PDF《arXiv:hep-ph/9704226v1 4 Apr 1997》

arXiv:hep-ph/9704226v1

4 Apr

1997 Probability of Color Rearrangement at Partonic Level in Hadronic W+ W? Decays Zong-Guo Sia , Qun Wanga , Qu-Bing Xieb,a a Department of Physics, Shandong University Jinan Shandong 250100, P.

R. China b Center of Theoretical Physics, CCAST(World Lab), Beijing 100080, P. R. China A strict method to calculate the color rearrangement probability at partonic level in hadronic W+W? decays is proposed. The color e?ective Hamiltonian Hc is constructed from invariant amplitude for the process e+e? → W+W? → q1q2q3q4 + ng (n = 0, 1, 2,and is used to calculate the cross sections of various color singlets and the color rearrangement probability. The true meaning of the color rearrangement is clari?ed and its di?erence from color interference is discussed. PACS number: 12.35E, 12.38, 13.65, 13.60H I. INTRODUCTION At LEP2 energy, the real W+ W? pair production through e+ e? annihilation becomes possible. More accurate measurements on W mass (MW ) and other properties can be made at this stage of LEP project. The success of the precision measurements of MW relies on accurate theoretical knowledge of the dynamics of the production and decay stages in e+ e? → W+ W? → q1q2q3q4. However, the possible Color Rearrangement (CR) may obscure the separate identities of two W bosons so that the ?nal hadronic state may no longer be

1 considered as a superposition of two separate W decays. Thus the W mass determination may be distorted by the color reconnection e?ect in hadronic W-pair decays. This e?ect was ?rst studied by Gustafson, Pettersson and Zerwas (GPZ) [1]. It attracts a lot of studies in recent years [2C5]. Considering that the two Ws decay into two quark pairs q1q2 and q3q4, GPZ assume that these two original color singlets can be rearranged into two new ones q1q4 and q3q2 with a probability (1

9 ), and then energetic gluons are emitted independently within each new singlet, which implies that the CR occurs before the parton shower process begins. But Sj¨ ostrand and Khoze do not regard this instantaneous scenario as a very likely one [2,3]. The reason is that the decay vertices of two W bosons are in general separated in space-time, and therefore the hard gluons (with Eg ≥ ΓW ) are produced incoherently by the two pairs q1q2 and q3q4 [2,6]. So there are two color singlets C1 and C2, each containing a qq pair and a set of gluons. Furthermore they argue that the CR in Perturbative QCD (PQCD) phase only comes from the color interference which should be very small. Hence they conclude that the non-perturbative contribution dominates the CR e?ect because the two color singlets C1 and C2 coexist later during the relatively larger space-time scale of hadronization compared with that of W± '

s life. The non-perturbative CR probability is controlled by the space- time overlaps of the color ?eld induced by two groups of partons in C1 and C2. Later on, Gustafson and H¨ akkinen stress that CR can only originate from the partonic level and argue that the hard gluon emission will enlarge the CR e?ect. Because there are increasing ways of color recombination between the partons of C1 and those of C2 with the growing number of emitted gluons, even though the rearrangement probability is only

1 N2 c for each way, the total probability at the partonic level may be greatly enhanced to the order ? (l + 1)(m + 1)/N2 c where l and m are the numbers of gluons in C1 and C2 respectively. The ?nal probability can in principle be modi?ed by multiplying the factor of order ? (l + 1)(m + 1)/N2 c by unknown functions of variables which characterize the space-time overlaps of the color ?elds induced by partons of C1 and C2. But Gustafson and H¨ akkinen'

s analysis on the total probability at the partonic level is only a qualitative one. It does not include many other ways of forming singlets and the calculation is not based on a strict formulation.

