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String model with baryons: Topology; classical motion
Nuclear Physics B85 (1975) 442-460. North-Holland Publishing Company
STRING MODEL WITH BARYONS: TOPOLOGY; CLASSICAL MOTION X. A R T R U ~ Laboratoire de Physique Th~orique et Hautes Energies ~, Universit~ de Paris-Sud, 91405 Orsay, F~ance
Received 27 June 1974
Abstract: We consider the model in which a meson is an open string with a quark at one end an anti-quark at the other end; a baryon is made of three strings joining at a point, and carrying quarks at their free ends. The triality condition is ensured by orienting the strings according to simple rules. These rules suggest the existence of an underlying magnetic monopole theory Independently of this "explanation", we investigate first the topological properties of the model, by looking at the duality diagrams: existence of exotic hadrons, of five basic interactions between strings. Some renormalization diagrams are assigned a negative power of the Veneziano coupling constant. Then, taking the same action (the world-sheet area) as in the conventional string model, we set down the equations of motion of a junction. We argue that the slope of the leading baryonic Regge trajectory is the same as that for the mesons. As an example of an application, we study the "sticking together" of two colliding strings; we find that it is classically forbidden at relative velocities greater than (~)~.
1. I n t r o d u c t i o n The string model of the hadrons [ 1 - 5 ]**offers an intuitive representation o f dual resonance models, and a powerful tool o f investigation in this field. More directly, w i t h o u t assuming a n y particular lagrangian, the string picture is a simple interpretation of the Harari-Rosner duality diagrams. A m e s o n is a string with a quark at one end and an antiquark at the other end. The "'rubber b a n d ; ' represents the two-dimensional world sheet swept out b y the strings in our 4-dimensional space-time. By looking at world sheets o f different topologies, we see that strings can j o i n b y the ends, with a quark-antiquark annihilation, split themselves, with a quark-antiquark creation, rearrange w h e n going through each other. T h e y can form closed strings w i t h o u t quarks called pomerons. Besides the conventional Harari-Freund duality (Regge o resonances and p o m e r o n o background), the string picture implies the absence o f free quarks, and the so-called " a b n o r m a l d u a l i t y " [6] ( p o m e r o n ~ reso:~ Present address: Theory Division, CERN, 1211 Geneva 23, Switzerland. * Laboratoire associ6 au Centre National de la Recherche Scientifique. ** For a review of the string model and a more complete list of references, see ref. [5].
X. Artru, String model
443
nances in the case of pomeron-hadron scattering). The string is supposed to be neutral except at the end (a continuous charge distribution would spoil the gauge invariances of the dual model, which are interpreted as the invariance of the theory with respect to a reparametrization of the world sheet), but oriented (to avoid strings with two quarks or two antiquarks). The analogy with an infinitely thin magnet, developed by Nambu [1 ], is very striking. When we want to include the baryons in the scheme, some problems arise immediately: (i) where must we put the quarks? (ii) how is triality satisfied (if we believe that quarks are fractionally charged)? (iii) we must introduce exotic states. The problems of quark spin, quark statistics will not be considered in this work, which stays at a classical level. In the next section, we expose our (rather subjective) choice for the shape of a baryonic string: three pieces of strings joining at a point and carrying the quarks at their free ends. A speculative explanation in terms of the existence of magnetic monopoles is given in sect. 3. According to this choice, the world sheets swept out by the strings are connected 3 by 3 along world lines of 3-string junctions. By looking at different topologies, 3 new basic interactions among strings emerge, in addition to the 2 basic interactions [ 7 - 9 ] of the purely mesonic model, which were the cutting (or the fusion) and the rearrangement. In sect. 4, we classify these interactions and discuss the formation of exotics. In the conventional model, one can assign a factor Xn-2 to the n-meson amplitude, X being the Veneziano coupling constant. This is equivalent to assigning a factor X to each cutting or fusion, and ),2 to a rearrangement. This correspondence will be extended to the new basic interactions, in sect. 5, giving an intriguing result. The equations of motion of strings with 3 by 3 junctions are set down in sect. 6. Examples of motions with Regge slope ~ and 1 (relative to mesons) are given. The evolution of the strings after a cutting or a rearrangement has been studied in ref. [7]. In this paper we consider (sect. 7) one of the new processes, the "sticking together" of two colliding strings, and solve the equations of motion for it.
2. Topology of the baryonic string Assuming that a baryon contains 3 quarks located somewhere on a neutral and non strange string, the simplest possibilities are: (A) meson = q - -
q
baryon -- q
qq
(B) meson = q - - q
baryon = q - - q - - q
(C) meson = q - -
baryon =
q
(D) meson = q,._./q
qYq q baryon = f q " x q~q
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X. Artru, String model
X
~, ),'
Ko
~)
b)
K-
P
n
P
K-
proton
p
n
p
Proton
Fig. 1. Duality diagram in model B. In (a) the n quark must take the place of the h quark at the extremity of the string (in O) before "emitting" the K°. There are 4 other topologically inequivalent diagrams for this reaction. In the model A, the baryon is made of a quark and a diquark which plays the role of the antiquark. There are 9 kinds o f diquarks : p n, p p, p ~, n n, etc. There is no duality diagram for the reaction proton + proton ~ ~ - + mesons + n u c l e o n , which is known to have a non-negligible cross section [ 10]. We therefore discard this possibility. In the model B, the quark in the middle is not treated on the same footing as the two others. One can restore the symmetry by allowing it to reach one end and dislodge the former occupant. This model has less duality than models C and D. Let us consider, for instance, the reaction K - + proton ~ K ÷ + K 0 + ~2- . All the quarks o f the p r o t o n must participate, therefore none can stay in the middle o f the string. There are 6 diagrams, such as those of fig. l a and l b , which are not topologically equivalent. By contrast, models C and D do not distinguish these 6 diagrams. Let us now consider the model C. There is a p r i o r i no limitation on the number o f 3-string junctions. The triality condition (absence of q q ~ states, for instance) is satisfied if (i) the strings are oriented from an antiquark (or a junction) towards a quark (or a junction), (ii) at a junction, all the 3 orientations either diverge or converge together. In the former case we call it a baryonic junction B.J., in the latter an anti-baryonic juncti_on B.J. The baryonic number is equal to the number o f B.J. minus the number o f B.J. Some configurations are shown in fig. 2. In the model D, the production or absorption o f a hadron occurs via a rearrangement (fig. 3). The triality condition cannot be cast into a rule as simple as in m o d e l C * • A closed string model, in which there are only 6 possible locations for the quarks and which satisfies the triality condition, has been considered, however [ 11 ].
445
X. Artru, String model
a
b
c
d
e
Fig. 2. String configurations in model C. (a) ordinary meson, (b) pomeron; (c) baryon; (d) exotic meson; (e) polymerized form of the 4He nucleus. > - - = antiquark; - - ~ = quark. The experimental value o f the slope of the baryonic Regge trajectories, which seems to be the same as for the mesonic trajectories (within 10% uncertainty) cannot a priori disfavour any of these 4 models. Most probably, models A, B and D lead to the same slope for the mesons and the baryons. But it will be shown in sect. 6 that model C also has (classically, at least) a leading baryonic trajectory o f the same slope as the mesonic ones. The slope of-~, which was used as an argument against this model, corresponds to a symmetric configuration o f the 3 branches, but this configuration does not give the highest angular m o m e n t u m for a given mass. F r o m now on, we shall consider only the model C, with 3 b y 3 connected strings.
3. Are t h e r e m a g n e t i c m o n o p o l e s ? A "raison d ' e t r e " f o r the orientation rules could be the existence o f magnetic monopoles. A standard procedure to build a "dual" theory * (here dual means symmetric with respect to electric and magnetic fields) is to introduce strings (the so-called "Dirac strings") which terminate at the electric or magnetic charges and which carry the necessary flux such that div E = div B = 0 everywhere. If the electric and magnetic charges e i and gi of the different species o f particles satisfy the generahzed Dirac condition ei g / -
e/gi = ½ n h ,
(n integer) ,
(3.1)
these strings are not observable and can be considered to be only a mathematical tool. What happens, instead, if the Dirac condition is not fulfilled? Let us consider,
O Fig. 3. Emission of a meson by a baryon in model D. - - e - A rearrangement occurs at R.
* For a review of dually charged particle theory, see ref. [12].
= quark.
e
= ant~uark.
