on each of the polarisation tensors are then "used up" in contractions between the two. As expected, if we set N = 0 in eq. (8) we recover the well-known Veneziano amplitude
A( s, t) = g'B( -a( s) ,- a(t)), where B(u,v) is the Euler beta function. To evaluate (8), one needs to pick a particular kinematical configuration. It is this problem we deal with in sect. 3.
3. The shape of the string We want to determine the shape of the massive string from a scattering experiment. This is done by extracting a form factor from the tachyon-string amplitude and Fourier transforming from momentum to position space. To simplify the calculation we take the incoming tachyon probe to travel along the spin axis of the massive string, and choose this to be the z-axis (fig. 2). The tachyon is deflected by an angle 0 and qu -- P l . + P4~
(9)
is the momentum transferred to the massive string. If p. is the momentum in the c.m. frame and E = V/p 2 - 2 is the tachyon energy, we have
Plu=(E,O,O,-p),
P4# = ( -- E, p sin 0,0, p cos 0 ) ,
q~,=(O, psinO,O,p(cosO-
1)),
q2=4p2sin2(O/2).
(10)
D. Mitchell, R Sundborg / Size and shape of strb~gs
[63
7.
.*y
Fig. 2. The kinematical configuration we are considering. The incoming tac~au ~p~~~ ~t~ered ~ an angle O in the x-z plane off lhe massive string wh~h is rotating in t ~ x~v p|~meo
The square of the momentum transfer in the plane of rotation is q~y =
p2
sin-' 0,
( t t)
The internal state of the massive string is specified by its polarization tensor which we take to be a tensor product of identical vectors g. W~th 1 -
=
. .
(o,
i
o)
each factor in the product excites one unit of spin along the z-axis. The state is only an eigenstate of S= but not of S 2. However, uncertainties in S~ and S: are small compared to the expectation value ( S x) - thus this state should represent a string rotating rigidly around the z-axis in the classical limit N --, ~. The physical state conditions (4) are satisfied for the above choice of polarization tensors only when the string is at rest• We would like to describe a process in which the string is close to being unaffected by the tachyon probe, but it cannot be at rest both before and after scattering. We take both initial and final polarization tensors to be given by eq. (12) in the respective rest frames for initial and final states. In a f'Lxed frame we therefore obtain polarization tensors satisfying the Virasoro conditions for the ingoing and outgoing strings by applying appropriate boosts to (12). However, it turns out that the effect of the boosts on the amplitude is suppressed by a factor of M - l , where M is the mass of the large string. Thus we may set =
3
=
To evaluate the amplitude in the limit M - > ~ we need the following scalar products which to leading order are 1 ~"3* " ~'2 --
1)
~'2"Pl
-- 0 '
~'2" q = ~ 3~" q = ~ ' f " P 4 = ~
p sin 0 •
(13)
1~
D. Mitct~dL B. Su
/ Size and shal~" o f strings
The kinematic factor Q of the amplitude (8) has the limit t"
Q = (ff2"P~
+
and the scattering amplitude ~ c o m e s A ---
£
:v dyy~
N ~.
