- Email: [email protected]
QCD argument on color flow model
14 November 1996
PHYSICS
ELSEVIER
LETTERS 6
Physics Letters B 388 (I 996) 346-352
QCD argument on color flow model Qun Wang a, Qu-bing Xie b*l, Zong-guo Si a a Department ofPhysics, Shandong University Jinan, Shandong 250100, PR China b Center of Theoretical Physics, CCAST(World Lab) Beijing 100080, PR China and Department of Physics, Shandong University, Jinan, Shandong 250100, PR China
Received 12 June 1996; revised manuscript received 8 August 1996 Editor: H. Georgi
Abstract
The true meaning of the color flow model (CFM) is elucidated in a strict PQCD formulation, where the model is shown to be only an approximation in describing the exact color structure of the multigluon system. PACS: 13.87.Fh; 12.38.B~; 12.40.-y; 13.65.+i
In most hadronic event generators, e.g. IETSET and HERWIG, the usual theoretical treatment of the hadronic process in various high energy collisions is divided into two distinct phases: the perturbative phase and the non-perturbative hadronization one. The perturbative phase is well described by perturbative QCD (PQCD) while the hadronization one cannot be described from first principle and can only be described by the phenomenological models. In order to deal with the transition from color-carrying partons in the first phase to color-free hadrons in the second phase, one need know the exact color singlet structure of the parton system. However, since the exact matrix element (ME) method of PQCD encounters great difficulties as the number of partons increases as the result of the vast number of Feynman diagrams involved and the complexity of the loop structures, the strict ME of PQCD is usually replaced by an approximate PQCD model at tree level, i.e. the parton shower model (PSM), which is based on the GLAP equation and
’ E-mail address: [email protected].
where the parton evolution is treated as a semiclassical Markov process. Thus the color indices of partons involved in each intermediate branching step are summed, averaged and lost as the need for modeling the probability rather than the amplitude. So it is impossible to retain the color structure of the final parton system. As a phenomenological alternative, the color flow model (CFM) is used to assign the color structure of the final parton system [ 11. In CFM, a gluon is considered as a bicolor object (a color nonet, in fact). For example, in e+e- -+ q7j + ng, the color flow begins at the quark, connects each gluon one by one in a certain order, and ends at the antiquark. In the two ends of each piece of the flow connecting two partons are one color charge and its anticharge so that the piece is color-neutral and can be treated in hadronization as the same way as q7j singlet system. The two commonly used hadronization schemes, the string model and the cluster model, are all based on this neutral color flow picture. Presumably this picture accounts for some of the success of these schemes in fitting the data. The study of large rapidity gap (LRG) physics in
0370-2693/%/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PI1 SO370-2693(96)0 1141-O
341
Q. Wang et al. / Physics Letters B 388 ( 19%) 346-352
e+e- annihilation [ 2-41 is stimulated by the discovery of large amount of LRG events in deep inelastic scattering (DIS) at HERA [ 5-71. One of the leading order PQCD processes which is responsible for LRG events is e+e- --, qTjglg2 where the two color singlets qij and glg2 are separated by a rapidity gap. Note that in these LRG events, there is no color connection between the singlet qg and glg2. This color configuration is not covered by the conventional CFM where all four partons should be connected by the neutral color flow: q-g1 -g2 -?j or q-g2 -gl --q. Although the percentage of the LRG events is small according to PQCD predictions, considering the very limited phase space allowed for LRG, the percentage of this kind of color separated (CS) configuration in the total phase space is not necessarily small. In 1982, Gustafson mentioned this CS configuration in a study of the color field in the process e+e- -, qTjglg2 and stated that this CS configuration should not cause problems to the CFM and that it cannot be subtracted from the total amplitude in an orthogonal way [8]. In the current letter, taking the process e+e- --$ qTj + ng as an example, we try to study this topic in a more strict and general PQCD formulation and reveal the underlying meaning of CFM in the context of PQCD. In Ref. [9 1, from PQCD, we have constructed a strict formulation to calculate the cross section of color singlets of a multiparton system at the tree level. For the process eie- * qZj + ng, the essential part of the formulation is to exploit the color effective Hamiltonian Z& to compute the amplitude (f] H, 10) of a certain color state If). The color effective Hamiltonian H, is found from the invariant amplitude IV;~~““~:
where i. j are color indices of the quark and antiquark; al, . , . , a, are those of II gluons with a, = 1,..., 8 (for u = 1,. . .,n); Tay = AaU/2 and Aa= is the Gell-Mann matrix for SU(3); the summation is over all permutations of ( 1,2, . . . n) ; = D(q,%gp(l)9gp(2)r..
H, = C(T~[T~~.
. . Ta”);DP@~jt@A~t
. . . A‘$
=~(l/~)‘Tr(Q’cfcl..-CI)DP P
(2)
where the repetition of two subscripts represents summing (we use this convention unless explicitly specified); *j = (R+, Y+, B+) is the color creation operator for quark; qj+ = (x+ ,$, Bt ) is the anticolor creation operator for antiquark; (Q+)i = ?j+qi is the nonet tensor operator; Gi of the gluon u is defined by GL = (l/&V”A~+ A”++ As+/fi =AIt + iAzt h [ A4t + iASt = W:“l’:
A’+_ iA2+
A4+- iA5+
-A3t + A8t/Jj
A”t _ iA7t
A@ + iA7+
-2A8t/d?
