Classical string phenomenology. How strings work

PHYSICS REPORTS (Review Section of Physics Letters) 97, Nos. 2 & 3 (1983) 147—171. North-Holland Publishing Company

CLASSICAL STRING PHENOMENOLOGY. HOW STRINGS WORK Xavier ARTRU Laboratoire de Physique Théorique et Hautes Energies, Université de Paris-Sud, 91405 ORSA Y, France Contents 1. Introduction 2. Topology of the String Model 3. Classical motion of the free string 3.1. The equations of motion 3.2. The classical solutions 4. Classical rules for the interactions 5. The yo-yo model 6. Summary References

147 148 150 150 153 161 164 169 170

1. Introduction The String Model of hadrons [1_5]*has a very tortuous history which may be far from over. It gives a simple explanation of confinement [1] and an intuitive interpretation of the Dual Resonance Model [6—71. This last has obtained many phenomenological successes (Regge behaviour, linear trajectories, etc.) but is still impeded by serious defects (tachyons, unphysical number of dimensions, lack of pointlike hard structure, etc.). Already at the classical limit, the String Model possesses some interesting features such as the relation J a’M2 between mass and spin. Independently of the Dual Resonance Model, the concept of strings is now used in many fields of hadronic physics: quark confinement, jets, heavy quark spectroscopy, exotic hadrons, etc. The Topological Expansion [8] is also a string model from the topological point of view. Our attitude is that the String Model may be more general than the currently existing Dual Resonance Model. It must have the same topology, but may start from a different Lagrangian [9] and different commutation rules. We hope it could thus acquire a hard structure. Nevertheless, the classical limit should be the same. Leaving aside the problem of quantization, we can try to exploit this classical limit as far as possible, and to develop a qualitative phenomenology which applies to as large an experimental domain as possible. The privileged field of application should be that of high energy, highly inelastic collisions because of the large kinematical-quantum numbers involved and the small degree of coherence of the final state. A first step in this program was made by G. Mennessier and the author [10]. The most striking features of low PT, non diffractive hadron—hadron collisions (cutoff in transverse momentum, leading particle effect, limiting fragmentation, pionization) were simply “explained” qualitatively at least by classical properties of the String Model. The basic picture is the following: By fusion or by crossingover, the two incident strings get into one or two very massive strings (called “darts” in ref. [10]);these —

*

For a review of the string model and a more complete list of references, see ref. [5].

0 370- 1573/83/0000—0000/$7.50 © 1983 North-Holland Publishing Company

—

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X. Ariru, Classical string phenomenology. How strings work

darts stretch in the longitudinal direction and then decay into the jets of low-PT final hadrons. This mechanism can be extended to hadron—nucleus collisions, and the “nuclear transparency” emerges quite naturally [lib]. Despite of severe limitations, part of them due to the classical limit, part of them due to the lack of high-PT events, these qualitative results were encouraging, and the author thought it worthwhile to illustrate them with a film [12]. Independently, but motivated by Quantum Chromodynamics, the Lund group arrived at an almost identical one-plus-one dimensional model for quark jet fragmentation [13].It was a great surprise, on both sides, to find nearly one’s figures in the other’s paper! Soon after this, the Lund group made a great improvement to the Classical String Model by introducing hard gluons as energetic kinks in the middle of the string [14].The bridge between short distance (perturbative QCD)- and long distance physics (string hadronization) could thus be achieved. The Lund group also introduced the tunnel effect to allow the creation of quark—antiquark pairs of nonzero mass, as a source of hadron transverse momenta and heavy flavor; it was thus able to offer to experimentalist a complete and realistic simulation program for deep inelastic reactions [15]. This first part of the report is an introduction to the dynamics of the classical relativistic strings which serves as a framework for the Classical String Model, presented after by the Lund group. The somewhat different development made by the present author [lib, c, d] will be published separately. First, in section 2, we consider the strings and their interactions from the topological point of view. Then, in section 3, we study the free motions of open and closed strings and give some examples. In section 4, we give the classical rules which govern the motions of interacting strings. The “yo-yo model”, which allows to incorporate massless partons in the strings, is presented in section 5. We briefly summarize the results in section 6.

2. Topology of the String Model Quark duality diagrams [16]such as that of fig. 1 suggest that mesons are strings with a quark at one end and an antiquark at the other end. These strings have to be oriented in order to avoid qq or ~ states. In this paper we take the convention to orient them from the antiquark to the quark.* They are similar to infinitely thin magnets, the quark and the antiquark playing the roles of the north and south pole respectively [1]. In addition to cutting and fusion, two strings can interact in the middle through a rearrangement [10,17, Sc] or “crossing-over”, as in fig. 2. Hnol strings

,)

ttin~ç~

Cu

Fusion

5. s-o

initial

strings

Fig. 1. Quark-duality diagram for meson—meson scattering. *

Fig. 2. String rearrangement.

The opposite convention is taken likewise; it corresponds to the colour flux tube interpretation.

X. Ariru, Classical string phenomenology. How strings work

149

In short, quark-duality diagrams are the analogues of Feynman diagrams as we pass from pointlike objects (e.g. electrons, photons) to stringlike objects. What is new is the property of duality: Diagram 1 alone, for instance, represents both the s-channel and the i-channel exchanges, in contrast to the case of pointlike particles (fig. 3). Diagram 2 represents t- and u-channel exchanges (the tu term of the Veneziano amplitude); the rearrangement is essentially a nonresonant process. Note that the history corresponding to the topology of diagram 2 is not necessarily a rearrangement: as indicated in fig. 4, it can also be a cutting followed by a fusion. By internal rearrangement or by closing a string onto itself (fig. 5a, b) we get a closed string, without quarks. The pomeron, or at least one component of it, is to be interpreted as a clOsed string; in fact, pomeron exchange is represented by a cylinder diagram (fig. 6).

