- Email: [email protected]
Applications of the quark parton model in one-particle inclusive leptonic-induced reactions
Nuclear
Physics
B51 (1973)
61 l-627.
North-Holland
Publishing
Company
APPLICATIONS OF THE QUARK PARTON MODEL IN ONE-PARTICLE INCLUSIVE LEPTONIC-INDUCED REACTIONS
*
M. GRONAU **, F. RAVNDAL and Y. ZARMI *** California Institute of Technology,
Pasadena, California 91 IO9
Received 31 July 1972 (Revised 25 September 1972)
Abstract: The quark parton model is applied to the deep-inelastic region of single-particle inclusive lepton-hadron processes. The current-fragmentation region is mainly discussed. Various conclusions concerning experimentally observable quantities such as single-particle distribution functions and average multiplicities are reached. Sum rules such as the Bjorken and Adler sum rules are generalized to this case. The average multiplicities of current-fragment pions in neutrino reactions are found to be independent of the target and of X. They are related to electroproduction multiplicities. A relationship is established between the x + 0 limit of the latter, and pion multiplicities in e+e- annihilation. Assuming SU(3) invariance, relations among n and K multiplicities are derived.
1. Introduction
The light cone and the parton models have been both applied to the analysis of the interactions of the weak and electromagnetic current with hadrons in the deepinelastic region [ 11. They have been shown to be equivalent in this aspect. In the near future, inclusive cross sections for single hadron production in deepinelastic lepton-nucleon scattering will be measured. It is, therefore, worthwhile to apply the two models to this type of process [2]. It has already been noticed that in deep-inelastic semi-inclusive reactions the region of applicability of the parton model is wider than that of the light-cone description. While the latter has been only applied [3] to the kinematical region of “target fragmentation” (i.e. a finite momentum carried by the observed hadron in the lab frame), the first can be applied [4] to both this region and to the “current fragmentation” region (i.e. in the lab frame, the momentum of the observed hadron is a finite fraction of the momentum carried by the current).
* Work supported in part by the U.S. Atomic Energy Commission. Prepared under contract AT(l l-1)-68 for the San Francisco Operations Office, U.S. Atomic Energy Commission. ** Permanent address: Physics Department, Technion, Haifa, Israel. *** On leave of absence from the Weizmann Institute, Rehovot, Israel.
M. Gronau et al., Quark parton model
612
In this paper we study the consequences of the quark parton model ideas, as suggested by Feynman [5], in semi-inclusive reactions. Testing of these ideas is of great importance, since if verified, they enable one to obtain properties of individual partons from experiment. We shall mainly treat the current fragmentation region, in which the model has its strongest predictive power. In sect. 2 we present the kinematics involved, and in sect. 3 the assumptions of the model [ 51 are presented. In sect. 4 we study the consequences of the model. We obtain relations among experimentally measurable quantities such as inclusive distributions and multiplicities. While some of these relations depend on the quark quantum numbers, others can be obtained in models in which the partons are not quarks. We conclude in sect. 5 with a few remarks concerning the tests we have presented.
2. Kinematics The process of interest, !Zt N + Iz’t h t anything is shown in fig. 1. Here II and Q’ are leptons carrying momenta k and k’ respectively. In the lab frame, where the nucleon momentum is p, = (M, 0); k, = (E,k) and k; = (E’,k’). The outgoing observed hadron carries four-momentum h, and the leptonic current carries four-momentum 4. The deep-inelastic region is defined by q2 -+-cm,
y,p’q,, M
The target-fragmentation e = h - p finite ,
2
’
x = $
finite .
(2.1)
region is defined by
K=h*q+t=‘,
u = if_ finite . Mu
in a frame where the target is moving fast this corresponds d(k) --‘(k’)
Fig. 1. The process P + N + P’ + h + anything.
(2.2)
to a detected hadron
M. Gronau et al., Quark parton model
613
h moving in the direction of p, and its momentum being a finite fraction, U, of it (except for a possible bounded transverse momentum hT). The current fragmentation region is defined by /(=h.q_,-m,
e=h.p-+m,
u = M% and ,$
are finite .
(2.3)
In the lab frame this corresponds to a detected hadron moving in the direction of the current, its momentum being a fraction -u/x of q (except for a possible bounded transverse momentum hT). The differential cross section for the inelastic electroproduction with one detected hadron in the final state is given in the lab frame by 202 d3k’ ye& Qpv lklq4E’2E,
do=
c (X,h X
X
IJp(0)lp, (X,h I.T(O)lp)* (2n)3d4)(k+p-k’-h-Px)
,
(2.4)
This can be written as d4c
- 27W2 I1p~p.m.
dq2dvdedK
q4 $I2
‘”
(2.5) ’
where E -e.m. = _; W
ep-e)6(h
~d4he(h,,)6(h2-m;)6(h
IJv
X c X
=
(X,h IJ&O)Ip)(X,h
*q-K)
iJv(O)ip)* (2n)36’4’(k+p-k’-h-P,)
q,qv ~i~*(q2S’&~E)
-gpv +----
q2 > ’
v
i3,“;;f’(q2,
(2.6)
Y, K, E)
!F’ is the leptonic tensor. The current matrix elements are averaged over the initial nucleon spin states. In neutriiro production the electromagnetic current is replaced by the weak current, and 4na2/q4 by G2/2n. zPv then has the form (neglecting the lepton mass)
.e
I - - we
2M2
paq4i$
.
(2.7)
614
hf. Gronau et al., Quark parton model
From eqs. (2.5)-(2.7)
one obtains
4rr(u2 E' (eN + ehX) = ~ -(cos2 4 E dq2dvdedrc 4 d40
dq~~~dedK(v(F)N+ pT hX) = $$
;BSZ~+2sin2:0~~),
(cos2 fr0 @TijN t 2 sin2 f 0 G’;‘“,”
T
E+E’sin2 1ie P$)N) . M
(2.8)
where i3;.n = i3; h(q2,v,K,E). IJ has been shown [3,4] that in the target fragmentation region VP?% and v2 W;t are functions of x,u, E. On the other hand, in the current fragmentation re"eNh that scale, and depend only on X,U [4]. Similar congion it is v2 $TNh and v3 W, clusions can be' reached for’the v(v)-N structure functions, Fy,fjN, with $“jN behaving like %$z)N. We shall find i’t more convenient to discuss a different cross section d4a
(eN -+ ehX) = ~
(2.9)
dq2dvdudh+ where in the deep inelastic limit
The reason for this choice will become apparent when we discuss the model in sect. J.
The neutrino cross section is_similarly defined. This cross section will have the same form as in eq. (2.8) with Wj replaced by Wi.From the scaling properties of gi in both target and current fragit follows that, for fixed finite u and h,,2 Wi,h scale mentation regions according to 'Wih -Fih(x,',h~)' i= 2, 3.
MW, n + Qn(xM;),
(2.10)
These structure functions scale with the same powers of v as the corresponding total structure functions. For neutrino reactions, in the approximation where the Cabibbo angle is zero, one finds from isospin invariance of the strong interactions wyP= I
p I
’
wi”P=wi”.
The analogous relations in our case are
(2.11)
M. Gronau et al., Quark parton model
615
@P = wyn, pn (2.12) l,h' ’ z,h ’ l,h l,h where h’ is the hadron obtained from h by a 180” isospin rotation around IY. By integrating over the kinematical variables of the observed hadron one has
WYP =
d4a
_Idq2dvdudh;
(QN -+ Q’hX)dudh$
(I1N + Q'X)
= (ah) sd7
,
(2.13)
where (nn) is the final hadron average multiplicity.
