- Email: [email protected]
Vector versus scalar gluon jets in electron-positron annihilation
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
VECTOR VERSUS SCALAR GLUON JETS IN ELECTRON-POSITRON ANNIHILATION S. RAJPOOT Physics Department, Imperial College, London SW7 2BZ, UK Received 25 September 1979
In electron-positron annihilation, modification of the 1 + cos20 angular distribution of two quark jets due to the possible production of two vector gluon jets and two scalar gluon jets is considered. It is shown that present data favour vector rather than scalar point-like gluons as constituents of hadronic matter.
1. In spite of the attractiveness of quantum chromodynamics (QCD) as the theory of strong interactions, a considerable amount of interest has been expended in alternative theories [1] as equally likely candidates. The primary difference between these "rival" theories is the nature of the gluons participating in strong interactions. These could be (a) non-abelian charged massive gauge particles [2,3], (b) abelian gauge particles, (c) abelian neutral scalars, (d) non-abelian neutral scalars, (e) non-abelian charged scalars. Comparison with experiment [11 of the moments of the structure function F2(x, Q~) in deep-inelastic neutrino-nucleon scattering as predicted by QCD and the rival theories (b), (c),. and (d) decide against (b), (c) and (d). QCD and (a) have identical predictions for deep-inelastic lepton-hadron scattering due to the suppression of colour quantum numbers expected to show up in (a). But there are differences expected in the ratio [4] of the longitudinal to the transverse cross s e c t i o n (tTL/OT) and in deep-inelastic Compton scattering [5]. In electron-positron annihilation no two gluon jets are possible in QCD due to their inertness to the electromagnetic current. On the other hand, the electron-positron energy can materialize into two gluons leading to two jets on fragmentation in cases (a) and (e). Thus the (1 + cos20) distribution expected of • 1 hadrons fragmenting off a spm-] quark-antiquark pair produced in the annihilation process would be altered to (1 + flcos20). The coefficient/3 provides a • 1 measure of the deviation from the pure spm-~ quark jet structure• For completeness a very brief outline of the theory (a) follows.
2. The basic ingredients are integer charged quarks and massive gluons (colour is globally conserved). The gauge group that unifies the basic interactions is assumed to be of the form G = Gflavour @Gcolour. Gflavour at present seems to be the weak SU(2)L X U(1) group of Weinberg and Salam [6] while Gcolour is SU(3) of strong interactions. In turn G could be part of a bigger unifying group [7] GG, i.e. SO(10) or SU(5) depending on whether one needs right-handed currents in the flavour sector. For simplicity G is taken to be SU(2) × U(1) × SU(3) c although the arguments apply equally to the bigger unifying group G G. The resulting mass lagrangian, after spontaneous symmetry breaking, leads to the following neutral eigenstates [1, 2] for the vector mesons: (1) The photon AU=A~cosa+AUcsina,
rnA = 0 .
(1)
(2) Its orthogonal combination U~ =Ac~cOs a - A ~ s i n a, m U = mg + O(g2/e2), (2) where 3//_ is the mass common to the octet of gluons
,r,-+ ~., p,. V ~ ,
vo,
V, Vc).