' +~v='~'( 1 - Y ) - ' - ' ~ " ' Eoo0 , (N-r)
!(r~.) z ( - q l ~ / 2 ) "
-=-B( N = , ( s), = ~( t ) } L x ( q ~,/9)- ~ B ( N - , ~~s )": - a ( t ) ) J 0 (¢Z-(:N + ..... l) q,,) e'#/"
(15)
where the limit of the Laguerre polynomial L N [7] is uniform for bounded momentum transfers. If % approaches O as N ~ ~ we may drop the exponential and only keep the Bessel function Y~. In fact this makes sense because the beta function ensures that the amplitude is ex~onentially suppressed for q2Ii I n ( M E ) > 1
(16)
To see this one may use Stirling's approximation in the beta function,
B{ N - a ( s ) , - a ( t ) ) ~ B ( - M E , -
a ( t ) ) ~ F( - a ( t ) ) ( - M E ) '-q:/z. (17)
The amplitude has a pole at t = 0 from the gamma function but the only other q2 dependence on scales smaller than In(ME)-~ is due to the Bessel function Jo- The pole can be viewed as reflecting the exchange of massless gauge bosons. In a conventional point-particle definition of form factors one takes advantage of this isolation of the effects of interactions. In our case only the Bessel function contributes to the form factor. The charge density is given by its Fourier transform. It is simpler to do the inverse transform from density to form factor: 1
2 rrl/2 ( N + 1)
-
1
f ax
¢2( N + 1) ) 6 ( x ± ) e tq'x
f d, exp[iq,,¢2(N +1)
= J0(¢2( N + 1) qll)"
(18)
D. Mitchdi~ B. S,mdb¢~ / S~e and shape ef sumg~
~65
We conclude that the endpoint traces out a circle of radius M just as expected classically (the tachyon is scattered off a stationary state) [2]. The appearance of N + 1 rather than N in the asymptotic expansion of the Laguerre ~ x ~ ~ prompts us to speculate that the tachyon ( N = O) is of nonzero exlent (~8)o logarithmic scale given by (16) translates to an uncer~ain~ ( a x ) 2 .. i n ( M E )
in position space, which increases with mass~ This scale governed ~, the ~ t a function is intrinsicai|y stringy and quantum mechanical. The amplitude is no longer dominated by processes where particles are exchanged ber~een a t a c h y ~ and a massive string that are well separated in space. Due to large z e r o - ~ n t fluctuations, the two strings do not appear as separate objects when observed on time scales as short as the interaction time - E-~, el. refs. [4, 5~. The fluc~uatk~ become observable because a large number of resonances are almost degee~erate, the energy difference between neighbouring levels being of order M-
4. The ~
U(I) am#Rmie
To obtain all the diagrams necessary to compute the full four-p~int am#trade one can consider holding strings l and 4 as fixed external states and aU~irig a|I possible orderings of the remaining strings. This produces eight diagrams - however, not all of them are distinct. A convenient way of counting the distinct diagrams involves using the twist operator, /2, which reverses the ends of the string. It is defined by ~ = ( - 1)N, where N is the level number operator. Thus mass eigenstates are either even or odd under the action of .O. The two distinct terms in the basic three-point amplitude then satisfy the following identity: (1[V213 > = ( -
1)N'+N'-+N~(lUV3t2).
20)
Using eq. (20) and denoting an insertion of the twist operator by the symbol × , one finds that only the four diagrams depicted in fig. 3 are distinct. The last two diagrams correspond to different regions of the integrand and combine to give a single beta function. It is now a simple matter to write down the
2
3
3
2
2
3
3
2
Fig. 3. The four distinct diagrams that combine to yield the full U(1) amplitude. N is the level number of the massive string and × denotes an insertion of the twist operator.
16(,
D. MitchdL R S u n d ~
/ Si:e and shaI~ of strings
full amplitude using the expression for the first diagram given in eq. (15),
A-~g ~(V~(_(N+I)q~I)iB(N-a(s )
a(t))+B(N
a(u),-a(t))
+B(N-a(s),N-a(u))].
(21)
From eq. (20) it is evident that photons cmmot couple to on-shell string states; take Na = 1, N 2 - N:~ and the two terms in eq. (20) then cancel. This is as one e x a c t s for an electromagnetic dipole. We should be able to see this decoupling in the full amplitude given by eq. (21) in the sense that processes corresponding to photon exchange should not contribute to the amplitude, i.e. the residue of the t = 0 ~3le should vanish. Only the first two beta functions in (21) have the possibiliD, of poles at a ( t ) = 0. 1 , 2 , . . . , ~ let us focus on these and define
Ap-- B( N - ~ ( s ) . - a ( t ) )
+ B( N - a ( u ) , - a ( t ) ) .
(22)
We want to calculate the residue of the pole at t = 0. Using the definition of
B(u,r), eq. (2) and the fact that F ( v ) o n l y has simple poles, we find, around t = 0
r(N-a(s)) F( N -
+
1 -~(s))
+ N-
F(a(s) + 2 - N )
r ( a ( s ) - N + 1)
I
-N+
II
=0. Hence the full amplitude given by eq. (21) does not have any singularities in the t = 0 region, showing that the photons do not couple to the dipole.
