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Photon production by gluon jets
Volume 84B, number 4
16 July 1979
PHYSICS LETTERS
PHOTON PRODUCTION BY GLUON JETS Thomas A. DeGRAND Department of Physics, University o f California, Santa Barbara, CA 93106, USA
Received 11 May 1979
The fragmentation function for photons from a gluon jet is uniquelycalculable in QCD. A phenomenologicalinvestigation of its properties indicates that the best place to observe it is in the continuum photon spectrum in the two-gluon decays of heavy Qt) P-waveor pseudo-scalarbound states. The fragmentation function may be observablein an experiment with sufficient efficiency in rejecting photons from *r° decay.
Structure functions and fragmentation functions involving photons have recently been the subject of much theoretical interest. At large momentum transfers the most important coupling of the photon to hadrons is not via vector mesons but directly to quark -antiquark pairs. Because this coupling simply involves the quark charge and is calculable in perturbation theory, and because the interactions of quarks and gluons are also completely understood in perturbation theory, all structure functions and fragmentation functions involving quarks, gluons and photons may be calculated in perturbative quantum chromodynamics with essentially no free parameters. Measurement of these structure and fragmentation functions thus will provide important tests of quantum chromodynamics. Four kinds of structure or fragmentation functions may be calculated. They are the structure function of a quark or a gluon in a photon and the fragmentation functions of a photon from a quark jet or a gluon jet. The structure function of a quark in a photon has been extensively discussed in the literature [ 1,2], the calculation first having been performed by Witten [1 ]. The fragmentation function of a photon in a quark jet may be seen via the inclusive production of photons at large transverse momentum in hadron-hadron scattering [3], but a better place appears to be in the reaction e+e - ~ 7X at center-of-mass energies in the ten-tohundred GeV range. The phenomenology of this reaction has been discussed recently in ref. [4]. The subject of this note is the fourth reaction listed above, the 478
fragmentation function for photons produced from gluon jets. QCD predicts that gluon jets may be produced in several reactions, the most important being in the Zweig-violating decays of heavy quark-antiquark systems [5] or as the away-side jet in the large p± reactions pp -+/a+/l- X, 7X [6]. The analysis which follows will be made with reference to the former case although it can easily be applied to the latter. The QQ decay system has the advantages over the large-pI reaction of higher counting rate and lower background, although there one cannot "tune" the Q2 which governs the color coupling constant at(Q2), as one can in the latter reaction (where Q2 ~ p2), In the leading logarithm approximation, the fragmentation function of the photon in the gluon is given by the convolution of the quark distribution in the gluon with the photon distribution in the quark: O 2 dk 2 A2
1
Xf
"z ~(k2, 02))O~/q(x/z) zdz G qi/gt,
(1)
X
where Gqi]q(Z, ~) is the fragmentation function for a quark of flavor i from a gluon beam (the factor of 2 counts both quarks and antiquarks), ~(k2, 0 2) = (4rib) -1 log[ac(Q2)/Otc(k2)] ,
(2)
governs the Q2 evolution of the quark and gluon structure functions,
D~ro/g(/"' ~(Q2, Q2)) =1
c%(Q2) = [b log(Q2/A2)]-I is the color coupling constant and
D.r/q(X ) =
~ Gqi/g~/, t" i=u,d
O2)D~r/qi (j, ~ = O)
+ Gg/g(/, Q2)O,~o/g(/, ~ = 0).
[1 + (1 -- x)Z]/x,
is the lowest-order QED fragmentation function for photons from quarks. In terms of moments, -- f x J - ' D ( x ) ~ , o
(8)
Here Dg/g is the fragmentation function for a gluon from a gluon jet Dg/g(j, ~) -
1
~)
16 July 1979
PHYSICS LETTERS
Volume 84B, number 4
(X+ - A n s ) e 2x+~ - (X_ - Ans ) e 2x- ~ ;k+ - •_ , (9)
eq. (1) becomes
and D~ro/g(/, 0) and DTro/q(/, 0) are the "primordial" fragmentation functions of quarks and gluons into rr0's. From eq. (7),
D.rlg(/' Q2)
D,ylnOlg(j, ~) = (2l])D~ro/g(j, ~) . t22
= 6 ~ e22-~-~ dk2~-Gqi/g(j, ~)D,.//q(j).
