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Jets from quark fragmentation: Treatment of flavour and new power counting rules at x → 1
Volume 73B, number 4, 5
PHYSICS LETTERS
13 March 1978
JETS FROM QUARK FRAGMENTATION: TREATMENT OF FLAVOUR AND NEW POWER COUNTING RULES AT x -+ 1 U.P. SUKHATME Laboratoire de Physique ThOorique et Particules Eldmentaires 1, Orsay, France Received 7 November 1977 Revised manuscript received 28 December 1977
Allowing for the possibility of many quark flavours in a crossing symmetric manner, the general problem of quark fragmentation into a jet of mesons via successive breakups (quark ~ meson + quark) is formulated and solved. Simple power counting rules for the behaviour of various fragmentation functions at x ~ 1 are derived and compared with experiment. Meson multiplicities are found to grow logarithmically with a slope independent of the jet-initiating quark, but dependent on the type of meson.
It is widely believed that quark jets are a consequence of confinement, and they are produced more or less in the manner of fig. 1. A quantitative description of such a recursive breakup process in terms of integral equations has been given by several authors [ 1 - 5 ] . The essential complication of having both pseudoscalar and vector mesons, and handling their subsequent decay into stable particles is nicely described in recent papers by Feynman and Field [4]. However, their simple treatment of quarks with several possible flavours is not symmetric under crossing of flavour indices. Other authors have taken the crossing property into account, but their results are not general and are presented in different physical frameworks [2,3]. The purpose of this article is to give a full general treatment of the problem of having an arbitrary number of quark flavours in jets, retaining symmetry under crossing, and deriving the consequences for physically observable quantities. Meson multiplicities in jets are found to grow logarithmically wifh energy with a slope which does not depend on the jet-initiating quark, but does depend on the detected meson type. Thus, for example, 7r and K multiplicities in e+e jets will not in general have the same coefficient of 1 Laboratoire associd au Centre National de la Recherche Scientifique. Postal address: Bfitiment 211, Universit~ Paris-Sud, 91405 Orsay, France. 478
~,JP)
M. = ~,-~,, (.P)
Fig. 1. The standard cascade model for quark jets consisting of a succession of breakups (quark ~ "meson" + quark). I n s at high energies. We also obtain a simple set of counting rules giving the power behaviour of various fragmentation functions at x ~ 1. These rules are substantially different from the predictions of dimensional counting(see e.g. ref. [6]), but as we shall show, they seem to be supported by experiment. N q u a r k f l a v o u r s . First let us recall [4,5] that jet formation is assumed to proceed via a succession of fundamental "breakups" of the form quark "meson" + quark. For example, the first breakup shown in fig. 1 is
Volume 73B, number 4, 5 %.(P) -+ MI(XP) + qi,(P - xP).
PHYSICS LETTERS (1)
Here, the initial quark qi with flavour i and momentum P undergoes a breakup into a "meson" M 1 (%. ~-,) with momentum xP and a leftover quark qi' with flavour i' and momentum (1 - x)P (see fig. 1). At each breakup, the quark momentum decreases. When it falls below a critical value/a "~ O(m~r), no more breakups occur and jet formation is complete. Neglecting masses, spins and transverse momenta, the breakup process is fully described by the "momentuna sharing function" f/i ,(x) [5,7] for which we have assumed scaling, fii'(x) dx is the probability that a breakup yields a "meson" with momentum fraction in the interval (x, x + dx). The "momentum sharing function" f/i'(x) describes the (ii'M1) vertex. At the present time, there is no fundamental theory for obtaining this breakup function, but two reasonable viewpoints can be taken. We shall take the conventional viewpoint [2,3] and treat the breakup as an ordinary vertex. Then, since the quarks qi and qi' appear on the same footing, fii,(x) is symmetric in the flavour indices i and i'. (An alternative viewpoint, based on qualitative arguments about q~ formation in colour fields, [4] is to take fii,(x) to be only a function o f / ' a n d x with no idependence.) It is convenient to define the quantities 1
Pii' =- f dx fii'(x) " 0
(2)
Pii' is the probability of getting an (it"M1) vertex, i.e. a connection between flavours i and i' in one breakup. Clearly, Pii' is symmetric, and since each breakup gives one "meson" 1
~., f dxfii,(x)= ~., Pii' = 1 , t
0
(3)
l
for all i. The symmetric N X N matrix P whose elements are 0 ~
13 March 1978
Given all the functions fii ,(x), any physical quantity for a jet can be derived by formulating and solving appropriate integral equations, and we shall now give some examples.
