Production of Jet Pairs at Large Relative Rapidity in Hadron-Hadron Collisions as a Probe of the Perturbative Pomeron

23 Pages·0.19 MB·English

by  W. J. Stirling

DTP/94/04
January 1994
Production of Jet Pairs at Large Relative Rapidity
in Hadron-Hadron Collisions as a Probe of the
4 Perturbative Pomeron
9
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n
a
W.J. Stirling
J

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1
Departments of Mathematical Sciences and Physics, University of Durham

1 Durham DH1 3LE, England
v
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Abstract
4
9
/ The production of jet pairs with small transverse momentum and large rel-
h
ative rapidity in high energy hadron-hadron collisions is studied. The rise of
p
- the parton-level cross section with increasing rapidity gap is a fundamental
p
e prediction of the BFKL ‘perturbative pomeron’ equation of Quantum Chro-
h modynamics. However, at fixed collider energy it is difficult to disentangle this
:
v effect from variations in the cross section due to the parton distributions. It
i
X is proposed to study instead the distribution in the azimuthal angle difference
r of the jets as a function of the rapidity gap. The flattening of this distribution
a
with increasing dijet rapidity gap is shown to be a characteristic feature of the
BFKL behaviour. Predictions for the Fermilab pp¯collider are presented.
1 Introduction
There is currently much interest in the QCD ‘perturbative pomeron’. This is the
phenomenon, obtained by resumming a certain type of soft gluon emission to all or-
dersintheleadinglogarithmapproximationusingtheBalitsky-Kuraev-Fadin-Lipatov
(BFKL) equation [1], which is supposed to produce a sharp rise at small x in deep
inelastic structure functions [2]. Recent measurements at HERA [3] may well show
the first evidence for this behaviour, although it is premature to draw any definitive
conclusions.
One of the difficulties with extracting information on the perturbative pomeron
from structure function measurements alone is that both perturbative and non-
perturbative effects are very likely intertwined in a non-trivial way, and therefore
obtaining information on the former requires some sort of model-dependent subtrac-
tion of the latter. For this reason, attempts have been made to find other quantities
which probe more directly the perturbative behaviour. In the context of deep inelas-
tic scattering, one can look for an associated jet of longitudinal momentum fraction
x′ x and comparable transverse momentum to the virtual photon momentum Q
Bj
≫
[4]. The BFKL behaviour is then reflected in the growth of the cross section with
log(x′/x ). Alternatively, Mueller and Navelet have shown [5] that in high-energy
Bj
hadron-hadron collisions one can utilize a two-jet inclusive cross section where the
jets have small transverse momentum and a large relative rapidity, ∆y. The rise in
the cross section with increasing ∆y is then controlled by the perturbative pomeron.
It is this latter process that we investigate here. We first define the cross section
introduced in Ref. [5], and show why it is difficult to measure under present experi-
mental conditions. We then consider an alternative but closely-related quantity, the
azimuthal angle correlation between jets at large relative rapidity, which we argue is
more promising from an experimental point of view. Predictions are presented for the
Fermilab pp¯ collider at √s = 1.8 TeV. Our results also apply to the corresponding
two-jet cross section in deep inelastic scattering as measured at HERA, although we
shall not pursue this issue here.
Consider the inclusive two-jet cross section in pp¯ (or pp) collisions at energy √s,
where each jet has a minimum transverse momentum M √s, and the jets are
≪
produced with equal and opposite large rapidity 1∆.1 Adapting the results of
±2
Ref. [5], the cross section for this can be written in the form
dσ
x G(x ,M2) x G(x ,M2) σˆ(α (M2),M2,∆) (1)
1 1 2 2 s
dy1dy2(cid:12)(cid:12)y1=−y2=12∆ ≃
(cid:12)
1The restriction(cid:12)(cid:12)to equal and opposite rapidities is not crucial and is made here only to simplify
the discussion.