3. The quark parton model We use the quark parton model in its standard version as presented in ref. [S]. It is assumed in this model that a fast-moving nucleon can be regarded as made up of constituents (partons) whose transverse momenta are bounded, and whose longitudinal momenta are finite fractions, x, of the total nucleon momentum, P. Only a finite number of partons have finite and small momenta (namely, x < l/P). These are the wee partons. The longitudinal momentum distribution for non-wee partons scales, i.e., it depends only on the fractions of momentum carried by the various types of partons and not on their absolute longitudinal momenta. The average number of partons of type i in the interval x, x + dx isfi(x)dx. For small x,4(x) g CJx. This translates into a plateau when the distribution is described in terms of the rapidity variable, y, of the partons. Assuming the partons are quarks, we shall denote their distribution functions by their names. Thus, a u-quark will have a distribution function u(x), a d-quark d(x), and similarly for the other types. The scaling functions, Pi(x), in the total inclusive cross sections are linear combinations of these functions. For example: FTp(x) = ; {?j(u(x)+ir(x))
+ ‘,(d(x)+d(x))
F,“p(x) = cos2eC(d(x)tG(x)) Fip(x)
= 2 cos2 BC(-d(x)+(x))
)
+ &(s(x)t$x))}
t sin2 ec(s(x)+G(x))
,
+ 2 sin2 e&s(x)+iY(x))
,
(3.1)
The neutron functions are obtained by substituting u ++d, U + d. The anti-neutrino functions are obtained by d + u, and ii + din the stangeness conserving part, and s + u, U + Yin the strangeness changing part. Spin f quarks imply F2(x) = 2xFI(x) L61. In order to study semi-inclusive reactions we have to say something about the fragmentation of partons into hadrons [4, 51. As suggested by Feynman [5], we do this in the current-parton Breit frame.
616
M. Gronau et al., Quark parton model
0 p=(P,o,o,P)
IF (xP,o,o,
q = (0, o,o, -2xP) XP)
(xP,o,o,-XP)
+--_ +--___ _ q--_ l _-_-_--_ *--
b ((l-x) -__,
P,
, o,(I-x)P) --+ --_-_+ -----* _-----•, ----_-_-_~ --A _----_)
T
--+
--,
Fig. 2. The photon-parton Breit frame description of deep inelastic interactions. (a) Before the current interacts with a quark. The nucleon momentum is pP = (P,O,O,P). The photon momentum is qP = (O,O,O, -2xP). x = -q’/2p . q is the momentum fraction carried by the parton. (b) After the interaction has occurred. The parton now moves to the left and fragments into leftmoving hadrons while the right-moving partons fragment into right-moving hadrons.
Fig. 2a describes the situation before the current, carrying a momentum 4 with a component in the -z direction only, interacts with one of the partons. The nucleon carries a momentum P in the +z direction. The parton hit by the current carries a longitudinal momentum XP (x = -q2/2q - p), and some small transverse momentum. Fig. 2b describes the process after the interaction has occurred. The interacting parton is now moving to the left with momentum XP (-z direction), and the rest of the partons move to the right with momentum (1 -x)P. Each of the two components (left and right-moving partons) now fragments into hadrons. It is assumed that there is no correlation between the fragmentation of the left-moving parton and that of the right-moving ones. For a finite 0
M. Gronou et al., Quark porton model
617
The fragmentation of the right-moving partons proceeds in a similar way. Only, in this case, the fragmentation function, denoted by E, depends both on the type of quark removed, and on itsx-value [5]. Thus, for example, when a u-quark is knocked outofaproton,En+ (pFU)(~,r,h%$)will denote the fragmentation function of the rightmoving component mto a rr+ with a momentum fraction r of the total (I-x)P. It is assumed that for small z(r) the D(E) functions behave like ~-f(r-~). One can easily show that left-moving hadrons are current fragments (as defined in sect. 2) with u=-zx.
(3.2)
In the lab frame z = h . p/q p is the fraction of current momentum carried by the observed hadron. Similarly, right-moving hadrons are target fragments with U = r(l-x) .
(3.3)
Applying isospin and charge conjugation
invariance
one finds
(3.4)
(3.5) We are now in a position d4a dq2dvdzdh;
=x
to calculate the differential
cross section
d4a (3.6)
dq2dvdudh$
for a current fragment. The corresponding
scaling functions,
denoted by Li,h satisfy
hf. Gronau et al., Quark parton model
618
Li,h(X,“,h~) =XF;:h(X,‘,h~)
(3.7)
)
where I;;,h are defined in eq. (2.10). The expression for these scaling functions are obtained by multiplying each term in eq. (3.1) for the total inclusive scaling functions by the appropriate D-functions. For example, Ly=+ (x,z,h;)
= ; { ‘?,uD;+ t $UD,“+t ‘,dD;+ t $jD$+t
~s~~+t $FLl;+}
,
which, by eq. (3.4) becomes {D;+(;ut#tD;+(+
LeP =; 1,?T+
.
t;d)t$D;(stS)}
Also L;pn+ = 2xLp
(3.8)
For the neutrino
case one finds
LyF*+ = cos2 Oc(d t U)Di+ t sin2 O,(sD”’u t iiD:> , Lifir+ = 2 cos2 Oc(-d t U)D:+ t 2 sin2 e,(-SD:+ Similar expressions can be obtained for a neutron various detected outgoing hadrons. In the BC = 0 approximation one has
Lspn+ = 2xLlp,+ ,
t iiD:> . target, anti-neutrinos
LyN = Dn’F!‘N Ul. z,n Therefore, in this approximation, d4a dq2dvdzdh$
(3.9) and for
(3.10)
(vN -+ I-(- nX) = ~ d2a (vN + p-X)D:, dq2dv
.
(3.11)
Notice that this relation as well as a similar relation for anti-neutrino reactions holds for all the three charge states of the pion. In the next section it will be convenient to use the normalized inclusive distribution functions: G&(x,z,
h;) =
d4a dq2dvdzdh+
(QN + Q’X) .
(3.12)
Notice that the explicit dependence on u and 0 is cancelled out in G due to the properties of the structure functions in this model, eqs. (3.8) and (3. IO). The same is true for the average multiplicity of left-moving hadrons. As a result, one obtains the following relation between the structure functions;
M, Gronau et al., Quark parton model
s
619
(3.13)
L’tN(x z h2)dzdh2 T = (n hQN ) FPN z z,h ’ ’ T
where is the average multiplicity of left-moving hadrons of type h. Scaling functions can also be written for right-moving hadrons with z replaced by r - the fraction of total right-moving momentum carried by the observed hadron, and the D-functions replaced by the E-functions. The latter also satisfy some equalities resulting from the application of isospin rotations, though these are not as restrictive as the relations satisfied by the D-functions. This is due to the dependence of the E-functions on the target. Thus we have, for example, (nh)QN
E&
= E ;L)
Egls)
’
= &,
.
(3.14)
In a manner analogous to eq. (3.6) we now obtain d40
=(l_x)
dq2dudrdh$ with corresponding
d4u dq2dududh;
9
(3.15)
scaling functions
Ri h(X,‘,h~) = (‘-X)‘i
h(X’“‘h.I.) )
(3.16)
where, in this region u = I( l-x).