Meson multiplicity. The equation for meson multiplicity in a jet is
nj~;i(P ) = 6ijP/k 1 t
r
+ ~., f dx'fii,(1 -x)nfic;i,(xP). t u/P
(4)
The quantity nj~;i(P ) is the number of (q/VZlk) "mesons" produced in a jet originating from quark qi(P). This process can occur in the first breakup only if the flavours i a n d j are the same and a (qkqk) pair is produced - hence the term 5ijpj k. The second term corresponds to the possibility that the leftover quark after the first breakup is qi'(x'P), and it produces the "meson" (qi~k) in subsequent breakups. Before solving eq. (4), we consider and solve the simpler equation obtained by summing over ] and k. This gives ni(P ) = total multiplicity from quark qi(P): 1
ni(P)= 1 + ~ . , f
dx'fii,(1 - x ' ) n i , ( x ' P ) .
(5t
u/P For large P (or equivalently, for small/a), the lower limit of integration is effectively zero. By analogy with the case of just one flavour [5], consider
ni(P) = ~i In P + C i .
(6)
Substitution in eq. (5) and equating the coefficients of In P and the constant terms yields the following constraint equations for the parameters o~i and C i" :
,
:
1 +
t
+
i'
ci,p.,
l
,
(7)
where, 1
gii' = f dx'f/i,(1 - x ' ) In x' .
(8)
0 Eqs. (3) and (7) can be satisfied only if all slopes oti are the same = c~(say). Summing eq. (7) over i yields
a=- N
i
1
gii'
,
(9) 479
Volume 73B, number 4, 5
PHYSICS LETTERS
a result which we quoted in ref. [5] without proof. Furthermore, eqs. (7) can be solved to find the constants Ci upto a single overall additive constant corresponding to an arbitrary mass scale; [5] this scale is/~ if we had properly kept the lower limit in eq. (5).
Meson distribution functions. Let Fjk;i (x) denote the inclusive distribution for "meson" (q/qk) from quark qi(P). Then, for P ~> gt, we have
F/i;;(x) = aq f/x(x)
N
Ci
= G ('I k=l
P)ik 1 (1
+ a ~"glk)" l
(10)
Note the very general result (independent of the choice of momentum sharing functions) that the total multiplicity ni(P ) increases logarithmically for large energies with a universal slope a. (This universality is not surprising since knowledge of the incident quark flavour is reduced with each successive breakup.) Now consider the more detailed multiplicity eq. (4). The k-dependence drops out ifnfi¢;i contains a factor Pjk" Thus, we are motivated to try the form
njk;i(P) = Pjk [aji In P + Cji I .
(11)
This solves eq. (4) provided the constants a/i and satisfy
Cji
13 March 1978
+.,
)dx' x
(14)
x') F/~;i,(x/x'),
--7 fii'(1
x
where, as usual, 6q,fjk(X) is the contribution from the first breakup, and the integral corresponds to s(lbsequent breakups. As expected, integrating eq. (14) over x gives the multiplicity integral eq. (4). Eqs. (14) are a set of coupled linear Volterra integral equations of the second kind, and a convergent solution can be obtained by successive iteration (see any good book on integral equations, e.g. ref. [8]). Instead of doing this directly, however, let us assume for simplicity that all momentum sharing functions have the same x-dependence:
f
fii,(x) =Pii,f(x),
1
dxf(x)= 1 .
