Angular distribution of lepton-pairs and QCD

Volume 80B, number 4,5

PHYSICS LETTERS

15 January 1979

ANGULAR DISTRIBUTION OF LEPTON-PAIRS AND QCD J. CLEYMANS and M. KURODA

Department of Theorettcal Physics, University of Bielefeld, Bielefeld, Germany Received 7 November 1978

The recently measured angular distribution of one of the leptons produced in n-N ~ #+#- + . is compared with theoretical calculations done in the framework of QCD.

It is widely recognized that the angular distribution of leptons produced in hadron-hadron collisions provides important information about their production mechanism [1]. For small values of the transverse momentum, QT, of the virtual photon one expects a 1 + cos20 distribution in the center-of-mass system of the lepton-pair if the Drell-Yan mechanism of quark-antiquark annihilation is correct [2]. For large values of QT hard gluon corrections become important [3] and substantial deviations from the simple 1 + cos20 distri-

_k g kl/g

bution are expected in some cases• These deviations can be calculated explicitly in the framework of QCD once the relevant hadronic structure function in terms of quark and gluon constituents are known. It is the purpose of this paper to compare directly these calculations with the recently measured angular distributions [4] m n - N -+ #+/a- . . . By choosing to work with the data having QT > 1 GeV/c for the leptonpair we avoid the problem of the infrared behaviour of the gluon corrections and can thus have a relatively clean comparison between theory and experiment. At large transverse momenta the angular distribution of the lepton-pair is dominated, within the framework of QCD, by the diagrams of fig. 1. The first two diagrams give the quark-antiquark annihilalaon contrlbution while the last two give the quark-gluon scattering contribution. Correspondingly we have for the angular distribution of the lepton-pair:

(la)

kl

dQ2dydQ2d(cos O)

z

do(q +-q-+ G + ~+1~-)

]

X dQ2dydQ2d( c°s O) + (q[° Uli) dx ldx 2 (1)

+ ~i "JJL-'((gai(Xl)G(x2)do(q+G~q+ ,~+~-) • dQ2dydQ2d(cos O)

Pz (lb)

Fig. 1. Diagramsused in calculating the angular distribution of the lepton for large values of QT"

+ (qz ¢~ G)] dx I dx 2 =N(1 + o~cos20)

(2) 385

Volume 80B, number 4,5

PHYSICS LETTERS

where Q 2, y and Q2 stand, respectively, for the invariant mass, the rapidity and the transverse momentum squared of the produced virtual photon while 0 is the angle between the direction of the lepton and the chosen z-axis in the rest frame of the virtual photon. The summation runs over all species of quarks. Because of the spin one nature of the photon at most terms proportmnal to cos20 can be present in (1). A straightforward standard calculation leads us to the following expression for the cross section in QCD. d o ( A + B -+~+~- +. )=

dQ2dydO2d(cos

JJffdxldX26(~+ i+a

Q2)

X ~l Q 2 [qi(xl)~t(x2) +qt(x2)gt(Xl)] X (Q2 _ / ) 2 + ( Q 2 _ 0)2 4 °t2as 4gia

9 Q2

× [1 + cos20 + (1]2 -

(3/2) cos20)Aq~]

+ f f a x , ax¢(e + i+ a - 0 5

o? l

1 a2°es

(Q2 _ g)2 + (Q2 _ 62

X qi(Xl)G(x2) 6 Q2

492[

X [1 + cos20 + (1/2 - (3/2) cos20)AgG] + ({*~ t~, x 1 *~x 2, AqG OAGq) ] ,

(3)

where t, ti and g are the Mandelstam variables relevant for the constituent-constituent scattering process (fig. 1): s = ( P l +P2 ) 2 = x l x 2 s ' [= (Pl - Q)2

=s [r-~r+x2/4x 1e-y]