I 0~~•

‘1

_~

C

0

C

+ A

1B

I

~

~

~f ~

0 >~.(_t

A

B

Fig. 3. Feynman diagrams for e~e scattering.

Fig. 4. The 1—u diagram interpreted as a cutting followed by a fusion.

0 I

I

I

(a)

(b)

(C)

Fig. 5. (a) Emission of a closed string by internal rearrangement, (b) closing of a string Onto itself, (c) splitting of a closed string into two by internal rearrangement.

I Fig. 6. Cylinder diagram for the pomeron.

A triple—pomeron coupling is implied by diagram 5c and a Reggeon—pomeron mixing by diagram Sb. Another process where a closed string may be present is the violation of the so-called O.Z.I. rule [18] (fig. 7). Closed strings should appear not only as exchanged objects (pomeron) or intermediate states (O.Z.I. rule violation) but also as physical states (“glue balls”). These are the objects of intense experimental investigation. It remains to describe the baryons. The most natural choice is a Y-shaped string [19]. Simple topological considerations then lead to the introduction of more and more complex objects such as baryonium, exotic baryons, dibaryons, etc. (fig. 8). The arrows defining the orientations of the strings either diverge from the junction J or converge to the “anti-junction” J. J and J can be considered as the carriers of the baryonic number (+1 and —1 respectively). They are created (and annihilated) in pairs via 3 new types of local string interaction [19a]which are shown in fig. 9.

150

X. Artru, Classical string phenomenology. How strings work

I

LU ting °n

4 Fig. 7. Mechanism for the violation of the O.Z.I. rule.

5

dibaryon

Fig. 8. String configuration for a baryon, an exotic meson or a baryonium M 4, an exotic baryon B5, a dibaryon.

2~c

_

(a)

(b)

(c)

Fig. 9. The three new basic local interactions introduced by the Y-shaped baryon model: (a) creation of a loop, (b) sticking-in-the-middle, (c) triple rearrangement.

3. Classical motion of the free string 3.1. The equations of motion The dynamics of the relativistic string is the simplest generalization of the dynamics of a point particle. We shall not give a full derivation of the equations of motion but only present them by analogy with those of the point particle. Let us first compare the action principles: point particle = ms s: proper_time

~particle

—

ds=\/1_v2dt

string ‘~‘strtng =

—

.~: invariant_area

swept by the string d~=V1—v~±dldt

(3.1)

v1 is the transverse velocity of the string (see fig. 10). The longitudinal component plays no role and is

Fig. 10. Motion of a piece of string.

X. Ariru, Classical string phenomenology. How strings work

151

in fact unobservable. This property, called “longitudinal invariance” in ref. [10],* is at the root of the rapidity plateau in high energy multiproduction Energy-momentum of a point particle E = Vi-

Linear density of energy momentum on the string dE =

v2

p=Ev

K

dl

(3.2)

dp=v±dE

is the linear energy density “at rest”. When we pull apart a quark and an antiquark by a distance dl, we store potential energy K dl in string. Thus, in the nonrelativistic limit, the string is equivalent to a linear potential of slope K. K is then the tension at rest. It is a universal constant, so that the forces between quarks are SU(nc) symmetric (flt being the number of flavors). For a string having a transverse velocity, the tension is diminished by the Lorentz factor, as is the attraction between an electron and a positron travelling together (fig. 11). K

ethje+ (a)

i

d

~.

i

Fig. 11. Comparison between electromagnetic attraction (a) and string tension (b): Coulomb force 2/4ird2)Vi~

F = (e

String tension T = K\/i~.

(3.3)

From now on, we choose units such that K = 1. The geometrical construction of T is given in fig. 12. For a rectilinear string the lines OA and OB are the paths of light along the string. The energy stored in the piece Oa is 1. Consider a point which lies on the string with zero longitudinal velocity. There is no energy transfer

Fig. 12. Geometrical construction of v 1 and T from the light paths (or

Fig. 13. Motion of the “energy cells” of the string.

generatrices) along the string. *

It is a part of the invariance under reparametrization, called “gauge invariance” in the Dual Resonance Model. It also implies that the middle

of the string is electrically neutral (see, however 120]).

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X. Artru, Classical string phenomenology. How strings work

across this point. Consider now a lot of such points; we divide the string into “energy cells” having no longitudinal velocities (but of variable length) (fig. 13). This suggests that the string should not be parametrized by its length, since it is not conserved, but instead by the partial energy, evaluated from one end, e.g. the antiquark end: X=X(w,t)

w=J

dl 2+E~, V1—v1

(3.4)

q

where Eq is the energy of the antiquark. Thus we have

t

=

(3.5a)

8~X=T.

(3.5b)

The light paths along the string have the velocity 3-vectors v1 ± T, thus correspond to the lines i w = constant. Equation of motion: point particle

“energy cell” ~j(v±dw)=dwt9~T

or =

a~x.