(3) VcU: another gluon orthogonal to Ac~, MV = M . (4) The neutral eigenstate Z~ of the SU(2)L × U~I) theory with mass MZo ~ 90 GeV and orthogonal to A~ where A~ is the photon of the standard SU(2)L × U(1) theory. Standard QCD corresponds to setting a = 0 and the octet of gluons massless (Mg = 0). In the above correc325
Volume
88B, number
3,4
PHYSICS
tions of order (mu/mz)* to the eigenstates I?’ and Z{ have been neglected. The currents coupling Af and A:, relevant for the discussion, are
Jp(Af) = efEype
c
+
eiq)qjYp9j 3
q=u,d,s,c i=-r,y,b Jy4,)
= ew~a%~aE;CY
c
X (1 + cos*O) +
C q=u,d,s,c i=-r,y,b
eiq)qjy’qi
2
+-A!!%
‘OS ’
Note the “suppression factor” [m,‘/(s - mi)] for the production of coloured states in eq. (5). In the limit s/m: 3 1, colour is suppressed and integer charged quarks behave as if they carried fractional charges in accordance with the observations of Pati-Salam and Roy-Rajasekaran [8]. Thus the differential cross section is the same as in QCD, i.e. 112 2
c q=u,d,s,c,
Q*,
qi
d(1
-9
2s
i=r,y,b
(1 + cos%) + (1 - co+-
4rni
g
(1_g3’*
Q; ,
(8)
($$k =:g;“‘i,)’ (1_q* (5 g
x (1
-
cos*e) c
t3) g
Q; ,
(9)
v;, vi and the total differential cross section (da/dcos e), is the sum of eqs. (8) and (9). In the absence of the factor [m,$(s - mi)] *, (doL/dcos O)g grows as s increases [9] . It is this part of the cross section that is responsible for the bad high energy behaviour of the conventional formula for e+e- + vector meson pair. With [mi/(s - mi)] * present (doT/dcos e), vanishes while eq. (9) reduces to
Q,2$%(I
- cos*@ ,
x (1 t pcos*e),
(7) The suppressionfactor is again operative. To make the discussion more transparent it is convenient to split the differential cross section into
(10)
in the limit s/mp”> 1. It is remarkable that while the factor [mg/(s - mz)] * prevents colour to emerge, it cures the unitarity problem of the conventional formula for e+e- + vector meson pair. Thus the overall angular distribution of the hadrons in the two jets is the sum of eqs. (6) with mi/s < 1 and eq. (10):
1.
Similarly the matrix element for the process (e’e+ V,‘V;, Vi Vi) due to one-photon and Up-gluon exchange is
326
1979
cy,eeqf.yPqf 1 (Ilavsour) s-6 2
(colour)
[
v;,vi
. (4)
(5)
x
$;fi
(dsl=; (3)
-e (electronic charge), ejq’ cos (Y= eQ{q) (flavour quark charge), eLqtg)sin (Y= eQAq,g) (charge of coloured quark or gluon), and VP@ is the YangMills triple gluon vertex. ET,2 are the polarisation vectors of the outgoing charged gluons. The matrix element for the process (e’e- -+ qq pair) is
Qf 2 4 [
17 December
pieces corresponding to longitudinal (L) and transverse (T) photon (gluon) polarisations. Thus
ef cos a =
e
LETTERS
P=(Bq.~acQ~i- ’ 7
i=y,y,b
,1
Q:)
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
Table 1 + _ Variation of the coefficient/3~ of the cosg0 term in the angular distribution of hadrons in e e ~ two jets. x is the ratio of production of a scalar pair ~ to a quark pair qq. x
0.05
0.1
0.2
0.25
0.3
0.4
t3~
0.97
0.94
0.89
0.86
0.83
0.79 0.74
With Q 2 . _ 10 q=u,d,s,c qt 3 ' i = 7,y,b
2
+
+
Qg-2,
%, vk (13)
/3 ~ 0.86 , falling within the experimental result [11 ] at x/~ = 7.4 GeV: /3 exp = 0.97 -+ 0.14.
(14)
3. The (1 - cos20) angular distribution is typical of production of a spin-zero pair of particles in electron-positron annihilation. If the vector gluons con+ + sidered previously form an octet of scalars (¢p, Ck, ¢O, ~0K, q~U, ¢ v, case (e)) then the differential cross + section for e+e - ~ ~bp,KCp,K is
(15)
assuming that the presently held view of point-like partons applies equally to scalar gluons./3 in this case is
\ q=u,d,s,c, "~t i=7,Y,b X( 2
~ n--u,d,s,c
Qq2.+
~
Q2) -1
(16)
'
i=3~,y,b With 2;e~,~ k Q~ ; 2,/3e is predicted to be /3~ ~ 0 . 5 4 ,
0.6
0.7
0.8
0.9
1
0.69
0.65
0.61
0.57
0.54
the experimental limits as shown in table 1 .There is no reason to support such a preferred production of qcl over 4~pairs. In conclusion, present experimental evidence does not support scalar gluon jets. More refined experimental data are needed to distinguish between theories with locally conserved colour (QCD) and globally conserved color (case (a)). The author would like to thank Professors B. Deo, D.P. Roy and Riazuddin for reading the manuscript and to Drs. H. Hussain, T. Sherry, V. Elias and to Professor Abdus Salam for discussions. Thanks are due to Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, where part of this work was done.