- ‘P’i,‘4$E/3
Iu
(3)
which is the gluon’s color octet operator; Ya and qtj are(fjt,?,Bt).and(R+,Yt,B+),respectively;Eis the unit matrix of 3 x 3; A?+ is the a,-th gluon color operator and is defined by
p=P( I ,2,...,n)
DP
P(l,Z. . . , n) has been used. The form of the invariant amplitude can be derived from the SU( 3) color structure of QCD. An early symmetry analysis for the color factor was given by Cvitanovic [ lo]. A modem convenient treatment was developed by Berends and Giele [ 111. They proposed a recursive method from which Bq. ( 1) can be readily obtained after including all Feynman diagrams at the tree level. We will confirm from the unitary properties of H, later that Bq. ( 1) is the exact result at the tree level where no approximation has been made. Then H, is built from Eq. (1) as follows:
, ,gpcn))
is the mo-
mentum function of partons where momentum indices are suppressed, and p denotes a certain permutation of (1,2,. . .,n); (Ta1Ta2...Tan)G is the ith row and the jth column element of the matrix ( T”PCI) T~PCZI . . . T~Pc~) ) where the notation p =
A’+ u = (YfE+ + R+y+),,/ti A;+ =
t/ii)
Azt = (B+y+ + Y+~+),/& Azt = (B+F+ - Y+E+),/(
hi)
Q. Wang et al./ PhysicsLettersB 388 (1996) 346-352
348
A;+ = (R@+ + Yty+ - 2B+p+) u/di
(4)
The color effective Hamiltonian is another expression of the S matrix, so it is not necessarily Hermitian. The validity of H, can be verified by the calculation of the matrix element for the process e+e- + q?j + ng. For the color initial state (0) and the final state [f) = Iqpqj’AyiA$ . . . A$), we sum over the color indices i’ and j’ of the quark and the antiquark and those ai, ai, . . . , a; of n gluons, we obtain
c Kfl HcWI2=(01H,HcIO) f
=
~y2-a”
. (,;a2-“)* (5)
We can see that the calculation of ordinary matrix elements via H, returns to the original form. A color state of the multiparton system qq + ng is composed of the color charges of q, Tjand n gluons. It belongs to the color space: &$3+8&
@3&8,@*..@&
(61
There are many ways of reducing this color space. Corresponding to each reduction way is one set of orthogonal singlet spaces whose bases make up a completeness orthogonal set of color singlets. If a color singlet set is denoted by {I fk), k = 1,2, . . .}, then we have Ifk) (fkl = 1,
(fk 1fk’) = akk’
(7)
and I(fki Hc io)i2 =
c
(01H! Ifk)
(fkl Hc 10)
k
This property implies that the sum of the cross sections over all color singlets in a completeness set of the system (q7j + ng) is equal to the total tree level cross section sue( e+e- --f qq + ng): c
gk = (+tr.%(e+e-
--) qij + ng) ?
u0.
k
This is the result of unitarity. This shows that the invariant amplitude from which the color effective Hamiltonian is derived includes all Feynman diagrams at the tree level.
We have recapitulated the formulation we constructed in Ref. [ 91. Usually the color flow is approximated by a color singlet string piece where the fragmentation model may be applied. This approximation is based on the assumption that a gluon is a color nonet (a bicolor object). In that paper, we showed that the accuracy of this approximation declines as the number of gluons increases. In this letter, we reexamine the true meaning of CPlvI in the strict PQCD context starting from treating a gluon as a color octet exactly. As an example, we look at two cases, n = 1 and 2, of the process e+e- + q?j + ng. For the process with only one gluon in the final state , i ** e e+e- + 4481, the color space is Jp$%3&
(9)
There is only one way of reducing the space which corresponds to the unique singlet of the system qqg. This singlet is I Tr(QGt )), where (Q); is the color nonet tensor of the quark pair and can be written as the irreducible octet tensor (Q’)$ plus its trace S, 3 Tr( Q) : Q = Q’ + iS@.
Then the singlet becomes
IWQG 1) = ITr[
1) (10)
where the last equality is due to the fact that the gluon octet tensor is traceless, i.e. Tr( G1) = 0. The Eq. (10) indicates that the color charges of the quark pair are always intertwined with those of the gluon to build the system’s sole singlet. We can verify this by expanding ( Tr( Q'Gl)) in terms of the color charge: Iqyi) = (IR) , IY) , IB)) and the anticolor one: I’Pi) = ( IX) , IF), IB)) of q, 4 and g, we will find that the net charge of any color or anticolor is zero in each term, i.e. each term is color neutral (never to be confused with the color singlet! It is the combination of all these neutral terms that makes up the singlet.). This is just what we see in CFM. So for e+e- -+ qZjgl, CFM is 100% precise. Note that there is a probability of $ for each neutral piece (which consists of the color charge of q and its anticharge of g (q-g) , or the color charge of g and its anticharge of if (4 - g) 1 to be in a singlet string state. However , for e+e- + qtjglgz, the situation is not very obvious. In this case, the color space is
Q. Wang et al./ Physics Letters B 388 (1996) 346-352
349
The corresponding completeness set of color singlets is
l(f3 ) H, IO)j* = 3( 1D’12’12+ 1D(21)12 _ @12)~(21)* _ ~U2)*@))
{If&k=
We sum over this three amplitude squares and obtain
1,2,3}=
&
Is, Tr(GiG2)) 9
_ 2. (‘2)Dw)*
(12) where {Gi, G2) and [ Gr,G2] are the traceless symmetric and antisymmetric octet built from Gr and G2; They are defined by
3D
(19)
_ 2D”2’*D’21’ 3
which is equal to
(21) ijal a2