4. Consequences and proposed tests of the model In this section, except for subsect. 4.7, we treat current fragments. For them, fragmentation functions depend only on z and ht. The equalities among various functions render simple forms for the scaling functions. As mentioned in sect. 3, it is assumed that for small z the D-functions behave l/z down to z - l/xP (xP is the momentum carried by the fragmenting parton). In the frame we are usingxP = f&$. Therefore, the integrated D-functions a logarithmic dependence on +2 s D:(z,ht)dz
- Rn G,
like have
(4.1)
where here, and from now on, z-integration implies h$ integration as well. A result of general character is the relation between no and rr+ production. eq. (3.4) one obtains LirrO = &Cia+ +J&),
the D-
i= 1,2,3,
Using
(4.2)
for all the processes considered. Hence, similar relations are satisfied by the cross sections, normalized distributions and average multiplicities for left-moving pions. Eq. (4.2) is a well-known result for the fragmentation of objects having isospin Gf [15].
hf. Gronau et al., Quark parton model
620
This result crucially depends on the isospin quantum let and not on their charges.
numbers of the quark trip-
4.1. Neu trino reactions In the BC = 0 approximation
G 79 --pi=
G;; = G;; = ;(G;;
"P
t
G;;)
(4.3)
Thus, though different, the fragmentation From this
the various inclusive and semi-inclusive cross sections are, in general, normalized distributions satisfy many equalities, and directly yield the functions of individual quarks. it follows that the normalized distributions of left-moving pions in neutrino reactions do not depend on x. Similar equalities will, of course, hold for the integrated functions, namely, the average multi licities of left-moving pions in these reactions. They will only be linear in In J4 -q2 (see eq. (4.1)). These equalities depend on the fact that the pion is a G-eigenstate and on the Gproperties of the quarks created by the strangeness concerving weak current, An important consequence obtained here is that the average multiplicity of leftmoving
pions
produced
in neutrino
and anti-neutn’no
reactions
does
not depend
on
the target
(in the BC= 0 approximation). The sum rules and inequalities obtained in deep inelastic neutrino-nucleon scattering can be generalized to our case. For example, the Bjorken and Adler [7] and the Gross-Llewellyn Smith [8] sum rules assume in the 8, = 0 approximation the form:
sCL;“,+ ,
- L;Pnf)dx
&I;“,_
- L’;:)dx
=D;+(z,h+.) ,
= - dj-(Lyn+ + L;;+)dx = - ;j-(L;Fn_
+ L;Pn_)dx
=D;+(z,h;)
.
(4.4)
One can also write down generalizations of these sum rules which do not depend on the BC = 0 approximation. For example,
s
](Ll;yn+ - L’;;*_) - CL’;“,+ - L;;_)]dx (Adler-Bjorken
s
= cos2 eC(D;‘-
sum rule)
[(L;p,+ t L;yn+) - (L;pn_ + L;fj7_)]dX = 6 (Gross-Llewellyn
D;+),
COS2 e,(D”,’
Smith sum rule)
-D;+)
.
(4.5)
hf. Gronau et al., Quark parton model
621
Obviously, the same relations hold after z-integration with the D-functions on the right-hand side replaced by the corresponding average pion multiplicities in neutrino reaction (current fragments, of course). The second sum rule can be tested directly in neutrino-deuteron reactions. 4.2. Relations between neutn’no and electro-production In the 19~= 0 approximation one has a generalized Llewellyn Smith [9] relation: (4.6) Another interesting (n,o$N
-
relation involving multiplicities
sts
bz,o jeN = ~
9 FfN(x)
s(D;’-
D;)dz
is
.
(4.7)
By eq. (4.2) a similar relation holds for (nn$ t (nn_) . From this relation we conclude that the current fragment pions average multiplicity in the neutrino reaction is larger or smaller than their multiplicity in e-N reactions depending on the sign of j(D;” ~ Dt)dz. This quantity is a measure of the extent to which it is “easier” or “more difficult” to produce pions from non-strange quarks then from strange quarks. Thus, measuring these linear combinations of pion multiplicities one will gain better insight into the fragmentation mechanism. Furthermore, assuming that this integral does not vanish, we obtain F?(x)
-=-= Fyyx)
F?(x)
(nno)vp- (nnJep
Ffy(x)
Tr4Il - %An
(4.8)
The quark model predicts i < Fzn/FsP< 4 [lo]. Experimentally this quantity is not larger than 1. Thus the ratio of these combinations of multiplicities should be bounded between a and 1. It is interesting to note that while current fragment multiplicities in neutrino reactions depend on q2 only, in the e-N reactions they depend on both q2 and x. The q2 dependence has, therefore to cancel in eq. (4.8). Our assumption that the integrated D-functions are linear in In &$ is consistent with this constraint. The result presented in eq. (4.8) does not depend on the quark charges. The following relations depend on this charge. The quark model inequalities JFfPdx > $, JFydx > f , satisfied by the total inclusive scaling functions have generalizations for the semi-inclusive functions which depend on the quark charges:
s
Lyrn+ dzdx = JCnw+), FFPdx > k(8(nll+)yp + (n
and a similar relation holds for n- production,
) ) , H- “P
Combining
the two we obtain
M. Gronau et al., Quark parton model
622
s“r”,+, sq*+ , +q+Lep
l,n-
)dzdx>
f(+r+$,
+(n
) ), 71- VP
and similarly,
Nzdx 2 :((nn+)“P t (r~_)~~) .
(4.9)
We would like to conclude this section with an amusing result, involving e-N quantities only. Using the definitions of the scaling functions one finds:
(4.10) Thus, unless 0:’ s
= Di’,
we obtain
‘LT”nt - pdx , =2
(4.11)
7 .
The right-hand side of the last equation is independent of z,h$. This result strongly depends on the quark charges. For example, if the partons were three trip let model quarks or Sakatons, this ratio would be 0~1. From the experimental point of view, the z-integrated version of eq. (4.11) may be more useful:
s
((n,,‘,, - (nn_)en) Fle”dx = 2.
7 ’
.i
(4.12)
Un,+Lp - (nn_jep)F,ePdx
Since FIti is already essentially known, one needs only the average multiplicities of current fragment pions in these processes at each value of x, in order to check this relation. Relations among pion multiplicities in eN and vN reactions should be, clearly, checked at the same q2 value in both reactions. 4.3. i%e x + 0 limit in the electroproduction Near x + 0 u(x), d(x), etc., are assumed to have a l/x dependence with coefticients c”, cd, etc. Borrowing ideas used in describing diffraction scattering in the parton model, one finds [5]
cu
=cE
‘Cd’c;iGC,
cs = cc .
(4.13)
M. Cronau ei al., Quark parron model
623
Equality of c, and c would imply that the Pomeranchuk singularity is an W(3) singlet, which is in disagreement with the existing difference between total K+p and nN cross sections. However, assuming c = cs, hopefully we do not commit a large error, since, for example, (4.14) Thus, the deviation of cJc from unity (which is probably of the order of - 20%) is damped by a factor of - 5~1. Let us, therefore, neglect it. This gives: (n,+)ep = (n,+)en= (n
_jep=(nn_)en = (nnojep=(rzJen = $nno)yp+ +‘;dz
71
That is, for small x the multiplicity of current pions in electro-production independent
. (4.15) becomes
of x, and also, of the target.
From eq. (4.15) we obtain (nJeN
2 ~(nmdyp ,
(4.16)
which depends on the quark charges. 4.4. Relations involving e+e- annihilation multiplicities The parton picture of e+e- annihilation into hadrons is shown in fig. 3. The e+epair annihilates into a photon which turns into a quark anti-quark pair. Each of these then fragments into hadrons. The total cross section for e+e- -+ anything is given by
ue+e_ =
‘$ ; CQ,', i
(4.17)
where the sum over i goes over all types of partons (quarks and anti-quarks). The semi-inclusive cross section for the production of a certain hadron, h, is
e+ Fig. 3. Hadron production
e-
in e+e- annihilation. The photon is converted pair, and these fragment into hadrone.
into a quark anti-quark
624
M. Gronau et al., Quark parton model
do dz (e’e-
-+ h + X) =s
c
QfJDf(z,h$)dh$
,
(4.18)
i
where z is the fraction of parton momentum (a@) From this one obtains the normalized distribution
G;+e_(z) =-!--
c
QfJD;(z,h;)dh;
carried by the detected hadron.
.
(4.19)
’
fCQf i
In the quark model G n+
=Gn-
e+e-
Integrating
=G#
e+e-
e+e-
=
;
s
[;(D;+ + D;+) + 3 D;] dh; > $D;+
+ D;+)dh; . (4.20)
over z one finds
(n,+)e+e_= (nn- )e+e_= (n77o)e+e- >
$unn+)vp +(y)“p) .