5. C o n c l u s i o n s
We have presented a calculation that uses a measuring "probe" to determine the extendedness of an object through its form factor. Our results bear out classical expectations- the string has a length that coincides asymptotically with the classical value. Also the concept of the U(1) string as a dipole is borne out in the large-mass limit as the Coulomb field vanishes. Above a momentum scale ~ i n ( M E ) - ~/2, the separation of the amplitude into a form factor and a point-particle factor becomes untenable. We cannot say a great deal about this regime as our procedure is not applicable, only that the difficulties are due to the quantum structure of the string. This structure is intimately related to duality and Regge behaviour. For simplicity we have only dealt with the least complicated excited states, those on the "leading trajectory". They are parametrized by the mass and a classical
D. Mitchell, R Sundbo~ / S~e and ~
of ~tr~g$
~:~
limit is obtained by taking it to infinity. We believe that shapes of more ~ n e ~ string states can be found as above, given a c|assificath~n of the ~ g ~ ~ terms of Regge trajectories. Another extensh3n of this work w ~ | d be to ~ r [nclastic proccsscs. Thc results a b ~ e concern a "time averaged" s~,r[n~ a ~ we cannot resolve the motion of the endp~nts. To find a t [ ~ ~ ~ n t p~ure could use a coherent-state description of mass~e s~r[:ngs~ but we h ~ e e ~ n t e t ~ technical problems in imposing physical sta|e condit~ns w i | ~ t m ~ n g t:~ calculation intractable. The work of D.Mo was supported by a UK SERC/NATO ~ d o c ' ~ ship and that of B.S. by the Swedish Natural Science Research ~ ~ i I contract F-PD 9275-100.
f e [ ~ ~u~r
References [I] [2] [3] [4] [5] [6]
A. Dabholkar. G. Gibbons, J.A. Harve3~and F. Ru~ R u ~ N~t. P~'~o B ~ J ~ | ~ 33 P. Goddard, J. Goldstone. C. Rebbi and C.B. Thorn, Ned. P~s. B56 ~|973) |09 B. Sundborg, Nucl. Phons. 13338 ~1990) | 0 | I_ Sussldnd, Phys. Rev. D | (i970) 1182 M. Karliner. i. Klebanov and L. Susskind, Int. J. Mad. P~s. A3 ~|c~SS]~|9S~ M.B. Green, J.H. Schwa~ and E. Witlen, Superstring theory.. Vo~s. L |I ( C ~ g e Ua~ ~. Press. Cambridge, 1987) [7] A. Erd~lyi, ed., Bateman manuscript project, Higher transcendental func'~o~. Vo[o I| ~McC_~'aw-B~ New York, i954) p. 199, eq. (2)
MEASURING THE SIZE A.ND SHAPE OF SWRJNGS David MITCHELL and i ~ S U N D B O R ~ *
Primers, N~ (~544, USA Receded 4~~ul~~~
We define s~e and shape in terms of ~ r ~ a b ~ e s . ioe° scattering a m # i t u ~ . E x ~ s~n~gs can be measured b~, scauefing ground-stale string p ~ e s ~ fi~temo We ~ k ~ e ~ e ~#~e~e, classically, the ends of the string trace out a circle. Our ~ l a t ~ s ~ s that t ~ r ~ o~ t ~ circle is proportional to the mass of li~e siting. M, m agreemen~ ~ i ~ ¢ l a s s ~ e ~ ~ , ~ . However, on distance scales less lhan ~ M . quamum f l u c t u a t ~ are ~ m ~ r ~ ' ~ ~ a ~ ~ space-time #c|ure becomes questkmab|e.