{G,,Gz}) =G~,G$j+G~,G:j-$~Tr(GrG2)
(13)
[G,,G*]f
(141
= Gf,G~j - G~kG:j
We can see from this completeness set that every two different singlets are orthogonal to each other. The singlets lf2) and Ifs) are CFM compatible while the singlet If,) is not covered by CFM. In If,), the quark pair forms a subsinglet which is color independent of the other subsinglet formed by gi and g2 in the same way as a glueball. This two subsinglet can also be separated in phase space, so ]fr) is part of the cause of LRG events in e+e- annihilation. Making use of the effective Hamiltonian Hc: H, = 1 Tr(QtG~G~)Do2’
+ i Tr(QtG~G~)Dc2’) (15)
we can compute the cross section of each singlet and estimate the percentage of the CFM-violating effect. The respective amplitude are
(f2IH, IO)= &D(12) + Dc2')) (16)
Then from the above equations, we have
= f(1@‘2’12 + [p(21)12 IV1 I HcI ON2 + @12)#21)* + @12)*#219 10-2 +
I Hc I OH2 = @wp)*
;&pj2 +
~w)*p9
+
M;“’ =
(77-a2)ijD(12)
+
(7-“2T=‘)ij$21’
qqglg2,
(22)
We see that the amplitudes of color singlets in a certain completeness set really satisfy the unitarity. The total cross section ou is the integral over the whole phase space at certain ycut value which is introduced to eliminate the infrared singularity: a0
= nde+e=
---t 4%x2)
IM(q4,gm)12d~ s .YNl
(23)
where fi denotes the phase space and corresponding kinematic factors. Given the total cross section, the percentages of the three color singlets are given by pi= L ao
J
[(fi I Hc 10)j2dR,
i= 1,273
(24)
Ycul
(f, 1H, IO)= &D'12' + Dc2'))
(f31H, IO)= x~(D('~) - Dc2'))
where IV;“~ is the matrix element of e+e- -+ and is written as
(17)
~p’21’~2
(18)
The result for the CS singlet jfr) is the same as that given in Refs. [3,4], but we provide here a rather more systematic approach from which the cross sections of all color states can be easily obtained. The results for other two singlets which are not calculated in Refs. [3,4] are also given. The results at various ycUtvalues are illustrated in Fig. 1. We can see that the percentage of If,), which is inconsistent with CFM, is about 6-8% of the total cross section. The CFM compatible color singlets have dominant percentage (92%). Recalling that in the case of efe- -+ qqgl , the only singlet of qqg1 is CFM compatible, hence, up
350
Q. Wang er ul. / Physics Letrers B 388 (I 996) 346-352
NCFM compatible singlet
( If,> )
. .
NCFM compatible singlet ( If,>
NS .
) .
10 CS singlet
.
. NC,
.
l
.
( If,>)
’ .
t0.002
0.000
0004
0.006
0.008
0.010
1
2
3
4
Ycut
5
6
7
8
9
10
11
of octets k
No.
Fig. I. The percentages of the CS singlet ( If,) ) and two CFM-compatible singlets (If2) and Ifs)) in the qqglg2 system at different .vcutvalues. This is the exact result at the tree level based on our formulation and is independent of collision energy.
Fig. 2. The number of total singlets and CS-type singlets Ns(k) and Ncs( k) from the reduction of k octets’ direct product.
to (~a at tree level, CFM is a good approximation to the true color structure of the parton system. We now turn to the process e+e- -+ q?j + ng where n > 2. Due to the great number of Feynman diagrams when n is large, it is almost impossible to directly compute the cross sections and then the percentages of the color separated (CS) singlets which violate CFM. Here we provide a heuristic method to estimate the percentages. The total singlets of multiparton system qq+ng is the sum of two groups of singlets: one group with the quark and antiquark forming the independent subsinglet l+ and the other group with their forming the suboctet S+. Obviously, the singlets in the former group are all CS; The latter one can be further divided into two subgroups: one is CS, the other is CFM compatible. Their cross sections are denoted by gcs (&), ucs (SF) and UC+I, respectively. Then we have
where Ncs (&) , Ncs (Q) and NCFMare the numbers of singlets in each of three groups respectively. Indeed
UCFM c( &FM
(26)
Ncs(&)
= Ns(&,,@&
Ncs(B,&
+ NCFM = Ns(!$$G,
= Ncs(n+
1) +NcFM(~+
+ msGq$
+ ~CF?vl
(25)
As n increases, the number of singlets in each of the three groups grows rapidly, so the contribution of a certain singlet to the total cross section is small. We assume that the contribution of a single singlet to the total cross section fluctuates around the average value, then we have QS(&)
0: Ncs(L&
~CS(Qj)
0: Ncs(8q7j)
@S,) = Ns(n) @...@&_I
1) = Ns(n+
BE,,) 1)
(27)
where Ns (n) is the number of singlets from the reduction of n octets’ direct product; Ncs (n + 1) and Ncm(n + 1) are the number of CS-type and CFM compatible singlets from (n + 1) octets’ product, respectively. When n is large, we have
pcs(Lj$= -(&j)bo
-
h(n) Ns(n
Pcs@qTj) = ~CS@qTj)l~O -
+
l)
+
Ncs(n +
Ns(n
ha4 = (+cFh4/uo = ms(&~>
@...@8,_,
+
l)
+
Ncm(n + 1) Ns(n + 1) + Ns(n)
Ns(n)
1) Ns(n)
(28)
where Ns (n + 1) + Ns( n) is the total number singlets. Obviously, Eq. (25) means
&uqq)
+ PCS@@> + hhl = 1
of
(29)
We use the Young Tableau method to calculate Ns ( k), NCS(k) and Nc~ (k) for k octets’ product. Fig. 2 shows Ns (k), NCS( k). One can see that they increase exponentially with growing k. Fig. 3 shows PCS(I+) and PCFM at the gluon number n = 2 to 9. We can see that Pcm is nearly saturated to about 70% as n
Q. Wang et ol./Physics
& d c m 2 (D =
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
P CFM .