This last inequality strongly depends on the quark charges. Naturally, the comparison should be made at a point where the momentum by the partons is the same in e+e- and in vp. That is,
(6_q21yp =(Q2)e+e-
(4.2 1) carried
(4.22)
Using eqs. (4.14) and (4.18) we obtain (4.23) which holds for all charge states of the pion. This relation is quite general, and does not depend on the quark charges. It essentially depends on the universal behavior of wee partons. 4.5. K-meson inclusive cross sections The less restrictive character of eq. (3.5) does not allow us to write so many simple relations for K-inclusive cross sections. However, many of the results involving pions may be applied to K-mesons by treating combinations of KK-pairs, the members of which are tranformed into each other under the G-parity operation. Thus, for example, in the 8, = 0 approximation, L”p t L;t, = (Df++ D$)F;p(x) . (4.24) l,K+ , Therefore, (nK+jvN + (nI&N is independent both of x and of the target. The same rule holds for any detected current fragment which is a G-eigenstate, or a pair of hadrons which transform into one another under G-parity.
M. Gronau et al., Quark parton model
625
4.6. SU(3) relations
If one assumes that the fragmentation of a quark into hadrons is SU(3) invariant, one finds that of all the possible D-functions only three are independent. We choose, for example, D;+, Dit and 0: For the meson octet Dy satisfy Dr
= w(D;+ - 0;)
+ W(D;+ - 0;)
t D; .
(4.25)
Here w(G) is the weight of the quark (anti-quark) qq representation of the meson m. For example D”+ = DK+ = DK-
lI
s ’
u
D”
s
=
Dft = DKd
’
D;
of type i(i)
=
in the non-relativistic
:_D;++ $D;‘-
$;.
(4.26)
Using these equalities one can obtain many new relations, of which we give a few examples. In the BC = 0 approximation (n&“n
(4.27)
+ (nKcJvp > (nn+jvp .
That is, in the SU(3) limit K-mesons should be produced as current fragments in neutrino reactions in quantities comparable to those of pions. In efe--annihilation, SU(3) symmetry gives directly (4.28) =(n n-+)e+e- ’ because the photon is a U-spin singlet and the rr+ and K+ belong to the same &pin doublet. The multiplicities of K” and K” are equal, but in general different from the multiplicities of K’. We can in this limit obtain the following relations: (Q),+,_
(n
)
76 e+e-
=
@n*+)+ ;b_) + $zKJ •t +Ko) t @zK_)t ~(nKo))vN
= ((nn+) (2 (nnt)+ (nKO))e+e_
)
+ (n=_) + $+)
(4.29)
+ (nKJ f (?zK_) + (nKO))“N .(4.30)
4.7. Target fragmentation Unlike the current fragmentation region, scaling for the target fragmentation region has already been predicted in the light cone approch [3]. In the model we use, the dependence of the E-functions on x, on the target and on the type of quark removed by the currents, limits the possibility of obtaining relations among various quantities. The E-functions are still partially related by isospin, as shown in eq. (3.14). From such equalities one can still obtain some relations among experimental quantities. For example, the generalized Llewellyn Smith [9] relation obtains the form: RYP 3 r-R”3”,~=12(R~p-R~$. ,n ,n The r + 0 region is of special interest.
(4.31)
Slow (wee) “right’‘-moving hadrons can be fragments of either the target or the current, just as slow “left’‘-moving hadrons can
626
hf. Gronau et al., Quark parton model
be fragments of either. Therefore, various properties shared by left-moving hadrons should be also shared by slow right-moving hadrons. Thus, for example, in neutrino reactions, the normalized distribution of wee right-moving hadrons is expected to be independent of the target, and should not depend on x (while for finite r they depend on both).
5. Remarks and conclusions In this paper we have presented relations which can test some of the assumptions of the quark parton model, as applied to semi-inclusive deep inelastic reactions. We would like to conclude with a few remarks. (a) The assumption about the smooth transition from right-moving wee hadrons to left-moving wee hadrons (for both, u - (-q2)-i) followed from the impossibility of distinguishing between them within the framework of the parton model. It is consistent with the apparent absence of a sudden break at the central region of the experimentally measured rapidity distributions in hadronic or real photon reactions ]111. The theoretical limitation mentioned above is a source of an ambiguity in the experimental evaluation of the separate average multiplicities of current and target fragments. However, since the number of wee hadrons is assumed to be finite, the effect of this uncertainty is negligible when compared to the In G term for sufficiently high 42. (b) The target independence of normalized distributions and multiplicities in neutrino reactions for all x is a striking phenomenon. In electroproduction processes it is true o&y (approximately) for x -+ 0, namely, finite q2. There, the result coincides with the conclusion of an analysis using the Mueller technique or the multiperipheral model [ 121; the reason being that for fixed q2 the diffraction phenomenon is expected to dominate the yp reaction. Since experiment indicates [ 121 that the Pomeranchuk singularity seems to factorize, all the dependence on the target drops out in the normalized distribution and in the average multiplicity. (c) If one assumes that the conclusions reached in the x + 0 limit of the deep inelastic region also apply to high v and fixed q2 photons, then the predictions presented in subsect. 4.3 and 4.4 can be checked in real photoproduction processes. (d) We have presented several relations which, within the context of the model, can determine whether it is “easier” to produce a pion from a non-strange quark than it is from a strange quark or vice versa. In a simple model in which a single quark fragments into a chain of quark anti-quark pairs one would expect D;+> 0:. The probability of producing a n+ from a u-quark is proportional to the probability of finding at least one dd pair in the chain. But the production of a rr+ from an squark will depend on the probability of finding at least one dd pair and one uii pair. (e) The ISR data 1131 indicate that K-meson production in hadron-hadron colli-
M. Gronau et al., Quark parton model
621
sions is much lower than pion production. The fragmentation of the nucleon in such a collision is probably related to the E-functions we have mentioned. However, if we assume a smooth joining of the plateau of “right” and “left” moving hadrons, current fragment K-mesons should be less abundant then current fragment n-mesons. This may be due to a large breaking of SU(3) by the D-functions, or due to a sharp cut-off in the mass of the products (if, for example, the natural variable is not h$ but the transverse mass: h$ + M;) [ 141. Professor Feynman’s lectures at Caltech were the motivation for this work. We would like to thank him for these and for many enlightening discussions. Helpful comments by A. Cisneros are also acknowledged.
References [l] [2] [3]
[4] [5]
[6] [7] [S] [9] [lo] [ 1 l]
[ 121 [ 131 [ 141 [15]
C.H. Llewellyn Smith, Talk presented at the Int. Conf. on high energy collisions, Oxford, April 1972. J.D. Bjorken and K. Berkelman in Proc. of the 1971 Int. Symposium on electron and photon interactions at high energies, Cornell University, 1972. 3. Ellis, Phys. Letters 35B (1971) 537; J.D. Stack, Phys. Rev. Letters 28 (1972) 57; 0. Nachtmann, IAS Princeton preprint, 1972. S.D. Drell and T.M. Yan, Phys. Rev. Letters 24 (1970) 855; P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B33 (1971) 221. R.P. Feynman, Photon-hadron interactions, to be published (W.A. Benjamin, New York, 1972) and talk presented at the Neutrino ‘72 Conf., Balatonfiired, Hungary, June 1972; A. Cisneros, Caltech preprint CALT-68-349. C.G. Callan and D.J. Gross, Phys. Rev. Letters 22 (1969) 156. J.D. Bjorken, Phys. Rev. 163 (1967) 1767; S.L. Adler, Phys. Rev. 143 (1966) 1144. D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B14 (1969) 337. C.H. Llewellyn Smith, Nucl. Phys. B17 (1970) 277. 0. Nachtmann, Nucl. Phys. B38 (1972) 397. B. Wiik in Proc. of the 1971 Int. Symposium on electron and photon interactions at high energies (Cornell University, 1972); W.R. Frazer et al., Rev. Mod. Phys. 44 (1972) 284. W.R. Frazer et al. in ref. [ 1 l] and references therein. J.C. Sens, talk presented at the Int. Conf. on high energy collisions, Oxford, April 1972. R. Hagedorn, Nucl. Phys. B24 (1970) 93; R.C. Arnold and E.L. Berger, ANL/HEP 7131 (1972). H.J. Lipkin and M. Peshkin, Phys. Rev. Letters 28 (1972) 862.