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X. Artru, String model
p
P~
~)
n
proton .. ...... i~' n
Fig. 4. The identity between Dirac strings and hadronic strings. Solid line : physical string; dashed line; fictitious string. for instance, a world in which there are purely magnetic charges g satisfying (3.1) with respect to the electrons: eg = ½h and quarks with purely electric charges ~e and + be. We know that a string which carries an electric flux Q not obeying (3.1) scatters a magnetic charge. We conjecture that it has other observable effects (for instance a linear mass distribution) which also depend on Q modulo e. We have to connect all quarks and antiquarks in the world by a network of strings (fig. 4). Branches that carry fluxes a multiple of e can be ignored and we are left with strings carrying fluxes of magnitude (n + ~) e and (n + 9) e (n positive or negative integer). Let us orient the strings so that the algebraic flux is (n + ~) e " we get the picture of the string model chosen in sect. 2 * Similarly, the magnetic charges must also be connected by strings (for reason of symmetry), with the same topology: we would have dual baryons, dual mesons of total magnetic charge 0 or + 3g, with quantitatively very different properties, however. In particular, the analogue of the "fine structure constant" would be (3g) 2 = 9 × 137/4 = 308. Therefore it would be hard to separate magnetic charges from magnetically neutral matter.
4. Interactions In our 4-dimensional space-time, strings sweep out 2-dimensional surfaces called "world sheets". Duality diagrams are topological drawings of these world sheets. The world sheet of a meson has the topology of a rubber band, the world sheet of a pomeron has the topology of a tube, and the history of a baryon is made of 3 sheets joining along the world line of the B.J. Correspondingly, given a set of world sheets bounded by (oriented) quark world lines and connected 3 by 3 by B.J. world lines **, we get the evolution of a system of strings by cutting it with successive hyperplanes t = t l , t2, t 3 etc. An "interaction'~ is a change in the topology of this system for a
* The possible connection between triality and the existence of magnetic charges has also been considered by Schiff [ 13 ]. ** It is convenient to orient the B.J. world line as an antiquark world line.
X. Artru, String model
447
particular value of t. We find 5 "basic interactions" plus 3 purely virtual processes in which strings disappear in (or are created out of) the vacuum (see figs. 5 and 6). (a) Fusion of 2 strings at the end-points with annihilation of a q ~ pair, and the inverse process: the cutting of a string with creation of a q ~ pair. It occurs, for instance, in M + M ~- M (M = meson, B = baryon). (b) Simple rearrangement between 2 strings. For instance M + M ~ M + M in an exotic channel. (c) Creation of a loop on a string, and the inverse process, for instance in M~B+B. (d) Sticking of 2 strings in the middle, and the inverse process, for instance in M+M-~B+B. (e) Triple rearrangement between 3 strings, for instance in M + M + M ~ B * B. When the 2-dimensional manifold has a relative minimum as in figs. 6, f, g, h, we have the processes of creation out of the vacuum shown in figs. 5f, g, h. The inverse processes holds for a maximum. initial
state
final state
strength
k X-1
vacuum
g
t , h
Q
k-2
k-'
X-3
Fig. 5. The 5 basic interactions (a to e) and the three virtual processes (f, g, h).
448
X. Artru, String model
t
fU Fig. 6. Duality diagrams illustrating the 5 basic interactions and the 3 virtual processes (see fig. 5) It should be kept in mind that the interaction points on the world sheet are not fixed but depend on the Lorentz frame. Even the nature of the interactions may change. For instance, the diagram of fig. 6b, after rotating sufficiently the axes, represents a cutting followed by a fusion. This is because our world sheets and world lines are not everywhere time-like. But for classical motions, the interactions points are angular points on the world sheets, which are independent of the frame. Let us consider the diagram of fig. 6d, which represents M + M ~ B + B. We see that there are intermediate states in all three channels: baryons in the t and u channels, and an exotic meson in the s channel as that of fig. 2d *. The same exotic configuration exists in the process M -~ B + B (fig. 6c) when one string of the loop has been cut but not yet the other one. By successive basic interactions, involving at least (c) or (d) or (e), the hadronic matter can polymerize itself to form more and more complicated exotic configurations (fig. 2e).
* The property that exotic mesons are "directly" coupled to M + M is not shared by all existing duality schem_es [14]. Generally, the simplest cases in which they are assumed to occur are B + B ~ B + B reactions (for instance in ~÷ + proton ~ ~, + 4÷÷).
X. Artru, String model
449
5. Implications in string field theory There exists the classical covariant theory o f the string which starts from the principle of the minimum action, the action being proportional to the covariant area of the world sheet. The quantification of this theory gives the spectrum of the dual resonance model [3]. A way to treat the n strings -+ rn strings processes, is to perform the "second quantization" o f the string [9]. If X(o) represents the shape o f the string for fixed t, the string field is an operator ~0 [ X ( ~ ; t] which is a functional o f the curve X(a), and acts on a Fock space of strings. The "free" lagrangian is quadratic in ~o. The interactions are introduced by adding a cubic term (symbolically k~03) which accounts for the cutting o f one string into two strings and the inverse process, and also a quartic term ()k2q04) to account for the rearrangement. These 2 interactions are sufficient if we consider only the meson sector. Taking into account the existence of the pomeron (P) sector, we have also a bilinear term for P ~ M (opening of a closed string), and cubic terms for M ~ M + P and P ~ P + P (via rearrangement). In the model with baryons, the situation becomes more complicated. A priori all kinds o f exotic configurations have their own free fields. The interaction lagrangian must have new terms to account for the processes (c), (d) and (e). The terms corresponding to (e) can be of 5th degree in the fields, as in the reaction M + M + M ~ B + B
a
b
g
c
h
d
e
i
f
j
Fig. 7. Renormalization diagrams: (a) unrenormalized world sheet of a meson; (b) virtual splitting and recombination of the same meson string; (c) emission and reabsorption of a pomeron; (d) formation of a virtual loop; (e) unrenormalized pomeron; (f) virtual 2-loop configuration of the pomeron (see fig. 5h); (g) and (h) mesonic and pomeronic bridge between 2 sheets of a baryon; (i) sticking of 2 sheets of a baryon; (j) additional connection of the 3 sheets of a baryon along a closed B.J. line.
450
x. Artru, String model
:. ,~.
/
8,
b
Fig. 8. More complicated renormalization diagrams with a closed B.J. line. One of the possible histories corresponding to a is the process 1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 of fig. 9. For the diagram b it is 1 --, 2 ~ 3 ~ 4 ~ 1 0 ~ 11 ~ 12. Let us now consider the renormalization diagrams. In the model without baryons, we start from an irreducible world sheet and get the higher order reducible diagrams b y making holes in it, or b y adding handles (fig. 7b and c). In the model including baryons, we must consider also the formation o f a virtual loop inside the string (fig. 7d). We have a bubble on the world sheet. In the case o f a pomeronic string, we get a two-loop virtual state, as in fig. 5h. If the initially small loop grows, then shrinks, we have a bubble as in fig. 7d. If it grows until the other loop shrinks completely, as in fig. (7f), we have an "obstructed tube". Some renormalization diagrams for baryons are drawn in fig. 7g, h, i, j. Diagram 7h is not irreducible in the sense that it is the iteration o f diagram 7g. There are more complicated histories which are difficult to draw at once on a sheet o f paper (fig. 8). It is helpful to decompose them in a movie, as in fig. 9. We learn from this figure that we are far from having made an exhaustive classification o f the "irreducible" renormalization diagrams. The problem o f the coupling constant assigned to each basic interaction seems to have a very intriguing solution. Let ~ be the coupling constant associated with the
t5 Fig. 9. Renormalization processes inside a string with a virtual loop (4) in the intermediate state.
2(. Artru, String model
451
interaction (a). It is the coupling constant of the conventional dual resonance model. Considered in the t channel, the diagram 6b is clearly of order 2, so that the rearrange ment interaction term is proportional to X2. Diagrams 6c, d, e are of order 1,2, 3 respectively. But there are 2 cuts in 6c, one cut in (6d), so that the basic interactions (c), (d) and (e) must be proportional to k -1, X and k 3 respectively. To be complete, we have to assign factors in X-1, X-2, X- 3 to the virtual processes of figs. 5f, g, h. With this power counting rule, the power of a duality diagram is the same in all possible channels and is a purely topological property of the world sheet * Renormalization diagrams with n bubbles are assigned a factor ~.-2n, as well as the tube obstructed n times. Therefore it is meaningless to make a perturbation theory, and it seems that the physical value of X has to be of the order of unity (perhaps, in some more "basic" formalism of the string model, X is not a free parameter). It seems plausible, however, that the diagrams that have the simplest topology dominate the n -+ m string amplitude, and that we have a convergent expansion in increasing complexity of the diagrams.
6. Classical motion of a baryonic string Let us recall the law that governs the motion of a string in the classical model without baryons. The action of a world sheet is proportional to its covariant area. Let t Xu(a , ~-) be a parametrization of the world sheet, Xu and Xu the partial derivatives with respect to o and r. We have Action = ~ f f
do dr [ ( X - X ' ) 2 _ X2X " ' 2] ~ = f f d o
dT .12.