(3)
Now we recall (cf. ref. [7])
Gqi/g(j', ~) = Aqg[e2X+~ =1
X+_ ~(Ans
+
- e2x-~]/(k+
- X_),
(4)
Agg)
1 + [(~(Ans - Agg)) 2 + 2fAgqAqg] 1/2
(5)
and the A0-'s are the appropriate anomalous dimensions listed, for example, in ref. [7]. Performing the t integral in eq. (3) then yields the QCD fragmentation function for the photon in a gluon jet,
=6 ~
The second background is found only in the decay of JP = 1 - QQ systems such as the T: the direct decay QQ -~ 3'gg. The inclusive photon spectrum from this reaction is [10]
D,,i/,r(x, Q2) = ~(ote~/Otc(QE))Dq/,r(x,
e2
i= udsc
a
D'r/q(/)Aqg(J)
(6)
at(Q2 ) (21rb - X_)(21rb - X+)
This expression may be inverted numerically using the method of Yndurain [8] to give D(x, Q2). Already apparent in eq. (6) is the factorized behavior characteristic of photon structure functions D(x, Q2) = log(Q 2) f(x). Two types of background compete with this production mechanism. The first is the photonic decay of hadronic systems, particularly n0's. The fragmentation function for a photon from a ~r° is flat:
D.y/uo(x ) = 2 .
(7)
The fragmentation function for 7r0's from a gluon jet in QCD is given by [9]
Q2),
(11)
where eQ is the charge of the heavy quark and Dg/,~(x) is given by the Ore-Powell [ 11 ] formula for orthopositronium DglT(X ) =
2
+
Dq,lg(j ' Q2)
(10)
mVgla rx(1 - x)
2(1 - x) 2 ln(1 - x) (2 x) 3
2 - x 2(1 - x) q + ln(1 - x).j x x2 "
(12)
This reaction peaks as x -+ 1 and if present will completely overwhelm both the direct g ~ 3' and indirect g ~ lr0 ~ ~, processes at large x. Of course, both these backgrounds are themselves tests of QCD. The reaction T ~ gg,y tests the hypothesis that the hadronic decays of T states proceed via three gluon intermediate states and the rr0 ~ ~, reaction allows one to measure the gluon fragmentation function into ~r0's. Its shape is uncertain, but its evolution in Q 2 is given by eq. (8). We have now assembled enough formulas to begin a discussion of phenomenology. We must first estimate the primordial quark and gluon fragmentation functions. A reasonable fit to the inclusive hadronic spectrum in e+e - annihilation is 479
Volume 84B, number 4
PHYSICS LETTERS
xDn/u(X ) = xD~/a(x ) ~ 0.5(1 - x ) .
(13)
I
I do not know any theoretically well-motivated formula for gluon fragmentation - its measurement is itself of great theoretical interest. For the purposes of estimating the n 0 background I will take
OTr/g(X ) = On/q(X) ,
I
i
[
i
2Q, )=2
2b, j=2
io-I
(14a)
or
152
DTr/g(X ) = (1 - x) D~r/q(X) ,
(lgb)
as "standard" guesses for the gluon fragmentation function into n0's. Two cases are of interest. The first is the decay of a 1 - Q(~ bound state. Here the gluons are produced with a decay distribution given by eq. (11) and then decay. One must convolute the distribution of gluons in the source over the distribution of photons in the gluon. An equivalent statement is that the moment of the photon distribution from the source is the product of moments of the two distributions. We can compare the three processes T ~ ggT, T ~ g g g ~ T X a n d T ~ g g g rr0X ~ 7X most efficiently via examining their moments, compared to the moments of the gluon decay distribution:
D,r/,r(j, a2)/Dg/,r(/. , 0
2) = ~-(o~/Otc(a2)) e ~ ,
(15a)
O3"/g/,r(j, a2)/Og/,r(i,
0 2) = D3"/g(j, a 2 ) ,
(15b)
D3"DrO/g/T(j ' Q2)/Dg/,r(j,
Q2) = D3"/lrO/g(j ' Q2).