Physics
B51 (1973)
61 l-627.
North-Holland
Publishing
Company
APPLICATIONS OF THE QUARK PARTON MODEL IN ONE-PARTICLE INCLUSIVE LEPTONIC-INDUCED REACTIONS
*
M. GRONAU **, F. RAVNDAL and Y. ZARMI *** California Institute of Technology,
Pasadena, California 91 IO9
Received 31 July 1972 (Revised 25 September 1972)
Abstract: The quark parton model is applied to the deep-inelastic region of single-particle inclusive lepton-hadron processes. The current-fragmentation region is mainly discussed. Various conclusions concerning experimentally observable quantities such as single-particle distribution functions and average multiplicities are reached. Sum rules such as the Bjorken and Adler sum rules are generalized to this case. The average multiplicities of current-fragment pions in neutrino reactions are found to be independent of the target and of X. They are related to electroproduction multiplicities. A relationship is established between the x + 0 limit of the latter, and pion multiplicities in e+e- annihilation. Assuming SU(3) invariance, relations among n and K multiplicities are derived.
1. Introduction
The light cone and the parton models have been both applied to the analysis of the interactions of the weak and electromagnetic current with hadrons in the deepinelastic region [ 11. They have been shown to be equivalent in this aspect. In the near future, inclusive cross sections for single hadron production in deepinelastic lepton-nucleon scattering will be measured. It is, therefore, worthwhile to apply the two models to this type of process [2]. It has already been noticed that in deep-inelastic semi-inclusive reactions the region of applicability of the parton model is wider than that of the light-cone description. While the latter has been only applied [3] to the kinematical region of “target fragmentation” (i.e. a finite momentum carried by the observed hadron in the lab frame), the first can be applied [4] to both this region and to the “current fragmentation” region (i.e. in the lab frame, the momentum of the observed hadron is a finite fraction of the momentum carried by the current).
* Work supported in part by the U.S. Atomic Energy Commission. Prepared under contract AT(l l-1)-68 for the San Francisco Operations Office, U.S. Atomic Energy Commission. ** Permanent address: Physics Department, Technion, Haifa, Israel. *** On leave of absence from the Weizmann Institute, Rehovot, Israel.
M. Gronau et al., Quark parton model
612
In this paper we study the consequences of the quark parton model ideas, as suggested by Feynman [5], in semi-inclusive reactions. Testing of these ideas is of great importance, since if verified, they enable one to obtain properties of individual partons from experiment. We shall mainly treat the current fragmentation region, in which the model has its strongest predictive power. In sect. 2 we present the kinematics involved, and in sect. 3 the assumptions of the model [ 51 are presented. In sect. 4 we study the consequences of the model. We obtain relations among experimentally measurable quantities such as inclusive distributions and multiplicities. While some of these relations depend on the quark quantum numbers, others can be obtained in models in which the partons are not quarks. We conclude in sect. 5 with a few remarks concerning the tests we have presented.
2. Kinematics The process of interest, !Zt N + Iz’t h t anything is shown in fig. 1. Here II and Q’ are leptons carrying momenta k and k’ respectively. In the lab frame, where the nucleon momentum is p, = (M, 0); k, = (E,k) and k; = (E’,k’). The outgoing observed hadron carries four-momentum h, and the leptonic current carries four-momentum 4. The deep-inelastic region is defined by q2 -+-cm,
y,p’q,, M
The target-fragmentation e = h - p finite ,
2
’
x = $
finite .
(2.1)
region is defined by
K=h*q+t=‘,
u = if_ finite . Mu
in a frame where the target is moving fast this corresponds d(k) --‘(k’)
Fig. 1. The process P + N + P’ + h + anything.
(2.2)
to a detected hadron
M. Gronau et al., Quark parton model
613
h moving in the direction of p, and its momentum being a finite fraction, U, of it (except for a possible bounded transverse momentum hT). The current fragmentation region is defined by /(=h.q_,-m,
e=h.p-+m,
u = M% and ,$
are finite .
(2.3)
In the lab frame this corresponds to a detected hadron moving in the direction of the current, its momentum being a fraction -u/x of q (except for a possible bounded transverse momentum hT). The differential cross section for the inelastic electroproduction with one detected hadron in the final state is given in the lab frame by 202 d3k’ ye& Qpv lklq4E’2E,
do=
c (X,h X
X
IJp(0)lp, (X,h I.T(O)lp)* (2n)3d4)(k+p-k’-h-Px)
,
(2.4)
This can be written as d4c
- 27W2 I1p~p.m.
dq2dvdedK
q4 $I2
‘”
(2.5) ’
where E -e.m. = _; W
ep-e)6(h
~d4he(h,,)6(h2-m;)6(h
IJv
X c X
=
(X,h IJ&O)Ip)(X,h
*q-K)
iJv(O)ip)* (2n)36’4’(k+p-k’-h-P,)
q,qv ~i~*(q2S’&~E)
-gpv +----
q2 > ’
v
i3,“;;f’(q2,
(2.6)
Y, K, E)
!F’ is the leptonic tensor. The current matrix elements are averaged over the initial nucleon spin states. In neutriiro production the electromagnetic current is replaced by the weak current, and 4na2/q4 by G2/2n. zPv then has the form (neglecting the lepton mass)
.e
I - - we
2M2
paq4i$
.
(2.7)
614
hf. Gronau et al., Quark parton model
From eqs. (2.5)-(2.7)
one obtains
4rr(u2 E' (eN + ehX) = ~ -(cos2 4 E dq2dvdedrc 4 d40
dq~~~dedK(v(F)N+ pT hX) = $$
;BSZ~+2sin2:0~~),
(cos2 fr0 @TijN t 2 sin2 f 0 G’;‘“,”
T
E+E’sin2 1ie P$)N) . M
(2.8)
where i3;.n = i3; h(q2,v,K,E). IJ has been shown [3,4] that in the target fragmentation region VP?% and v2 W;t are functions of x,u, E. On the other hand, in the current fragmentation re"eNh that scale, and depend only on X,U [4]. Similar congion it is v2 $TNh and v3 W, clusions can be' reached for’the v(v)-N structure functions, Fy,fjN, with $“jN behaving like %$z)N. We shall find i’t more convenient to discuss a different cross section d4a
(eN -+ ehX) = ~
(2.9)
dq2dvdudh+ where in the deep inelastic limit
The reason for this choice will become apparent when we discuss the model in sect. J.
The neutrino cross section is_similarly defined. This cross section will have the same form as in eq. (2.8) with Wj replaced by Wi.From the scaling properties of gi in both target and current fragit follows that, for fixed finite u and h,,2 Wi,h scale mentation regions according to 'Wih -Fih(x,',h~)' i= 2, 3.
MW, n + Qn(xM;),
(2.10)
These structure functions scale with the same powers of v as the corresponding total structure functions. For neutrino reactions, in the approximation where the Cabibbo angle is zero, one finds from isospin invariance of the strong interactions wyP= I
p I
’
wi”P=wi”.
The analogous relations in our case are
(2.11)
M. Gronau et al., Quark parton model
615
@P = wyn, pn (2.12) l,h' ’ z,h ’ l,h l,h where h’ is the hadron obtained from h by a 180” isospin rotation around IY. By integrating over the kinematical variables of the observed hadron one has
WYP =
d4a
_Idq2dvdudh;
(QN -+ Q’hX)dudh$
(I1N + Q'X)
= (ah) sd7
,
(2.13)
where (nn) is the final hadron average multiplicity.