1
.
.
l
l
l
,
I
I
.
.
l
=
0 PCJ’,?
I
0.1
.
2
3
4
5
the number
1 .
. t
6
B 7
.
1
8
.
I
9
10
of gluons
Fig. 3. The percentage of CFM compatible singlets &FM and that
of CS-type singlets with qFjforming a subsinglet Pcs(l+).
increases and PCS(l+) decreases slowly from 20% of n=3to 15.2%ofn=9. We have constructed a systematic approach to calculating the cross section of various color configurations for a multigluon system &j + ng. We can add more qq pairs to the system, write down its H,, and calculate the cross sections of different color singlets following the same procedure as in the q?j+ng case. Some points should be noted concerning multi-quark production. Compared with the probability of e+e- + (m 1) qq+ (n - 2) g at the same order, generally the probability of e+e- + mqTj+ng where there is one more qZj pair is suppressed by a factor of about 1/ 10 [ 121. Special attention should be paid to the lowest order processes responsible for LRG events: e+e- -+ qlif, q$& and e+e- + qifgg. In the phase space which allows for large rapidity gaps, the differential cross section of qlTj,q2ij2 outweighs that of qijgg. In other words, the process e+e- ---fqlZf,q2Tjzcontributes more to the LRG events than efe- -+ q?jgg does. The reason for this “anomaly” lies in that there is an unique way of forming two CS singlets for each case: ( q17j2) ( q2ql ) and (qif) (gg), where the round brackets denote the CS singlets. In the latter case, the singlet qQ must be separated by rapidity gaps from the other one (gg) . So both of the two gluons should be hard enough for a rapidity gap, which makes the differential cross section free of the infrared enhancement existing in the whole phase space. However, the total cross section of ( q1Fj2)( q2Fj1> in the whole phase space is much lower than that of (qTj) (gg), which is still within the normal expectation [4]. For higher order processes, we
Letters %388 (1996) 346-352
351
expect that no clear signals of LRG events exist because there are more overlaps among the phase spaces of different singlets. Here we do not intend to study LRG events, a limited phase space phenomenon, but are aiming at the study of the color singlet structure in the whole phase space. The main goal of this letter is to use e+e- -+ qij + ng, the simplest and most important process of hadronic events, as an ideal example to elucidate the true meaning of widely used CPM in a strict PQCD formulation. So for this goal, the process of multi-quark production is less important than the multigluon one. Now we will have a discussion and the concluding remarks. CPM is used in two widespread QCD event generators as an alternative phenomenological method to determine the color structure of multiparton system. In each branching step during the parton showering process, the corresponding color flow is recorded. When the process terminates, the final neutral color flow is obtained. The flow begins at the quark and ends at the antiquark connecting all partons one by one in a certain order. Such a procedure of recording the neutral color flow excludes the emerging possibility of a CS configuration which is part of the resources of LRG events. We point out in this letter that this discrepancy is caused by that the CFM is a semiclassical method in describing the color structure which is of quantum nature. We provide here a strict formulation derived from tree level PQCD to analyze the color structure of a multiparton system. In this formulation, we point out that each piece of neutral color flow cannot be equivalent to a color singlet though there is a certain probability for it to be in the singlet state. When the multiparton system is composed of color separated singlet subsystems, the neutral color flow is only confined within each subsystem. We give the completeness sets of color singlets for q?jg and q4gg and the corresponding cross sections of each singlet. We show that up to LYEat tree level, CFM is a good approximation to the exact color structure of the multiparton system. The percentages of CS-type and CFM compatible singlets is estimated by the color symmetry analysis when the gluon number is large. We find that as the gluon number grows, the percentage of CS-type singlets decreases and that of CFM compatible ones increases very slowly. In our study range of the gluon number (n = 2 to 9>, the percentage of CFM-type singlets is around 70%, which means the conventional
352
Q. Wang et al. / Physics Letters B 388 (1996) 346-352
CFM can partly reflect the system’s color structure.
References [ 11 T. Sjostrand, Int. J. Mod. Phys. A 3 ( 1988) 751; in: Z Physics at LEP 1, Proceedings of the Workshop, Geneva, Switzerland, 1989, edited by G. Altamlli, R. Kleiss and C. Verzegnassi (CERN Report No. 89-09, Geneva, 1989) Vol. 3. [2] J. Randa, Phys. Rev. D 21 (1980) 1795. 131 J.D. Bjorken, S.J. Brodsky and H.J. Lu, Phys. Lett. B 286 (1992) 153.
[4] J. Ellis and D.A. Ross, 2. Phys. C 70 (1996) 115.
I51 ZEUS Collab., M. Derrick et al., Phys. Lett. B 315 (1993) 481; B 332 (1994) 228; B 346 (1995) 399. [6] Hl Collab., T. Ahmed et al., Nucl. Phys. B 429 ( 1994) 477. [7] HI Collab., T. Ahmed et al., Phys. Lett. B 348 (1995) 681. [8] G. Gustafson, Z. Phys. C 15 (1982) 155. [9] Qun Wang and Qu-bing Xie, Phys. Rev. D 52 (1995) 1469. [IO] Predrag Cvitanovic, Phys. Rev. D 14 (1976) 1536. [ 111 EA. Berends and W.T. Giele, Nucl. Phys. B 306 ( 1988) 759; EA. Berends, W.T. Giele and H. Kuijf, Nucl. Phys. B 321 ( 1989) 39. 1121 G. Gustafson, Phys. Lett. B 175 ( 1986) 453; Nucl. Phys. B 392 (1993) 251.