References
3/2
~:%,~K
0.5
(17)
well outside the limits of the experimental result in eq. (14). Here it is assumed that in deriving eq. (16) the virtual photon produces q~ pairs and q~Tpairs with equal probability. However, if this were not the case and the q~Tpair production was x times that of qcl pair production then the values of/3~ in eq. (17) can lie within
[ 1] E. Reya, Invited talk presented at the Intern. Meeting on Probing hadrons with leptons (Erice, 1979); DESY preprint DESA 79•30. [2] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240. [3] J.C. Pati and A. Salam, Phys. Rev, D10 (1974) 275. [4] V. Elias, J.C. Pati, A. Salam and J. Strathdee, Pramana 8 (1977) 303. [5] H.K. Lee and J.K. Kim, Phys. Rev. Lett. 40 (1978) 485. [6] S. Weinberg, Phys. Rev. Lett. 19 (1:967) 1264; A. Salam, in: Elementary particle theory: relativistic group., and analyticity, Nobel Symp. No. 8, ed. N. Svartholm (Almquist & WikseU, Stockholm, 1968) p. 367. [7] H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193; H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [8] J.C. Pati and A. Salam, Phys. Rev. Lett. 36 (1976) 11; 37 (1976) 1312E. G. Rajasekaran and P. Roy, Pramana 5 (1975) 303;. Phys. Rev. Lett. 36 (1976) 355,689E. [9] N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577; M. Krammer and T.F. Walsh, Z. Phys. 263 (1973) 361. [10] G. Rajesekaran and S.S. Rindani, Phys. Lett. 83B (1979) 107. [ 11 ] G.G. Hanson, in: Proc. 13th Rencontre de Moriond, ed. J. Tran Thanh Van, Vol. II, p. 15; SLAC PUB-2118 (1978). 327
PHYSICS LETTERS
17 December 1979
VECTOR VERSUS SCALAR GLUON JETS IN ELECTRON-POSITRON ANNIHILATION S. RAJPOOT Physics Department, Imperial College, London SW7 2BZ, UK Received 25 September 1979
In electron-positron annihilation, modification of the 1 + cos20 angular distribution of two quark jets due to the possible production of two vector gluon jets and two scalar gluon jets is considered. It is shown that present data favour vector rather than scalar point-like gluons as constituents of hadronic matter.
1. In spite of the attractiveness of quantum chromodynamics (QCD) as the theory of strong interactions, a considerable amount of interest has been expended in alternative theories [1] as equally likely candidates. The primary difference between these "rival" theories is the nature of the gluons participating in strong interactions. These could be (a) non-abelian charged massive gauge particles [2,3], (b) abelian gauge particles, (c) abelian neutral scalars, (d) non-abelian neutral scalars, (e) non-abelian charged scalars. Comparison with experiment [11 of the moments of the structure function F2(x, Q~) in deep-inelastic neutrino-nucleon scattering as predicted by QCD and the rival theories (b), (c),. and (d) decide against (b), (c) and (d). QCD and (a) have identical predictions for deep-inelastic lepton-hadron scattering due to the suppression of colour quantum numbers expected to show up in (a). But there are differences expected in the ratio [4] of the longitudinal to the transverse cross s e c t i o n (tTL/OT) and in deep-inelastic Compton scattering [5]. In electron-positron annihilation no two gluon jets are possible in QCD due to their inertness to the electromagnetic current. On the other hand, the electron-positron energy can materialize into two gluons leading to two jets on fragmentation in cases (a) and (e). Thus the (1 + cos20) distribution expected of • 1 hadrons fragmenting off a spm-] quark-antiquark pair produced in the annihilation process would be altered to (1 + flcos20). The coefficient/3 provides a • 1 measure of the deviation from the pure spm-~ quark jet structure• For completeness a very brief outline of the theory (a) follows.