(15)
0
aji = ~i' aji' Pii' ' Cji = 6 ij + "~i aji'gii' + ~I"' Pii' Cji"
This simplifies eq. (14): (12)
Fj~;i(x) = 6q Pik f(x) 1
Effectively, these equations are the same as eqs. (7), since different/' values do not mix. Thus, we again find that a/i does not depend on i o r / a n d equals a/N, where a is given by eq. (9). Likewise, the constants C]i can be obtained from solving the linear eqs. (12). The full result is
nj~;i(P) = (aPjk/N) In P + PjkCji .
f t
,
-~f(1
x')F/k;i,(x/x'
) .
(16)
X
X
Clearly, P/k is a factor in F/k;i(x ), and it eliminates the k-dependence. The general solution obtained by iteration is
(13)
It is not hard to show that this is the most general solution to integral eq. (4) [8]. Summation overj and k yields the solution ni(P ) obtained before. Note the reasonable result that the slope (apjk/N) is proportional to the vertex probability Pjk but does not depend on the initial flavour i. In principle, the slope is experimentally obtainable, and a good way to determine the parameters P/k" However, it must be remembered that the presence of vector mesons will affect the slope in a calculable way [4], although the logarithmic behaviour will remain.
480
+
Fl'k'i(x) = Pjk ~ '
q=l
F(q)(x) [pq-11 if,
(17)
where F(1)(x) = f ( x ) , (18) (y J f 1 - Y ) F(q)(y), F(q+l)(x) = ; I d--~-
(q = 1,2,...).
X
Note that the quantities F(q)(x) are the same as those present in the case of one flavour only [5]. The only additional complication due to the presence of an at-
Volume 73B, number 4, 5
PHYSICS LETTERS
bitrary number of flavours, are the powers p q - I of the probability connectedness matrix p. The physical meaning of the qth term is that it is the contribution to Fjk;i(x ) coming from the qth breakup. Typically, in phenomenological applications for x >~ 0.3, only 3 or 4 terms need to be retained. Let us now focus on some very general results at x ~ 1. The contributions to the distribution function coming from successive breakups each involve an additional integral and it is not hard to check that each integration produces a factor of (1 - x ) at x ~ 1; that is F(q)(x)/F(1)(x) is proportional to (1 - x)q -1 . Thus, if f(x) =(1 +/3)(1 - x ) ~ (/3 ~> 0), then a t x -+ 1
Ffic;i(x ) -+ p/k(1 +/3)(1 -+p/kPij(1
- x) # ,
i = f,
+/3)(1 - - x ) t3+l,
i--/=j.
Therefore, the distribution function for a meson which cannot be formed in the first breakup is always one power of (1 - x) greater than the distribution function for a meson which can be formed from the jet-initiating quark in the first breakup. This is a new and simple rule for the behaviour of various fragmentation functions at x ~ 1. In particular, as an explicit example, we find that F~r+;a/FTr-;c~-+ Purl (1 - - x ) at x ~ 1. This result is very different from dimensional power counting [6], where a ratio (1 - x) 4 would be q,
f (a)
0 - ~ ) ¢s
m (b)
( I - x ) t~÷J
f ~")
O - x) ~"j
(~)
13 March 1978
expected. However, phenomenological analysis of e+e- data seems to give a power of unity [3,4] which is fully consistent with our prediction. For completeness, we mention that if the detected meson comes from the two-body decay of a resonance, then the decay introduces an integral over the resonance momentum [4] which also gives rise to an extra factor of (1 - x) at x ~ 1. Some illustrative examples are shown in fig. 2.