(6.1)
Considering small variations fiX, and writing 5 (action) = 0, we get an "interior equation of motion" a ~° a~ 0r aX:'- + aa ~ = 0
.
(6.2)
At an end-point, to which we can assign a fixed o, say oi, we get the boundary condition a/2 I
=0 .
(6.3)
aX~ [°=°i The energy-momentum is distributed along the string according to d E = K ds (1 - v 2 ) - ~ ,
(6.4)
* The exponent of ~ is the sum of the exponents corresponding to the one-string world sheets. This last number is a kind of an Euler-Poincard characteristic of the world sheet. In fact, in the tube model (the model with only closed strings), we have: exponent of ~.= - 2 × the Euler-Poincar6 characteristic.
452
X. Artru, String model
dp = v d E
,
(6.5)
where s is the arc length and v the transverse velocity. The tension of the string is T=t¢/(1 -v2) ~ ,
(6.6)
where l is the unit tangent vector. The energy EAB between two points which have no longitudinal motion on the string (Vn./h = VB'/B = 0) is constant. A convenient choice of parameters is therefore B
o(B) = u = / ¢ - 1 E A B =
f ds (1 - v 2 ) -~
,
(6.7)
A r=X o=t
.
(6.8)
This choice satisfies the conditions X " J( = 0
(transversality of the m o t i o n ) ,
X,2 +j(2 = 1 ,
(6.9) (6.10)
which means 2 = v,
(6.11)
X ' =to -1 T .
(6.12)
The equation of motion (6.2) becomes X=X"
(6.13)
An end-point has a fixed u. The boundary condition becomes T=0
,
(6.14)
which implies that the end of the string moves with the velocity of light. The general solution of(6.13), (6.9) and (6.10) is 2X(t, u) = Y ( t + u) + Z ( t - u) ,
(6.15)
with [~'[ -- 1 = [Z[ .
(6.16)
At an end-point, we have, from T = 0 , ~;(t + u) = Z ( t - u) (= quark velocity).
(6.17)
Eqs. (6.15) - (6.17) give us explicit solutions in the case of strings without B.J., and, by studying the simplest ones, one can get familiar with the behavior of the strings [7].
X. Artru, String model
453
In the model with baryons, the simplest assumption is that the action is the sum of the covariant areas of the world sheets, which are 3 by 3 connected along the B.J. world lines. The "interior equation of motion" (6.2), as well as the boundary condition (6.3) at an end point are unmodified. The essential difference is that the energy of a branch is not constant and the parameter u of a B.J. is not fixed. Let Xa(u, t), Xb(u, t),Xc(u, t) represent the 3 strings around a junction, according to the parametrization (6.7) - (6.8). Let Ua(t), Ub(t ) and ue(t ) be the parameter u of the junction on the strings a, b, and c respectively. In the following, we will take • = 1. The velocity of the vertex is, from (6.11) - (6.12), du a
V = v a +--~- Ta du b = vb + ~
du c T h = v c + --d-i- T c
.
(6.18)
Energy conservation requires du _ _a+ du b + du c dt T ~ =0 .
(6.19)
There remains to see what momentum conservation implies. Let us consider, in the neighborhood of the junction, 3 points of fixed u, one on each string, and the mechan ical system bounded by these 3 points. The external forces are F = - T a - Tb - Tc. During the time dt, the change o f momentum is, from (6.5), dp = v a d u a + v b d u b + v c d u c . Writing d p = F dt gives du .__&a
dt va + Ta + (same with b and c) = 0 .
(6.20)
eqs. (6.18) - (6.20) are the equations of motion of the junction. A covariant derivation, using the action principle, is given in an appendix. Considering the quantity
du i we get
[ i=a, b,c
~dui~21Ti-
1-\dr]
_J
-0
,
(6.21)
[ aUa;]
which tells us that the strings are (locally) coplanar. Furthermore, from the relations 1 -
V 2 = (1 - va 2)
1- \ ~
= same with b and c,
(6.22)
X. Artru, String model
454 we can rewrite (6.21) as ~.2 = 0 .
(6.23)
Thus, the strings cannot be on the same side of a given straight line. Let us show that the junction does not move faster than light: Suppose I VI > 1. This implies Idui/dtl > 1, for instance dUa/dt < - 1 , dUb/dt and duc/dt > 1. We can take a frame where the string a is at rest du a dt v b = V sin 0ab > sin 0 ab , du b dr" =(1 - v b 2 ) -~ V c o S 0 a b > V , thus eq. (6.19) cannot be satisfied. There exists therefore a rest frame for the junction. In this frame, we have vi = 0, dui/dt = O, I Til = 1, and Y, Ti = 0. The three strings are at angles -~zragainst each other We can say that the world sheets make a covariant angle of ~zr between them. This is as in the junction of surfaces separating bubbles of soap (in that case, we also minimize the area). If only one string is at rest, it is the bisectrix of the other two. Due to Lorentz con traction their opening angle 20 is given by tg 0 = ( 1 ~ 3 V 2 ) ~
(6.24)
A simple solution of the equations of motion of the baryonic string is the analogu~ of the revolving segment in the mesonic case. It consists of 3 segments in a plane, making angles ]lr between them, and revolving around the junction so that their extremities travel at the speed of light. L being the common length of the segments, their energies are
e(L)=~Tr tc L , and their angular momenta
](L ) = lrr t¢L 2 The total energy and angular momentum are M = 3 e(L), J = 3 j(L), and the classical Regge slope is therefore J _1 1 M2 3 ~'K '
X. Artru, String model
455
whereas for a revolving mesonic segment we have J M2
1 1 7rK (= a' ~ 1 G e V - 2) . 2
(6.25)
However, this symmetric configuration does not give the maximum value of J for a given M. Let us consider, for instance, a big mesonic revolving segment and a small 3 revolving segment-baryon, and make a fusion-at-the-end of these 2 hadrons. We get a baryon with about the same mass and spin as the original meson, which satisfies (6.25). We can conclude that, classically at least, the leading trajectories of the baryons and the mesons ( and the exotics), have the same slope. It would be interesting to get more classical solutions. This seems a priori not so easy as for mesonic or pomeronic strings, due to the constraints at the junction. Are all the solutions periodic? If this is the case, what kind of Fourier decomposition do we have? These questions are of importance in finding the spectrum of the quantized baryonic string.
7. Application : sticking together of two colliding strings The classical theory cannot tell us whether or not one of the five basic interactions occurs, when, for their part, the kinematic conditions are satisfied. One can only speak o f the probability of a given process, as in the case of radioactive decay. But it allows us to study the evolution of the strings after such an interaction. For the cutting and the rearrangement, this has been studied in ref. [7]. The fusion can be considered as a limiting case of the rearrangement, when the crossing point is very close to one end of each string. The formation of a loop (process c) in a "quiet" part of the string is energetically forbidden, and occurs only at the quantum level. The triple rearrangement (process e) has a vanishing classical cross section. For this reason, we shall not consider it here. There remains the process (d), the "sticking together", which has the particular interest of having a non-vanishing classical cross section, and may compete with the rearrangement. We are mainly interested in the evolution just after sticking. Therefore we shall consider, without loss of generality, the case of two infinitely long straight strings, parallel to the x y plane and moving with velocities + t3 and - / 3 in the z direction (fig. 10). The angle between the strings is 2a and they meet each other in 0 at time t = 0. The configuration after the interaction is the following one: (a) 4 unperturbed parts of the initial strings. (b) The part which results from the sticking together of the 2 strings. For symmetry reasons, this part is at rest and lies on the bisectrix of the initial strings It is bounded by:
X. Artru, String model
456
z T cos c~ l
I I -//~ ~ - - k
) o ....
i
V
\
I I
\ \
P
I \
)y
\q I \
od, Fig. 10. Configuration of the strings after the sticking together of two straight line strings in uniform translation. The figure displays the projections on the xy, yz, and zx planes. (i) The b a r y o n i c j u n c t i o n B.J. and the B.J. which recede from the origin at speeds - V and + V, respectively. (ii) 4 rectilinear pieces of string connecting the vertices to the u n p e r t u r b e d parts at angular points which propagate at the speed of light *. Let T = X/1 - / 3 2 be the tension o f the initial strings. Fig. 10 represents the strings at t = 1. The angular points are at the distance 1 "from the origin, therefore Op = T, Oq = (1 - On2)~ = (1 - T 2 cos2a) ~. Let 20 be the opening angle at the vertex: tg 0 - Oq _ (1 - T 2 c o s 2 a ) ~- . mn TcosaV ' 0 is also given b y (6.24). We get (1 -- V 2) (1 - T 2 cos 2 a) = 3 ( T cos a - V) 2 , with T cos a > V, or (2 + V) (2 - T cos a) = 3 .