2. The basic ingredients are integer charged quarks and massive gluons (colour is globally conserved). The gauge group that unifies the basic interactions is assumed to be of the form G = Gflavour @Gcolour. Gflavour at present seems to be the weak SU(2)L X U(1) group of Weinberg and Salam [6] while Gcolour is SU(3) of strong interactions. In turn G could be part of a bigger unifying group [7] GG, i.e. SO(10) or SU(5) depending on whether one needs right-handed currents in the flavour sector. For simplicity G is taken to be SU(2) × U(1) × SU(3) c although the arguments apply equally to the bigger unifying group G G. The resulting mass lagrangian, after spontaneous symmetry breaking, leads to the following neutral eigenstates [1, 2] for the vector mesons: (1) The photon AU=A~cosa+AUcsina,
rnA = 0 .
(1)
(2) Its orthogonal combination U~ =Ac~cOs a - A ~ s i n a, m U = mg + O(g2/e2), (2) where 3//_ is the mass common to the octet of gluons
,r,-+ ~., p,. V ~ ,
vo,
V, Vc).
(3) VcU: another gluon orthogonal to Ac~, MV = M . (4) The neutral eigenstate Z~ of the SU(2)L × U~I) theory with mass MZo ~ 90 GeV and orthogonal to A~ where A~ is the photon of the standard SU(2)L × U(1) theory. Standard QCD corresponds to setting a = 0 and the octet of gluons massless (Mg = 0). In the above correc325
Volume
88B, number
3,4
PHYSICS
tions of order (mu/mz)* to the eigenstates I?’ and Z{ have been neglected. The currents coupling Af and A:, relevant for the discussion, are
Jp(Af) = efEype
c
+
eiq)qjYp9j 3
q=u,d,s,c i=-r,y,b Jy4,)
= ew~a%~aE;CY
c
X (1 + cos*O) +
C q=u,d,s,c i=-r,y,b
eiq)qjy’qi
2
+-A!!%
‘OS ’
Note the “suppression factor” [m,‘/(s - mi)] for the production of coloured states in eq. (5). In the limit s/m: 3 1, colour is suppressed and integer charged quarks behave as if they carried fractional charges in accordance with the observations of Pati-Salam and Roy-Rajasekaran [8]. Thus the differential cross section is the same as in QCD, i.e. 112 2
c q=u,d,s,c,
Q*,
qi
d(1
-9
2s
i=r,y,b
(1 + cos%) + (1 - co+-
4rni
g
(1_g3’*
Q; ,
(8)
($$k =:g;“‘i,)’ (1_q* (5 g
x (1
-
cos*e) c
t3) g
Q; ,
(9)
v;, vi and the total differential cross section (da/dcos e), is the sum of eqs. (8) and (9). In the absence of the factor [m,$(s - mi)] *, (doL/dcos O)g grows as s increases [9] . It is this part of the cross section that is responsible for the bad high energy behaviour of the conventional formula for e+e- + vector meson pair. With [mi/(s - mi)] * present (doT/dcos e), vanishes while eq. (9) reduces to
Q,2$%(I
- cos*@ ,
x (1 t pcos*e),
(7) The suppressionfactor is again operative. To make the discussion more transparent it is convenient to split the differential cross section into
(10)
in the limit s/mp”> 1. It is remarkable that while the factor [mg/(s - mz)] * prevents colour to emerge, it cures the unitarity problem of the conventional formula for e+e- + vector meson pair. Thus the overall angular distribution of the hadrons in the two jets is the sum of eqs. (6) with mi/s < 1 and eq. (10):
1.