Jets with baryons. So far we have only discussed meson jets, since each breakup proceeded via the formation of a q~ pair. By analogy, for baryon production, it is reasonable to assume the formation of diquark pairs (qq)(~i]) in a breakup (see fig. 3). This guarantees baryon number conservation and does not alter the formalism which we have presented, since the (qq) and ( ~ systems act like new "effective" flavours and simply give additional rows and columns in the probability matrix p. Note that the production of a baryon will be followed by an antibaryon at the next breakup; however, as was emphasized in ref. [5], this does not make them consecutive in rapidity. In conclusion, it is clear that the complete treatment of flavour given in this paper can be applied to phenomenology in any hadronic reaction involving jets. Since our results (eqs. (13) and (17)) using a crossing symmetric treatment of flavour are different from those of Feynman and Field [4] it would be interesting to see how this will affect their extensive phenomenological Monte Carlo calculations. Qualitatively, it seems that several substantial changes will occur. For example, the probability matrix PFF used in ref. [4] and the corresponding p which is crossing symmetric, are
q,
O - x ) ~÷z
Fig. 2. The power behaviour of the meson fragmentation function at x ~ 1 when the meson (M) comes from (a) the first breakup, (b) the second breakup, (c) the first breakup and vector meson (V) decay, (d) the second breakup and vector meson decay.
Fig. 3. The production of baryons and antibaryons in jets. 481
Volume 73B, number 4, 5
PFF =
p=
d
s
0.4 0.4
0.4 0.4
0.21u 0.2 d ,
0.4
0.4
0.2
d
0.4 0.4
0.4 0.4
0.21u 0.2 d
LO.2
0.2
0.6
s
s
s
Although the parameters could be varied somewhat, it would seem that jets from strange quarks in the two formalisms would be quite different. In particular, the matrix p will substantially enhance the formation of sg vertices, hence increase r/and 77' production, and consequently the final number of produced photons. It is a pleasure to thank Prof. J.C. Polkinghorne and Dr. P.V. Landshoff for crucial suggestions and helpful discussions.
482
13 March 1978
References
u
u
I
PHYSICS LETTERS
[1] A. Krzywicki and B. Petersson, Phys. Rev. D6 (1972) 924; J. Finkelstein and R.D. Peccei, Phys. Rev. D6 (1972) 2606; F. Niedermayer, Nucl. Phys. B79 (1974) 355. [2] S. Matsuda, Phys. Rev. D12 (1975) 1940; H. Fukuda and C. Iso, Progr. Theoret. Phys. 57 (1977) 483; R. Kronefeld and R.D. Peccei, Phys. Rev. D7 (1973) 1556. [3] O. Sawada and G. Takeda, Tohoku Univ. preprints TU/77/166 (1977) and TU/76/142 (1976). [4] R.P. Feynman, in: Proc. VIIIth Internat. Syrup. on Multiparticle dynamics (Kaysersberg, 1977); R.D. Field and R.P. Feynman, Caltech preprint (1977). [5] U.P. Sukhatme, Cambridge Univ. DAMTP preprint 77/25 (1977). [6] S.J. Brodsky and J.F. Gunion, SLAC preprint 1939 (1977). [7] A. Seiden, Phys. Lett. 68B (1977) 157. [8] W. Pogorzelski, Integral equations, Vol. I (Pergamon, 1966).
PHYSICS LETTERS
13 March 1978
JETS FROM QUARK FRAGMENTATION: TREATMENT OF FLAVOUR AND NEW POWER COUNTING RULES AT x -+ 1 U.P. SUKHATME Laboratoire de Physique ThOorique et Particules Eldmentaires 1, Orsay, France Received 7 November 1977 Revised manuscript received 28 December 1977
Allowing for the possibility of many quark flavours in a crossing symmetric manner, the general problem of quark fragmentation into a jet of mesons via successive breakups (quark ~ meson + quark) is formulated and solved. Simple power counting rules for the behaviour of various fragmentation functions at x ~ 1 are derived and compared with experiment. Meson multiplicities are found to grow logarithmically with a slope independent of the jet-initiating quark, but dependent on the type of meson.