(7.1)
The condition V > 0 requires T cos a = cos a X/1 - / 3 2 > 1 .
(7.2)
(7.2) means that the tensions o f the u n p e r t u r b e d parts not only c o m p e n s a t e that o f the sticked part but also serve to increase the m o m e n t u m o f the 4 intermediate parts When/3 is too big, this is not possible. We must have at least * Such angular points fulfill the equations of motion. They are obtained by choosing, in (6.15), a straight line for the trajectory Y(t) and a broken line for Z(t).
X. Artru, String model
T =
1/ch~>~
457
,
or
~=
th~p~<~x/3 .
(7.3)
In the rest frame of one of the strings, the velocity of the other one is therefore restricted by 6
1
v = th 2~ ~< (~-)~ .
(7.4)
For higher speeds, the sticking process is classically forbidden. This is unfortunate, because the threshold for the reaction B+B~3B+B
,
which could proceed via a sticking together, is just given by ch ¢ = 2! Of course, this process is not forbidden quantum mechanically. We expect however that the sticking has less and less importance at high energy, in contrast to the rearrangement.
8. Summary and conclusions We have considered, among different string models for the baryon, the one in which 3 strings are connected by a 3-string junction. In this model, the rules for excluding non-zero triality states are very simple. They are suggestive of an underlying magnetic monopole theory. By studying the topology of the world sheets, we have classified the interactions of the strings into five basic ones, two of which (the cutting or-fusion and the rearrangement) were already present in the model without baryons. They allow the construction of exotic states with any number of quarks + antiquarks. In particular an exotic meson can be exchanged in the t channel for the reaction M + B ~ M + B, and is dual to baryons in both the s and u channels. For instance it must contribute to the forward K - p ~ 7r+':- amplitude (other models do not require such a direct M - M - exotic coupling). In a field theory of strings we must have interaction terms from the 2nd to the 5th degree in the fields. A naive generalization of the power counting rule, which works in the model without baryons, gives the appropriate power of )t associated with each term, X being the Veneziano coupling constant. One of the interaction terms is found to be associated with X- I , suggesting that X is not a free parameter. The renormalization diagrams are enriched with bubbles, "obstructed pipes" and more complicated objects. A systematic topological study of the irreducible renormalization diagrams is to be made. Starting from the same action principle as in the model without baryons, we have written the equations of motion of a junction. They are equivalent to the conservation laws for the energy-momentum. In the rest frame of the junction, the angles are -27r 3 , as expected. The slope of the leading baryonic trajectory is the same as for the mesons.
458
X. Artru, String model
As an application, we have solved the sticking together of straight line strings in uniform translation; the classical solution exists only if the relative velocity of the strings and the angle between them satisfy the condition (7.2), which suggests that this process is suppressed in high energy collisions (Elab/m >~7). A phenomenological application of this duality scheme consists of writing finite energy sum rules relating s-channel resonances and t-channel exotic trajectories, as in the reaction mentioned above. The classical solutions, for their part, can have their own phenomenological interest, as was shown in ref. [7]. For instance, the sticking together process studied in sect. 7 may be relevant at low energy. A study of the periodicity of the classical solu tions could help finding the quantized levels. From a theoretical point of view, this string model provides a framework which can be at least useful but is perhaps necessary, for constructing a dual resonance model including baryons. After the quantization of a string with 3 branches, the main steps towards a physically acceptable model would be: the inclusion of quark spins, the (para ?) statistics of the quarks and to implement a parton-like behavior at high momentum transfer I am indebted to E. Cremmer for his interest and advice about this work and to J.L. Gervais and C. Montonen for reading the manuscript.
Appendix. Equation of motion in the presence of a junction Let X ( a , r) (we omit the Lbrentz indices for clarity) describe a piece of world sheet bounded by: the initial and final configurations of the strings, B.J. world lines and quark world lines. Let us assume that this piece is simply connected; the boundary is represented by a closed curve C in the (o, r) plane. We write the action as A =
ffdo dr ~?
[J((a, r ) , X ' ( a , r)] ,
(A.1)
with (setting K = 1) = [(j(. X ' ) 2 -- X 2 X '2] ~
(A.2)
We consider small variations 8 X = 8 [X(o, r)] of the interior of the world sheet. The variation of the action is 8A =
-~
~,
• 6X'
do dr ;
(A.3)
* The conventional dual resonance model is too "soft" in the sense that it leads to an exponential decrease of the elastic scattering amplitude at large t as well as of the inclusive spectrum at large p ±. In the string picture, this can be explained by the fact that the quarks carry no energy-momentum by themselves.
459
X. Artru, String model
upon integrating by parts, this becomes
fsx.
C
~ - ~ dr - - - do aX
12 ar 0ox
-ff
+ 0o
5Y'
do d r .
(A.4)
The principle that the action is stationary 5A = 0 , world sheets gives (i) the "interior equation o f motion"
(A.5)
0 ~ ? + ~ ° 022 Or 0 X ~-X' = 0 .
(A.6)
(ii) At a free and, we have necessarily .C = 0 (to derive it, it is sufficient to consider variations of the (o, r) domain instead of variations of X); the world sheet is tangent to the light cone. There is no constraint on 6X in (A.4), therefore 0.I2 dr = ~.67 do 0X' 0X
(A.7)
on the trajectory of the end point. For an arbitrary parametrization, both sides of (A.7) are infinite. With a suitable choice o f gauge, however, everything is finite and one can show that the end-point moves with the velocity of light [3]. (iii) Let us now consider the junction. The three strings are described by three 4-vector Xa(o, r), Xb (o, r) and X c ( e , T). The contribution to 5A is \8-~ i=a, b, c Ci
dr-
. OXi
(A.8)
]
But here 6Xa, 6Xb and 6Xc are not independent: each X + fiX must lie on the new B.J. world line. To insure this, we can impose (A.9)
5Xa = 5Xb = 6Xc .
We can take these constraints into account by adding to (A.8) the Lagrange multiplier terms
fd~ [pb(~)" (6Xb -
6Xa) + pc(£)" (5Xc - 5Xa)]
(A.10)
(we :take a common parameter £ to describe the world line of the junction). Requiring that the coefficient of 6Xi be zero in (A.8) + (A.10) gives a Z9 dr i OX'i d£
~ aXi
with pa defined by
doi d£ = - P i ( J ~ ) '
(A.11)
460
X. Artru, String model / ~ +/ab +/~c = 0 .
(A.12)
The 4-vector/~ plays the role o f a tension [see also (A.7)]. Let us now consider the particular "gauge" (6.7) - (6.8) and choose £ = t. We have dr/d£ = 1; do/d~ = d u / d t ; X = (1 , v ) ; X ' = (0, T); due to (6.9) - (6.10) we have also 0 ~?/03f = X; ~ .6?/OX' = - X ' (we take tile metric + ------). Putting these expres sions in (A.11) gives ~--
(~
du ) ~,T+~-v
(A.12) is nothing but (6.19) - (6.20).
References [ 1] Y. Nambu, Talk presented at the American Physical Society Meeting in Chicago, January 1970, EFI 70-07. [2] L. Susskind, Phys. Rev. D1 (1970) 1182. [3] P. Goddard, J. Goldstone, C. Rebbi and C. Thorn, Nucl. Phys. B56 (t973) 109. [4] J.L. Gervais and B. Sakita, Phys. Rev. Letters 30 (1973) 716. [5] J. Scherk, Lectures given at New York University, NYU/TR3/74. [6] M.B. Einhorn, M.B. Green and M.A. Virasoro, Phys. Letters 37B (1971) 292. [7] X. Artru and G. Mennessier, Nucl. Phys. B70 (1974) 93. [8] S. Mandelstam, Nucl. Phys. B69 (1974) 77; J.L. Gervais and E. Cremmer, Nucl. Phys. B76 (1974) 209. [9] M. Kaku and K. Kikkawa, Phys. Rev. D10 (1974) 110. [10] B.Y. Oh and G.A. Smith, Nucl. Phys. B49 (1972) 13; J. Badier et al., Phys. Letters 39B (1972) 414. [11] S. Ellis, P.H. Frampton and P.G.O. Freund, Nucl. Phys. B24 (1970) 465. [12] V.I. Strazhev and L.M. Tomil'chik, Soy. J. Particles and Nuclei 4 (1973) 78. [13] L.I. Schiff, Phys. Rev. 160 (1967) 1257. [14] P.G.O. Freund, R. Waltz and J. Rosner, Nucl. Phys. B13 (1969) 237.