Similarly the matrix element for the process (e’e+ V,‘V;, Vi Vi) due to one-photon and Up-gluon exchange is
326
1979
cy,eeqf.yPqf 1 (Ilavsour) s-6 2
(colour)
[
v;,vi
. (4)
(5)
x
$;fi
(dsl=; (3)
-e (electronic charge), ejq’ cos (Y= eQ{q) (flavour quark charge), eLqtg)sin (Y= eQAq,g) (charge of coloured quark or gluon), and VP@ is the YangMills triple gluon vertex. ET,2 are the polarisation vectors of the outgoing charged gluons. The matrix element for the process (e’e- -+ qq pair) is
Qf 2 4 [
17 December
pieces corresponding to longitudinal (L) and transverse (T) photon (gluon) polarisations. Thus
ef cos a =
e
LETTERS
P=(Bq.~acQ~i- ’ 7
i=y,y,b
,1
Q:)
Volume 88B, number 3,4
PHYSICS LETTERS
17 December 1979
Table 1 + _ Variation of the coefficient/3~ of the cosg0 term in the angular distribution of hadrons in e e ~ two jets. x is the ratio of production of a scalar pair ~ to a quark pair qq. x
0.05
0.1
0.2
0.25
0.3
0.4
t3~
0.97
0.94
0.89
0.86
0.83
0.79 0.74
With Q 2 . _ 10 q=u,d,s,c qt 3 ' i = 7,y,b
2
+
+
Qg-2,
%, vk (13)
/3 ~ 0.86 , falling within the experimental result [11 ] at x/~ = 7.4 GeV: /3 exp = 0.97 -+ 0.14.
(14)
3. The (1 - cos20) angular distribution is typical of production of a spin-zero pair of particles in electron-positron annihilation. If the vector gluons con+ + sidered previously form an octet of scalars (¢p, Ck, ¢O, ~0K, q~U, ¢ v, case (e)) then the differential cross + section for e+e - ~ ~bp,KCp,K is
(15)
assuming that the presently held view of point-like partons applies equally to scalar gluons./3 in this case is
\ q=u,d,s,c, "~t i=7,Y,b X( 2
~ n--u,d,s,c
Qq2.+
~
Q2) -1
(16)
'
i=3~,y,b With 2;e~,~ k Q~ ; 2,/3e is predicted to be /3~ ~ 0 . 5 4 ,
0.6
0.7
0.8
0.9
1
0.69
0.65
0.61
0.57
0.54
the experimental limits as shown in table 1 .There is no reason to support such a preferred production of qcl over 4~pairs. In conclusion, present experimental evidence does not support scalar gluon jets. More refined experimental data are needed to distinguish between theories with locally conserved colour (QCD) and globally conserved color (case (a)). The author would like to thank Professors B. Deo, D.P. Roy and Riazuddin for reading the manuscript and to Drs. H. Hussain, T. Sherry, V. Elias and to Professor Abdus Salam for discussions. Thanks are due to Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, where part of this work was done.
References
3/2
~:%,~K
0.5
(17)
well outside the limits of the experimental result in eq. (14). Here it is assumed that in deriving eq. (16) the virtual photon produces q~ pairs and q~Tpairs with equal probability. However, if this were not the case and the q~Tpair production was x times that of qcl pair production then the values of/3~ in eq. (17) can lie within
[ 1] E. Reya, Invited talk presented at the Intern. Meeting on Probing hadrons with leptons (Erice, 1979); DESY preprint DESA 79•30. [2] J.C. Pati and A. Salam, Phys. Rev. D8 (1973) 1240. [3] J.C. Pati and A. Salam, Phys. Rev, D10 (1974) 275. [4] V. Elias, J.C. Pati, A. Salam and J. Strathdee, Pramana 8 (1977) 303. [5] H.K. Lee and J.K. Kim, Phys. Rev. Lett. 40 (1978) 485. [6] S. Weinberg, Phys. Rev. Lett. 19 (1:967) 1264; A. Salam, in: Elementary particle theory: relativistic group., and analyticity, Nobel Symp. No. 8, ed. N. Svartholm (Almquist & WikseU, Stockholm, 1968) p. 367. [7] H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193; H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [8] J.C. Pati and A. Salam, Phys. Rev. Lett. 36 (1976) 11; 37 (1976) 1312E. G. Rajasekaran and P. Roy, Pramana 5 (1975) 303;. Phys. Rev. Lett. 36 (1976) 355,689E. [9] N. Cabibbo and R. Gatto, Phys. Rev. 124 (1961) 1577; M. Krammer and T.F. Walsh, Z. Phys. 263 (1973) 361. [10] G. Rajesekaran and S.S. Rindani, Phys. Lett. 83B (1979) 107. [ 11 ] G.G. Hanson, in: Proc. 13th Rencontre de Moriond, ed. J. Tran Thanh Van, Vol. II, p. 15; SLAC PUB-2118 (1978). 327