/~ = (P2 - 0)2 = sir - ' ~ x - 2 / 4

,

frame) or target axis (u-channel frame [4]). Another poss:ble choice (s-channel frame) is to take the direction of the actually observed hadromc momentum recoiling against the virtual photon (which is different from the recoil momentum at the constituent level). If the photon is going off perpendicular to the beam axas, this last frame corresponds to taking the z-axis perpendicular to the beam axis (in the hadron-hadron cms system). It is therefore not surprising that m this frame we will have an angular distribution differing strongly from the distributions in the two previously mentioned frames. One last possibility considered in ref. [4] is the frame proposed by Collins and Soper [5], in which the z-axxs bisects the angle between the beam and reversed target momentum directions Tins frame is interesting because a simple expression can be found for the effect of the so-called primordial transverse momentum of the constituents. In this frame the angular distribution wall again be similar to the two first ones. The exphcit forms for Aq~, AqG and AGq appearing in eq. (3) can be obtained from a straightforward calculation starting from the diagrams in fig. 1. The expressions are gaven in table 1 for each one of the different frames mentioned in the previous paragraph. For the numerical evaluation of eq. (3) we used the parametrizations for the quark number densities in the proton given by Barger and Phillips [6]. Since for rrN scattering valence quarks and gluons will dominate the cross section, the small sea in their parametrization does not bother us. For the gluon number density in the proton we took

xG(x) = 3(1

- x) 5.

Other parametrizations were also tried and their effect wall be briefly discussed below. For the pion we used the following simple parametrization suggested by the data of ref. [4] (n refers to rr- here):

x~(x)

= 0.717 (1 -

x)~/x.

x2eY] ,

and, as usual, r = Q2/s while x T = 2QT/X/s. At this point it is necessary to discuss the choice of z-axis in the rest frame of the lepton-pair. The following have been presented in ref. [4] : the beam axis (tins frame is called t-channel or Gottfried-Jackson 386

15 January 1979

The coefficient was fitted from the value of the integral [4] fl.zsX~(x ) dx= 0.14. For the gluon distribution in the pion we used

xG(x)

= 2.47(1 - x) 3.

The power of (1 - x) was taken from the power count-

Volume 80B, number 4,5

PHYSICS LETTERS

15 January 1979

I0 5 m

I

+ +

~r- N ~ , u *

10 ~

~2

,u - X

i

i I o~

10 3

<

xTlxJ +

mr

ol ol

,0

45<

C

i

%

rt

+ +

me,

10

%2

,

I

i

1

QI t~

I;

+

+

m I

~ [-.-~ N

I

%

+

+

~7 ~N

I

+ N

m r

I

%

e~

©

a

I 3

i

IX 4

t 5

GeV/c

I

mJ

I==

I 2

m

o

Fig. 2, Transverse momentum distribution as compared with the data of ref. [4]. ing rules [7] while the coefficient was chosen to balance the missing m o m e n t u m . Sea-quarks were neglected in the pmon. The QT distribution, obtained after integrating eq. (3) o v e r y > 0, and cos 0 is shown in fig. 2 for r = 0.071, corresponding to Q = 5.5 GeV and beam energy 225 GeV. Adjusting for overall normalization, the agreement is good for QT > 1 GeV. It must be mentioned, however, that, if we were to include the primordial quark transverse momentum, the curve in fig. 2 would sluft to the right, thus indicating that the observed cross section at large QT is presumably bigger than the QCD calculation in the present range of QT values. In our calculations we noted that the value o f or is strongly dependent on the value QT and ~', even becoming negative in some cases. This behaviour is indicated m fig. 3, where the results for different choices 387

Volume 80B, number 4,5

I0

PHYSICS LETTERS

I

08

i

i

,

frame

$ - chclnnel

,

t-

06

r

Mu#>3.5 GeV/c 2 QT>I. 0 G eV/c

i

(hamnel

01

15 January 1979

frame

1~=02

"G

04

(" 3

02

T:OO5

0

5

~'=01

E

-r=o2

-

e~

10 (%

I,. CS

0 -1

-frame

O8

i o C O S 4~s

I 0

cos

Ot

fl) 1"=02

O6

O

T=OI

u X7

04

1

+

"13

OZ

I

0 I

0

011

0.2

013 014

I

0.5 XT

I

I

011 0i2 013 (14 0.5 XT I

Fig 3 The coefficient a as a funcnon oft andx T aty = 0 in the four chfferent reference frames of z-axis are shown for zero rapidity. Especially noteworthy is that for small values o f x T , a becomes negative in the s-channel frame. Fig. 4 compares the measured angular distributions'with our formula (3) integrated overy > 0 and over QT for QT > 1 GeV at one fixed value of the invariant mass of the lepton pair (Q = 5.5 GeV). One notes that in particular in the s-channel frame, where the z-axis is given as the actually observed recoil momentum, substantial deviations from a 1 + cos20 distribution occur. In the other three frames the values of a (eq. (2)) lie close to each other around a = 0.7 so that the distinction with 1 + cos20 is less pronounced here. This is in qualitatwe agreement with the trend of the data. In the different frames we find for a: s-channel t-channel u-channel Collins-Soper

: c~= 0.34 (expt. : a = 0.66 (expt. : a = 0.73 (expt. : a = 0.79 (expt.