PHYSICS
ELSEVIER
LETTERS 6
Physics Letters B 388 (I 996) 346-352
QCD argument on color flow model Qun Wang a, Qu-bing Xie b*l, Zong-guo Si a a Department ofPhysics, Shandong University Jinan, Shandong 250100, PR China b Center of Theoretical Physics, CCAST(World Lab) Beijing 100080, PR China and Department of Physics, Shandong University, Jinan, Shandong 250100, PR China
Received 12 June 1996; revised manuscript received 8 August 1996 Editor: H. Georgi
Abstract
The true meaning of the color flow model (CFM) is elucidated in a strict PQCD formulation, where the model is shown to be only an approximation in describing the exact color structure of the multigluon system. PACS: 13.87.Fh; 12.38.B~; 12.40.-y; 13.65.+i
In most hadronic event generators, e.g. IETSET and HERWIG, the usual theoretical treatment of the hadronic process in various high energy collisions is divided into two distinct phases: the perturbative phase and the non-perturbative hadronization one. The perturbative phase is well described by perturbative QCD (PQCD) while the hadronization one cannot be described from first principle and can only be described by the phenomenological models. In order to deal with the transition from color-carrying partons in the first phase to color-free hadrons in the second phase, one need know the exact color singlet structure of the parton system. However, since the exact matrix element (ME) method of PQCD encounters great difficulties as the number of partons increases as the result of the vast number of Feynman diagrams involved and the complexity of the loop structures, the strict ME of PQCD is usually replaced by an approximate PQCD model at tree level, i.e. the parton shower model (PSM), which is based on the GLAP equation and
’ E-mail address: [email protected].
where the parton evolution is treated as a semiclassical Markov process. Thus the color indices of partons involved in each intermediate branching step are summed, averaged and lost as the need for modeling the probability rather than the amplitude. So it is impossible to retain the color structure of the final parton system. As a phenomenological alternative, the color flow model (CFM) is used to assign the color structure of the final parton system [ 11. In CFM, a gluon is considered as a bicolor object (a color nonet, in fact). For example, in e+e- -+ q7j + ng, the color flow begins at the quark, connects each gluon one by one in a certain order, and ends at the antiquark. In the two ends of each piece of the flow connecting two partons are one color charge and its anticharge so that the piece is color-neutral and can be treated in hadronization as the same way as q7j singlet system. The two commonly used hadronization schemes, the string model and the cluster model, are all based on this neutral color flow picture. Presumably this picture accounts for some of the success of these schemes in fitting the data. The study of large rapidity gap (LRG) physics in
0370-2693/%/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PI1 SO370-2693(96)0 1141-O
341
Q. Wang et al. / Physics Letters B 388 ( 19%) 346-352
e+e- annihilation [ 2-41 is stimulated by the discovery of large amount of LRG events in deep inelastic scattering (DIS) at HERA [ 5-71. One of the leading order PQCD processes which is responsible for LRG events is e+e- --, qTjglg2 where the two color singlets qij and glg2 are separated by a rapidity gap. Note that in these LRG events, there is no color connection between the singlet qg and glg2. This color configuration is not covered by the conventional CFM where all four partons should be connected by the neutral color flow: q-g1 -g2 -?j or q-g2 -gl --q. Although the percentage of the LRG events is small according to PQCD predictions, considering the very limited phase space allowed for LRG, the percentage of this kind of color separated (CS) configuration in the total phase space is not necessarily small. In 1982, Gustafson mentioned this CS configuration in a study of the color field in the process e+e- -, qTjglg2 and stated that this CS configuration should not cause problems to the CFM and that it cannot be subtracted from the total amplitude in an orthogonal way [8]. In the current letter, taking the process e+e- --$ qTj + ng as an example, we try to study this topic in a more strict and general PQCD formulation and reveal the underlying meaning of CFM in the context of PQCD. In Ref. [9 1, from PQCD, we have constructed a strict formulation to calculate the cross section of color singlets of a multiparton system at the tree level. For the process eie- * qZj + ng, the essential part of the formulation is to exploit the color effective Hamiltonian Z& to compute the amplitude (f] H, 10) of a certain color state If). The color effective Hamiltonian H, is found from the invariant amplitude IV;~~““~:
where i. j are color indices of the quark and antiquark; al, . , . , a, are those of II gluons with a, = 1,..., 8 (for u = 1,. . .,n); Tay = AaU/2 and Aa= is the Gell-Mann matrix for SU(3); the summation is over all permutations of ( 1,2, . . . n) ; = D(q,%gp(l)9gp(2)r..
H, = C(T~[T~~.
. . Ta”);DP@~jt@A~t
. . . A‘$
=~(l/~)‘Tr(Q’cfcl..-CI)DP P
(2)
where the repetition of two subscripts represents summing (we use this convention unless explicitly specified); *j = (R+, Y+, B+) is the color creation operator for quark; qj+ = (x+ ,$, Bt ) is the anticolor creation operator for antiquark; (Q+)i = ?j+qi is the nonet tensor operator; Gi of the gluon u is defined by GL = (l/&V”A~+ A”++ As+/fi =AIt + iAzt h [ A4t + iASt = W:“l’:
A’+_ iA2+
A4+- iA5+
-A3t + A8t/Jj
A”t _ iA7t
A@ + iA7+
-2A8t/d?