STRING MODEL WITH BARYONS: TOPOLOGY; CLASSICAL MOTION X. A R T R U ~ Laboratoire de Physique Th~orique et Hautes Energies ~, Universit~ de Paris-Sud, 91405 Orsay, F~ance
Received 27 June 1974
Abstract: We consider the model in which a meson is an open string with a quark at one end an anti-quark at the other end; a baryon is made of three strings joining at a point, and carrying quarks at their free ends. The triality condition is ensured by orienting the strings according to simple rules. These rules suggest the existence of an underlying magnetic monopole theory Independently of this "explanation", we investigate first the topological properties of the model, by looking at the duality diagrams: existence of exotic hadrons, of five basic interactions between strings. Some renormalization diagrams are assigned a negative power of the Veneziano coupling constant. Then, taking the same action (the world-sheet area) as in the conventional string model, we set down the equations of motion of a junction. We argue that the slope of the leading baryonic Regge trajectory is the same as that for the mesons. As an example of an application, we study the "sticking together" of two colliding strings; we find that it is classically forbidden at relative velocities greater than (~)~.
1. I n t r o d u c t i o n The string model of the hadrons [ 1 - 5 ]**offers an intuitive representation o f dual resonance models, and a powerful tool o f investigation in this field. More directly, w i t h o u t assuming a n y particular lagrangian, the string picture is a simple interpretation of the Harari-Rosner duality diagrams. A m e s o n is a string with a quark at one end and an antiquark at the other end. The "'rubber b a n d ; ' represents the two-dimensional world sheet swept out b y the strings in our 4-dimensional space-time. By looking at world sheets o f different topologies, we see that strings can j o i n b y the ends, with a quark-antiquark annihilation, split themselves, with a quark-antiquark creation, rearrange w h e n going through each other. T h e y can form closed strings w i t h o u t quarks called pomerons. Besides the conventional Harari-Freund duality (Regge o resonances and p o m e r o n o background), the string picture implies the absence o f free quarks, and the so-called " a b n o r m a l d u a l i t y " [6] ( p o m e r o n ~ reso:~ Present address: Theory Division, CERN, 1211 Geneva 23, Switzerland. * Laboratoire associ6 au Centre National de la Recherche Scientifique. ** For a review of the string model and a more complete list of references, see ref. [5].
X. Artru, String model
443
nances in the case of pomeron-hadron scattering). The string is supposed to be neutral except at the end (a continuous charge distribution would spoil the gauge invariances of the dual model, which are interpreted as the invariance of the theory with respect to a reparametrization of the world sheet), but oriented (to avoid strings with two quarks or two antiquarks). The analogy with an infinitely thin magnet, developed by Nambu [1 ], is very striking. When we want to include the baryons in the scheme, some problems arise immediately: (i) where must we put the quarks? (ii) how is triality satisfied (if we believe that quarks are fractionally charged)? (iii) we must introduce exotic states. The problems of quark spin, quark statistics will not be considered in this work, which stays at a classical level. In the next section, we expose our (rather subjective) choice for the shape of a baryonic string: three pieces of strings joining at a point and carrying the quarks at their free ends. A speculative explanation in terms of the existence of magnetic monopoles is given in sect. 3. According to this choice, the world sheets swept out by the strings are connected 3 by 3 along world lines of 3-string junctions. By looking at different topologies, 3 new basic interactions among strings emerge, in addition to the 2 basic interactions [ 7 - 9 ] of the purely mesonic model, which were the cutting (or the fusion) and the rearrangement. In sect. 4, we classify these interactions and discuss the formation of exotics. In the conventional model, one can assign a factor Xn-2 to the n-meson amplitude, X being the Veneziano coupling constant. This is equivalent to assigning a factor X to each cutting or fusion, and ),2 to a rearrangement. This correspondence will be extended to the new basic interactions, in sect. 5, giving an intriguing result. The equations of motion of strings with 3 by 3 junctions are set down in sect. 6. Examples of motions with Regge slope ~ and 1 (relative to mesons) are given. The evolution of the strings after a cutting or a rearrangement has been studied in ref. [7]. In this paper we consider (sect. 7) one of the new processes, the "sticking together" of two colliding strings, and solve the equations of motion for it.
2. Topology of the baryonic string Assuming that a baryon contains 3 quarks located somewhere on a neutral and non strange string, the simplest possibilities are: (A) meson = q - -
q
baryon -- q
(B) meson = q - - q
baryon = q - - q - - q
(C) meson = q - -
baryon =
q
(D) meson = q,._./q
qYq q baryon = f q " x q~q
444
X. Artru, String model
X
~, ),'
Ko
~)
b)
K-
P
n
P
K-
proton
p
n
p
Proton
Fig. 1. Duality diagram in model B. In (a) the n quark must take the place of the h quark at the extremity of the string (in O) before "emitting" the K°. There are 4 other topologically inequivalent diagrams for this reaction. In the model A, the baryon is made of a quark and a diquark which plays the role of the antiquark. There are 9 kinds o f diquarks : p n, p p, p ~, n n, etc. There is no duality diagram for the reaction proton + proton ~ ~ - + mesons + n u c l e o n , which is known to have a non-negligible cross section [ 10]. We therefore discard this possibility. In the model B, the quark in the middle is not treated on the same footing as the two others. One can restore the symmetry by allowing it to reach one end and dislodge the former occupant. This model has less duality than models C and D. Let us consider, for instance, the reaction K - + proton ~ K ÷ + K 0 + ~2- . All the quarks o f the p r o t o n must participate, therefore none can stay in the middle o f the string. There are 6 diagrams, such as those of fig. l a and l b , which are not topologically equivalent. By contrast, models C and D do not distinguish these 6 diagrams. Let us now consider the model C. There is a p r i o r i no limitation on the number o f 3-string junctions. The triality condition (absence of q q ~ states, for instance) is satisfied if (i) the strings are oriented from an antiquark (or a junction) towards a quark (or a junction), (ii) at a junction, all the 3 orientations either diverge or converge together. In the former case we call it a baryonic junction B.J., in the latter an anti-baryonic juncti_on B.J. The baryonic number is equal to the number o f B.J. minus the number o f B.J. Some configurations are shown in fig. 2. In the model D, the production or absorption o f a hadron occurs via a rearrangement (fig. 3). The triality condition cannot be cast into a rule as simple as in m o d e l C * • A closed string model, in which there are only 6 possible locations for the quarks and which satisfies the triality condition, has been considered, however [ 11 ].
445
X. Artru, String model
a
b
c
d
e
Fig. 2. String configurations in model C. (a) ordinary meson, (b) pomeron; (c) baryon; (d) exotic meson; (e) polymerized form of the 4He nucleus. > - - = antiquark; - - ~ = quark. The experimental value o f the slope of the baryonic Regge trajectories, which seems to be the same as for the mesonic trajectories (within 10% uncertainty) cannot a priori disfavour any of these 4 models. Most probably, models A, B and D lead to the same slope for the mesons and the baryons. But it will be shown in sect. 6 that model C also has (classically, at least) a leading baryonic trajectory o f the same slope as the mesonic ones. The slope of-~, which was used as an argument against this model, corresponds to a symmetric configuration o f the 3 branches, but this configuration does not give the highest angular m o m e n t u m for a given mass. F r o m now on, we shall consider only the model C, with 3 b y 3 connected strings.
3. Are t h e r e m a g n e t i c m o n o p o l e s ? A "raison d ' e t r e " f o r the orientation rules could be the existence o f magnetic monopoles. A standard procedure to build a "dual" theory * (here dual means symmetric with respect to electric and magnetic fields) is to introduce strings (the so-called "Dirac strings") which terminate at the electric or magnetic charges and which carry the necessary flux such that div E = div B = 0 everywhere. If the electric and magnetic charges e i and gi of the different species o f particles satisfy the generahzed Dirac condition ei g / -
e/gi = ½ n h ,
(n integer) ,
(3.1)
these strings are not observable and can be considered to be only a mathematical tool. What happens, instead, if the Dirac condition is not fulfilled? Let us consider,
O Fig. 3. Emission of a meson by a baryon in model D. - - e - A rearrangement occurs at R.
* For a review of dually charged particle theory, see ref. [12].
= quark.
e
= ant~uark.