1. l n t r o d ~ n
The fundamental difference between string theory and point partic|e theory lies in its spectrum of massive states. Duality, modular invariance and u i t r a ~ e t finiteness are all consequences of the properties of this spectrum. In some respects strings behave as "soliton-like'" objects of the low-energy field theory [t~ but they are probably more important than sol|tons as they already appear at lowest order in the perturbative expansion of the full theory. It is clear from classical comiderations [2] that strings are extended, their size increasing with mass. Strings therefore have the potential of modifying not only ultraviolet properties, but also infrared physics [3]. As a first attempt to understand massive strings one may want to take the classical limit by studying solutions to the classical string equations. ~ e r e are, however, certain limitations to this approach. Firstly, splitting and joining interactions cannot be properly taken into account. Secondly, it may be objected that violent vacuum fluctuations ruin attempts to define classical trajectories (string coordinates X~(tr, ¢) are massless quantum fields in two dimensions) [4,5]. In this paper we work directly with quantum amplitudes in order to find a space-time picture from first principles. The string coordinates X-~((r, ~') are not themselves observables, but appear in vertex operators used in calculating string amplitudes. * Present address: Institute of Theoretical Physics, University of Stockholm, Vanadi~'~igen 9. S-113 46, Stockholm, Sweden. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
D. Mitchell, B. Sundborg / Size and shape of strings
160
Therefore we can get information about the string's embedding in spacetime indirectly by studying the true observables of the theory, i.e. scattering amplitudes. On top of this we find to what extent the vacuum fluctuations are regularized automatically in this measurement process. The paper is organized as follows. In sect. 2 we calculate a diagram for the scattering of a tachyon off an excited open string, which in the classical limit corresponds to a rotating rod. In sect. 3 we pick a convenient kinematic configuration so that we can easily extract a form factor and determine the shape of the string. In sect. 4 we discuss additional diagrams that contribute to the full amplitude for the case of a U(1) gauge group. Strings are then dipoles and we verify that the Coulomb field vanishes. All calculations are very basic and just slight extensions of what can be found in ref. [6]. In the conclusion we summarize our results and what remains to be done.
2. The elastic string-tachyon amplitude This section contains some well-known physics but we include it here in order to make our discussion self-contained and to introduce our notation. We will use units where the string tension T is set equal to rr-~, and our convention is such that all incoming momenta are positive and the metric is of the form d i a g ( - , + , + , + ). Also we will choose our strings to be described by a U(1) gauge group, so we may think of the ends of the string as carrying opposite charges. Thus in terms of the standard Mandelstam invariants, S=
--(Pl
+/02) 2 ,
t = - ( p 2 + p 3 ) 2,
u= -(p,
+ p 3 ) 2,
(1)
one can define the following trajectories: !
a ( s ) = 5s + 1, and similarly for t and u. They then satisfy the identity 1
4
a(s) +a(t) +a(u)=-~ E M 2 + 3 .
(2)
i=1
For the open bosonic string we have M 2 = 2 ( N - 1), where N is the eigenvalue of the level number operator and M is the mass. The process we will be calculating is shown diagrammatically below (fig. 1). Here states 1 and 4 are tachyonic and states 2 and 3 describe equally massive strings. This is not the only diagram that contributes to the overall amplitude; we shall discuss the full amplitude in sect. 4. For ease of calculation we shall take the massive states to lie on the leading Regge trajectory. Classically they correspond to
D. Mitchell, B. Sundborg / Size and shape of strings
t6i
4
|
Fig. 1. The four-point string amplitude where states 1 and 4 are tachyonk: and states 2 ~ massive.
3
rigidly rotating rods. The normalized oscillator representation of one of these states at level N is 1
N indices
(3)
v~N! _~,.p... a~_,a~_,aP_,....__ -..-_. [0, p ) . N oscillators
Here ~: is a totally symmetric polarisation tensor. In order that (3) be physical ~: must be transverse and traceless: =
=0,
= 0.