(15c) These three ratios are shown in fig. 1 for/. = 2 and 4 for the range 10 < Q < 100 GeV, using the nominal values A = 0.5 GeV, Q2 = 1.8 GeV 2 , and eQ = 1/3. We see that the inclusive photon spectrum is dominated at low/" by n 0 decay and at large / by direct gg7 production. Clearly a 1 - Q(~ state is a poor place to look for the prompt g ~ 3' reaction, and even the direct gg7 production mechanism is small compared to n 0 decay except at large x. Next we turn to gluon jets produced in other reactions, such as di-gluon decays of heavy QQ P-wave or pseudoscalar states, or in large p± reactions. The former seems to be the best place to measure the fragmentation function because counting rates are probably higher and because of the absence of " p r o m p t " photon backgrounds such as T -~ 7gg or (in large p± reactions) subprocesses which produce photons without the necessity 480
16 July 1979
t31
I,j=2
r'a
2a, j=4
2b, j=4
154
~
'
-
I, j=4
I
I
I
I
I
I0
30
50
70
90
I00
Q (GeV)
Fig. 1. Second and fourth moments of photon distributions divided by moments of gluon distributions in the source (eq. (15)). Curves 1 are for the prompt g ~ 3"reaction eq. (1), curves 2a and 2b for cascade g ~ rro ~ 3' decays, with the two choices ofDlr/g(X, Q~) of eq. (14a) and (14b), and the dotted line shows the size of all moments of the direct gluon reaction "r ~ gg3",eq. (15a). of fragmentation. The major photon background in the QQ ssytem will be monochromatic_photons from radiative transitions among the heavy QQ states. Now it is most convenient to invert the moment equations and to display the fragmentation functions directly. Fig. 2 shows the results of this inversion: D3"/g(X, Q2) and D3"DrO/g(X ' Q2) for Q = 10 and 30 GeV (appropriate to the T system and to a possible heavier QQ system). Only at very large x does the prompt 3' signal exceed the photons from n 0 decay. The ability of experimentalists to see the prompt QCD photons is clearly dependent on their efficiency in removing the n 0 background at large x. If no attempt is made to subtract n 0's, the QCD signal and the n 0 signal cross over at x ~ 0.7 to 0.9. However, if one can remove 90 of the n 0 background, the signal to noise ratio exceeds unity for x >~ 0 . 4 - 0 . 5 . To extrapolate our results to other Q2's, it is convenient to give approximate formulas for the fragmentation functions which are valid at large x. The QCD
Volume 84B, number 4 I
,o o
PHYSICS LETTERS I
16 July 1979
I
I0 0
'1
16'
1
0.6
0.8
id 2 O
2t
0 x a x
I
Id'
2o
162
I
,g
t:) x
td a
16 3
16'
164
,6 s
0
,
a2
,
a4
,
×
0.6
,
O.a
~ll
,6 5
1.0
0
0.2
0.4
X
' I~
1.0
(a) (b) Fig. 2. xD,.r/g(X, Q2) from (1) the prompt g ~ ~"reaction eq. (1) and (2a) and (2b) from the cascade decay g ~ Ir° ~ "r, for the two choices ofDlrO/g(X, Q~) ofeq. (14a) and (14b). (a) Q = 10 GeV, (b) Q = 30 GeV. _ x)4N~+2
D.rfitO/g(X ' Q2).,. eA2~((14N~ + 3)
counting rules o f ref. [7] give [I 2] e Al ~(1 - x)4C: ~
Cq/g(X, ~) ~ ~. r(1 + 4C2~)
(16) '
+1 ~ 2~ eA1;~l -- x)4C2~ +3 2 i=u,d, fi, d + 4C2~ )
(19)
Gg/g(X, ~) ~ eA2~(1 -- x)4N~- 1 r(gN~)
'
(17)