- ‘P’i,‘4$E/3
Iu
(3)
which is the gluon’s color octet operator; Ya and qtj are(fjt,?,Bt).and(R+,Yt,B+),respectively;Eis the unit matrix of 3 x 3; A?+ is the a,-th gluon color operator and is defined by
p=P( I ,2,...,n)
DP
P(l,Z. . . , n) has been used. The form of the invariant amplitude can be derived from the SU( 3) color structure of QCD. An early symmetry analysis for the color factor was given by Cvitanovic [ lo]. A modem convenient treatment was developed by Berends and Giele [ 111. They proposed a recursive method from which Bq. ( 1) can be readily obtained after including all Feynman diagrams at the tree level. We will confirm from the unitary properties of H, later that Bq. ( 1) is the exact result at the tree level where no approximation has been made. Then H, is built from Eq. (1) as follows:
, ,gpcn))
is the mo-
mentum function of partons where momentum indices are suppressed, and p denotes a certain permutation of (1,2,. . .,n); (Ta1Ta2...Tan)G is the ith row and the jth column element of the matrix ( T”PCI) T~PCZI . . . T~Pc~) ) where the notation p =
A’+ u = (YfE+ + R+y+),,/ti A;+ =
t/ii)
Azt = (B+y+ + Y+~+),/& Azt = (B+F+ - Y+E+),/(
hi)
Q. Wang et al./ PhysicsLettersB 388 (1996) 346-352
348
A;+ = (R@+ + Yty+ - 2B+p+) u/di
(4)
The color effective Hamiltonian is another expression of the S matrix, so it is not necessarily Hermitian. The validity of H, can be verified by the calculation of the matrix element for the process e+e- + q?j + ng. For the color initial state (0) and the final state [f) = Iqpqj’AyiA$ . . . A$), we sum over the color indices i’ and j’ of the quark and the antiquark and those ai, ai, . . . , a; of n gluons, we obtain
c Kfl HcWI2=(01H,HcIO) f
=
~y2-a”
. (,;a2-“)* (5)
We can see that the calculation of ordinary matrix elements via H, returns to the original form. A color state of the multiparton system qq + ng is composed of the color charges of q, Tjand n gluons. It belongs to the color space: &$3+8&
@3&8,@*..@&
(61
There are many ways of reducing this color space. Corresponding to each reduction way is one set of orthogonal singlet spaces whose bases make up a completeness orthogonal set of color singlets. If a color singlet set is denoted by {I fk), k = 1,2, . . .}, then we have Ifk) (fkl = 1,
(fk 1fk’) = akk’
(7)
and I(fki Hc io)i2 =
c
(01H! Ifk)
(fkl Hc 10)
k
This property implies that the sum of the cross sections over all color singlets in a completeness set of the system (q7j + ng) is equal to the total tree level cross section sue( e+e- --f qq + ng): c
gk = (+tr.%(e+e-
--) qij + ng) ?
u0.
k
This is the result of unitarity. This shows that the invariant amplitude from which the color effective Hamiltonian is derived includes all Feynman diagrams at the tree level.
We have recapitulated the formulation we constructed in Ref. [ 91. Usually the color flow is approximated by a color singlet string piece where the fragmentation model may be applied. This approximation is based on the assumption that a gluon is a color nonet (a bicolor object). In that paper, we showed that the accuracy of this approximation declines as the number of gluons increases. In this letter, we reexamine the true meaning of CPlvI in the strict PQCD context starting from treating a gluon as a color octet exactly. As an example, we look at two cases, n = 1 and 2, of the process e+e- + q?j + ng. For the process with only one gluon in the final state , i ** e e+e- + 4481, the color space is Jp$%3&
(9)
There is only one way of reducing the space which corresponds to the unique singlet of the system qqg. This singlet is I Tr(QGt )), where (Q); is the color nonet tensor of the quark pair and can be written as the irreducible octet tensor (Q’)$ plus its trace S, 3 Tr( Q) : Q = Q’ + iS@.