(4)
Massive physical states are then created by the following v e r t e x operators: V ( p ) =~rr..." P~P~"" e i P x :
(5)
N factors
the • - denote normal ordering and the X ~ and P~ are the usual mode expansions for the string position and momentum operator respectively. By introducing the dummy variable A, we can rewrite eq. (5) as 0
V(p)
0
= ~:,~... 0A,~ 0Ar " " ' e i p X + a P : "
(6)
The amplitude we are interested in is then formally given by A = g2(0, - P 4 I :I/3(P3) :A( Pm + P2):V2( P2)" 10, P l ) ,
(7)
where A is the usual string propagator and g is the string coupling constant. The algebra involved in calculating (7) is greatly facilitated by the use of (6) - indeed, using only a slight generalisation of the coherent state techniques employed in
162
D. Mitchell, B. &mdborg / Size and shape of strings
calculating the Veneziano amplitude, one finds A = f01 d y y - " - ' / ' ( 1
N
x E
_y)-2-,/2
N!
!(
r!)
,y x_,~(l _y)2~Q({p}, y, r,{_~}, N ) ,
[
Q-- - p 4 - ( p , + P 4 ) ~ _ y
Nr I
r'~'* " ~¢2"r P I + ( P ~ + P 4 )
y
®r
,8,
The notation used in the definition of Q in eq. (8) is somewhat condensed and needs a little explanation. There are r contractions between each of the polarization tensors and the respective term in square brackets. The remaining ( N - r) indices on each of the polarisation tensors are then "used up" in contractions between the two. As expected, if we set N = 0 in eq. (8) we recover the well-known Veneziano amplitude
A( s, t) = g'B( -a( s) ,- a(t)), where B(u,v) is the Euler beta function. To evaluate (8), one needs to pick a particular kinematical configuration. It is this problem we deal with in sect. 3.
3. The shape of the string We want to determine the shape of the massive string from a scattering experiment. This is done by extracting a form factor from the tachyon-string amplitude and Fourier transforming from momentum to position space. To simplify the calculation we take the incoming tachyon probe to travel along the spin axis of the massive string, and choose this to be the z-axis (fig. 2). The tachyon is deflected by an angle 0 and qu -- P l . + P4~
(9)
is the momentum transferred to the massive string. If p. is the momentum in the c.m. frame and E = V/p 2 - 2 is the tachyon energy, we have
Plu=(E,O,O,-p),
P4# = ( -- E, p sin 0,0, p cos 0 ) ,
q~,=(O, psinO,O,p(cosO-
1)),
q2=4p2sin2(O/2).
(10)
D. Mitchell, R Sundborg / Size and shape of strb~gs
[63
7.
.*y
Fig. 2. The kinematical configuration we are considering. The incoming tac~au ~p~~~ ~t~ered ~ an angle O in the x-z plane off lhe massive string wh~h is rotating in t ~ x~v p|~meo
The square of the momentum transfer in the plane of rotation is q~y =
p2
sin-' 0,
( t t)
The internal state of the massive string is specified by its polarization tensor which we take to be a tensor product of identical vectors g. W~th 1 -
=
. .
(o,
i
o)
each factor in the product excites one unit of spin along the z-axis. The state is only an eigenstate of S= but not of S 2. However, uncertainties in S~ and S: are small compared to the expectation value ( S x) - thus this state should represent a string rotating rigidly around the z-axis in the classical limit N --, ~. The physical state conditions (4) are satisfied for the above choice of polarization tensors only when the string is at rest• We would like to describe a process in which the string is close to being unaffected by the tachyon probe, but it cannot be at rest both before and after scattering. We take both initial and final polarization tensors to be given by eq. (12) in the respective rest frames for initial and final states. In a f'Lxed frame we therefore obtain polarization tensors satisfying the Virasoro conditions for the ingoing and outgoing strings by applying appropriate boosts to (12). However, it turns out that the effect of the boosts on the amplitude is suppressed by a factor of M - l , where M is the mass of the large string. Thus we may set =
3
=
To evaluate the amplitude in the limit M - > ~ we need the following scalar products which to leading order are 1 ~"3* " ~'2 --
1)
~'2"Pl
-- 0 '
~'2" q = ~ 3~" q = ~ ' f " P 4 = ~
p sin 0 •
(13)
1~
D. Mitct~dL B. Su
/ Size and shal~" o f strings
The kinematic factor Q of the amplitude (8) has the limit t"
Q = (ff2"P~
+
and the scattering amplitude ~ c o m e s A ---
£
:v dyy~
N ~.