1
where
4
G(x,µ2) = g(x,µ2) + (q(x,µ2)+q¯(x,µ2)) ,
9
q
X
2M M
x = x = cosh(1∆) e∆/2 ,
1 2 √s 2 ≃ √s
α C 2 π3
σˆ(α ,M2,∆) = s A 1+ a (α ∆)n +... . (2)
s π 2M2  n s 
(cid:18) (cid:19) n≥1
X
 
Note the use of the ‘effective subprocess approximation’, appropriate here because of
the dominance of small momentum transfer in the subprocess tˆchannel. The ... in
(2) refers to corrections outside the leading logarithm approximation implicit in (1),
i.e. terms of order αn∆n−1, αn∆n−2, ... and power-correction terms suppressed by
s s
powers of e−∆.
According to the BFKL ‘perturbative pomeron’ analysis, the subprocess cross
section M2σˆ is expected to rise at large ∆. The asymptotic behaviour, ignoring
possible ‘parton saturation’ effects [5], is predicted to be
M2σˆ eλ∆, as ∆ , (3)
∼ → ∞
with λ a number of order 0.5. This is the analogue of the expected x−λ growth of the
F structure function at small x [2].
2
Now with √s = 1.8 TeV and M 10 GeV, the maximum value of ∆ is of
∼
order 10, which might at first sight appear sufficiently large to test the predicted
BFKL behaviour.2 The problem, however, is the additional ∆ dependence in the
cross section induced by the x dependence of the parton distributions in (1). At large
∆, the behaviour of the cross section will be completely dominated by the x 1
→
suppression of the parton distributions. This can only be avoided by increasing the
energy √s of the collider as ∆ is increased, in such a way that x and x remain fixed,
1 2
which is not an easy proposition in practice.
There are at least two ways around this difficulty. First, one could argue that the
parton distributions at medium-to-large x are now sufficiently well known that they
can be factored out of the measured cross section with sufficient accuracy. While this
might be true for the quark component of G(x,µ2), the gluon component is much less
well known at large x. Even if it was, there is still a problem with scale dependence
— the cross section in (1) is not known beyond leading logarithm accuracy, and so
the choice of scale in the parton distributions is somewhat arbitrary.
A second possibility is to use the distribution in the azimuthal angle difference
between the two jets as a signature for BFKL behaviour. When ∆ is small, the cross
2In fact, the predicted asymptotic behaviour appears to set in at relatively modest values of ∆,
see Section 2 and Ref. [5].
2
section is dominated by the lowest order 2 2 scattering subprocesses and the jets
→
are back-to-back in the transverse plane. As ∆ increases, more and more soft gluons
with transverse momentum M are emitted in the rapidity interval between the two
∼
fast jets, and the azimuthal correlation is gradually lost until, asymptotically, there
is no correlation at all. The change in the overall weighting, due to variations in the
parton distributions as ∆ increases at fixed √s, has no effect on the shape of the
azimuthal distribution.
The study of the azimuthal correlation between the two jets at large rapidity is
the subject of the present analysis. We first of all derive the basic formulae, which
is an extension of the treatment in [5], and then make numerical predictions for
experimentally measurable quantities. We shall demonstrate that the weakening of
the correlation should already be observable at the Fermilab pp¯collider.
2 BFKL formalism for dijet production
We start by considering the subprocess cross section for inclusive two-jet (i.e. par-
ton) production. We are interested in jets produced with equal and opposite large
longitudinal energy, k E where E is the parton beam energy in the parton-parton
L
∼
centre-of-mass, and small transverse momentum k > M where M E is a fixed
T
≪
cut-off. The rapidity gap between the jets is then ∆y ∆ 2log(E/M) 1. In
≡ ≃ ≫
practice, M will be a number of order 10 GeV and the maximum value of E is set
by the kinematic limit √s/2.
At the Fermilab collider, a large fraction of such jet pairs are produced in gluon-
gluon collisions, and so in what follows we focus on the subprocess gg gg + X.