[4]: 0.16 -+ 0.19), [4]: 0.65 +- 0.17), [4] : 1.42 +- 0.39), [4] : 1.47 + 0.39).

In the s and t-channel frames the agreement between theory and experiment xs very good, while in the uchannel and Colhns-Soper frames a is below the experimentally measured values. Because of the large experimental uncertainty and because we neglected the 388

I

o cos

-1

Ou

o COS Ocs

Fig. 4. The angular distribution of the lepton in the s-channel (top left), t-channel (top right), u-channel (bottom left) and m the Collins-Soper frame (bottom right). The full curve is the result of our calculation for Q = 5.5 GeV and QT > 1 0 GeV The data points are taken from ref. [4] and have Q > 3.5 GeV and QT > 1.0 GeV. primordial transverse m o m e n t u m of constituents (whose effect disappears at large QT) this qualitative agreement cannot at present be taken as definite proof of the QCD result. It would be interesting at this point to have data for fixed r and x T since a has a very specific behaviour with respect to these variables, as can be seen from fig. 3. This could provide more definite evidence for gluon effects in the production of lepton pairs. We have tried to see the influence on our results of different choices of quark distribution functions. In general, these do not produce different results. We have also looked at the effect of choosing a hard gluon distribution function of protons, proporUonal to (1 x) 3 as suggested by the recent BEBC analysis [8]. At large x T (say x T = 0.9) this results in an increase of the q u a r k - g l u o n scattering cross section term of about two orders of magnitude compared to the more standard parametrlzation using (1 - x) 5. This, however, results m a change of the value of a o f only about

Volume 80B, number 4,5

PHYSICS LETTERS

10% and clearly shows the stabihty of the numerical value of a wath respect to different parametrizations of the proton structure functions. It is a straightforward matter to calculate the expected azimuthal (~) distribution in the QCD framework. This will be done ia a subsequent paper [9]. In conclusion we have shown that the recent data on n - N -+ t~+~t- + agree well with the QCD expectation for the shape of the QT dependence of the cross section for QT > 1 GeV/c and also qualitatively well with the angular distributions calculated with different choices for z-axes. Since the angular distribution of the lepton is strongly dependent on the value of r and x T, measurements for fixed r and x T are highly desirable, as these could contribute to clarify the production mechanism of high mass lepton pairs.

References

15 January 1979

[2] S.D. Drell and T M. Yan, Phys. Rev. Lett. 25 (1970) 316. [3] G Altarelh, G. Pans1 and R Petronzio, Phys. Lett. 76B (1978) 351,356, H. Fntzsch and P Mmkowski, Phys. Lett. 73B (1978) 80, F. Halzen and D M. Scott, Umv. of Wisconsin preprint (00-881-21 6 1978); D. Politzer, Nucl. Phys. B129 (1977) 301, C.T. Sachrajda, Phys Lett. 73B (1978) 185; J. Hinchliffe and C.H. Llewellyn-Smith, Phys. Lett. 65B (1977) 281, K. KaIantle and R. Raitlo, Nucl. Phys. B139 (1978) 72. [4] K.J. Anderson et al., Enrico Fermi Institute preprmt No. EFI 78-38 (revised), submitted to the XIX Intern. Conf on High Energy Physics, Tokyo, Japan, 1978. [5] J.C. Collins and D.E. Soper, Phys. Rev. D16 (1977) 2219. [6] V. Barger and R Phdhps, Nucl. Phys. B73 (1974) 269. [7] S. Brodsky and G. Farrar, Phys. Rev. Lett. 31 (1973) 1153; V. Matveevet al., Nuovo Clmento Lett. 7 (1973) 719. [8] Aachen-Bonn-CERN-London-Oxford-SaclayCollaboration, P.C. Bosette et al., Oxford Umv preprmt ref. 16/78. [9] J. Cleymans and M. Kuroda, m preparaUon.

[1] K Kajantle, J Lmdfors and R. Raltlo, Phys. Lett. 74B (1978) 384.

389