(3.6)

This is the equation of motion of vibrating strings. It must be supplemented by the subsidiary conditions 3~X.8~Xuev~.T=0 (3.7a) 2 + (8~X)2 v~.+ T2 = 1 (3.7b) (3,X) which come from our choice of parametrization. These equations govern the inner part of the string. In the mesonic case, the equations of motion for the quarks will provide the boundary conditions. We have two possibilities: (a) The quarks carry the internal quantum numbers, but do not participate to the action and have no proper ernergy-momentum. Eq = E~= 0 and w ranges from 0 to E, the energy of the hadron. The tension has to vanish at the ends of the string: a~X(0,t) = a~X(E,t) = 0.

(3.8)

This is the case of the conventional Dual Resonance Model. From (3.7b), the quarks travel at the velocity of light.

X. Artru, Classical string phenomenology. How strings work

153

(b) In a more realistic model, the quarks must assume the role of partons, i.e. carry a finite fraction of the energy-momentum of the hadron. This can be done classically by adding two mass terms mqfdsq and mJ ds~to the action of the string [21].The energy of the quarks are functions of time Eq(t) and E~(t),and the interval for w is no longer [0,E] but [Eq(t),E Eq(t)]. The position and velocity of the quark (for instance) are given by —

VqVj~T~1T.

Xq”~X[EEq(t),t], The usual formula EqV1 TEqV1

—

=

—

mq, together with (3.7), gives the relation

(dEq!dt)2 = mq.

(3.9)

The equations of motion for the quarks yield nonlinear boundary conditions. In this introduction we shall not develop this model further, except in the limit mq = 0 (Eq being finite). This limit will be shown to be equivalent to the first model, restricted to a particular type of solutions. For closed strings, the boundary conditions are replaced by the periodicity condition X(w,

t)

=

X(w + E, t).

(3.10)

If we now consider baryons, or more complex objects with 3-string junctions, we have the write boundary conditions at the vertex. They can be found in refs. [19]and [22].Here again, the problem is nonlinear. Let us only say that in the rest frame of the junction, the 3 strings are at 120°from each other. This is to balance the three tensions. In this paper, we shall consider only mesonic and closed strings. Let us finally mention new approaches of the classical string by differential geometry [23],which lead to a (nonlinear) Liouville equation. 3.2. The classical solutions Let us first ignore the boundary conditions and consider only equations (3.6) and (3.7). The general solution is /t+w\ ft—w\ X(w, t)y~—~--_)+z~_)

(3.11)

and the subsidiary conditions are satisfied by =

~-z(r)~

=

1

(which express the fact that t ±w = constant describe light paths). These equations can be written in four-vector form, with X°(w,t) X(w, (dy)2

t)

=

y

/t+w\ (,,—~-—)

(dz)2 = 0.

+

/t—w\ z~

(3.12)

=

t,

y°(r) r

z°(r): (3.13) (3.14)

X. Artru, Classical string phenomenology. How strings work

154

We see from (3.13) that the light lines t + w = constant and t w = constant on the world sheet of the string form two families of parallel curves (fig. 14). We call these curves “generatrices”. [See fig. 14.] To a piece AB of the string, we associate the “causality diamond” ACBD limited by the generatrices passing through A and B. The history of the string inside this diamond is completely determined from the shape and transverse velocities of the part AB only. The energy-momentum of AB is —

PAB=

or: PAB=CD

(3.15)

(we use the notation CD XD Xe). This relation allows us to construct the generatrices from the dynamical state of the string at a given time: —

AD=~(AB+PAB),

AC=~(AB—PAB).

Conversely, the momentum transfer across the line w side,

QCD=

=

cst joining C to D is, from the B side to the A

=

JdtT= ~ YD

—

Yc + Z~

—

=

AB

or, including the time components, (3.16)

QCD=AB.

D~’ -

~L

5..

~Se

Fig. 14. Generatrices and causality diamond on the world sheet of a string.

X. Artru, Classical string phenomenology. How strings work

155

The causality diamond ABCD can also be defined when A and B are not at equal time, but joined by a spacelike curve drawn on the sheet. In this case, (3.15) and (3.16) are still meaningful, PAB (QcD) represent the energy-momentum flow across any spacelike curve AB (any timelike curve CD) drawn on the sheet. From the causality diamonds of two adjacent pieces of string, AB and BC, we can construct the causality diamond of the piece AC by parallel displacement of the generatrices, as indicated in fig. 15. The resulting diamond will be called the “fusion” of the first two, and the inverse operation will be called “fission”. We can divide the world sheet into infinitesimal diamonds as in fig. 16. We shall call “genes” the elements dy and dz of generatrices. Then the string may be viewed as a 2-way road where genes circulate at the velocity of light. A gene carries the elementary momentum dy or dz and an elementary message: A gene dy “tells” every gene dz it meets to make a translation by dy, and vice-versa. A generatrix y(T) or z(T) can possess a discontinuity in its first derivative. This results in an angular point which propagates on the string at the speed of light (see fig. 17).

_1~_~

- -

Fig. 15. Fusion of two causality diamonds (hatched areas) into the single one AECF.

Fig. 16. Division of a world sheet in infinitesimal causality diamonds, whose sides are the “genes”.

S

Fig. 17. Propagation of an angular point on the string.

Let us now take account of the boundary conditions (3.8). We first consider only one boundary (as for a semi-infinite string), for instance that of the antiquark, which we write in four-dimensional form: 3~4~X(0, t) = 0. Integrating, we get y(T)= z(T)+ C.

(3.17)

We can choose the four-vector C to be zero. The trajectory of the antiquark is then Y(t) X(0, t) = 2y(t/2).