It is widely believed that quark jets are a consequence of confinement, and they are produced more or less in the manner of fig. 1. A quantitative description of such a recursive breakup process in terms of integral equations has been given by several authors [ 1 - 5 ] . The essential complication of having both pseudoscalar and vector mesons, and handling their subsequent decay into stable particles is nicely described in recent papers by Feynman and Field [4]. However, their simple treatment of quarks with several possible flavours is not symmetric under crossing of flavour indices. Other authors have taken the crossing property into account, but their results are not general and are presented in different physical frameworks [2,3]. The purpose of this article is to give a full general treatment of the problem of having an arbitrary number of quark flavours in jets, retaining symmetry under crossing, and deriving the consequences for physically observable quantities. Meson multiplicities in jets are found to grow logarithmically wifh energy with a slope which does not depend on the jet-initiating quark, but does depend on the detected meson type. Thus, for example, 7r and K multiplicities in e+e jets will not in general have the same coefficient of 1 Laboratoire associd au Centre National de la Recherche Scientifique. Postal address: Bfitiment 211, Universit~ Paris-Sud, 91405 Orsay, France. 478
~,JP)
M. = ~,-~,, (.P)
Fig. 1. The standard cascade model for quark jets consisting of a succession of breakups (quark ~ "meson" + quark). I n s at high energies. We also obtain a simple set of counting rules giving the power behaviour of various fragmentation functions at x ~ 1. These rules are substantially different from the predictions of dimensional counting(see e.g. ref. [6]), but as we shall show, they seem to be supported by experiment. N q u a r k f l a v o u r s . First let us recall [4,5] that jet formation is assumed to proceed via a succession of fundamental "breakups" of the form quark "meson" + quark. For example, the first breakup shown in fig. 1 is
Volume 73B, number 4, 5 %.(P) -+ MI(XP) + qi,(P - xP).
PHYSICS LETTERS (1)
Here, the initial quark qi with flavour i and momentum P undergoes a breakup into a "meson" M 1 (%. ~-,) with momentum xP and a leftover quark qi' with flavour i' and momentum (1 - x)P (see fig. 1). At each breakup, the quark momentum decreases. When it falls below a critical value/a "~ O(m~r), no more breakups occur and jet formation is complete. Neglecting masses, spins and transverse momenta, the breakup process is fully described by the "momentuna sharing function" f/i ,(x) [5,7] for which we have assumed scaling, fii'(x) dx is the probability that a breakup yields a "meson" with momentum fraction in the interval (x, x + dx). The "momentum sharing function" f/i'(x) describes the (ii'M1) vertex. At the present time, there is no fundamental theory for obtaining this breakup function, but two reasonable viewpoints can be taken. We shall take the conventional viewpoint [2,3] and treat the breakup as an ordinary vertex. Then, since the quarks qi and qi' appear on the same footing, fii,(x) is symmetric in the flavour indices i and i'. (An alternative viewpoint, based on qualitative arguments about q~ formation in colour fields, [4] is to take fii,(x) to be only a function o f / ' a n d x with no idependence.) It is convenient to define the quantities 1
Pii' =- f dx fii'(x) " 0
(2)
Pii' is the probability of getting an (it"M1) vertex, i.e. a connection between flavours i and i' in one breakup. Clearly, Pii' is symmetric, and since each breakup gives one "meson" 1
~., f dxfii,(x)= ~., Pii' = 1 , t
0
(3)
l
for all i. The symmetric N X N matrix P whose elements are 0 ~
13 March 1978
Given all the functions fii ,(x), any physical quantity for a jet can be derived by formulating and solving appropriate integral equations, and we shall now give some examples.
Meson multiplicity. The equation for meson multiplicity in a jet is
nj~;i(P ) = 6ijP/k 1 t
r
+ ~., f dx'fii,(1 -x)nfic;i,(xP). t u/P
(4)
The quantity nj~;i(P ) is the number of (q/VZlk) "mesons" produced in a jet originating from quark qi(P). This process can occur in the first breakup only if the flavours i a n d j are the same and a (qkqk) pair is produced - hence the term 5ijpj k. The second term corresponds to the possibility that the leftover quark after the first breakup is qi'(x'P), and it produces the "meson" (qi~k) in subsequent breakups. Before solving eq. (4), we consider and solve the simpler equation obtained by summing over ] and k. This gives ni(P ) = total multiplicity from quark qi(P): 1
ni(P)= 1 + ~ . , f
dx'fii,(1 - x ' ) n i , ( x ' P ) .
(5t
u/P For large P (or equivalently, for small/a), the lower limit of integration is effectively zero. By analogy with the case of just one flavour [5], consider
ni(P) = ~i In P + C i .