→
The quark initiated processes are reinstated afterwards using the effective subprocess
approximation. If we assume, to begin with, that we can ignore the running of the
strong coupling and take α = α (M2) as fixed, then the subprocess cross section of
s s
Eq. (1) can be written as [5]
α C 2 π3 π
σˆ(α ,M2,∆) = s A dφ F(φ,∆) . (4)
s (cid:18) π (cid:19) 2M2 Z−π
Here we have introduced the variable φ = π φ where φ is the azimuthal angle
jj jj
−
difference between the two jets. Thus φ = 0 corresponds to back-to-back jets in the
transverse plane. In what follows we will be particularly interested in the differential
distribution dσˆ/dφ, which is proportional to F.
The function F in Eq. (4) is simply the quantity f(k ,k ,∆) defined in Ref. [5],
T1 T2
integrated over M2 < k2 < at fixed φ. The latter function satisfies the BFKL
Ti ∞
equation (see Appendix). The solution for F is
1 +∞ α C
F(φ,∆) = einφ C (t) , t = s A ∆ , (5)
n
2π π
n=−∞
X
3
where
1 +∞ dz
C (t) = e2tχn(z),
n 2π −∞ z2 + 1
Z 4
χ (z) = Re[ψ(1) ψ(1(1+ n )+iz)] , (6)
n − 2 | |
and ψ(x) is the logarithmic derivative of the gamma function. Note that this re-
sult corresponds to a perturbative expansion in powers of α ∆ α log(E2/M2).
s s
∼
It is these leading logarithms which have been resummed by the BFKL equation.
Substituting back in the original expression for the cross section gives
dσ α C 2 π3
x G(x ,M2) x G(x ,M2) s A F(φ,∆) , (7)
1 1 2 2
dy1dy2dφ(cid:12)(cid:12)(cid:12)y1=−y2=21∆ ≃ (cid:18) π (cid:19) 2M2
(cid:12)
(cid:12)
with x = x = 2M cosh(1∆)/√s. Before performing a full numerical calculation
1 2 2
of this cross section, we discuss several important analytic results which follow from
Eqs. (5,6).
2.1 Comparison with exact lowest-order calculation
Application of the BFKL formalism only makes sense in a kinematic region where
the exact leading order (2 2) cross section is well-approximated by the first term
→
in the perturbation series on the right-hand side of Eq. (7). The latter is obtained
by setting t = 0 in Eq. (6), which gives
1 +∞ dz
C (0) = = 1
n 2π −∞ z2 + 1
Z 4
F(φ,∆) = δ(φ) . (8)
⇒
When this is substituted in Eq. (4), we obtain
M2 σˆ = 1πα2C2 . (9)
|LO 2 s A
Note that in the ∆ limit the subprocess cross section scales as 1/M2. In
→ ∞
this section we rederive this result starting from the exact (2 2) lowest order
→
cross section, which will enable us to assess how rapidly the asymptotic behaviour is
approached.
The two-jet inclusive cross section in leading order is given by
dσ 1
= x−1f (x ,µ2) x−1f (x ,µ2) (ab cd) 2 , (10)
dy dy dp2 16π2s 1 a 1 2 b 2 |M → |
1 2 T a,b,c,d=q,g
X X
4
where denotes the appropriate sums and averages over colours and spins. To begin
with, we restrict our attention to the gg gg subprocess. Setting y = y = 1∆
P → 1 − 2 2
gives
dσ xg(x,µ2) xg(x,µ2)
= (gg gg) 2 , (11)
dy1dy2dp2T(cid:12)(cid:12)(cid:12)y1=−y2=21∆ 256πp4T cosh4(12∆) X |M → |
where (cid:12)
(cid:12)
2p
x = T cosh(1∆) , (12)
√s 2
and the subprocess matrix element is evaluated at
tˆ 1 uˆ 1
= 1 tanh(1∆) , = 1+tanh(1∆) ;. (13)
sˆ −2 − 2 sˆ −2 2
(cid:16) (cid:17) (cid:16) (cid:17)
Integrating over the jet transverse momentum p > M then gives
T
dσ = |M(gg → gg)|2 p2T(max) dp2T [xg(x,µ2)]2 , (14)
dy1dy2(cid:12)(cid:12)(cid:12)y1=−y2=12∆ P256π cosh4(12∆) ZM2 p4T
(cid:12)
with p2(max) =(cid:12) 1scosh−2(1∆). By extracting a factor of M−2, and ignoring loga-
T 4 2
rithmic scaling violations in the parton distributions, the integral on the right-hand
side becomes a function of the dimensionless quantity X = 2M cosh(1∆)/√s.
