Transverse momentum distributions for Drell-Yan pairs in QCD

Volume 106B, number 3

PHYSICS LETTERS

5 November 1981

TRANSVERSE MOMENTUM DISTRIBUTIONS FOR DRELL-YAN PAIRS IN QCD P. CHIAPPETTA 1 Centre de Physique Thdorique, CNRS, 13288 Marseille Cedex 2, France

and M. GRECO Laboratori Nazionali INFN, Frascati, Italy and Centre de Physique Thdorique, Section 11, CNRS, 13288 Marseille Cedex 2, France and Facult~ des Sciences de Luminy, Marseille, France

Received 7 July 1981

Absolute Pi-distributions for Drell Yan pairs produced in rrN and pN collisions are studied in perturbative QCD and found in excellent agreement with data. Soft gluon effects, implemented by exact kinematics considerations, play a very important role in the analysis. A moderate value of the paxton intrinsic ~ 0.4 GeV2 is found.

Much theoretical and experimental work has been recently devoted to the study o f massive lepton-pair production in hadronic collisions [1]. It is by now well known that the absolute Px-integrated cross section for the basic model o f Drell and Yan [2] is found [1] to be renormalized by the so-called "K factor" when one uses quark densities defined in deep inelastic leptoproduction. This K factor, which is approximately constant over the kinematical region explored so far, is well understood [3] in perturbative quantum chromodynamics (QCD) and this fact can be considered as a good success of the theory. The important role played by soft gluon effects in the evaluation o f those corrections to the parton model results has been emphasized b y various authors [4]. Furthermore it has been shown recently [5] that the inclusion o f higher order soft effects slightly modifies the first-order results in the kinematical regions reasonably accessible to experiments. On the other hand the transverse properties of the observed spectra, which should appear as the most spectacular effects o f QCD, have not yet found a satisfactory theoretical explanation [1,6] +1. In fact, although the 1 Boursier CEA. *I For a successful non-QCD analysis, see ref. [7]. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1981 North-Holland

predicted increase o f (p2> with S and M 2 is qualitatively in agreement with data, one requires a quite large value of the parton intrinsic p±, say ~ 1 GeV 2. Furthermore, the shape of the distributions from first-order diagrams is wrong at small and intermediate P i , it is better for large p2 ~ O(M2), but still the theoretical predictions lie somewhat lower than the data. As is well known, the failure of first-order results in the small and intermediate p± regions is not so dramatic. In fact, for A 2 ~ p2 ~ M 2 perturbation theory breaks down to the appearance of large a sn ln2n(M2/ p2) terms which have to be summed to all orders. This task has been essentially accomplished by Dokshitser et al. [8] in the leading double logarithmic approximation. A further improvement has been suggested by Parisi and Petronzio [9] by going into impact parameter space rather than in m o m e n t u m space. More re cent theoretical analyses [10] have confirmed this result, but no phenomenological use o f it has been made so far. On the other hand the same result was independently obtained by C u r d et al. [11] in e+e - annihilation. A recent analysis [12] o f total transverse momentum distributions o f e+e - jets at PETRA energies by the PLUTO collaboration strongly supports this resummed QCD formula. 219

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In the present letter we study the absolute p±-distributions of lepton pairs produced in 7rN and pN collisions in the same general framework. We fend excellent agreement between data [13] and the resummed QCD formula for small and intermediate Pi, with a much more reasonable value of the global quark intrinsic Pi, i.e. (p2)intr ~ 0.4 GeV 2. At large p± and in case of pN collisions the first-order Compton contribution nicely adds to obtain the observed tail in the pl-distribution, much the same way as the hard gluon bremsstrahlung affects the large K T e+e - qq jet distributions. Such a term is quite small for 7rN collisions. The observed features of (p2) dependence on the c.m. energy and the mass of the lepton pair is well reproduced also. We stress that our results are essentially independent of free parameters. The use of exact kinematics in the transverse momentum phase space for multigluon emission is of crucial importance for our findings. This is somewhat similar to what has been found elsewhere in next-to4eading QCD corrections [4,14]. The starting formula for the full differential cross section in the soft multigluon approximation is [9,11 ] do ~ do dp dM dy dp 2 dM dy dp2 '

(1)

with do -

87ra 2

dM dy -

--

9MS

5 November 1981

point later. A few remarks are in order here to clarify our starting formulae. The approximate factorization in eq. (1) is a consequence of the soft gluon approximation ,2 and allows for a great simplification in the problem. The factor K which, as said above, is quite well understood in perturbative QCD, depends also on the parametrization of the structure functions. An approximate constant value K = 1.8 + 0.2 has been used below in comparing with all experiments with n and p beams at various energies, having chosen a unique parametrization of the parton densities. The upper limit of the transverse phase space is

z)/2X/~, z = M2/(XlX2S ) is the

q±ma.x = M(1 -

where squared ratio of the lepton pair mass to the effective invariant energy (~)1/2 = (XlX2S) 1/2 of the subprocess qO-+ 3,(M 2) g. In the usual double leading approximation one takes q±max ~ M , but it is very important to keep trace of the finite terms left. Finally the running coupling constant has been chosen to freeze at very small values of k, namely a ( k 2) = 12rr/25 in [(k 2 + X2)/A 2 ], in agreement with previous phenomenology at low k 2 [9,11,16]. From the distribution (3), which is properly normalized to unity, one also gets [11,12] a soft contribution to (p2} given by

K

4 f

(p2)s°ft : ~ 0

2 q±max

dq21n(M2) a(q±). \q2 !