3. The quark parton model We use the quark parton model in its standard version as presented in ref. [S]. It is assumed in this model that a fast-moving nucleon can be regarded as made up of constituents (partons) whose transverse momenta are bounded, and whose longitudinal momenta are finite fractions, x, of the total nucleon momentum, P. Only a finite number of partons have finite and small momenta (namely, x < l/P). These are the wee partons. The longitudinal momentum distribution for non-wee partons scales, i.e., it depends only on the fractions of momentum carried by the various types of partons and not on their absolute longitudinal momenta. The average number of partons of type i in the interval x, x + dx isfi(x)dx. For small x,4(x) g CJx. This translates into a plateau when the distribution is described in terms of the rapidity variable, y, of the partons. Assuming the partons are quarks, we shall denote their distribution functions by their names. Thus, a u-quark will have a distribution function u(x), a d-quark d(x), and similarly for the other types. The scaling functions, Pi(x), in the total inclusive cross sections are linear combinations of these functions. For example: FTp(x) = ; {?j(u(x)+ir(x))
+ ‘,(d(x)+d(x))
F,“p(x) = cos2eC(d(x)tG(x)) Fip(x)
= 2 cos2 BC(-d(x)+(x))
)
+ &(s(x)t$x))}
t sin2 ec(s(x)+G(x))
,
+ 2 sin2 e&s(x)+iY(x))
,
(3.1)
The neutron functions are obtained by substituting u ++d, U + d. The anti-neutrino functions are obtained by d + u, and ii + din the stangeness conserving part, and s + u, U + Yin the strangeness changing part. Spin f quarks imply F2(x) = 2xFI(x) L61. In order to study semi-inclusive reactions we have to say something about the fragmentation of partons into hadrons [4, 51. As suggested by Feynman [5], we do this in the current-parton Breit frame.
616
M. Gronau et al., Quark parton model
0 p=(P,o,o,P)
IF (xP,o,o,
q = (0, o,o, -2xP) XP)
(xP,o,o,-XP)
+--_ +--___ _ q--_ l _-_-_--_ *--
b ((l-x) -__,
P,
, o,(I-x)P) --+ --_-_+ -----* _-----•, ----_-_-_~ --A _----_)
T
--+
--,
Fig. 2. The photon-parton Breit frame description of deep inelastic interactions. (a) Before the current interacts with a quark. The nucleon momentum is pP = (P,O,O,P). The photon momentum is qP = (O,O,O, -2xP). x = -q’/2p . q is the momentum fraction carried by the parton. (b) After the interaction has occurred. The parton now moves to the left and fragments into leftmoving hadrons while the right-moving partons fragment into right-moving hadrons.
Fig. 2a describes the situation before the current, carrying a momentum 4 with a component in the -z direction only, interacts with one of the partons. The nucleon carries a momentum P in the +z direction. The parton hit by the current carries a longitudinal momentum XP (x = -q2/2q - p), and some small transverse momentum. Fig. 2b describes the process after the interaction has occurred. The interacting parton is now moving to the left with momentum XP (-z direction), and the rest of the partons move to the right with momentum (1 -x)P. Each of the two components (left and right-moving partons) now fragments into hadrons. It is assumed that there is no correlation between the fragmentation of the left-moving parton and that of the right-moving ones. For a finite 0
M. Gronou et al., Quark porton model
617
The fragmentation of the right-moving partons proceeds in a similar way. Only, in this case, the fragmentation function, denoted by E, depends both on the type of quark removed, and on itsx-value [5]. Thus, for example, when a u-quark is knocked outofaproton,En+ (pFU)(~,r,h%$)will denote the fragmentation function of the rightmoving component mto a rr+ with a momentum fraction r of the total (I-x)P. It is assumed that for small z(r) the D(E) functions behave like ~-f(r-~). One can easily show that left-moving hadrons are current fragments (as defined in sect. 2) with u=-zx.
(3.2)
In the lab frame z = h . p/q p is the fraction of current momentum carried by the observed hadron. Similarly, right-moving hadrons are target fragments with U = r(l-x) .
(3.3)
Applying isospin and charge conjugation
invariance
one finds
(3.4)
(3.5) We are now in a position d4a dq2dvdzdh;
=x
to calculate the differential
cross section
d4a (3.6)
dq2dvdudh$
for a current fragment. The corresponding
scaling functions,
denoted by Li,h satisfy
hf. Gronau et al., Quark parton model
618
Li,h(X,“,h~) =XF;:h(X,‘,h~)
(3.7)
)
where I;;,h are defined in eq. (2.10). The expression for these scaling functions are obtained by multiplying each term in eq. (3.1) for the total inclusive scaling functions by the appropriate D-functions. For example, Ly=+ (x,z,h;)
= ; { ‘?,uD;+ t $UD,“+t ‘,dD;+ t $jD$+t
~s~~+t $FLl;+}
,
which, by eq. (3.4) becomes {D;+(;ut#tD;+(+
LeP =; 1,?T+
.
t;d)t$D;(stS)}
Also L;pn+ = 2xLp
(3.8)
For the neutrino
case one finds
LyF*+ = cos2 Oc(d t U)Di+ t sin2 O,(sD”’u t iiD:> , Lifir+ = 2 cos2 Oc(-d t U)D:+ t 2 sin2 e,(-SD:+ Similar expressions can be obtained for a neutron various detected outgoing hadrons. In the BC = 0 approximation one has
Lspn+ = 2xLlp,+ ,
t iiD:> . target, anti-neutrinos
LyN = Dn’F!‘N Ul. z,n Therefore, in this approximation, d4a dq2dvdzdh$
(3.9) and for
(3.10)
(vN -+ I-(- nX) = ~ d2a (vN + p-X)D:, dq2dv
.
(3.11)
Notice that this relation as well as a similar relation for anti-neutrino reactions holds for all the three charge states of the pion. In the next section it will be convenient to use the normalized inclusive distribution functions: G&(x,z,
h;) =
d4a dq2dvdzdh+
(QN + Q’X) .
(3.12)
Notice that the explicit dependence on u and 0 is cancelled out in G due to the properties of the structure functions in this model, eqs. (3.8) and (3. IO). The same is true for the average multiplicity of left-moving hadrons. As a result, one obtains the following relation between the structure functions;
M, Gronau et al., Quark parton model
s
619
(3.13)
L’tN(x z h2)dzdh2 T = (n hQN ) FPN z z,h ’ ’ T
where is the average multiplicity of left-moving hadrons of type h. Scaling functions can also be written for right-moving hadrons with z replaced by r - the fraction of total right-moving momentum carried by the observed hadron, and the D-functions replaced by the E-functions. The latter also satisfy some equalities resulting from the application of isospin rotations, though these are not as restrictive as the relations satisfied by the D-functions. This is due to the dependence of the E-functions on the target. Thus we have, for example, (nh)QN
E&
= E ;L)
Egls)
’
= &,
.
(3.14)
In a manner analogous to eq. (3.6) we now obtain d40
=(l_x)
dq2dudrdh$ with corresponding
d4u dq2dududh;
9
(3.15)
scaling functions
Ri h(X,‘,h~) = (‘-X)‘i
h(X’“‘h.I.) )
(3.16)
where, in this region u = I( l-x).
4. Consequences and proposed tests of the model In this section, except for subsect. 4.7, we treat current fragments. For them, fragmentation functions depend only on z and ht. The equalities among various functions render simple forms for the scaling functions. As mentioned in sect. 3, it is assumed that for small z the D-functions behave l/z down to z - l/xP (xP is the momentum carried by the fragmenting parton). In the frame we are usingxP = f&$. Therefore, the integrated D-functions a logarithmic dependence on +2 s D:(z,ht)dz
- Rn G,
like have
(4.1)
where here, and from now on, z-integration implies h$ integration as well. A result of general character is the relation between no and rr+ production. eq. (3.4) one obtains LirrO = &Cia+ +J&),
the D-
i= 1,2,3,
Using
(4.2)
for all the processes considered. Hence, similar relations are satisfied by the cross sections, normalized distributions and average multiplicities for left-moving pions. Eq. (4.2) is a well-known result for the fragmentation of objects having isospin Gf [15].
hf. Gronau et al., Quark parton model
620
This result crucially depends on the isospin quantum let and not on their charges.