446
X. Artru, String model
p
P~
~)
n
proton .. ...... i~' n
Fig. 4. The identity between Dirac strings and hadronic strings. Solid line : physical string; dashed line; fictitious string. for instance, a world in which there are purely magnetic charges g satisfying (3.1) with respect to the electrons: eg = ½h and quarks with purely electric charges ~e and + be. We know that a string which carries an electric flux Q not obeying (3.1) scatters a magnetic charge. We conjecture that it has other observable effects (for instance a linear mass distribution) which also depend on Q modulo e. We have to connect all quarks and antiquarks in the world by a network of strings (fig. 4). Branches that carry fluxes a multiple of e can be ignored and we are left with strings carrying fluxes of magnitude (n + ~) e and (n + 9) e (n positive or negative integer). Let us orient the strings so that the algebraic flux is (n + ~) e " we get the picture of the string model chosen in sect. 2 * Similarly, the magnetic charges must also be connected by strings (for reason of symmetry), with the same topology: we would have dual baryons, dual mesons of total magnetic charge 0 or + 3g, with quantitatively very different properties, however. In particular, the analogue of the "fine structure constant" would be (3g) 2 = 9 × 137/4 = 308. Therefore it would be hard to separate magnetic charges from magnetically neutral matter.
4. Interactions In our 4-dimensional space-time, strings sweep out 2-dimensional surfaces called "world sheets". Duality diagrams are topological drawings of these world sheets. The world sheet of a meson has the topology of a rubber band, the world sheet of a pomeron has the topology of a tube, and the history of a baryon is made of 3 sheets joining along the world line of the B.J. Correspondingly, given a set of world sheets bounded by (oriented) quark world lines and connected 3 by 3 by B.J. world lines **, we get the evolution of a system of strings by cutting it with successive hyperplanes t = t l , t2, t 3 etc. An "interaction'~ is a change in the topology of this system for a
* The possible connection between triality and the existence of magnetic charges has also been considered by Schiff [ 13 ]. ** It is convenient to orient the B.J. world line as an antiquark world line.
X. Artru, String model
447
particular value of t. We find 5 "basic interactions" plus 3 purely virtual processes in which strings disappear in (or are created out of) the vacuum (see figs. 5 and 6). (a) Fusion of 2 strings at the end-points with annihilation of a q ~ pair, and the inverse process: the cutting of a string with creation of a q ~ pair. It occurs, for instance, in M + M ~- M (M = meson, B = baryon). (b) Simple rearrangement between 2 strings. For instance M + M ~ M + M in an exotic channel. (c) Creation of a loop on a string, and the inverse process, for instance in M~B+B. (d) Sticking of 2 strings in the middle, and the inverse process, for instance in M+M-~B+B. (e) Triple rearrangement between 3 strings, for instance in M + M + M ~ B * B. When the 2-dimensional manifold has a relative minimum as in figs. 6, f, g, h, we have the processes of creation out of the vacuum shown in figs. 5f, g, h. The inverse processes holds for a maximum. initial
state
final state
strength
k X-1
vacuum
g
t , h
Q
k-2
k-'
X-3
Fig. 5. The 5 basic interactions (a to e) and the three virtual processes (f, g, h).
448
X. Artru, String model
t
fU Fig. 6. Duality diagrams illustrating the 5 basic interactions and the 3 virtual processes (see fig. 5) It should be kept in mind that the interaction points on the world sheet are not fixed but depend on the Lorentz frame. Even the nature of the interactions may change. For instance, the diagram of fig. 6b, after rotating sufficiently the axes, represents a cutting followed by a fusion. This is because our world sheets and world lines are not everywhere time-like. But for classical motions, the interactions points are angular points on the world sheets, which are independent of the frame. Let us consider the diagram of fig. 6d, which represents M + M ~ B + B. We see that there are intermediate states in all three channels: baryons in the t and u channels, and an exotic meson in the s channel as that of fig. 2d *. The same exotic configuration exists in the process M -~ B + B (fig. 6c) when one string of the loop has been cut but not yet the other one. By successive basic interactions, involving at least (c) or (d) or (e), the hadronic matter can polymerize itself to form more and more complicated exotic configurations (fig. 2e).
* The property that exotic mesons are "directly" coupled to M + M is not shared by all existing duality schem_es [14]. Generally, the simplest cases in which they are assumed to occur are B + B ~ B + B reactions (for instance in ~÷ + proton ~ ~, + 4÷÷).
X. Artru, String model
449
5. Implications in string field theory There exists the classical covariant theory o f the string which starts from the principle of the minimum action, the action being proportional to the covariant area of the world sheet. The quantification of this theory gives the spectrum of the dual resonance model [3]. A way to treat the n strings -+ rn strings processes, is to perform the "second quantization" o f the string [9]. If X(o) represents the shape o f the string for fixed t, the string field is an operator ~0 [ X ( ~ ; t] which is a functional o f the curve X(a), and acts on a Fock space of strings. The "free" lagrangian is quadratic in ~o. The interactions are introduced by adding a cubic term (symbolically k~03) which accounts for the cutting o f one string into two strings and the inverse process, and also a quartic term ()k2q04) to account for the rearrangement. These 2 interactions are sufficient if we consider only the meson sector. Taking into account the existence of the pomeron (P) sector, we have also a bilinear term for P ~ M (opening of a closed string), and cubic terms for M ~ M + P and P ~ P + P (via rearrangement). In the model with baryons, the situation becomes more complicated. A priori all kinds o f exotic configurations have their own free fields. The interaction lagrangian must have new terms to account for the processes (c), (d) and (e). The terms corresponding to (e) can be of 5th degree in the fields, as in the reaction M + M + M ~ B + B
a
b
g
c
h
d
e
i
f
j
Fig. 7. Renormalization diagrams: (a) unrenormalized world sheet of a meson; (b) virtual splitting and recombination of the same meson string; (c) emission and reabsorption of a pomeron; (d) formation of a virtual loop; (e) unrenormalized pomeron; (f) virtual 2-loop configuration of the pomeron (see fig. 5h); (g) and (h) mesonic and pomeronic bridge between 2 sheets of a baryon; (i) sticking of 2 sheets of a baryon; (j) additional connection of the 3 sheets of a baryon along a closed B.J. line.
450
x. Artru, String model
:. ,~.
/
8,
b
Fig. 8. More complicated renormalization diagrams with a closed B.J. line. One of the possible histories corresponding to a is the process 1 ~ 2 ~ 3 ~ 4 ~ 5 ~ 6 of fig. 9. For the diagram b it is 1 --, 2 ~ 3 ~ 4 ~ 1 0 ~ 11 ~ 12. Let us now consider the renormalization diagrams. In the model without baryons, we start from an irreducible world sheet and get the higher order reducible diagrams b y making holes in it, or b y adding handles (fig. 7b and c). In the model including baryons, we must consider also the formation o f a virtual loop inside the string (fig. 7d). We have a bubble on the world sheet. In the case o f a pomeronic string, we get a two-loop virtual state, as in fig. 5h. If the initially small loop grows, then shrinks, we have a bubble as in fig. 7d. If it grows until the other loop shrinks completely, as in fig. (7f), we have an "obstructed tube". Some renormalization diagrams for baryons are drawn in fig. 7g, h, i, j. Diagram 7h is not irreducible in the sense that it is the iteration o f diagram 7g. There are more complicated histories which are difficult to draw at once on a sheet o f paper (fig. 8). It is helpful to decompose them in a movie, as in fig. 9. We learn from this figure that we are far from having made an exhaustive classification o f the "irreducible" renormalization diagrams. The problem o f the coupling constant assigned to each basic interaction seems to have a very intriguing solution. Let ~ be the coupling constant associated with the
t5 Fig. 9. Renormalization processes inside a string with a virtual loop (4) in the intermediate state.
2(. Artru, String model
451
interaction (a). It is the coupling constant of the conventional dual resonance model. Considered in the t channel, the diagram 6b is clearly of order 2, so that the rearrange ment interaction term is proportional to X2. Diagrams 6c, d, e are of order 1,2, 3 respectively. But there are 2 cuts in 6c, one cut in (6d), so that the basic interactions (c), (d) and (e) must be proportional to k -1, X and k 3 respectively. To be complete, we have to assign factors in X-1, X-2, X- 3 to the virtual processes of figs. 5f, g, h. With this power counting rule, the power of a duality diagram is the same in all possible channels and is a purely topological property of the world sheet * Renormalization diagrams with n bubbles are assigned a factor ~.-2n, as well as the tube obstructed n times. Therefore it is meaningless to make a perturbation theory, and it seems that the physical value of X has to be of the order of unity (perhaps, in some more "basic" formalism of the string model, X is not a free parameter). It seems plausible, however, that the diagrams that have the simplest topology dominate the n -+ m string amplitude, and that we have a convergent expansion in increasing complexity of the diagrams.
6. Classical motion of a baryonic string Let us recall the law that governs the motion of a string in the classical model without baryons. The action of a world sheet is proportional to its covariant area. Let t Xu(a , ~-) be a parametrization of the world sheet, Xu and Xu the partial derivatives with respect to o and r. We have Action = ~ f f
do dr [ ( X - X ' ) 2 _ X2X " ' 2] ~ = f f d o
dT .12.