2
dσ (gg gg) 2 1 X−2 du2 2
= |M → | xg(x,µ2) . (15)
dy1dy2(cid:12)(cid:12)(cid:12)y1=−y2=21∆ P256π cosh4(12∆) M2 Z1 u4 h i (cid:12)(cid:12)x=Xu
Next, cons(cid:12)(cid:12)ider the limit (cid:12)(cid:12)
√s
1 ∆ log . (16)
≪ ≪ M !
The first term on the right-hand side of Eq. (15) tends to a finite value of
(gg gg) 2
|M → | 1πC2α2 . (17)
256π cosh4(1∆) → 2 A s
P
2
The integral over the parton distributions is dominated by the contribution from the
lower limit, i.e.
X−2 du2 2 2
xg(x,µ2) Xg(X,µ2) (18)
u4 →
Z1 h i (cid:12)(cid:12)x=Xu h i
(cid:12)
Combining the results from Eqs. (17,18) g(cid:12)ives
dσ 1 2
1πC2α2 Xg(X,µ2) (19)
dy1dy2(cid:12)(cid:12)(cid:12)y1=−y2=12∆ ≃ 2 A s M2 h i
(cid:12)
(cid:12)
5
in agreement with the leading-order contribution in Eq. (7).
The next step is to investigate quantitatively how rapidly the asymptotic result
is attained in practice. We focus first on the matrix element part, Eq. (17). Figure 1
shows the ratio of the left-hand side (the exact result) to the right-hand side (the
asymptotic result), as a function of the rapidity gap ∆. Evidently, the two agree
to better than 10% for ∆ > 3.3. This is not unexpected, since we are testing here
the size of the ‘power correction’ terms of order e−∆. We can also test the effective
subprocess approximation, by comparing the qg qg and qq¯ qq¯ amplitudes,
→ →
scaled by 9/4 and (9/4)2 respectively, to the asymptotic gg gg result. These ratios
→
are shown as the dashed (qg) and dash-dotted (qq¯) lines in Fig. 1. The approach to
the limiting form is very similar to the gg amplitude, indicating that in the large ∆
region the effective subprocess approximation is valid.
We have already discussed how when M and √s are fixed the bulk of the ∆
dependence comes from the parton distributions. This is illustrated in Fig. 2, where
the cross section of Eq. (15), scaled by M2, is shown as a function of ∆. We have
chosen√s = 1.8TeV,andusedthelatest‘MRS(H)’partons[8]withΛ(4) = 230MeV.
MS
The scales in the running coupling and in the parton distributions are both set equal
to M. The solid curves correspond to the exact gg gg cross section, for the
→
representative values M = 10 GeV and M = 30 GeV. Also shown (dashed lines)
are the asymptotic cross sections defined by Eq. (19). There is a broad range of ∆
where the shape of the exact cross sections is reasonably well approximated. The
normalization, however, is too high by a factor of order two.3 This can be traced to
the fact that Eq. (18) is only valid when both X is small and the function xg(x,µ) is
slowly varying. If xg x−λ in the relevant x region, then an error of order (1+λ) is
∼
made in the normalization.