Then the singlet becomes
IWQG 1) = ITr[
1) (10)
where the last equality is due to the fact that the gluon octet tensor is traceless, i.e. Tr( G1) = 0. The Eq. (10) indicates that the color charges of the quark pair are always intertwined with those of the gluon to build the system’s sole singlet. We can verify this by expanding ( Tr( Q'Gl)) in terms of the color charge: Iqyi) = (IR) , IY) , IB)) and the anticolor one: I’Pi) = ( IX) , IF), IB)) of q, 4 and g, we will find that the net charge of any color or anticolor is zero in each term, i.e. each term is color neutral (never to be confused with the color singlet! It is the combination of all these neutral terms that makes up the singlet.). This is just what we see in CFM. So for e+e- -+ qZjgl, CFM is 100% precise. Note that there is a probability of $ for each neutral piece (which consists of the color charge of q and its anticharge of g (q-g) , or the color charge of g and its anticharge of if (4 - g) 1 to be in a singlet string state. However , for e+e- + qtjglgz, the situation is not very obvious. In this case, the color space is
Q. Wang et al./ Physics Letters B 388 (1996) 346-352
349
The corresponding completeness set of color singlets is
l(f3 ) H, IO)j* = 3( 1D’12’12+ 1D(21)12 _ @12)~(21)* _ ~U2)*@))
{If&k=
We sum over this three amplitude squares and obtain
1,2,3}=
&
Is, Tr(GiG2)) 9
_ 2. (‘2)Dw)*
(12) where {Gi, G2) and [ Gr,G2] are the traceless symmetric and antisymmetric octet built from Gr and G2; They are defined by
3D
(19)
_ 2D”2’*D’21’ 3
which is equal to
(21) ijal a2
{G,,Gz}) =G~,G$j+G~,G:j-$~Tr(GrG2)
(13)
[G,,G*]f
(141
= Gf,G~j - G~kG:j
We can see from this completeness set that every two different singlets are orthogonal to each other. The singlets lf2) and Ifs) are CFM compatible while the singlet If,) is not covered by CFM. In If,), the quark pair forms a subsinglet which is color independent of the other subsinglet formed by gi and g2 in the same way as a glueball. This two subsinglet can also be separated in phase space, so ]fr) is part of the cause of LRG events in e+e- annihilation. Making use of the effective Hamiltonian Hc: H, = 1 Tr(QtG~G~)Do2’
+ i Tr(QtG~G~)Dc2’) (15)
we can compute the cross section of each singlet and estimate the percentage of the CFM-violating effect. The respective amplitude are
(f2IH, IO)= &D(12) + Dc2')) (16)
Then from the above equations, we have
= f(1@‘2’12 + [p(21)12 IV1 I HcI ON2 + @12)#21)* + @12)*#219 10-2 +
I Hc I OH2 = @wp)*
;&pj2 +
~w)*p9
+
M;“’ =
(77-a2)ijD(12)
+
(7-“2T=‘)ij$21’
qqglg2,
(22)
We see that the amplitudes of color singlets in a certain completeness set really satisfy the unitarity. The total cross section ou is the integral over the whole phase space at certain ycut value which is introduced to eliminate the infrared singularity: a0
= nde+e=
---t 4%x2)
IM(q4,gm)12d~ s .YNl
(23)
where fi denotes the phase space and corresponding kinematic factors. Given the total cross section, the percentages of the three color singlets are given by pi= L ao
J
[(fi I Hc 10)j2dR,
i= 1,273
(24)
Ycul
(f, 1H, IO)= &D'12' + Dc2'))
(f31H, IO)= x~(D('~) - Dc2'))
where IV;“~ is the matrix element of e+e- -+ and is written as
(17)
~p’21’~2
(18)
The result for the CS singlet jfr) is the same as that given in Refs. [3,4], but we provide here a rather more systematic approach from which the cross sections of all color states can be easily obtained. The results for other two singlets which are not calculated in Refs. [3,4] are also given. The results at various ycUtvalues are illustrated in Fig. 1. We can see that the percentage of If,), which is inconsistent with CFM, is about 6-8% of the total cross section. The CFM compatible color singlets have dominant percentage (92%). Recalling that in the case of efe- -+ qqgl , the only singlet of qqg1 is CFM compatible, hence, up
350
Q. Wang er ul. / Physics Letrers B 388 (I 996) 346-352
NCFM compatible singlet
( If,> )
. .
NCFM compatible singlet ( If,>
NS .
) .
10 CS singlet
.
. NC,
.
l
.
( If,>)
’ .
t0.002
0.000
0004
0.006
0.008
0.010
1
2
3
4
Ycut
5
6
7
8
9
10
11
of octets k
No.
Fig. I. The percentages of the CS singlet ( If,) ) and two CFM-compatible singlets (If2) and Ifs)) in the qqglg2 system at different .vcutvalues. This is the exact result at the tree level based on our formulation and is independent of collision energy.
Fig. 2. The number of total singlets and CS-type singlets Ns(k) and Ncs( k) from the reduction of k octets’ direct product.
to (~a at tree level, CFM is a good approximation to the true color structure of the parton system. We now turn to the process e+e- -+ q?j + ng where n > 2. Due to the great number of Feynman diagrams when n is large, it is almost impossible to directly compute the cross sections and then the percentages of the color separated (CS) singlets which violate CFM. Here we provide a heuristic method to estimate the percentages. The total singlets of multiparton system qq+ng is the sum of two groups of singlets: one group with the quark and antiquark forming the independent subsinglet l+ and the other group with their forming the suboctet S+. Obviously, the singlets in the former group are all CS; The latter one can be further divided into two subgroups: one is CS, the other is CFM compatible. Their cross sections are denoted by gcs (&), ucs (SF) and UC+I, respectively. Then we have
where Ncs (&) , Ncs (Q) and NCFMare the numbers of singlets in each of three groups respectively. Indeed
UCFM c( &FM
(26)
Ncs(&)
= Ns(&,,@&
Ncs(B,&
+ NCFM = Ns(!$$G,
= Ncs(n+
1) +NcFM(~+
+ msGq$
+ ~CF?vl
(25)
As n increases, the number of singlets in each of the three groups grows rapidly, so the contribution of a certain singlet to the total cross section is small. We assume that the contribution of a single singlet to the total cross section fluctuates around the average value, then we have QS(&)
0: Ncs(L&
~CS(Qj)
0: Ncs(8q7j)
@S,) = Ns(n) @...@&_I
1) = Ns(n+
BE,,) 1)
(27)
where Ns (n) is the number of singlets from the reduction of n octets’ direct product; Ncs (n + 1) and Ncm(n + 1) are the number of CS-type and CFM compatible singlets from (n + 1) octets’ product, respectively. When n is large, we have
pcs(Lj$= -(&j)bo
-
h(n) Ns(n
Pcs@qTj) = ~CS@qTj)l~O -
+
l)
+
Ncs(n +
Ns(n
ha4 = (+cFh4/uo = ms(&~>
@...@8,_,
+
l)
+
Ncm(n + 1) Ns(n + 1) + Ns(n)
Ns(n)
1) Ns(n)
(28)
where Ns (n + 1) + Ns( n) is the total number singlets. Obviously, Eq. (25) means
&uqq)
+ PCS@@> + hhl = 1
of
(29)
We use the Young Tableau method to calculate Ns ( k), NCS(k) and Nc~ (k) for k octets’ product. Fig. 2 shows Ns (k), NCS( k). One can see that they increase exponentially with growing k. Fig. 3 shows PCS(I+) and PCFM at the gluon number n = 2 to 9. We can see that Pcm is nearly saturated to about 70% as n
Q. Wang et ol./Physics
& d c m 2 (D =
0.8
-
0.7
-
0.6
-
0.5
-
0.4
-
0.3
-
0.2
-
P CFM .