(6.1)
Considering small variations fiX, and writing 5 (action) = 0, we get an "interior equation of motion" a ~° a~ 0r aX:'- + aa ~ = 0
.
(6.2)
At an end-point, to which we can assign a fixed o, say oi, we get the boundary condition a/2 I
=0 .
(6.3)
aX~ [°=°i The energy-momentum is distributed along the string according to d E = K ds (1 - v 2 ) - ~ ,
(6.4)
* The exponent of ~ is the sum of the exponents corresponding to the one-string world sheets. This last number is a kind of an Euler-Poincard characteristic of the world sheet. In fact, in the tube model (the model with only closed strings), we have: exponent of ~.= - 2 × the Euler-Poincar6 characteristic.
452
X. Artru, String model
dp = v d E
,
(6.5)
where s is the arc length and v the transverse velocity. The tension of the string is T=t¢/(1 -v2) ~ ,
(6.6)
where l is the unit tangent vector. The energy EAB between two points which have no longitudinal motion on the string (Vn./h = VB'/B = 0) is constant. A convenient choice of parameters is therefore B
o(B) = u = / ¢ - 1 E A B =
f ds (1 - v 2 ) -~
,
(6.7)
A r=X o=t
.
(6.8)
This choice satisfies the conditions X " J( = 0
(transversality of the m o t i o n ) ,
X,2 +j(2 = 1 ,
(6.9) (6.10)
which means 2 = v,
(6.11)
X ' =to -1 T .
(6.12)
The equation of motion (6.2) becomes X=X"
(6.13)
An end-point has a fixed u. The boundary condition becomes T=0
,
(6.14)
which implies that the end of the string moves with the velocity of light. The general solution of(6.13), (6.9) and (6.10) is 2X(t, u) = Y ( t + u) + Z ( t - u) ,
(6.15)
with [~'[ -- 1 = [Z[ .
(6.16)
At an end-point, we have, from T = 0 , ~;(t + u) = Z ( t - u) (= quark velocity).
(6.17)
Eqs. (6.15) - (6.17) give us explicit solutions in the case of strings without B.J., and, by studying the simplest ones, one can get familiar with the behavior of the strings [7].
X. Artru, String model
453
In the model with baryons, the simplest assumption is that the action is the sum of the covariant areas of the world sheets, which are 3 by 3 connected along the B.J. world lines. The "interior equation of motion" (6.2), as well as the boundary condition (6.3) at an end point are unmodified. The essential difference is that the energy of a branch is not constant and the parameter u of a B.J. is not fixed. Let Xa(u, t), Xb(u, t),Xc(u, t) represent the 3 strings around a junction, according to the parametrization (6.7) - (6.8). Let Ua(t), Ub(t ) and ue(t ) be the parameter u of the junction on the strings a, b, and c respectively. In the following, we will take • = 1. The velocity of the vertex is, from (6.11) - (6.12), du a
V = v a +--~- Ta du b = vb + ~
du c T h = v c + --d-i- T c
.
(6.18)
Energy conservation requires du _ _a+ du b + du c dt T ~ =0 .
(6.19)
There remains to see what momentum conservation implies. Let us consider, in the neighborhood of the junction, 3 points of fixed u, one on each string, and the mechan ical system bounded by these 3 points. The external forces are F = - T a - Tb - Tc. During the time dt, the change o f momentum is, from (6.5), dp = v a d u a + v b d u b + v c d u c . Writing d p = F dt gives du .__&a
dt va + Ta + (same with b and c) = 0 .
(6.20)
eqs. (6.18) - (6.20) are the equations of motion of the junction. A covariant derivation, using the action principle, is given in an appendix. Considering the quantity
du i we get
[ i=a, b,c
~dui~21Ti-
1-\dr]
_J
-0
,
(6.21)
[ aUa;]
which tells us that the strings are (locally) coplanar. Furthermore, from the relations 1 -
V 2 = (1 - va 2)
1- \ ~
= same with b and c,
(6.22)
X. Artru, String model
454 we can rewrite (6.21) as ~.2 = 0 .
(6.23)
Thus, the strings cannot be on the same side of a given straight line. Let us show that the junction does not move faster than light: Suppose I VI > 1. This implies Idui/dtl > 1, for instance dUa/dt < - 1 , dUb/dt and duc/dt > 1. We can take a frame where the string a is at rest du a dt v b = V sin 0ab > sin 0 ab , du b dr" =(1 - v b 2 ) -~ V c o S 0 a b > V , thus eq. (6.19) cannot be satisfied. There exists therefore a rest frame for the junction. In this frame, we have vi = 0, dui/dt = O, I Til = 1, and Y, Ti = 0. The three strings are at angles -~zragainst each other We can say that the world sheets make a covariant angle of ~zr between them. This is as in the junction of surfaces separating bubbles of soap (in that case, we also minimize the area). If only one string is at rest, it is the bisectrix of the other two. Due to Lorentz con traction their opening angle 20 is given by tg 0 = ( 1 ~ 3 V 2 ) ~
(6.24)
A simple solution of the equations of motion of the baryonic string is the analogu~ of the revolving segment in the mesonic case. It consists of 3 segments in a plane, making angles ]lr between them, and revolving around the junction so that their extremities travel at the speed of light. L being the common length of the segments, their energies are
e(L)=~Tr tc L , and their angular momenta
](L ) = lrr t¢L 2 The total energy and angular momentum are M = 3 e(L), J = 3 j(L), and the classical Regge slope is therefore J _1 1 M2 3 ~'K '
X. Artru, String model
455
whereas for a revolving mesonic segment we have J M2
1 1 7rK (= a' ~ 1 G e V - 2) . 2
(6.25)
However, this symmetric configuration does not give the maximum value of J for a given M. Let us consider, for instance, a big mesonic revolving segment and a small 3 revolving segment-baryon, and make a fusion-at-the-end of these 2 hadrons. We get a baryon with about the same mass and spin as the original meson, which satisfies (6.25). We can conclude that, classically at least, the leading trajectories of the baryons and the mesons ( and the exotics), have the same slope. It would be interesting to get more classical solutions. This seems a priori not so easy as for mesonic or pomeronic strings, due to the constraints at the junction. Are all the solutions periodic? If this is the case, what kind of Fourier decomposition do we have? These questions are of importance in finding the spectrum of the quantized baryonic string.
7. Application : sticking together of two colliding strings The classical theory cannot tell us whether or not one of the five basic interactions occurs, when, for their part, the kinematic conditions are satisfied. One can only speak o f the probability of a given process, as in the case of radioactive decay. But it allows us to study the evolution of the strings after such an interaction. For the cutting and the rearrangement, this has been studied in ref. [7]. The fusion can be considered as a limiting case of the rearrangement, when the crossing point is very close to one end of each string. The formation of a loop (process c) in a "quiet" part of the string is energetically forbidden, and occurs only at the quantum level. The triple rearrangement (process e) has a vanishing classical cross section. For this reason, we shall not consider it here. There remains the process (d), the "sticking together", which has the particular interest of having a non-vanishing classical cross section, and may compete with the rearrangement. We are mainly interested in the evolution just after sticking. Therefore we shall consider, without loss of generality, the case of two infinitely long straight strings, parallel to the x y plane and moving with velocities + t3 and - / 3 in the z direction (fig. 10). The angle between the strings is 2a and they meet each other in 0 at time t = 0. The configuration after the interaction is the following one: (a) 4 unperturbed parts of the initial strings. (b) The part which results from the sticking together of the 2 strings. For symmetry reasons, this part is at rest and lies on the bisectrix of the initial strings It is bounded by:
X. Artru, String model
456
z T cos c~ l
I I -//~ ~ - - k
) o ....
i
V
\
I I
\ \
P
I \
)y
\q I \
od, Fig. 10. Configuration of the strings after the sticking together of two straight line strings in uniform translation. The figure displays the projections on the xy, yz, and zx planes. (i) The b a r y o n i c j u n c t i o n B.J. and the B.J. which recede from the origin at speeds - V and + V, respectively. (ii) 4 rectilinear pieces of string connecting the vertices to the u n p e r t u r b e d parts at angular points which propagate at the speed of light *. Let T = X/1 - / 3 2 be the tension o f the initial strings. Fig. 10 represents the strings at t = 1. The angular points are at the distance 1 "from the origin, therefore Op = T, Oq = (1 - On2)~ = (1 - T 2 cos2a) ~. Let 20 be the opening angle at the vertex: tg 0 - Oq _ (1 - T 2 c o s 2 a ) ~- . mn TcosaV ' 0 is also given b y (6.24). We get (1 -- V 2) (1 - T 2 cos 2 a) = 3 ( T cos a - V) 2 , with T cos a > V, or (2 + V) (2 - T cos a) = 3 .