numbers of the quark trip-
4.1. Neu trino reactions In the BC = 0 approximation
G 79 --pi=
G;; = G;; = ;(G;;
"P
t
G;;)
(4.3)
Thus, though different, the fragmentation From this
the various inclusive and semi-inclusive cross sections are, in general, normalized distributions satisfy many equalities, and directly yield the functions of individual quarks. it follows that the normalized distributions of left-moving pions in neutrino reactions do not depend on x. Similar equalities will, of course, hold for the integrated functions, namely, the average multi licities of left-moving pions in these reactions. They will only be linear in In J4 -q2 (see eq. (4.1)). These equalities depend on the fact that the pion is a G-eigenstate and on the Gproperties of the quarks created by the strangeness concerving weak current, An important consequence obtained here is that the average multiplicity of leftmoving
pions
produced
in neutrino
and anti-neutn’no
reactions
does
not depend
on
the target
(in the BC= 0 approximation). The sum rules and inequalities obtained in deep inelastic neutrino-nucleon scattering can be generalized to our case. For example, the Bjorken and Adler [7] and the Gross-Llewellyn Smith [8] sum rules assume in the 8, = 0 approximation the form:
sCL;“,+ ,
- L;Pnf)dx
&I;“,_
- L’;:)dx
=D;+(z,h+.) ,
= - dj-(Lyn+ + L;;+)dx = - ;j-(L;Fn_
+ L;Pn_)dx
=D;+(z,h;)
.
(4.4)
One can also write down generalizations of these sum rules which do not depend on the BC = 0 approximation. For example,
s
](Ll;yn+ - L’;;*_) - CL’;“,+ - L;;_)]dx (Adler-Bjorken
s
= cos2 eC(D;‘-
sum rule)
[(L;p,+ t L;yn+) - (L;pn_ + L;fj7_)]dX = 6 (Gross-Llewellyn
D;+),
COS2 e,(D”,’
Smith sum rule)
-D;+)
.
(4.5)
hf. Gronau et al., Quark parton model
621
Obviously, the same relations hold after z-integration with the D-functions on the right-hand side replaced by the corresponding average pion multiplicities in neutrino reaction (current fragments, of course). The second sum rule can be tested directly in neutrino-deuteron reactions. 4.2. Relations between neutn’no and electro-production In the 19~= 0 approximation one has a generalized Llewellyn Smith [9] relation: (4.6) Another interesting (n,o$N
-
relation involving multiplicities
sts
bz,o jeN = ~
9 FfN(x)
s(D;’-
D;)dz
is
.
(4.7)
By eq. (4.2) a similar relation holds for (nn$ t (nn_) . From this relation we conclude that the current fragment pions average multiplicity in the neutrino reaction is larger or smaller than their multiplicity in e-N reactions depending on the sign of j(D;” ~ Dt)dz. This quantity is a measure of the extent to which it is “easier” or “more difficult” to produce pions from non-strange quarks then from strange quarks. Thus, measuring these linear combinations of pion multiplicities one will gain better insight into the fragmentation mechanism. Furthermore, assuming that this integral does not vanish, we obtain F?(x)
-=-= Fyyx)
F?(x)
(nno)vp- (nnJep
Ffy(x)
Tr4Il - %An
(4.8)
The quark model predicts i < Fzn/FsP< 4 [lo]. Experimentally this quantity is not larger than 1. Thus the ratio of these combinations of multiplicities should be bounded between a and 1. It is interesting to note that while current fragment multiplicities in neutrino reactions depend on q2 only, in the e-N reactions they depend on both q2 and x. The q2 dependence has, therefore to cancel in eq. (4.8). Our assumption that the integrated D-functions are linear in In &$ is consistent with this constraint. The result presented in eq. (4.8) does not depend on the quark charges. The following relations depend on this charge. The quark model inequalities JFfPdx > $, JFydx > f , satisfied by the total inclusive scaling functions have generalizations for the semi-inclusive functions which depend on the quark charges:
s
Lyrn+ dzdx = JCnw+), FFPdx > k(8(nll+)yp + (n
and a similar relation holds for n- production,
) ) , H- “P
Combining
the two we obtain
M. Gronau et al., Quark parton model
622
s“r”,+, sq*+ , +q+Lep
l,n-
)dzdx>
f(+r+$,
+(n
) ), 71- VP
and similarly,
Nzdx 2 :((nn+)“P t (r~_)~~) .
(4.9)
We would like to conclude this section with an amusing result, involving e-N quantities only. Using the definitions of the scaling functions one finds:
(4.10) Thus, unless 0:’ s
= Di’,
we obtain
‘LT”nt - pdx , =2
(4.11)
7 .
The right-hand side of the last equation is independent of z,h$. This result strongly depends on the quark charges. For example, if the partons were three trip let model quarks or Sakatons, this ratio would be 0~1. From the experimental point of view, the z-integrated version of eq. (4.11) may be more useful:
s
((n,,‘,, - (nn_)en) Fle”dx = 2.
7 ’
.i
(4.12)
Un,+Lp - (nn_jep)F,ePdx
Since FIti is already essentially known, one needs only the average multiplicities of current fragment pions in these processes at each value of x, in order to check this relation. Relations among pion multiplicities in eN and vN reactions should be, clearly, checked at the same q2 value in both reactions. 4.3. i%e x + 0 limit in the electroproduction Near x + 0 u(x), d(x), etc., are assumed to have a l/x dependence with coefticients c”, cd, etc. Borrowing ideas used in describing diffraction scattering in the parton model, one finds [5]
cu
=cE
‘Cd’c;iGC,
cs = cc .
(4.13)
M. Cronau ei al., Quark parron model
623
Equality of c, and c would imply that the Pomeranchuk singularity is an W(3) singlet, which is in disagreement with the existing difference between total K+p and nN cross sections. However, assuming c = cs, hopefully we do not commit a large error, since, for example, (4.14) Thus, the deviation of cJc from unity (which is probably of the order of - 20%) is damped by a factor of - 5~1. Let us, therefore, neglect it. This gives: (n,+)ep = (n,+)en= (n
_jep=(nn_)en = (nnojep=(rzJen = $nno)yp+ +‘;dz
71
That is, for small x the multiplicity of current pions in electro-production independent
. (4.15) becomes
of x, and also, of the target.
From eq. (4.15) we obtain (nJeN
2 ~(nmdyp ,
(4.16)
which depends on the quark charges. 4.4. Relations involving e+e- annihilation multiplicities The parton picture of e+e- annihilation into hadrons is shown in fig. 3. The e+epair annihilates into a photon which turns into a quark anti-quark pair. Each of these then fragments into hadrons. The total cross section for e+e- -+ anything is given by
ue+e_ =
‘$ ; CQ,', i
(4.17)
where the sum over i goes over all types of partons (quarks and anti-quarks). The semi-inclusive cross section for the production of a certain hadron, h, is
e+ Fig. 3. Hadron production
e-
in e+e- annihilation. The photon is converted pair, and these fragment into hadrone.
into a quark anti-quark
624
M. Gronau et al., Quark parton model
do dz (e’e-
-+ h + X) =s
c
QfJDf(z,h$)dh$
,
(4.18)
i
where z is the fraction of parton momentum (a@) From this one obtains the normalized distribution
G;+e_(z) =-!--
c
QfJD;(z,h;)dh;
carried by the detected hadron.
.
(4.19)
’
fCQf i
In the quark model G n+
=Gn-
e+e-
Integrating
=G#
e+e-
e+e-
=
;
s
[;(D;+ + D;+) + 3 D;] dh; > $D;+
+ D;+)dh; . (4.20)
over z one finds
(n,+)e+e_= (nn- )e+e_= (n77o)e+e- >
$unn+)vp +(y)“p) .