with A 1 = (3 - 4TE) C2,
A2 = (!~ _ 4.),E)N _ ½f,
where ?E = 0.577 is Euler's constant, and C 2 = 4]3, N = 3 in color SU(3). Inserting these distributions in the convolution formulae above give
xD. r/g (x, Q 2) 6 ~,e.2
a 2(1 - x) a t ( Q 2 ) [41rb + 4 C 2 log(1 - x) -1 - A 1 1 2 ' (18)
The first formula is accurate to within a factor of 2 over the entire range o f x . The latter formula assumes the primordial gluon fragmentation function of eq. (14a). It is accurate to within a factor o f 2 for x > 0.5 at Q = 30 GeV. The different Q2 dependence of the two reactions indicates that at very large Q2, D~,/g (X, Q2)/D~/ItO/g(X , Q2) can be o f order unity even at low x, but explicit calculation shows that even at Q = 90 GeV the crossover point is still x ~ 0 . 6 - 0 . 8 . Our conclusions may be summarized briefly: The fragmentation function of a photon from a gluon jet is an interesting object to measure because its functional form is uniquely predicted by QCD. The best place to measure it will be in the continuum photon 481
Volume 84B, number 4
PHYSICS LETTERS
spectrum o f the decaYs o f heavy QQ P-wave and pseudoscalar systems. The ability to subtract the substantial n 0 background will be a strong component o f an experiment which will hope to successfully test QCD by measuring the p h o t o n fragmentation function from a gluon - although measurement o f the inclusive n 0 spectrum is itself o f interest, too. It seems possible that the experiment may be feasible for heavy QQ systems whose mass is in the 1 0 - 3 0 GeV range.
[3] [4] [5]
[6]
References [7] [1 ] E. Witten, Nucl. Phys. B120 (1977) 189. [2] S. Brodsky, T. DeGrand, J. Gunion and J. Weis, Phys. Rev. Lett. 41 (1978) 672; Phys. Rev. D19 (1979) 1418, SLAC-PUB-2199; W. Fraser and J. Gunion, preprint UCD-78-5 (1978); C.H. Llewellyn-Smith, Phys. Lett. 79B (1978) 83; C.T. Hill and G. Ross, CALT-68-659 (1978); R.J. DeWitt, L. Jones, J. Sullivan, D. WiUen and H.Wyld, ILL-(TH)-78-54. Recently, higher-order in c~c calculations of the photon structure function have been computed by
482
[8] [9] [10] [11 ] [12]
16 July 1979
A. Buras and W. Bardeen, Fermilab preprint 78/91-THY (1978). Cf. S. Brodsky, J. Gunion and R. Riickl, SLAC-PUB-2115 (1978); D. Jones and R. Rtickt, Davis preprint (1979). K. KoUer, T. Walsh and P. Zerwas, DESY preprint 78/77 (1978). Cf. T. DeGrand, Y.J. Ng and S.-H. Tye, Phys. Rev. D16 (1977) 3251; K. KoUer and T. Walsh, Phys. Lett. 72B (1977) 227; H. Frisch and K. Steng, Phys. Lett. 74B (1978) 90. Cf. J. Brucker, J. Husser, M. Fontannaz, D. Schiff and B. Pire, preprint LPTHE 78/14 (1978). T. DeGrand, preprint UCSB-TH-15 (1978); Nucl. Phys. B, to be published. F.J. Yndurain, Phys. Lett. 74B (1978) 68, J. Owens, Phys. Lett. 76B (1978) 85. CL S.J. Brodsky, D. Coyne, T. DeGrand and R. Horgan, Phys. Lett. 73B (1978) 203; K. Koller and T. Walsh, ref. [5]. A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. See also Y. Dokshitser, D. Dyakonov and S. Troyan, Proc. 13th Winter School of the Leningrad B.P. Konstantinov Institute of Nuclear Physics (1978), SLACTRANS-183.