2.2 Comparison with exact O(α3) calculation for φ = 0
s
6
Expanding the exponential inside the integral in Eq. (6) allows us to read off the
differential φ-distribution at next-to-lowest order, i.e. O(α3):
s
dσˆ α C 3 π3
M2 = s A ∆ f(φ) , (20)
dφ(cid:12) π 2
(cid:12)NLO (cid:18) (cid:19)
(cid:12)
(cid:12)
where (cid:12)
1 +∞
f(φ) = einφ c , (21)
n
2π
n=−∞
X
3In the range2<∆<6 forM =10GeV the ratioofthe exactto the approximatecrosssection
always lies in the range 0.45 to 0.55.
6
and the Fourier coefficients are
1 +∞ dz
c = χ (z) = 2 [ ψ(1) ψ(1+ 1 n ) ] . (22)
n 2π −∞ z2 + 1 n − 2| |
Z 4
The coefficients can be calculated explicitly by contour integration in the complex z
plane:
4
c = c , c = c (n 0) ,
n −n n+2 n
− 2+n ≥
c = 0 , c = 4(log2 1) . (23)
0 1
−
The first point to note is that since c = 0, the integral of f(φ) vanishes, i.e.
0
M2 σˆ = 0 . (24)
|NLO
To calculate the φ distribution we have to substitute the c coefficients into Eq. (21)
n
and sum the Fourier series. We have found it easier, however, to start from the
solution to the BFKL equation in transverse momentum space, where for φ = 0 (see
6
Appendix),
M2 ∞ ∞ dk2 dk2
f(φ) = T1 T2
2π ZM2ZM2 kT21kT22(kT21 +kT22 −2kT1kT2cosφ)
1
= [log(2(1 cosφ)) + (π φ)cotφ] for 0 < φ π , (25)
π − − ≤
and f( φ) = f(φ). The singular behaviour f φ−1 as φ 0, which corresponds
− ∼ →
to almost back-to-back jets, arises from soft emission of the third (gluon) jet [1]. It
is cancelled in the total cross section by a virtual gluon contribution proportional to
δ(φ). This can be taken into account by invoking the standard ‘plus prescription’,
i.e. f(φ) [f(φ)] where
+
→
π π
dφ g(φ) [f(φ)] = dφ (g(φ) g(0)) f(φ) . (26)
+
Z−π Z−π −
It is important to note that the distribution we have derived is a leading loga-
rithm result, in that it corresponds to retaining only the leading α ∆ contribution
s
and ignoring corrections of order α , α e−∆ etc. To study the validity of this approx-
s s
imation, we can compare the result in Eq. (25) with an exact calculation based on
the complete gg ggg matrix element. The analogue of the 2 2 cross section
→ →
Eq. (14) is
1
dσ 1 ∆
= dk2 dk2 2 dy
dy1dy2dφ(cid:12)(cid:12)(cid:12)y1=−y2=12∆ 512π4 ZM2 T1 ZM2 T2 Z−12∆ 3
(cid:12) [x g(x ,µ2) x g(x ,µ2)]
(cid:12) 1 1 2 2
×
sˆ−2 (gg ggg) 2 (27)
× |M → |
X
7
where
k2 = k2 +k2 2k k cosφ
T3 T1 T2 − T1 T2
1 1
x1 = (kT1e2∆ +kT2e−2∆ +kT3ey3)/√s
1 1
x2 = (kT1e−2∆ +kT2e2∆ +kT3e−y3)/√s
sˆ = k2 +k2 +k2 +2k k cosh∆
T1 T2 T3 T1 T2
+2k k cosh(1∆ y )+2k k cosh(1∆+y ) .
T1 T3 2 − 3 T2 T3 2 3
(28)
According to the BFKL analysis, this cross section should have the asymptotic limit
dσ α C 3 π3
= s A ∆ f(φ) [Xg(X,µ2)]2 . (29)
dy1dy2dφ(cid:12)(cid:12)(cid:12)y1=−y2=12∆ (cid:18) π (cid:19) 2M2
(cid:12)
We can see how this(cid:12)behaviour arises: the matrix element in (27) is dominated by
configurations where the third gluon jet is produced centrally, i.e. y 1∆. With
| 3| ≪ 2
the matrix element approximated by its value at y = 0, the y integral gives the
3 3
overall factor of ∆, and the remaining k integrals give the function f(φ), as in
Ti
Eq. (25). The parton distributions are again dominated by their values at x = x =
1 2
X = 2M cosh(1∆)/√s, as for the leading order cross section.