(3.18)

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X. Artru, Classical string phenomenology. How strings work

We recall that the antiquark travels at the velocity of light: =1.

(3.19)

The other points of the string are given by X(w, t) =

~{Y(t + w) +

Y(t

—

w)}.

(3.20)

They are obtained by the geometrical construction of fig. 18. The antiquark trajectory Y completely determines the motion of the whole string; we call it the “directrix”. There is now only one family of curves which serves both as y-generatrices and z-generatrices. When a dy-gene meets the antiquark it is reflected as a dz-gene, as shown in fig. 19.

(I-sw) Fig. 18. Geometrical construction of the string from the antiquark trajectory Y(t) (directrix). The 4-momentum of 4X isp = AX = ~AB.

Fig. 19. Reflection of the genes at the extremity of the string.

Alternatively, one can consider eq. (3.20) as describing a string with no antiquark at w = 0 but folded onto itself (extending the w range to negative values). The knowledge of the shape and the velocities in the part ~X of the string determines the history in the “folded causality diamond” associated with the equivalent folded piece X~X(dark area in fig. 18). This is a larger domain than the normal diamond of ~X. The four-momentum of i~Xis half that of the folded piece, i.e. P~Pqx=~AB’AX.

(3.21)

When we come close to the antiquark, at a distance d, the linear energy density increases like ~ where R is the radius of curvature of the directrix. What happens if a part of the directrix is linear (R = co)? This is the situation depicted in fig. 20. A finite energy cell lies on top of ~, as can be shown from the geometrical construction of fig. 18. This “energy grain” simulates a massless parton. Its four-momentum is Pgrain =

Inf(I~,4F).

(3.22)

It is “accelerated” by the string in the first half part of IF, then “decelerated”. It satisfies relation (3.9) with mq = 0. Thus, if we choose only broken line directrices, we have equivalence with the limit mq—*O of the model with massive quarks, as was stated before. This submodel, called the “yo-yo model” will be studied in more details in section 5.

X. Ariru, Classical string phenomenology. How strings work

157

1~~Y

String Fig. 20. Directrix having a straight-line portion, resulting in an accumulation of energy-momentum at the endpoint of the string

Fig. 21. Quark and antiquark trajectory for a free string.

(‘energy grain”).

It remains to take account of the boundary condition for the other extremity, a~X(E,t)=0. By integration, we get ft+E\ Y~)

ft—E\ Z~,,—~-_)+P~

where P is a constant four-vector. With C= 0 in eq. (3.17), we find y(T+E)=y(T)+P.

(3.23)

The generatrices are thus periodic (like a cork-screw) in space-time. P is the four-momentum of the~ string: For the time component indeed, we have y°(r) r, thus E = P°.For the space components, it suffices to say that, according to (3.23), P~is collinear to the mean four-velocity of the string. The directrix, given by (3.18), is periodic with a pitch four-vector 2P. Instead of being the trajectory of the antiquark, the directrix could have been defined as the trajectory of the quark. The two trajectories are shifted by P (fig. 21). As for the string itself, it can be considered as having a 4-period P if we forget its orientation (do not distinguish the quark from the antiquark), or 2P if we orient it: X(w, t) = X(E

—

w, t + E)— P = X(w, t + 2E)— 2P.

P” will be referred to as the 4-semi-period. The periodicity can be considered as a consequence of the back-and-forth motion of the genes between the quark and the antiquark. This is illustrated by mapping the history on the w, t-plane (fig. 22). The complete history of the string will be given either by one causality diamond or by one period of the directrix. In a meson at rest, the quark and the antiquark travel on the same closed orbit Y(t), of length 2m. The motion of the same meson after a boost can be obtained by applying a Lorentz transformation to the space-time directrix Y. This increases the time period (time dilation) from 2m to 2E. At very large

158

X. Artru, Classical string phenomenology. How strings work

2E~’

Fig. 22. Mapping of the string history on the w, I plane. The set of genes a meets the set of genes

boos~~

(/~i

I

periodically.

S.

~

—*

b

7

.

meson at rest Fig. 23. Time dilation and Lorentz contraction of a fast moving string.

rapidity, the space projection Y(t) of Y is stretched in the direction of the boost as illustrated on fig. 23. The quark velocities — and the genes point nearly in the forward direction. From the geometrical construction of fig. 12, we see that the string itself is nearly perpendicular to the boost (Lorentz contraction). The angular momentum of a meson at rest is —

J=JdWXAÔIX=~YAdY,

(3.24a)

J=

(3.24b)

swept by the string in a semi-revolution

(see fig. 24). The same vector area also appears in the expression for the magnetic flux which crosses the surface, in the case of a uniform field: —+

=

B area.

The quark and antiquark circulating along the directrix create a magnetic dipole moment

Fig. 24. Angular momentum of the string.

Fig. 25. The revolving rod.

X. Ariru, Classical string phenomenology. How strings work

~=~—area

159

(3.25)

where e is the total charge of the meson (2m is the revolution period). Thus the gyromagnetic ratio 2m4~/eJof the meson is always two. Actually one must add the spins and magnetic moments of the quarks themselves. Let us study the two simplest examples of string motion: The revolving rod, which is obtained by taking a circular directrix (fig. 25). This maximizes J for a given m and thus is interpreted as the “leading Regge trajectory”. We have J=lTr2/2,

m=irr,

thus, putting the right power of

K

for dimensionality, (3.26)

m

2ITK

It is the experimental value of the universal Regge slope a’ K

=

0.18 GeV2/h

=

0.9 GeV/Fermi

=

0.9h GeV2 which serves to fix ~:

14 tons.