2 We should keep in mind that the color ?elds stretched between partons must be restricted in a precon?ned state, i.e. a singlet. Thus in PQCD phase, CR means the transformation from a set of original singlets to that of new ones. Hence the CR probability from C1 (con- taining q1q2) and C2(containing q3q4) to new recombined ones where q1q4 and q3q2 belong to di?erent singlets should and can be estimated by PQCD with more reason and accuracy. In this paper, we try to calculate the rearrangement probability from a strict systematic approach of PQCD. This approach is based on the color e?ective Hamiltonian [7] Hc which is built from the recursive form [8C10] of the invariant amplitude M for the process: e+ e? → W+ W? → q1q2q3q4 + ng, n = 0, 1, 2,1) The color e?ective Hamiltonian Hc is used to calculate the cross sections and the probabilities of various rearranged color singlets formed by ?nal partons. The physical signi?cance of our approach lies in that it includes all of e?ects caused by the di?erent space-time intervals between the decay vertices of two W-bosons. The CR probability we obtain shows that the CR of PQCD stage is not negligible. This seems di?erent from what Sj¨ ostrand and Khoze conclude in ref. [2,3]. In our approach, the meaning of CR can be clearly de?ned and the di?erence from color interference can be easily elucidated. The outline of this paper is as follows: in section II, we give the invariant amplitude Mn for the process (1) in the recursive form;

then the color e?ective Hamiltonian Hc is abstracted from Mn in section III;

thirdly, we use Hc to analyze the color singlet structure of the parton states q1q2q3q4 + ng with n = 0, 1,

2 and give the rearrangement probabilities in section IV;

?nally, a summary is given in the last section. II. CROSS SECTION The di?erential cross section dσn for the process (1) is dσn = Φ|Mn|2 d?n+4(Q1, Q2, Q3, Q4, K1,Kn), (2)

3 where Φ =

1 8s is the ?ux factor multiplied by a spin average factor, and the phase space factor d?n+4 is de?ned by d?n+4 = (2π)4 δ4 (P+ + P? ?

4 i=1 Qi ? n j=1 Kj)

4 i=1 d3 → Qi (2π)32Q0 i n j=1 d3 → Kj (2π)32K0 j , here P+, P?, Qi (i = 1,4) and Kj (j = 1,n) denote the 4-momenta of e+ , e? , quarks and gluons respectively. According to Feynman rules, the matrix element Mn of the process (1) (see ?g.1) can be written as Mn = n m=0 V =νe,γ?,Z0 ?V ν?Dνν′ (W +2 )D??′ (W ?2 ) * ? Sν′ (Q1;

K1,Km;

Q2) ? S?′ (Q3;

Km+1,Kn;

Q4), (3) where W± denotes the 4-momenta of W± boson, and Dν? (W±2 ) is the propagator of W± boson;

?V ν? is the polarization tensor of leptons;

? Sν is the current containing quarks and gluons which depends on the 4-momenta, helicities and color indices of the outgoing partons. In the recursive form, the current ? Sν can be expressed by ? Sν(Q1;

K1, K2,Km;

Q2) = iegm s P (1,2,・・・,m) (Ta1 Ta2 ・ ・ ・ Tam )i jSν(Q1;

1, 2,m;

Q2), (4) where gs is QCD coupling constant;

Ta = λa

2 and λa is Gell-Mann matrix for SUc(3), and Sν is the spinor current (for detail, see refs. [8,9]). Substituting eq. (4) into eq. (3), we obtain Mn = n m=0 P (1,・・・,m) P (m+1,・・・,n) (Ta1 ・ ・ ・Tam )i j(Tam+1 ・ ・ ・ Tan )k l X(q1g1・・・gmq2)(q3gm+1・・・gnq4), (5) where the indices i, k (j, l) denote the color (anticolor) of outgoing quark (antiquark), and X(q1g1・・・gmq2)(q3gm+1・・・gnq4) = ?e2 gn s V =νe,γ?,Z0 ?V ν?Dνν′ (W +2 )D??′ (W ?2 ) *Sν′ (Q1;

K1,Km;

Q2)S?′ (Q3;

Km+1,Kn;

Q4). ?V ν? can be written as follows ? ? ? ? ? ? ? ?V ν? = v(P+)[ieΓW ν ]iQαγα Q2 [ieΓW ? ]u(P?), V = νe, ?V ν? = v(P+)[ieΓV α ]u(P?) ? D(V )αβ (Q′2 )[ieFV βν?(Q′ , W+ , W? )], V = γ? , Z0 , (6)

4 where the repetition of indices represents summing (we use this convention unless explicitly speci?ed);

ΓB α is the fermion-boson vertice........