16 July 1979
PHYSICS LETTERS
PHOTON PRODUCTION BY GLUON JETS Thomas A. DeGRAND Department of Physics, University o f California, Santa Barbara, CA 93106, USA
Received 11 May 1979
The fragmentation function for photons from a gluon jet is uniquelycalculable in QCD. A phenomenologicalinvestigation of its properties indicates that the best place to observe it is in the continuum photon spectrum in the two-gluon decays of heavy Qt) P-waveor pseudo-scalarbound states. The fragmentation function may be observablein an experiment with sufficient efficiency in rejecting photons from *r° decay.
Structure functions and fragmentation functions involving photons have recently been the subject of much theoretical interest. At large momentum transfers the most important coupling of the photon to hadrons is not via vector mesons but directly to quark -antiquark pairs. Because this coupling simply involves the quark charge and is calculable in perturbation theory, and because the interactions of quarks and gluons are also completely understood in perturbation theory, all structure functions and fragmentation functions involving quarks, gluons and photons may be calculated in perturbative quantum chromodynamics with essentially no free parameters. Measurement of these structure and fragmentation functions thus will provide important tests of quantum chromodynamics. Four kinds of structure or fragmentation functions may be calculated. They are the structure function of a quark or a gluon in a photon and the fragmentation functions of a photon from a quark jet or a gluon jet. The structure function of a quark in a photon has been extensively discussed in the literature [ 1,2], the calculation first having been performed by Witten [1 ]. The fragmentation function of a photon in a quark jet may be seen via the inclusive production of photons at large transverse momentum in hadron-hadron scattering [3], but a better place appears to be in the reaction e+e - ~ 7X at center-of-mass energies in the ten-tohundred GeV range. The phenomenology of this reaction has been discussed recently in ref. [4]. The subject of this note is the fourth reaction listed above, the 478
fragmentation function for photons produced from gluon jets. QCD predicts that gluon jets may be produced in several reactions, the most important being in the Zweig-violating decays of heavy quark-antiquark systems [5] or as the away-side jet in the large p± reactions pp -+/a+/l- X, 7X [6]. The analysis which follows will be made with reference to the former case although it can easily be applied to the latter. The QQ decay system has the advantages over the large-pI reaction of higher counting rate and lower background, although there one cannot "tune" the Q2 which governs the color coupling constant at(Q2), as one can in the latter reaction (where Q2 ~ p2), In the leading logarithm approximation, the fragmentation function of the photon in the gluon is given by the convolution of the quark distribution in the gluon with the photon distribution in the quark: O 2 dk 2 A2
1
Xf
"z ~(k2, 02))O~/q(x/z) zdz G qi/gt,
(1)
X
where Gqi]q(Z, ~) is the fragmentation function for a quark of flavor i from a gluon beam (the factor of 2 counts both quarks and antiquarks), ~(k2, 0 2) = (4rib) -1 log[ac(Q2)/Otc(k2)] ,
(2)
governs the Q2 evolution of the quark and gluon structure functions,
D~ro/g(/"' ~(Q2, Q2)) =1
c%(Q2) = [b log(Q2/A2)]-I is the color coupling constant and
D.r/q(X ) =
~ Gqi/g~/, t" i=u,d
O2)D~r/qi (j, ~ = O)
+ Gg/g(/, Q2)O,~o/g(/, ~ = 0).
[1 + (1 -- x)Z]/x,
is the lowest-order QED fragmentation function for photons from quarks. In terms of moments, -- f x J - ' D ( x ) ~ , o
(8)
Here Dg/g is the fragmentation function for a gluon from a gluon jet Dg/g(j, ~) -
1
~)
16 July 1979
PHYSICS LETTERS
Volume 84B, number 4
(X+ - A n s ) e 2x+~ - (X_ - Ans ) e 2x- ~ ;k+ - •_ , (9)
eq. (1) becomes
and D~ro/g(/, 0) and DTro/q(/, 0) are the "primordial" fragmentation functions of quarks and gluons into rr0's. From eq. (7),
D.rlg(/' Q2)
D,ylnOlg(j, ~) = (2l])D~ro/g(j, ~) . t22
= 6 ~ e22-~-~ dk2~-Gqi/g(j, ~)D,.//q(j).