The yo-yo [10, 24, 13]. The directrix is a straight line segment, along which the quark and the antiquark move back and forth: ~ c This is a pure 1 + 1-dimensional motion, which we depict in fig. 26a. The angular momentum is zero. As for any broken-line directrix, there are “energy grains” at the quark and antiquark locations. If we boost the yo-yo collinearly along its axis (fig. 26b), we have -

P=AB 1 P÷ueE+p=~AD~V2 ~ P...ueE_p=IACIV2 J

(3.27)

m2 = P+P_

(3.28)

=

2 x area(ADBC).

If a noncollinear boost is made, the yo-yo sweeps a succession of diamonds, as indicated in fig. 27. Each diamond is a causality diamond of the whole yo-yo, projected on the 3-dimensional space. Conversely, any parallelepipedic causality diamond defines a yo-yo motion.

2.

A. boost

L..~

--:_-----—

(a) Fig. 26. Yo-yo motion of a string (a) at rest, (b) after a longitudinal boost.

160

X. Ariru, Classical string phenomenology. How strings work

~X2

Fig. 27. Space motion of a yo-yo boosted in an arbitrary direction.

Fig. 28. Motion of the “pulsating circle” in 2 + 1 dimensional spacetime.

For closed strings, instead of boundary conditions, we have written a periodicity condition (3.10). Applying it to the general solution (3.13) gives (3.29)

y(T++E!2)—y(T÷)= z(r..)—z(T_—E/2)=P/2,

where r~= (t ±w)/2 and P is a constant four-vector which again we identify with the four-momentum (this is obvious for the time component). As for open strings, the generatrices are periodic curves. But the y-family and the z-family generally differ in shape; only their pitch four-vector must be the same. A simple example of closed string motion is the pulsating circle (fig. 28). Its equation is Xi(w, t) + iX2(w, t) = a cos(t/a) e”° while the generatrices are given by 2~ria.

y1 + iy2

=

—

iz2 = ~a e

In table 1 we give the global kinematical quantities for open and closed strings. For closed strings, we can also have y z. In this case the string is folded onto itself and has the same motion as the open string of generatrix y. Quarks are replaced by the folding points (fig. 29). These strings are invariant under reversal of the intrinsic orientation, i.e. under charge conjugation. The simplest one is the “folded revolving rod” (fig. 29b). It is the closed string which has the maximum J at a given m, its Regge slope being half the mesonic one. It appears as the best classical candidate for the unrenormalized pomeron. Table 1 Kinematical quantities for open an closed strings

periodicity of the generatrices periodicity of the motion spin action integral

Open string of momentum P

Closed string of momentum P

P 1 revolution: 2P ~ysdy 1 revolution: m2

P12 1 pulsation: P12 ~ysdy+~zsdz

1 pulsation: m2/4

X. Artru, Classical string phenomenology. How strings work

161

~ ~2Z~ (a)

(b)

Fig. 29. Folded closed strings: (a) arbitrary shape, (b) “folded revolving rod”, lying on the unrenormalized pomeron trajectory.

4. Classical rules for the interactions The classical solutions of the equations of motion for a meson or a closed string are periodic and “nothing happens”. This is the same situation as in the classical mechanics of a pointlike particle (it will travel indefinitely without decaying). However, decay (cutting, in the string case) is not forbidden by the classical action principle (which is more fundamental than the equations of motion). To illustrate this, let us consider the decay of one pointlike particle into two (fig. 30). The action principle simply tells us to minimize the quantity

~i=—JmAds—JmBds—Jmcds where ds2 = dt2 (dx)2. It does not give the relative probability decay/nondecay. This is the job of quantum mechanics. Requiring .s~1to be stationary with respect to small variations of the trajectories, first with D fixed, tells us to take rectilinear uniform motions in AiD, DO 1 and DD1. Then, allowing for small variations of D, we get —

(4.1) 2 = 1). Eq. (4.1) expresses the conservation of energy-momentum and also where (u D of the decay. serves uto isfixthe thefour-velocity space-time point We now take the same approach for the string decay. The problem is to span a world sheet between the initial string A 1, in fig. 31, and the final strings B1 and Cf, with a stationary area. If such a sheet mAuA

=

mBUB +

mcuc

Fl na I

Fig. 30. History of a decay in the case of pointlike particles.

Fig. 31. History of a string decay.

162

X. Artru, Classical string phenomenology. How strings work

exists, it must satisfy two conditions: (i) the motions of string A before cutting, and of strings B and C after cutting, are those of free strings (ii) if one continues the motion of A at t> t~and the motions of B and C at t < t~,then A coincides with B and C in the causality diamonds of 4! and Iq respectively (grey areas in fig. 31). We conjecture that (i) and (ii) are not only necessary but also sufficient conditions to have a stationary action. Instead of giving us the shapes of the initial and final strings, suppose that we know the motion of the decaying string, A, and the space-time point I where it breaks. Then the motion of the decay products is completely determined by (i) and (ii): we have only to make the fission of the causality diamond of 4q into those of 41 and Iq. The genes see their two-way traffic interrupted at the cut. They are reflected back by the newly created quarks and make two independent sets. Let us see what happens locally in the neighborhood of the cutting point. Let ulu’ and vlv’ be the generatrices of the initial string passing through I. The new generatrix for the left-hand string in fig. 31 is ulv’. (At an endpoint we have to consider only one generatrix, which is of the y-type for t < t1 and of the z-type for t> t5.) It has an angular point at I. As was shown in fig. 17, this results in an angular point on the string, which is a kind of shock wave coming from the cutting. This “cicatrix” of the cutting will move indefinitely back-and-forth along the string. The quark created at I follows the trajectory obtained from Iv’ by an homothethy of factor 2. The motion of the antiquark remains unchanged until the cicatrix reaches it. This gives the pattern of fig. 32: The new period of the directrix is the arch AB of the old one. At B, the antiquark suddenly again takes the direction it had at A. In fact the genes emitted by the antiquark just after A have been reflected by the cut and are now received again by the antiquark. t~ect~iX -

Sc.