(5)

X G 6,2 [o~l)(~/Te,Y) 1~[~2)(N/fTe-Y) + (1 ~ 2)1, i

@ dp 2

-

1

(2)

7

J b dbJo(bPz ) exp[A(b,

20

qzmax)] ,

(3)

where

In comparing with actual experiments one has to add to eq. (5) the hard contribution (p2)hard, given by the first-order Compton process, namely qg ~ 7(M 2) q, and the parton intrinsic component (p2)intr. We therefore write for the full (p2) of the lepton pair (p2) = (p2)sof t + (p2)har d + (p2)intr .

A(b, q±max) 16 f ± m a x dq± l n ( M ~

3rr

-~l

\-q~±/a(q±)[Jo(bq± ) -

1] , (4)

0

The intrinsic p± distribution of the partons can be included also in eq. (3), which is therefore modified as [9]

dp' _ 1 f b dbJo(bp±)e-bZ/aA 2N0

exp[A(b,q±max)] , (7)

7"= M2/S, a(q±) is the rurming coupling constant, K = K(y,M, 7")is the K factor, and q±max is the phase

dp2

space limit for the emitted gluons. So far we have neglected the effect of an intrinsic p± in the parton distributions. We will come back to this

corresponding to an average value (p2)intr =

220

(6)

#2 For a detailed derivation see ref. [15].

1/.4. The

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5 November 1981

factor N properly normalized the new distribution (7), namely dp' dp 2 = 1.

i0 q

So far we have not completely specified qzmax, which plays an important role in the analysis. Then we estimate the average
'10-2

',?

c

ly=O

, 10-3

fr1 dz (2%/z) l X/z[1 -- (r/z)l/2]~(r/z) ~/2 , (8)

f~ dz(2x/~) 111 - ( r l z ) l l 2 l ~ ( r l z ) n / 2 having parametrized the parton densities as ld 5

q(x)~x~,(1-

x)~, ,

(l(X)~xr~z(1

x)~2 ,

(9)

with ~ = r/1 + f/2 and ~ = ~1 + ~2- This leads to an explicit dependence of qsmax, and therefore of the full distribution (7), upon r. Although eq. (8) gets slightly modified for y =~ 0, we have used it for simplicity for all values o f y needed for comparison with data. We have then used the NA3 [13] parametrizations for the pion and the proton structure functions:

10-6 0

= o.12(1

r

xSP(x) = 0.37(1

i

i

~. 280 G e V / c • 200 o 150

x) 5 , L

xuP(x) = Bx 0"5 (1 - x) 3"2 ,

xdP(x) = Cx°.5(1 -

6

Fig. 1. Differential cross section p21 d2o/dpz dM in nN collisions at Plab = 150 GeV. The data are from ref. [13]. The theoretical curves are obtained for y = 0. N refers to 60% neutrons and 40% protons.

x Vrr(x) = A x 0"4(1 - x) 0"9 ,

xS.(x)

2 L P.I. ( G e V / c )

x ) 4.2 ,

x) 9"4 ,

(10)

w i t h A , B, C determined by the normalization conditions

-2 tD

f V ' ( x ) dx = l ,

f uP(x) dx = 2 ,

fdP(x)dx=l.

~/"2 o24

For the gluon distributions, which enter in the Compton term, we have taken

xG(x) = 3.4(1 - x ) 5 .

(11)

Finally we have used for c~(k) A = 0.35 GeV and X = 1.1 GeV and (p2)intr = 0.4 GeV 2. Our results are shown in figs. 1 - 4 for 7rN and pN collisions, respectively, and compared with the experimental data. Let us discuss them in detail. The transverse m o m e n t u m distributions shown in fig. 1 are absolutely normalized through eqs. (1), (2)

l

0

I

0.2

I

I

0.L

I

I

0.6

I

0.8

Fig. 2. Predictions for (p12}from eq. (6) in nN collisions at Plab = 150,200 and 280 GeV/c. The data are from ref. [13 ]. 221

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I

PHYSICS LETTERS

I

I

5
= 103

10~

I

. ~ - -

: -~ i

i

(b)

i

--

l0 L

i

\

• ~ "t~ i ~ ' ~ X

|Z~ 7 _ 0.21

102

I

I

t a5
ioZ.
10

%

5 November 1981

(C)

' c l ' 8 < H <9 GeV 1

10:

02

v

<% Ill

10

lO

1

%

0

I

;

%

0

%

Fig. 3. Invariant cross section E d 3 o/d3p in pN collisions at Plab = 200 (a), 300 (b) and 400 GeV/c (c). The data are from ref. [ 13[. The theoretical curves are: soft + intrinsic (full line); soft + intrinsic + hard (dashed line).

and (7). The hard Compton term, obtained from ref. [17] without any intrinsic pz effects, gives a very small contribution at the tail of the distributions for low M values, which is obtained from ref. [18], if of the order In fig. 2
summing soft gluons effects also in the D r e l l - Y a n process. To conclude, we have shown that the observed features of the transverse m o m e n t u m distributions o f D r e l l - Y a n pairs are very well described in perturbative QCD. The multiple gluon soft contributions play a very important role in this respect. Finally a quite reasonable value of the partonic intrinsic (p2) is suggested from our analysis. I

!

% 2.o A c-i

c~ 1.0

v

• z,00GeV • 300 GeV • 200 GeV '

g

'

'b

l '

'

m p+.p- ( G e V )

Fig. 4. Predictions for
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O n e o f us (M.G.) w o u l d fike t o t h a n k E. de Rafael for the w a r m h o s p i t a l i t y e x t e n d e d to h i m at CPT, Marseille.

References

[7] [8]

[1] For an extensive review, see: Moriond Workshop on Lepton pairs (Les Arcs, Savoie, France, 1981); [2] S.D. Drell and T.M. Yan, Phys. Rev. Lett. 25 (1970) 316. [3] J. Kubar-Andr~ and F.E. Paige, Phys. Rev. D19 (1979) 221. G. Altarelli, R.K. Ellis and G. MartineUi, Nucl. Phys. B157 (1979) 461; J. Abad and B. Humpert, Phys. Lett. 78B (1978) 627; 80B (1979) 286; K. Harada, T. Kaneko and N. Sakai, Nucl. Phys. B155 (1979) 169; A.P. Contogouris and K. Kripfganz, Phys. Lett. 84B (1979) 473; Phys. Rev. D19 (1979) 2207; J. Kubar, M. Le Bellac, J.L. Meunier and G. Plaut, Nucl. Phys. B175 (1980) 251. [4] G. Parisi, Phys. Lett. 90B (1980) 295; G. Curci and M. Greco, Phys. Lett. 92B (1980) 175; D. Amati, A. Bessetto, M. Ciafaloni, G. Marchesini and G. Veneziano, Nuel. Phys. B173 (1980) 429. [5] G. Curei and M. Greco, Phys. Lett. 102B (1981) 280. [6] H.D. PoLitzer, Nucl. Phys. B129 (1977) 301; H. Fritzsch and P. Minkowski, Phys. Lett. 73B (1978) 80; G. Altarelfi, G. Parisi and R. Petronzio, Phys. Lett. 76B (1978) 351,356;

[9] [10]

[11] [12] [13]

[14] [15] [16] [17] [18]

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K. Kajantie and R. Raitio, Nucl. Phys. B139 (1978) 72; F. Halzen and D.M. Scott, Phys. Rev. D18 (1978) 3378; E. Berger, Vanderbilt Conf. (1978); ANL-HEP-PR-78-12 (1978). C. Bourrely, P. Chiappetta and J. Softer, Marseille preprint 81/P. 1267 (Jan. 1981). Yu.L. Dokshitzer, D.I. Dyakonov and S.I. Troyan, Phys. Lett. 78B (1978) 290; 79B (1978) 269. G. Parisi and R. Petronzio, Nucl. Phys. B154 (1979) 427. C.Y. Lo and J.D. Sullivan, Phys. Lett. 86B (1979) 327; S.D. Ellis and W.J. Stirling, Phys. Rev. D23 (1981) 214; J.C. Collins and D.E. Soper, Oregon Univ. preprint OITS155 (1981). G. Curci, M. Greco and Y. Srivastava, Phys. Rev. Lett. 43 (1979) 834; Nucl. Phys. B159 (1979) 451. Pluto Collab., Ch. Berger et al., Phys. Lett. 100B (1981) 351. CFS CoUab., J.K. Yoh et al., Phys. Rev. Lett. 41 (1978) 684; A.S. Ito et al., Phys. Rev. D23 (1981) 604; NA3 Collab., J. Badier et al., Phys. Lett. 89B (1979) 145 J. Badier et al., CERN-EP/80-150 (1980). M. Ciafaloni and G. Curci, Phys. Lett. 102B (1981) 352. M.f1. Friedman, G. Pancheri-Srivastava and Y. Srivastava, Boston preprint NUB-2486 (1981). R.M. Barnett, D. Schlatter and L. Trentaduc, SLAC-PUB2714 (1981). K. Kajantie and R. Raito, Nucl. Phys. B139 (1978) 72. J. Kubar, M. Le BeUac, J.L. Meunier and G. Plaut, Nucl. Phys. B175 (1980) 251.

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