(7.1)
The condition V > 0 requires T cos a = cos a X/1 - / 3 2 > 1 .
(7.2)
(7.2) means that the tensions o f the u n p e r t u r b e d parts not only c o m p e n s a t e that o f the sticked part but also serve to increase the m o m e n t u m o f the 4 intermediate parts When/3 is too big, this is not possible. We must have at least * Such angular points fulfill the equations of motion. They are obtained by choosing, in (6.15), a straight line for the trajectory Y(t) and a broken line for Z(t).
X. Artru, String model
T =
1/ch~>~
457
,
or
~=
th~p~<~x/3 .
(7.3)
In the rest frame of one of the strings, the velocity of the other one is therefore restricted by 6
1
v = th 2~ ~< (~-)~ .
(7.4)
For higher speeds, the sticking process is classically forbidden. This is unfortunate, because the threshold for the reaction B+B~3B+B
,
which could proceed via a sticking together, is just given by ch ¢ = 2! Of course, this process is not forbidden quantum mechanically. We expect however that the sticking has less and less importance at high energy, in contrast to the rearrangement.
8. Summary and conclusions We have considered, among different string models for the baryon, the one in which 3 strings are connected by a 3-string junction. In this model, the rules for excluding non-zero triality states are very simple. They are suggestive of an underlying magnetic monopole theory. By studying the topology of the world sheets, we have classified the interactions of the strings into five basic ones, two of which (the cutting or-fusion and the rearrangement) were already present in the model without baryons. They allow the construction of exotic states with any number of quarks + antiquarks. In particular an exotic meson can be exchanged in the t channel for the reaction M + B ~ M + B, and is dual to baryons in both the s and u channels. For instance it must contribute to the forward K - p ~ 7r+':- amplitude (other models do not require such a direct M - M - exotic coupling). In a field theory of strings we must have interaction terms from the 2nd to the 5th degree in the fields. A naive generalization of the power counting rule, which works in the model without baryons, gives the appropriate power of )t associated with each term, X being the Veneziano coupling constant. One of the interaction terms is found to be associated with X- I , suggesting that X is not a free parameter. The renormalization diagrams are enriched with bubbles, "obstructed pipes" and more complicated objects. A systematic topological study of the irreducible renormalization diagrams is to be made. Starting from the same action principle as in the model without baryons, we have written the equations of motion of a junction. They are equivalent to the conservation laws for the energy-momentum. In the rest frame of the junction, the angles are -27r 3 , as expected. The slope of the leading baryonic trajectory is the same as for the mesons.
458
X. Artru, String model
As an application, we have solved the sticking together of straight line strings in uniform translation; the classical solution exists only if the relative velocity of the strings and the angle between them satisfy the condition (7.2), which suggests that this process is suppressed in high energy collisions (Elab/m >~7). A phenomenological application of this duality scheme consists of writing finite energy sum rules relating s-channel resonances and t-channel exotic trajectories, as in the reaction mentioned above. The classical solutions, for their part, can have their own phenomenological interest, as was shown in ref. [7]. For instance, the sticking together process studied in sect. 7 may be relevant at low energy. A study of the periodicity of the classical solu tions could help finding the quantized levels. From a theoretical point of view, this string model provides a framework which can be at least useful but is perhaps necessary, for constructing a dual resonance model including baryons. After the quantization of a string with 3 branches, the main steps towards a physically acceptable model would be: the inclusion of quark spins, the (para ?) statistics of the quarks and to implement a parton-like behavior at high momentum transfer I am indebted to E. Cremmer for his interest and advice about this work and to J.L. Gervais and C. Montonen for reading the manuscript.
Appendix. Equation of motion in the presence of a junction Let X ( a , r) (we omit the Lbrentz indices for clarity) describe a piece of world sheet bounded by: the initial and final configurations of the strings, B.J. world lines and quark world lines. Let us assume that this piece is simply connected; the boundary is represented by a closed curve C in the (o, r) plane. We write the action as A =
ffdo dr ~?
[J((a, r ) , X ' ( a , r)] ,
(A.1)
with (setting K = 1) = [(j(. X ' ) 2 -- X 2 X '2] ~
(A.2)
We consider small variations 8 X = 8 [X(o, r)] of the interior of the world sheet. The variation of the action is 8A =
-~
~,
• 6X'
do dr ;
(A.3)
* The conventional dual resonance model is too "soft" in the sense that it leads to an exponential decrease of the elastic scattering amplitude at large t as well as of the inclusive spectrum at large p ±. In the string picture, this can be explained by the fact that the quarks carry no energy-momentum by themselves.
459
X. Artru, String model
upon integrating by parts, this becomes
fsx.
C
~ - ~ dr - - - do aX
12 ar 0ox
-ff
+ 0o
5Y'
do d r .
(A.4)
The principle that the action is stationary 5A = 0 , world sheets gives (i) the "interior equation o f motion"
(A.5)
0 ~ ? + ~ ° 022 Or 0 X ~-X' = 0 .
(A.6)
(ii) At a free and, we have necessarily .C = 0 (to derive it, it is sufficient to consider variations of the (o, r) domain instead of variations of X); the world sheet is tangent to the light cone. There is no constraint on 6X in (A.4), therefore 0.I2 dr = ~.67 do 0X' 0X
(A.7)
on the trajectory of the end point. For an arbitrary parametrization, both sides of (A.7) are infinite. With a suitable choice o f gauge, however, everything is finite and one can show that the end-point moves with the velocity of light [3]. (iii) Let us now consider the junction. The three strings are described by three 4-vector Xa(o, r), Xb (o, r) and X c ( e , T). The contribution to 5A is \8-~ i=a, b, c Ci
dr-
. OXi
(A.8)
]
But here 6Xa, 6Xb and 6Xc are not independent: each X + fiX must lie on the new B.J. world line. To insure this, we can impose (A.9)
5Xa = 5Xb = 6Xc .
We can take these constraints into account by adding to (A.8) the Lagrange multiplier terms
fd~ [pb(~)" (6Xb -
6Xa) + pc(£)" (5Xc - 5Xa)]
(A.10)
(we :take a common parameter £ to describe the world line of the junction). Requiring that the coefficient of 6Xi be zero in (A.8) + (A.10) gives a Z9 dr i OX'i d£
~ aXi
with pa defined by
doi d£ = - P i ( J ~ ) '
(A.11)
460
X. Artru, String model / ~ +/ab +/~c = 0 .
(A.12)
The 4-vector/~ plays the role o f a tension [see also (A.7)]. Let us now consider the particular "gauge" (6.7) - (6.8) and choose £ = t. We have dr/d£ = 1; do/d~ = d u / d t ; X = (1 , v ) ; X ' = (0, T); due to (6.9) - (6.10) we have also 0 ~?/03f = X; ~ .6?/OX' = - X ' (we take tile metric + ------). Putting these expres sions in (A.11) gives ~--
(~
du ) ~,T+~-v
(A.12) is nothing but (6.19) - (6.20).
References [ 1] Y. Nambu, Talk presented at the American Physical Society Meeting in Chicago, January 1970, EFI 70-07. [2] L. Susskind, Phys. Rev. D1 (1970) 1182. [3] P. Goddard, J. Goldstone, C. Rebbi and C. Thorn, Nucl. Phys. B56 (t973) 109. [4] J.L. Gervais and B. Sakita, Phys. Rev. Letters 30 (1973) 716. [5] J. Scherk, Lectures given at New York University, NYU/TR3/74. [6] M.B. Einhorn, M.B. Green and M.A. Virasoro, Phys. Letters 37B (1971) 292. [7] X. Artru and G. Mennessier, Nucl. Phys. B70 (1974) 93. [8] S. Mandelstam, Nucl. Phys. B69 (1974) 77; J.L. Gervais and E. Cremmer, Nucl. Phys. B76 (1974) 209. [9] M. Kaku and K. Kikkawa, Phys. Rev. D10 (1974) 110. [10] B.Y. Oh and G.A. Smith, Nucl. Phys. B49 (1972) 13; J. Badier et al., Phys. Letters 39B (1972) 414. [11] S. Ellis, P.H. Frampton and P.G.O. Freund, Nucl. Phys. B24 (1970) 465. [12] V.I. Strazhev and L.M. Tomil'chik, Soy. J. Particles and Nuclei 4 (1973) 78. [13] L.I. Schiff, Phys. Rev. 160 (1967) 1257. [14] P.G.O. Freund, R. Waltz and J. Rosner, Nucl. Phys. B13 (1969) 237.