(6)
Substitution in eq. (5) and equating the coefficients of In P and the constant terms yields the following constraint equations for the parameters o~i and C i" :
,
:
1 +
t
+
i'
ci,p.,
l
,
(7)
where, 1
gii' = f dx'f/i,(1 - x ' ) In x' .
(8)
0 Eqs. (3) and (7) can be satisfied only if all slopes oti are the same = c~(say). Summing eq. (7) over i yields
a=- N
i
1
gii'
,
(9) 479
Volume 73B, number 4, 5
PHYSICS LETTERS
a result which we quoted in ref. [5] without proof. Furthermore, eqs. (7) can be solved to find the constants Ci upto a single overall additive constant corresponding to an arbitrary mass scale; [5] this scale is/~ if we had properly kept the lower limit in eq. (5).
Meson distribution functions. Let Fjk;i (x) denote the inclusive distribution for "meson" (q/qk) from quark qi(P). Then, for P ~> gt, we have
F/i;;(x) = aq f/x(x)
N
Ci
= G ('I k=l
P)ik 1 (1
+ a ~"glk)" l
(10)
Note the very general result (independent of the choice of momentum sharing functions) that the total multiplicity ni(P ) increases logarithmically for large energies with a universal slope a. (This universality is not surprising since knowledge of the incident quark flavour is reduced with each successive breakup.) Now consider the more detailed multiplicity eq. (4). The k-dependence drops out ifnfi¢;i contains a factor Pjk" Thus, we are motivated to try the form
njk;i(P) = Pjk [aji In P + Cji I .
(11)
This solves eq. (4) provided the constants a/i and satisfy
Cji
13 March 1978
+.,
)dx' x
(14)
x') F/~;i,(x/x'),
--7 fii'(1
x
where, as usual, 6q,fjk(X) is the contribution from the first breakup, and the integral corresponds to s(lbsequent breakups. As expected, integrating eq. (14) over x gives the multiplicity integral eq. (4). Eqs. (14) are a set of coupled linear Volterra integral equations of the second kind, and a convergent solution can be obtained by successive iteration (see any good book on integral equations, e.g. ref. [8]). Instead of doing this directly, however, let us assume for simplicity that all momentum sharing functions have the same x-dependence:
f
fii,(x) =Pii,f(x),
1
dxf(x)= 1 .
(15)
0
aji = ~i' aji' Pii' ' Cji = 6 ij + "~i aji'gii' + ~I"' Pii' Cji"
This simplifies eq. (14): (12)
Fj~;i(x) = 6q Pik f(x) 1
Effectively, these equations are the same as eqs. (7), since different/' values do not mix. Thus, we again find that a/i does not depend on i o r / a n d equals a/N, where a is given by eq. (9). Likewise, the constants C]i can be obtained from solving the linear eqs. (12). The full result is
nj~;i(P) = (aPjk/N) In P + PjkCji .
f t
,
-~f(1
x')F/k;i,(x/x'
) .
(16)
X
X
Clearly, P/k is a factor in F/k;i(x ), and it eliminates the k-dependence. The general solution obtained by iteration is
(13)
It is not hard to show that this is the most general solution to integral eq. (4) [8]. Summation overj and k yields the solution ni(P ) obtained before. Note the reasonable result that the slope (apjk/N) is proportional to the vertex probability Pjk but does not depend on the initial flavour i. In principle, the slope is experimentally obtainable, and a good way to determine the parameters P/k" However, it must be remembered that the presence of vector mesons will affect the slope in a calculable way [4], although the logarithmic behaviour will remain.
480
+
Fl'k'i(x) = Pjk ~ '
q=l
F(q)(x) [pq-11 if,
(17)
where F(1)(x) = f ( x ) , (18) (y J f 1 - Y ) F(q)(y), F(q+l)(x) = ; I d--~-
(q = 1,2,...).