' +~v='~'( 1 - Y ) - ' - ' ~ " ' Eoo0 , (N-r)
!(r~.) z ( - q l ~ / 2 ) "
-=-B( N = , ( s), = ~( t ) } L x ( q ~,/9)- ~ B ( N - , ~~s )": - a ( t ) ) J 0 (¢Z-(:N + ..... l) q,,) e'#/"
(15)
where the limit of the Laguerre polynomial L N [7] is uniform for bounded momentum transfers. If % approaches O as N ~ ~ we may drop the exponential and only keep the Bessel function Y~. In fact this makes sense because the beta function ensures that the amplitude is ex~onentially suppressed for q2Ii I n ( M E ) > 1
(16)
To see this one may use Stirling's approximation in the beta function,
B{ N - a ( s ) , - a ( t ) ) ~ B ( - M E , -
a ( t ) ) ~ F( - a ( t ) ) ( - M E ) '-q:/z. (17)
The amplitude has a pole at t = 0 from the gamma function but the only other q2 dependence on scales smaller than In(ME)-~ is due to the Bessel function Jo- The pole can be viewed as reflecting the exchange of massless gauge bosons. In a conventional point-particle definition of form factors one takes advantage of this isolation of the effects of interactions. In our case only the Bessel function contributes to the form factor. The charge density is given by its Fourier transform. It is simpler to do the inverse transform from density to form factor: 1
2 rrl/2 ( N + 1)
-
1
f ax
¢2( N + 1) ) 6 ( x ± ) e tq'x
f d, exp[iq,,¢2(N +1)
= J0(¢2( N + 1) qll)"
(18)
D. Mitchdi~ B. S,mdb¢~ / S~e and shape ef sumg~
~65
We conclude that the endpoint traces out a circle of radius M just as expected classically (the tachyon is scattered off a stationary state) [2]. The appearance of N + 1 rather than N in the asymptotic expansion of the Laguerre ~ x ~ ~ prompts us to speculate that the tachyon ( N = O) is of nonzero exlent (~8)o logarithmic scale given by (16) translates to an uncer~ain~ ( a x ) 2 .. i n ( M E )
in position space, which increases with mass~ This scale governed ~, the ~ t a function is intrinsicai|y stringy and quantum mechanical. The amplitude is no longer dominated by processes where particles are exchanged ber~een a t a c h y ~ and a massive string that are well separated in space. Due to large z e r o - ~ n t fluctuations, the two strings do not appear as separate objects when observed on time scales as short as the interaction time - E-~, el. refs. [4, 5~. The fluc~uatk~ become observable because a large number of resonances are almost degee~erate, the energy difference between neighbouring levels being of order M-
4. The ~
U(I) am#Rmie
To obtain all the diagrams necessary to compute the full four-p~int am#trade one can consider holding strings l and 4 as fixed external states and aU~irig a|I possible orderings of the remaining strings. This produces eight diagrams - however, not all of them are distinct. A convenient way of counting the distinct diagrams involves using the twist operator, /2, which reverses the ends of the string. It is defined by ~ = ( - 1)N, where N is the level number operator. Thus mass eigenstates are either even or odd under the action of .O. The two distinct terms in the basic three-point amplitude then satisfy the following identity: (1[V213 > = ( -
1)N'+N'-+N~(lUV3t2).
20)
Using eq. (20) and denoting an insertion of the twist operator by the symbol × , one finds that only the four diagrams depicted in fig. 3 are distinct. The last two diagrams correspond to different regions of the integrand and combine to give a single beta function. It is now a simple matter to write down the
2
3
3
2
2
3
3
2
Fig. 3. The four distinct diagrams that combine to yield the full U(1) amplitude. N is the level number of the massive string and × denotes an insertion of the twist operator.