1...,

~.......~i~rxtr~ecIory

Fig. 32. Evolution of one fragment of string after a cutting.

As an example, let us consider the decay of a yo-yo. The history, which is all in a 1 + 1-dimensional subspace, is represented in fig. 33. Each rectangle, corresponding to one half oscillation, is by itself a causality diamond of the string. The fission of a rectangular diamond gives two rectangular diamonds, thus two yo-yos. It will be convenient, in a 1 + 1-dimensional space, to introduce the dual momentum P obtained from P by interchanging the space and time components. Thus, energy-momentum conservation is visualized by AB = AC + CB. The opening of a closed string proceeds with the same kind of rules as the cutting of a mesonic string. If the resulting open string does not decay, it has an infinite number of opportunities to close again, because of its periodic motion.

X. Artru, Classicalstring phenomenology. How strings work

163

V~I

Fig. 33. Decay of a yo-yo.

Fig. 34. Evolution of a string resulting from a fusion.

Fusion The rules for fusion are obtained from the rules for cutting by time reversal: we have to make the fusion of two causality diamonds, which gathers the genes of the initial strings. But, generally, the final state of a fusion looks different from the initial state of a cutting and vice-versa: After a cutting, we find cicatrices, whereas before a fusion the strings have a priori no angular point. Let us see what happens locally in the latter case. On fig. 34 we have drawn the initial trajectories of the quark and of the antiquark which annihilate, and the initial generatrices ulu’ and vlv’ passing through the contact point, obtained by homothethy of ratio 1/2. The final motion has ulv’ and vlu’ for y- and z-generatrices respectively and the final string consists of three parts: — an unperturbed part A’a, which ignores the interaction, a mixed part ab where dz-genes of A meet dy-genes of B, an unperturbed part bB’. The angular point I of the generatrices ulv’ and vlu’ gives the angular points a and b respectively, which we again call cicatrices. The construction of the new directrix from the two initial ones is given in fig. 35. Roughly speaking, one gets one new period by attaching two old ones end to end. Cutting and fusion can also be analysed with the help of the w, t plane (fig. 36). It displays regions where the final state keeps memory of the initial state, i.e. the causality diamonds A0 and B0 and their — —

periodic repetitions A±1and B±3.

( b

A,, B

~

,

b

-.

/

.~ ~‘

.

Fig. 35. Construction of the new directrix after a fusion. áI’ is the arch of the antiquark trajectory of A which serves to construct A (see fig. 18). bb’ is the analogue for B, but with the quark trajectory. By translating bb’ into a’b”, one gets a period ab” of the new directrix.

A

\~ —2

Fusion

—1

__

/‘

A0

\\ ,‘

,‘

B1

A

,~

.—

cutting

Fig. 36. w, t mapping of the history A + B ~C.

‘..

2

±t

164

X. Artru, Classical string phenomenology. How strings work

Rearrangement From the geometrical point of view, the rearrangement of “cross-over” aq + a’q’ equivalent to the two cuttings aq-*aR+Rq,

—*

a’q + aq’ (fig.

37)

is

a’q’-*a’R+Rq’

followed immediately by the two fusions aR+Rq’-~aq’,

a’R+Rq-*a’q.

Thus there is a rearrangement of genes resulting in a cross-over of the y-generatrices passing through R and a similar cross-over for the z-generatrices. Go_\

q’

~,oG

2c

q

rea.

G~

...oQ’

q’

a~ ~a

q ‘~

Fig. 37. The rearrangement and its equivalent 2 cutting + 2 fusion process.

Here again, the final strings generally have cicatrices (two in each). Thus a swarm of interacting strings will have more and more cicatrices. This kind of entropy increase is expected to be stopped by quantum, mechanics; because the angular points requires high quantum numbers (high-order Fourier components)).

5. The yo-yo model In section 3, we have noticed that, when the extremity of a string travels through a rectilinear part of the directrix, a finite part of the string energy is sitting on it (see fig. 20 and eq. (3.22)), as if there was a massless parton. On this basis, we shall develop in this section a model for “partoned” strings, by considering only broken-line directrices and attributing the “energy grain” to the present quark or antiquark. In fact, in 1 + 1-dimensions, broken-line directrices are the only solutions of (3.19). The yo-yo presented in fig. 26 is the simplest example (whence the name “yo-yo” model). The multiple decay of a very massive yo-yo (generalizing fig. 33) gives a very cheap model of jet in 1 + 1-dimension [10, 13, 25], satisfying all conservation laws (energy, momentum, charges). The general free string motion in 1+ 1-dimension has been studied in detail in ref. [241. The string can be folded several times, and one has energy grains not only at the extremities (quark—partons), but also at the folding points and anywhere in the middle of the string, all travelling at the velocity of light. This was compared in [24a]to a set of interacting yo-yo’s. We shall see below how these features can be generalized to the 3+ 1-dimensional case. Let us first take the example shown in fig. 38. From (3.20) it is clear that the string itself is broken-line. In fig. 39, we analyse the string motion in the vicinity of the segment AB (fig. 39). The

X. Artru, Classical string phenomenology. How strings work

Fig. 38. String with a broken line directrix. The momentum sitting on top of ~ is k = min(A~,~B).