1
.
.
l
l
l
,
I
I
.
.
l
=
0 PCJ’,?
I
0.1
.
2
3
4
5
the number
1 .
. t
6
B 7
.
1
8
.
I
9
10
of gluons
Fig. 3. The percentage of CFM compatible singlets &FM and that
of CS-type singlets with qFjforming a subsinglet Pcs(l+).
increases and PCS(l+) decreases slowly from 20% of n=3to 15.2%ofn=9. We have constructed a systematic approach to calculating the cross section of various color configurations for a multigluon system &j + ng. We can add more qq pairs to the system, write down its H,, and calculate the cross sections of different color singlets following the same procedure as in the q?j+ng case. Some points should be noted concerning multi-quark production. Compared with the probability of e+e- + (m 1) qq+ (n - 2) g at the same order, generally the probability of e+e- + mqTj+ng where there is one more qZj pair is suppressed by a factor of about 1/ 10 [ 121. Special attention should be paid to the lowest order processes responsible for LRG events: e+e- -+ qlif, q$& and e+e- + qifgg. In the phase space which allows for large rapidity gaps, the differential cross section of qlTj,q2ij2 outweighs that of qijgg. In other words, the process e+e- ---fqlZf,q2Tjzcontributes more to the LRG events than efe- -+ q?jgg does. The reason for this “anomaly” lies in that there is an unique way of forming two CS singlets for each case: ( q17j2) ( q2ql ) and (qif) (gg), where the round brackets denote the CS singlets. In the latter case, the singlet qQ must be separated by rapidity gaps from the other one (gg) . So both of the two gluons should be hard enough for a rapidity gap, which makes the differential cross section free of the infrared enhancement existing in the whole phase space. However, the total cross section of ( q1Fj2)( q2Fj1> in the whole phase space is much lower than that of (qTj) (gg), which is still within the normal expectation [4]. For higher order processes, we
Letters %388 (1996) 346-352
351
expect that no clear signals of LRG events exist because there are more overlaps among the phase spaces of different singlets. Here we do not intend to study LRG events, a limited phase space phenomenon, but are aiming at the study of the color singlet structure in the whole phase space. The main goal of this letter is to use e+e- -+ qij + ng, the simplest and most important process of hadronic events, as an ideal example to elucidate the true meaning of widely used CPM in a strict PQCD formulation. So for this goal, the process of multi-quark production is less important than the multigluon one. Now we will have a discussion and the concluding remarks. CPM is used in two widespread QCD event generators as an alternative phenomenological method to determine the color structure of multiparton system. In each branching step during the parton showering process, the corresponding color flow is recorded. When the process terminates, the final neutral color flow is obtained. The flow begins at the quark and ends at the antiquark connecting all partons one by one in a certain order. Such a procedure of recording the neutral color flow excludes the emerging possibility of a CS configuration which is part of the resources of LRG events. We point out in this letter that this discrepancy is caused by that the CFM is a semiclassical method in describing the color structure which is of quantum nature. We provide here a strict formulation derived from tree level PQCD to analyze the color structure of a multiparton system. In this formulation, we point out that each piece of neutral color flow cannot be equivalent to a color singlet though there is a certain probability for it to be in the singlet state. When the multiparton system is composed of color separated singlet subsystems, the neutral color flow is only confined within each subsystem. We give the completeness sets of color singlets for q?jg and q4gg and the corresponding cross sections of each singlet. We show that up to LYEat tree level, CFM is a good approximation to the exact color structure of the multiparton system. The percentages of CS-type and CFM compatible singlets is estimated by the color symmetry analysis when the gluon number is large. We find that as the gluon number grows, the percentage of CS-type singlets decreases and that of CFM compatible ones increases very slowly. In our study range of the gluon number (n = 2 to 9>, the percentage of CFM-type singlets is around 70%, which means the conventional
352
Q. Wang et al. / Physics Letters B 388 (1996) 346-352
CFM can partly reflect the system’s color structure.
References [ 11 T. Sjostrand, Int. J. Mod. Phys. A 3 ( 1988) 751; in: Z Physics at LEP 1, Proceedings of the Workshop, Geneva, Switzerland, 1989, edited by G. Altamlli, R. Kleiss and C. Verzegnassi (CERN Report No. 89-09, Geneva, 1989) Vol. 3. [2] J. Randa, Phys. Rev. D 21 (1980) 1795. 131 J.D. Bjorken, S.J. Brodsky and H.J. Lu, Phys. Lett. B 286 (1992) 153.
[4] J. Ellis and D.A. Ross, 2. Phys. C 70 (1996) 115.
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