(3)
Now we recall (cf. ref. [7])
Gqi/g(j', ~) = Aqg[e2X+~ =1
X+_ ~(Ans
+
- e2x-~]/(k+
- X_),
(4)
Agg)
1 + [(~(Ans - Agg)) 2 + 2fAgqAqg] 1/2
(5)
and the A0-'s are the appropriate anomalous dimensions listed, for example, in ref. [7]. Performing the t integral in eq. (3) then yields the QCD fragmentation function for the photon in a gluon jet,
=6 ~
The second background is found only in the decay of JP = 1 - QQ systems such as the T: the direct decay QQ -~ 3'gg. The inclusive photon spectrum from this reaction is [10]
D,,i/,r(x, Q2) = ~(ote~/Otc(QE))Dq/,r(x,
e2
i= udsc
a
D'r/q(/)Aqg(J)
(6)
at(Q2 ) (21rb - X_)(21rb - X+)
This expression may be inverted numerically using the method of Yndurain [8] to give D(x, Q2). Already apparent in eq. (6) is the factorized behavior characteristic of photon structure functions D(x, Q2) = log(Q 2) f(x). Two types of background compete with this production mechanism. The first is the photonic decay of hadronic systems, particularly n0's. The fragmentation function for a photon from a ~r° is flat:
D.y/uo(x ) = 2 .
(7)
The fragmentation function for 7r0's from a gluon jet in QCD is given by [9]
Q2),
(11)
where eQ is the charge of the heavy quark and Dg/,~(x) is given by the Ore-Powell [ 11 ] formula for orthopositronium DglT(X ) =
2
+
Dq,lg(j ' Q2)
(10)
mVgla rx(1 - x)
2(1 - x) 2 ln(1 - x) (2 x) 3
2 - x 2(1 - x) q + ln(1 - x).j x x2 "
(12)
This reaction peaks as x -+ 1 and if present will completely overwhelm both the direct g ~ 3' and indirect g ~ lr0 ~ ~, processes at large x. Of course, both these backgrounds are themselves tests of QCD. The reaction T ~ gg,y tests the hypothesis that the hadronic decays of T states proceed via three gluon intermediate states and the rr0 ~ ~, reaction allows one to measure the gluon fragmentation function into ~r0's. Its shape is uncertain, but its evolution in Q 2 is given by eq. (8). We have now assembled enough formulas to begin a discussion of phenomenology. We must first estimate the primordial quark and gluon fragmentation functions. A reasonable fit to the inclusive hadronic spectrum in e+e - annihilation is 479
Volume 84B, number 4
PHYSICS LETTERS
xDn/u(X ) = xD~/a(x ) ~ 0.5(1 - x ) .
(13)
I
I do not know any theoretically well-motivated formula for gluon fragmentation - its measurement is itself of great theoretical interest. For the purposes of estimating the n 0 background I will take
OTr/g(X ) = On/q(X) ,
I
i
[
i
2Q, )=2
2b, j=2
io-I
(14a)
or
152
DTr/g(X ) = (1 - x) D~r/q(X) ,
(lgb)
as "standard" guesses for the gluon fragmentation function into n0's. Two cases are of interest. The first is the decay of a 1 - Q(~ bound state. Here the gluons are produced with a decay distribution given by eq. (11) and then decay. One must convolute the distribution of gluons in the source over the distribution of photons in the gluon. An equivalent statement is that the moment of the photon distribution from the source is the product of moments of the two distributions. We can compare the three processes T ~ ggT, T ~ g g g ~ T X a n d T ~ g g g rr0X ~ 7X most efficiently via examining their moments, compared to the moments of the gluon decay distribution:
D,r/,r(j, a2)/Dg/,r(/. , 0
2) = ~-(o~/Otc(a2)) e ~ ,
(15a)
O3"/g/,r(j, a2)/Og/,r(i,
0 2) = D3"/g(j, a 2 ) ,
(15b)
D3"DrO/g/T(j ' Q2)/Dg/,r(j,
Q2) = D3"/lrO/g(j ' Q2).