16(,
D. MitchdL R S u n d ~
/ Si:e and shaI~ of strings
full amplitude using the expression for the first diagram given in eq. (15),
A-~g ~(V~(_(N+I)q~I)iB(N-a(s )
a(t))+B(N
a(u),-a(t))
+B(N-a(s),N-a(u))].
(21)
From eq. (20) it is evident that photons cmmot couple to on-shell string states; take Na = 1, N 2 - N:~ and the two terms in eq. (20) then cancel. This is as one e x a c t s for an electromagnetic dipole. We should be able to see this decoupling in the full amplitude given by eq. (21) in the sense that processes corresponding to photon exchange should not contribute to the amplitude, i.e. the residue of the t = 0 ~3le should vanish. Only the first two beta functions in (21) have the possibiliD, of poles at a ( t ) = 0. 1 , 2 , . . . , ~ let us focus on these and define
Ap-- B( N - ~ ( s ) . - a ( t ) )
+ B( N - a ( u ) , - a ( t ) ) .
(22)
We want to calculate the residue of the pole at t = 0. Using the definition of
B(u,r), eq. (2) and the fact that F ( v ) o n l y has simple poles, we find, around t = 0
r(N-a(s)) F( N -
+
1 -~(s))
+ N-
F(a(s) + 2 - N )
r ( a ( s ) - N + 1)
I
-N+
II
=0. Hence the full amplitude given by eq. (21) does not have any singularities in the t = 0 region, showing that the photons do not couple to the dipole.
5. C o n c l u s i o n s
We have presented a calculation that uses a measuring "probe" to determine the extendedness of an object through its form factor. Our results bear out classical expectations- the string has a length that coincides asymptotically with the classical value. Also the concept of the U(1) string as a dipole is borne out in the large-mass limit as the Coulomb field vanishes. Above a momentum scale ~ i n ( M E ) - ~/2, the separation of the amplitude into a form factor and a point-particle factor becomes untenable. We cannot say a great deal about this regime as our procedure is not applicable, only that the difficulties are due to the quantum structure of the string. This structure is intimately related to duality and Regge behaviour. For simplicity we have only dealt with the least complicated excited states, those on the "leading trajectory". They are parametrized by the mass and a classical
D. Mitchell, R Sundbo~ / S~e and ~
of ~tr~g$
~:~
limit is obtained by taking it to infinity. We believe that shapes of more ~ n e ~ string states can be found as above, given a c|assificath~n of the ~ g ~ ~ terms of Regge trajectories. Another extensh3n of this work w ~ | d be to ~ r [nclastic proccsscs. Thc results a b ~ e concern a "time averaged" s~,r[n~ a ~ we cannot resolve the motion of the endp~nts. To find a t [ ~ ~ ~ n t p~ure could use a coherent-state description of mass~e s~r[:ngs~ but we h ~ e e ~ n t e t ~ technical problems in imposing physical sta|e condit~ns w i | ~ t m ~ n g t:~ calculation intractable. The work of D.Mo was supported by a UK SERC/NATO ~ d o c ' ~ ship and that of B.S. by the Swedish Natural Science Research ~ ~ i I contract F-PD 9275-100.
f e [ ~ ~u~r
References [I] [2] [3] [4] [5] [6]
A. Dabholkar. G. Gibbons, J.A. Harve3~and F. Ru~ R u ~ N~t. P~'~o B ~ J ~ | ~ 33 P. Goddard, J. Goldstone. C. Rebbi and C.B. Thorn, Ned. P~s. B56 ~|973) |09 B. Sundborg, Nucl. Phons. 13338 ~1990) | 0 | I_ Sussldnd, Phys. Rev. D | (i970) 1182 M. Karliner. i. Klebanov and L. Susskind, Int. J. Mad. P~s. A3 ~|c~SS]~|9S~ M.B. Green, J.H. Schwa~ and E. Witlen, Superstring theory.. Vo~s. L |I ( C ~ g e Ua~ ~. Press. Cambridge, 1987) [7] A. Erd~lyi, ed., Bateman manuscript project, Higher transcendental func'~o~. Vo[o I| ~McC_~'aw-B~ New York, i954) p. 199, eq. (2)