165

Fig. 39. Example of string motion, on the antiquark side, in the yo-yo model. We have taken a directnx with only two corners, for simplicity. The size of the ball symbolizes the energy of the antiquark.

antiquark momentum in fig. 38 is, according to eq. (3.22), -+

k

=

-+

min(A4, 4B)

(5.1)

and its energy k0 = Ik~.During the first half (Al) of~e path AB, the antiquark is “accelerated”, i.e. Its momentum increases linearly with time from 0 to Al, at a rate dIkI/dt = K = 1. In the second half, it is “decelerated” in a symmetrical way. The angle between the antiquark velocity v and the string is acute in the former case, obtuse in the latter. It may be surprising that, although the string tension (in this example) is not collinear to the quark path AB, this one is not curved. In fact, by a suitable Lorentz transformation, one can put the local transverse velocity of the string equal to zero, as in the 1 + 1-dimensional case. There, we know that the quark motion is uniform. But we can also understand the string—antiquark interaction in the following way (see ref. [141). Let T be the tension of the string near the antiquark. We have (see fig. 40 for the notations; cf. also fig. 12) V.L

=

sin a

T=V1_v21.=Icosal

(5.2)

v=v1±T. The + and — signs correspond to the acceleration and deceleration phases respectively. The increase of momentum parallel to the string is dk11/dt = T.

(5.3)

The string itself has a linear density of energy-momentum

“acceleration oF ~

s”cleceleraiion”

Fig. 40. Antiquark—string interaction.

X. Artru, Classical string phenomenology. How strings work

166

dw/dl

=

iiVi

—

v~= 1/Icos a!,

dp/dl

=

v±/IcosaI.

(5.4)

In acceleration (deceleration) phase, the antiquark “swallows” (rejects) the string at speed v11 = cos a. It thus gains (loses) energy and transverse momentum at a rate (k0, kj = ±(1,v±).

(5.5)

Adding (5.5) and (5.3) gives dk/dt= ±(1,v)

(5.6)

in accordance with (5.1). The angular points of the string also travel at the velocity of light along special “generatrices” (see fig. 17 and text before). These lines are obtained from the directrix by homothetic transformations of ratio ~and centered at the corners (e.g. A and B in fig. 39). The simple “yo-yo” is given by a directrix with only 2 corners within one period (fig. 43a). It is the only one to have no angular point. Gluon as energy grains in the middle of the string. Energy grains not attached to the extremities of the string were found in the 1+ 1-dimensional model [24]. The Lund group has generalized them to noncollinear string motion, and proposed them as a model for gluons [14]. To see how they can be obtained, we consider the general solution* ft+w\

ft—w\

X(w,t)=yl~,,—~--—)+zI~,,-—~--—), (5.7) 2= (t9z)2=0, (5.8) (8y) which works also for closed strings. Let G(t) be the trajectory of a gluon. We have in a domain 9~of the w, t plane of nonzero measure: G(t).

(5.9)

Let us differentiate (2.12) twice with respect to w. We get (5.10)

c9y—öz=82y+32z=0.

The solution of (5.8)—(5.10) is then t+ y(T)= VT+

Yo,

w

ras—~-----E[r 1, T2]

z(o-)=ur+zo,

o-as~-~E[o5,o2]

(5.11)

G(t)= vt+y0+z0, tEa1+r1,o2+r2 2 = 0. v Thus, the condition for the existence of a gluon is that both y- and z-generatrices have rectilinear *

Here X, y and z denote four vectors. The symbol 8 means that we take the derivative with respect to the whole argument, e.g. + w)12) = (d/8t)y(r)~*—(t+*)12.

X. Artru, Classical string phenomenology. How strings work

167

pieces parallel to a common lightlike vector v. The gluon travels at the velocity of light. In the case of open strings, y as z = ~Y, the directrix must have two parallel rectilinear parts not related by periodicity. The domain ~ is a rectangle, shown in fig. 41. Thus the gluon trajectory is a squeezed “causality diamond” of the string history. It has generally 3 phases (see fig. 41): 2K, i.e. twice the rate for an tA < t < tB, the gluon energy increases linearly at the rate dE/dt = accelerating quark. In fact the gluon receives energy from each side of the string at the rate K. tB < t < t~,the gluon energy is constant, the gluon receives energy from one side at the rate K and gives energy to the other side at the same rate. t~< t < t~, the gluon energy goes down to zero at the rate dE/dt = —2K.

Fig. 41. Reciprocal image of the gluon history in the w, t plane.

In fig. 42 we have drawn the corresponding string history using the generatrices. The three phases of the gluon trajectory are AB,,C and CD. In AB, the gluon (g) is pulled forward by both sides of the string. Its momentum is 2Ag. In BC, the gluon (~)Jspulled forward by the left-hand string and backward by the right-hand one. Its momentum is2AB. In CD, the gluon (g”) is pulled backward by both sides of the string. Its momentum is 2g”D. In our terminology, dy-genes (element of yN

I\ I “ I

-

‘_—=

N ~

-5 I

Fig. 42. Gluon history. ABCD is the image of the domain ABCD of fig. 41. AB: acceleration phase; BC: constant energy phase; CD: deceleration phase. The string history is made of flat “causality diamonds” limited by the rectilinear pieces of generatrices. S, 5’ and S” represent the string at three different times, one for each phase. The dotted lines represent trajectories of a dz-gene.