(15c) These three ratios are shown in fig. 1 for/. = 2 and 4 for the range 10 < Q < 100 GeV, using the nominal values A = 0.5 GeV, Q2 = 1.8 GeV 2 , and eQ = 1/3. We see that the inclusive photon spectrum is dominated at low/" by n 0 decay and at large / by direct gg7 production. Clearly a 1 - Q(~ state is a poor place to look for the prompt g ~ 3' reaction, and even the direct gg7 production mechanism is small compared to n 0 decay except at large x. Next we turn to gluon jets produced in other reactions, such as di-gluon decays of heavy QQ P-wave or pseudoscalar states, or in large p± reactions. The former seems to be the best place to measure the fragmentation function because counting rates are probably higher and because of the absence of " p r o m p t " photon backgrounds such as T -~ 7gg or (in large p± reactions) subprocesses which produce photons without the necessity 480
16 July 1979
t31
I,j=2
r'a
2a, j=4
2b, j=4
154
~
'
-
I, j=4
I
I
I
I
I
I0
30
50
70
90
I00
Q (GeV)
Fig. 1. Second and fourth moments of photon distributions divided by moments of gluon distributions in the source (eq. (15)). Curves 1 are for the prompt g ~ 3"reaction eq. (1), curves 2a and 2b for cascade g ~ rro ~ 3' decays, with the two choices ofDlr/g(X, Q~) of eq. (14a) and (14b), and the dotted line shows the size of all moments of the direct gluon reaction "r ~ gg3",eq. (15a). of fragmentation. The major photon background in the QQ ssytem will be monochromatic_photons from radiative transitions among the heavy QQ states. Now it is most convenient to invert the moment equations and to display the fragmentation functions directly. Fig. 2 shows the results of this inversion: D3"/g(X, Q2) and D3"DrO/g(X ' Q2) for Q = 10 and 30 GeV (appropriate to the T system and to a possible heavier QQ system). Only at very large x does the prompt 3' signal exceed the photons from n 0 decay. The ability of experimentalists to see the prompt QCD photons is clearly dependent on their efficiency in removing the n 0 background at large x. If no attempt is made to subtract n 0's, the QCD signal and the n 0 signal cross over at x ~ 0.7 to 0.9. However, if one can remove 90 of the n 0 background, the signal to noise ratio exceeds unity for x >~ 0 . 4 - 0 . 5 . To extrapolate our results to other Q2's, it is convenient to give approximate formulas for the fragmentation functions which are valid at large x. The QCD
Volume 84B, number 4 I
,o o
PHYSICS LETTERS I
16 July 1979
I
I0 0
'1
16'
1
0.6
0.8
id 2 O
2t
0 x a x
I
Id'
2o
162
I
,g
t:) x
td a
16 3
16'
164
,6 s
0
,
a2
,
a4
,
×
0.6
,
O.a
~ll
,6 5
1.0
0
0.2
0.4
X
' I~
1.0
(a) (b) Fig. 2. xD,.r/g(X, Q2) from (1) the prompt g ~ ~"reaction eq. (1) and (2a) and (2b) from the cascade decay g ~ Ir° ~ "r, for the two choices ofDlrO/g(X, Q~) ofeq. (14a) and (14b). (a) Q = 10 GeV, (b) Q = 30 GeV. _ x)4N~+2
D.rfitO/g(X ' Q2).,. eA2~((14N~ + 3)
counting rules o f ref. [7] give [I 2] e Al ~(1 - x)4C: ~
Cq/g(X, ~) ~ ~. r(1 + 4C2~)
(16) '
+1 ~ 2~ eA1;~l -- x)4C2~ +3 2 i=u,d, fi, d + 4C2~ )
(19)
Gg/g(X, ~) ~ eA2~(1 -- x)4N~- 1 r(gN~)
'
(17)
with A 1 = (3 - 4TE) C2,
A2 = (!~ _ 4.),E)N _ ½f,