2
We can study the approach to the asymptotic result in two stages. First, at the
subprocess level, we can compare the cross section
1
dσˆ 1 ∞ ∆
= dk2 dk2 2 dy sˆ−2 (gg ggg) 2
dy1dy2dφ(cid:12)(cid:12)(cid:12)y1=−y2=12∆ 512π4 ZM2 T1 T2 Z−21∆ 3 X |M → |
(cid:12) (30)
(cid:12)
with the asymptotic form
dσˆ α C 3 π3
s A
= ∆ f(φ) . (31)
dy1dy2dφ(cid:12)(cid:12)(cid:12)y1=−y2=21∆ (cid:18) π (cid:19) 2M2
(cid:12)
Figure 3 compares the φ d(cid:12)istribution calculated from Eq. (30) with the function
f(φ), for ∆ = 4,8,12. The exact calculation has been scaled by the same factors
multiplying f(φ)ontheright-handsideofEq.(31),sothattheexactandapproximate
distributions should coincide in the limit ∆ . The results confirm the approach
→ ∞
to the asymptotic distribution. At small φ, the asymptotic behaviour is already a
good approximation for ∆ = 4, while the convergence is slower at large φ. This
is presumably because at large φ the third gluon can have significant transverse
momentum and energy, thus invalidating the approximations under which the BFKL
equation is derived and leading to sizeable sub-asymptotic corrections.
8
Figure 4 makes the same comparison at the cross-section level, for pp¯ collisions
with √s = 1.8 TeV, M = 10 GeV and ∆ = 4,6,8. The curves correspond to the
exactandasymptoticcrosssectionsofEqs.(27)and(29)respectively. Theconstraints
x ,x 1 now give upper limits on the transverse momentum integrals. As for the
1 2
≤
leading-order case, the normalization is overestimated by the asymptotic form, but
evidently the shape of the φ distribution is reasonably well approximated even at
moderate ∆. The rapid change in the distribution with increasing ∆ supports our
assertionthattheazimuthaldistributionofthejetsshouldbeamorereliableindicator
of BFKL behaviour than the overall φ-integrated cross section.
2.3 All-orders φ distribution
Through next-to-lowest order, then, we have
dσˆ(E,M) α C 2 π3
M2 = s A F(φ,∆)
dφ π 2
(cid:18) (cid:19)
α C
s A
F(φ,∆) = δ(φ) + ∆ [f(φ)] + ... , (32)
+
π
(cid:18) (cid:19)
where the ... represent terms O((α ∆)n) with n 2. Formally, the first few terms
s
≥
in this series will be a good approximation to the all-orders distribution provided
∆ 1 and α ∆ 1. As the second of these inequalities is relaxed, higher order
s
≫ ≪
terms become more and more important. The inclusion of all terms of the form
(α ∆)n, via Eqs. (5,6), requires a numerical calculation and will be discussed below.
s
First, we consider the limit α ∆ 1, where an analytic approximation can again be
s
≫
obtained.
To calculate the distribution in the asymptotic limit α ∆ , we return to
s
→ ∞
Eq. (6) and use a saddle-point method [1] to evaluate the Fourier coefficients in the
large t = α C ∆/π limit. We expand the χ (z) about the saddle point at z = 0,
s A n
χ (z) = a b z2 +... , (33)
n n n
−
where
2
a = 2log2, a = 0, a = a , (n 0)
0 1 n+2 n
− 1+n ≥
8
b = 7ζ(3), b = ζ(3), b = b , (n 0) (34)
0 1 n+2 n
− (1+n)3 ≥
from which the asymptotic t behaviour follows,
→ ∞
1
C (t) e2ant . (35)
n
∼ 1πb t
2 n
q
9