X. Artru, Classical string phenomenology. How strings work

168

generatrices carrying momentum dy) are trapped during a finite time (equal to IACD in th~~uonin their travel from right to left. dz-genes, travelling from left to right, are trapped for a time ABI. Let us give some simple examples of “gluoned” strings: The closed simple yo-yo, obtained from the simple yo-yo by replacing the quarks by two gluons, and doubling the string (fig. 43).

~

;‘~L’,’~7:

(U) ‘~ (b) Fig. 43. Motions (a) of a simple yo-yo, (b) of a closed simple yo-yo.

The gluon triangle (fig. 44). We take for y and z generatrices two triangles symmetric about the origin, but with opposite orientation: y(t) = —z(t). The generalization to a n-gluon star is straightforward. A quark—antiquark—gluon string can be obtained by taking a crossed trapezoid for directrix (fig. 45). The fusion of two yo-yo’s, according to the rules of section 4, replaces the annihilating quarks by a pair of gluons (fig. 46). The gluon momenta just after collision are equal to the quark momenta just before. We have thus a “forward reaction” q+4—*g+g. Other kinds of two-by-two reactions (g+g-+ g + g, etc.) can be simulated likewise, always a zero scattering angle.

~

Fig. 44. Motion of a gluon triangle. At timej,,,= 0,e string~s gathered in 0. 3 gluons emerge with momenta 2 OA, 2 OB and 2 OC. They disappear in A, B and C, each giving two angular points. The gluons reappear in A’, B’ and C’. OBC’ is a y-generatrix, OCB’ is a z-generatrix.

Fig. 45. A quark—antiquark—gluon string. The directrix is ABCDA (in this order). The gluon path is IJ. Acceleration phase: positions 1—2—3; constant momentum: position 4; deceleration phase: position 6. In 7 the gluon has disappeared. We do not draw the subsequent string shapes of one period of motion, for clarity. The fact that the gluon collides with the quarks (in 3 and 5) is due to Z-shaped parts in the directrix.

To sum up, we have now a classical model with quarks, continuous glue (pieces of string with finite linear energy density) and gluons. There is some analogy with electrodynamics: electrons classical electromagnetic field (Coulomb potential) (transverse) real photons

quarks (end points) continuous glue (string segments) gluons (energy grains)

X. Artru, Classical string phenomenology. How strings work

169

/

~

,‘q

‘I

,.:~/-~*

Fig. 46. Fusion of two yo-yo’s, producing a pair of gluons.

The separation between “continuous glue” and “gluons” is perhaps an artificial classical way to take into account the particle-wave duality of the colour field in QCD. Nevertheless it will prove to be a good classical model incorporating both confinement and parton ideas. Remarks (1) It does not seem possible to generalize the yo-yo model to the case of a curved space-time; this would destroy the periodicity of the string motion, which guarantees that the energy grains are indefinitely renewed. Neither to the case of a classical external electromagnetic field. However, in a quantum theory, due to the uncertainty principle, the quark momentum cannot vanish, and we do not have this problem. (2) As yet formulated, the yo-yo model possesses only massless partons. Massive quarks can be introduced, at the price of loosing the mathematical simplicity of the solutions. (3) The partons interact only at zero impact parameter and zero momentum transfer. This second limitation is removed in the preceding contribution to this report.

6. Summary We have studied the classical motion of free nonbaryonic (open or closed) strings, and presented the classical rules governing their interactions: cutting, fusion and rearrangement. The string can be viewed as a two-way road where “genes” circulate at the velocity of light. These carry elementary momenta and information which serves to determine the string motion. When they reach an extremity (quark or antiquark), they are reflected back. This gives rise to the periodic motion of the string, whose space-time pinch is equal to the four-momentum. For an open string, the motion can be equivalently given by the directrix (i.e. the quark or antiquark trajectory). In the case of a broken-line directrix, the quark and the antiquark are accompanied by “energy grains” which simulate massless partons; a particularly interesting example is the one-plus-one dimensional “yo-yo” motion. The rules for fusion, cutting and rearrangement follow from causality and are simply given in terms of “causality diamonds” or subsets of genes. The interaction generally produces “cicatrices”, i.e. angular points which move indefinitely back and forth along the final strings. The “yo-yo model” constructed with broken-line generatrices gives also energy grains in the middle of the string which simulate the gluons. These last, like the energy grains at the extremities, live finite times but are renewed by the periodicity of the string motion. Parton reactions can take place, but only at zero scattering angle.

170

X. Ariru, Classical string phenomenology. How strings work

As it is now, the theory is not a complete one; given a string, the theory does not predict whether the string will break and if so when and where. If two strings come into contact, the theory does not tell whether or not they interact (through rearrangement or fusion). This shows the need for a quantum treatment. Nevertheless, it is possible to apply the classical model to hadronic reactions, getting as much information as one can from the classical rules and supplementing these rules by an interaction probability law (as one usually does for radioactivity).

Acknowledgements I wish to thank Dr. U. Sukhatme for carefully reading the manuscript and for his interesting suggestions.

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/

Fig. 47. Decay of a revolving rod (picture taken from “Le Ballet des Hadrons” [12]).

171