where ?E = 0.577 is Euler's constant, and C 2 = 4]3, N = 3 in color SU(3). Inserting these distributions in the convolution formulae above give
xD. r/g (x, Q 2) 6 ~,e.2
a 2(1 - x) a t ( Q 2 ) [41rb + 4 C 2 log(1 - x) -1 - A 1 1 2 ' (18)
The first formula is accurate to within a factor of 2 over the entire range o f x . The latter formula assumes the primordial gluon fragmentation function of eq. (14a). It is accurate to within a factor o f 2 for x > 0.5 at Q = 30 GeV. The different Q2 dependence of the two reactions indicates that at very large Q2, D~,/g (X, Q2)/D~/ItO/g(X , Q2) can be o f order unity even at low x, but explicit calculation shows that even at Q = 90 GeV the crossover point is still x ~ 0 . 6 - 0 . 8 . Our conclusions may be summarized briefly: The fragmentation function of a photon from a gluon jet is an interesting object to measure because its functional form is uniquely predicted by QCD. The best place to measure it will be in the continuum photon 481
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spectrum o f the decaYs o f heavy QQ P-wave and pseudoscalar systems. The ability to subtract the substantial n 0 background will be a strong component o f an experiment which will hope to successfully test QCD by measuring the p h o t o n fragmentation function from a gluon - although measurement o f the inclusive n 0 spectrum is itself o f interest, too. It seems possible that the experiment may be feasible for heavy QQ systems whose mass is in the 1 0 - 3 0 GeV range.
[3] [4] [5]
[6]
References [7] [1 ] E. Witten, Nucl. Phys. B120 (1977) 189. [2] S. Brodsky, T. DeGrand, J. Gunion and J. Weis, Phys. Rev. Lett. 41 (1978) 672; Phys. Rev. D19 (1979) 1418, SLAC-PUB-2199; W. Fraser and J. Gunion, preprint UCD-78-5 (1978); C.H. Llewellyn-Smith, Phys. Lett. 79B (1978) 83; C.T. Hill and G. Ross, CALT-68-659 (1978); R.J. DeWitt, L. Jones, J. Sullivan, D. WiUen and H.Wyld, ILL-(TH)-78-54. Recently, higher-order in c~c calculations of the photon structure function have been computed by
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[8] [9] [10] [11 ] [12]
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A. Buras and W. Bardeen, Fermilab preprint 78/91-THY (1978). Cf. S. Brodsky, J. Gunion and R. Riickl, SLAC-PUB-2115 (1978); D. Jones and R. Rtickt, Davis preprint (1979). K. KoUer, T. Walsh and P. Zerwas, DESY preprint 78/77 (1978). Cf. T. DeGrand, Y.J. Ng and S.-H. Tye, Phys. Rev. D16 (1977) 3251; K. KoUer and T. Walsh, Phys. Lett. 72B (1977) 227; H. Frisch and K. Steng, Phys. Lett. 74B (1978) 90. Cf. J. Brucker, J. Husser, M. Fontannaz, D. Schiff and B. Pire, preprint LPTHE 78/14 (1978). T. DeGrand, preprint UCSB-TH-15 (1978); Nucl. Phys. B, to be published. F.J. Yndurain, Phys. Lett. 74B (1978) 68, J. Owens, Phys. Lett. 76B (1978) 85. CL S.J. Brodsky, D. Coyne, T. DeGrand and R. Horgan, Phys. Lett. 73B (1978) 203; K. Koller and T. Walsh, ref. [5]. A. Ore and J.L. Powell, Phys. Rev. 75 (1949) 1696. See also Y. Dokshitser, D. Dyakonov and S. Troyan, Proc. 13th Winter School of the Leningrad B.P. Konstantinov Institute of Nuclear Physics (1978), SLACTRANS-183.