This last inequality strongly depends on the quark charges. Naturally, the comparison should be made at a point where the momentum by the partons is the same in e+e- and in vp. That is,
(6_q21yp =(Q2)e+e-
(4.2 1) carried
(4.22)
Using eqs. (4.14) and (4.18) we obtain (4.23) which holds for all charge states of the pion. This relation is quite general, and does not depend on the quark charges. It essentially depends on the universal behavior of wee partons. 4.5. K-meson inclusive cross sections The less restrictive character of eq. (3.5) does not allow us to write so many simple relations for K-inclusive cross sections. However, many of the results involving pions may be applied to K-mesons by treating combinations of KK-pairs, the members of which are tranformed into each other under the G-parity operation. Thus, for example, in the 8, = 0 approximation, L”p t L;t, = (Df++ D$)F;p(x) . (4.24) l,K+ , Therefore, (nK+jvN + (nI&N is independent both of x and of the target. The same rule holds for any detected current fragment which is a G-eigenstate, or a pair of hadrons which transform into one another under G-parity.
M. Gronau et al., Quark parton model
625
4.6. SU(3) relations
If one assumes that the fragmentation of a quark into hadrons is SU(3) invariant, one finds that of all the possible D-functions only three are independent. We choose, for example, D;+, Dit and 0: For the meson octet Dy satisfy Dr
= w(D;+ - 0;)
+ W(D;+ - 0;)
t D; .
(4.25)
Here w(G) is the weight of the quark (anti-quark) qq representation of the meson m. For example D”+ = DK+ = DK-
lI
s ’
u
D”
s
=
Dft = DKd
’
D;
of type i(i)
=
in the non-relativistic
:_D;++ $D;‘-
$;.
(4.26)
Using these equalities one can obtain many new relations, of which we give a few examples. In the BC = 0 approximation (n&“n
(4.27)
+ (nKcJvp > (nn+jvp .
That is, in the SU(3) limit K-mesons should be produced as current fragments in neutrino reactions in quantities comparable to those of pions. In efe--annihilation, SU(3) symmetry gives directly (4.28) =(n n-+)e+e- ’ because the photon is a U-spin singlet and the rr+ and K+ belong to the same &pin doublet. The multiplicities of K” and K” are equal, but in general different from the multiplicities of K’. We can in this limit obtain the following relations: (Q),+,_
(n
)
76 e+e-
=
@n*+)+ ;b_) + $zKJ •t +Ko) t @zK_)t ~(nKo))vN
= ((nn+) (2 (nnt)+ (nKO))e+e_
)
+ (n=_) + $+)
(4.29)
+ (nKJ f (?zK_) + (nKO))“N .(4.30)
4.7. Target fragmentation Unlike the current fragmentation region, scaling for the target fragmentation region has already been predicted in the light cone approch [3]. In the model we use, the dependence of the E-functions on x, on the target and on the type of quark removed by the currents, limits the possibility of obtaining relations among various quantities. The E-functions are still partially related by isospin, as shown in eq. (3.14). From such equalities one can still obtain some relations among experimental quantities. For example, the generalized Llewellyn Smith [9] relation obtains the form: RYP 3 r-R”3”,~=12(R~p-R~$. ,n ,n The r + 0 region is of special interest.
(4.31)
Slow (wee) “right’‘-moving hadrons can be fragments of either the target or the current, just as slow “left’‘-moving hadrons can
626
hf. Gronau et al., Quark parton model
be fragments of either. Therefore, various properties shared by left-moving hadrons should be also shared by slow right-moving hadrons. Thus, for example, in neutrino reactions, the normalized distribution of wee right-moving hadrons is expected to be independent of the target, and should not depend on x (while for finite r they depend on both).
5. Remarks and conclusions In this paper we have presented relations which can test some of the assumptions of the quark parton model, as applied to semi-inclusive deep inelastic reactions. We would like to conclude with a few remarks. (a) The assumption about the smooth transition from right-moving wee hadrons to left-moving wee hadrons (for both, u - (-q2)-i) followed from the impossibility of distinguishing between them within the framework of the parton model. It is consistent with the apparent absence of a sudden break at the central region of the experimentally measured rapidity distributions in hadronic or real photon reactions ]111. The theoretical limitation mentioned above is a source of an ambiguity in the experimental evaluation of the separate average multiplicities of current and target fragments. However, since the number of wee hadrons is assumed to be finite, the effect of this uncertainty is negligible when compared to the In G term for sufficiently high 42. (b) The target independence of normalized distributions and multiplicities in neutrino reactions for all x is a striking phenomenon. In electroproduction processes it is true o&y (approximately) for x -+ 0, namely, finite q2. There, the result coincides with the conclusion of an analysis using the Mueller technique or the multiperipheral model [ 121; the reason being that for fixed q2 the diffraction phenomenon is expected to dominate the yp reaction. Since experiment indicates [ 121 that the Pomeranchuk singularity seems to factorize, all the dependence on the target drops out in the normalized distribution and in the average multiplicity. (c) If one assumes that the conclusions reached in the x + 0 limit of the deep inelastic region also apply to high v and fixed q2 photons, then the predictions presented in subsect. 4.3 and 4.4 can be checked in real photoproduction processes. (d) We have presented several relations which, within the context of the model, can determine whether it is “easier” to produce a pion from a non-strange quark than it is from a strange quark or vice versa. In a simple model in which a single quark fragments into a chain of quark anti-quark pairs one would expect D;+> 0:. The probability of producing a n+ from a u-quark is proportional to the probability of finding at least one dd pair in the chain. But the production of a rr+ from an squark will depend on the probability of finding at least one dd pair and one uii pair. (e) The ISR data 1131 indicate that K-meson production in hadron-hadron colli-
M. Gronau et al., Quark parton model
621
sions is much lower than pion production. The fragmentation of the nucleon in such a collision is probably related to the E-functions we have mentioned. However, if we assume a smooth joining of the plateau of “right” and “left” moving hadrons, current fragment K-mesons should be less abundant then current fragment n-mesons. This may be due to a large breaking of SU(3) by the D-functions, or due to a sharp cut-off in the mass of the products (if, for example, the natural variable is not h$ but the transverse mass: h$ + M;) [ 141. Professor Feynman’s lectures at Caltech were the motivation for this work. We would like to thank him for these and for many enlightening discussions. Helpful comments by A. Cisneros are also acknowledged.
References [l] [2] [3]
[4] [5]
[6] [7] [S] [9] [lo] [ 1 l]
[ 121 [ 131 [ 141 [15]
C.H. Llewellyn Smith, Talk presented at the Int. Conf. on high energy collisions, Oxford, April 1972. J.D. Bjorken and K. Berkelman in Proc. of the 1971 Int. Symposium on electron and photon interactions at high energies, Cornell University, 1972. 3. Ellis, Phys. Letters 35B (1971) 537; J.D. Stack, Phys. Rev. Letters 28 (1972) 57; 0. Nachtmann, IAS Princeton preprint, 1972. S.D. Drell and T.M. Yan, Phys. Rev. Letters 24 (1970) 855; P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B33 (1971) 221. R.P. Feynman, Photon-hadron interactions, to be published (W.A. Benjamin, New York, 1972) and talk presented at the Neutrino ‘72 Conf., Balatonfiired, Hungary, June 1972; A. Cisneros, Caltech preprint CALT-68-349. C.G. Callan and D.J. Gross, Phys. Rev. Letters 22 (1969) 156. J.D. Bjorken, Phys. Rev. 163 (1967) 1767; S.L. Adler, Phys. Rev. 143 (1966) 1144. D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B14 (1969) 337. C.H. Llewellyn Smith, Nucl. Phys. B17 (1970) 277. 0. Nachtmann, Nucl. Phys. B38 (1972) 397. B. Wiik in Proc. of the 1971 Int. Symposium on electron and photon interactions at high energies (Cornell University, 1972); W.R. Frazer et al., Rev. Mod. Phys. 44 (1972) 284. W.R. Frazer et al. in ref. [ 1 l] and references therein. J.C. Sens, talk presented at the Int. Conf. on high energy collisions, Oxford, April 1972. R. Hagedorn, Nucl. Phys. B24 (1970) 93; R.C. Arnold and E.L. Berger, ANL/HEP 7131 (1972). H.J. Lipkin and M. Peshkin, Phys. Rev. Letters 28 (1972) 862.