X
Note that the quantities F(q)(x) are the same as those present in the case of one flavour only [5]. The only additional complication due to the presence of an at-
Volume 73B, number 4, 5
PHYSICS LETTERS
bitrary number of flavours, are the powers p q - I of the probability connectedness matrix p. The physical meaning of the qth term is that it is the contribution to Fjk;i(x ) coming from the qth breakup. Typically, in phenomenological applications for x >~ 0.3, only 3 or 4 terms need to be retained. Let us now focus on some very general results at x ~ 1. The contributions to the distribution function coming from successive breakups each involve an additional integral and it is not hard to check that each integration produces a factor of (1 - x ) at x ~ 1; that is F(q)(x)/F(1)(x) is proportional to (1 - x)q -1 . Thus, if f(x) =(1 +/3)(1 - x ) ~ (/3 ~> 0), then a t x -+ 1
Ffic;i(x ) -+ p/k(1 +/3)(1 -+p/kPij(1
- x) # ,
i = f,
+/3)(1 - - x ) t3+l,
i--/=j.
Therefore, the distribution function for a meson which cannot be formed in the first breakup is always one power of (1 - x) greater than the distribution function for a meson which can be formed from the jet-initiating quark in the first breakup. This is a new and simple rule for the behaviour of various fragmentation functions at x ~ 1. In particular, as an explicit example, we find that F~r+;a/FTr-;c~-+ Purl (1 - - x ) at x ~ 1. This result is very different from dimensional power counting [6], where a ratio (1 - x) 4 would be q,
f (a)
0 - ~ ) ¢s
m (b)
( I - x ) t~÷J
f ~")
O - x) ~"j
(~)
13 March 1978
expected. However, phenomenological analysis of e+e- data seems to give a power of unity [3,4] which is fully consistent with our prediction. For completeness, we mention that if the detected meson comes from the two-body decay of a resonance, then the decay introduces an integral over the resonance momentum [4] which also gives rise to an extra factor of (1 - x) at x ~ 1. Some illustrative examples are shown in fig. 2.
Jets with baryons. So far we have only discussed meson jets, since each breakup proceeded via the formation of a q~ pair. By analogy, for baryon production, it is reasonable to assume the formation of diquark pairs (qq)(~i]) in a breakup (see fig. 3). This guarantees baryon number conservation and does not alter the formalism which we have presented, since the (qq) and ( ~ systems act like new "effective" flavours and simply give additional rows and columns in the probability matrix p. Note that the production of a baryon will be followed by an antibaryon at the next breakup; however, as was emphasized in ref. [5], this does not make them consecutive in rapidity. In conclusion, it is clear that the complete treatment of flavour given in this paper can be applied to phenomenology in any hadronic reaction involving jets. Since our results (eqs. (13) and (17)) using a crossing symmetric treatment of flavour are different from those of Feynman and Field [4] it would be interesting to see how this will affect their extensive phenomenological Monte Carlo calculations. Qualitatively, it seems that several substantial changes will occur. For example, the probability matrix PFF used in ref. [4] and the corresponding p which is crossing symmetric, are
q,
O - x ) ~÷z
Fig. 2. The power behaviour of the meson fragmentation function at x ~ 1 when the meson (M) comes from (a) the first breakup, (b) the second breakup, (c) the first breakup and vector meson (V) decay, (d) the second breakup and vector meson decay.
Fig. 3. The production of baryons and antibaryons in jets. 481
Volume 73B, number 4, 5
PFF =
p=
d
s
0.4 0.4
0.4 0.4
0.21u 0.2 d ,
0.4
0.4
0.2
d
0.4 0.4
0.4 0.4
0.21u 0.2 d
LO.2
0.2
0.6
s
s
s
Although the parameters could be varied somewhat, it would seem that jets from strange quarks in the two formalisms would be quite different. In particular, the matrix p will substantially enhance the formation of sg vertices, hence increase r/and 77' production, and consequently the final number of produced photons. It is a pleasure to thank Prof. J.C. Polkinghorne and Dr. P.V. Landshoff for crucial suggestions and helpful discussions.
482
13 March 1978
References
u
u
I
PHYSICS LETTERS
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