NLO QCD corrections to Drell-Yan processes in the SANC framework

0.24 MB·English

by  A. Andonov

NLO QCD corrections to Drell-Yan processes in the
SANC framework
9
A. Andonov1, A. Arbuzov2,3, S. Bondarenko2,3, P. Christova3, V. Kolesnikov3,
0
0 G. Nanava4, R. Sadykov3
2

n 1Bishop Konstantin Preslavsky University, Shoumen, Bulgaria
a
J 2Bogoliubov Laboratory of Theoretical Physics, JINR

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1 3Dzhelepov Laboratory of Nuclear Problems, JINR

ul. Joliot-Curie 6, RU-141980 Dubna, Russia
]
h
4Physikalisches Institut der Universit¨at Bonn,
p
-
p Nussallee 12, D-53115, Bonn, Germany
e
h
[

Abstract
1
v
5 NLO QCD corrections to charged and neutral current Drell-Yan processes and their
8 implementation in the computer system SANC are considered. On the partonic level
7
both quark-antiquark and quark-gluon scattering channels are taken into account. Sub-
2
. tractions of the collinear singularities in the massive case are compared with ones in
1
0 the MS scheme. Results of SANC on the hadronic level are presented. Comparison with
9 results of the MCFM package is shown.
0
:
v
i 1 Introduction
X
r
a
Charged and neutral current Drell–Yan (DY) processes [1] on the eve of the first proton
collisions at the LHC become very important for precision tests of the Standard Model.
They are easily detected and will provide standard candles for detector calibration during
the first stage of LHC running. They will be also used for extraction of partonic density
functions (PDF)inthekinematical regionwhich hasnot beenaccessed by earlier experiments.
Therefore it is crucial to control the theoretical predictions for production cross sections and
kinematic distributions of these processes.
In the previous paper [2] we presented a part of the QCD sector of our computer system
SANC [3] (http://sanc.jinr.ru/ and http://pcphsanc.cern.ch/ ) where the NLO QCD processes
are treated. There we considered the implementation into SANC the calculation of the charged
(CC) and neutral (NC) current quark–antiquark Drell–Yan processes on the partonic level
and briefly presented some numerical results for the hadronic level. The QCD corrections to
DY processes are known in the literature for many years, see Refs. [5, 6, 8]. Recently the
corresponding NNLO corrections for differential distributions have been received [9, 10].
1
In this paper with respect to Ref. [2] we add into consideration quark–gluon and gluon-
antiquark Drell–Yan processes on the partonic level side by side with the quark–antiquark
ones. We implemented into SANC the calculation of QCD corrections to DY processes on the
hadronic level. Working with massive quarks, we regularized the collinear singularities by
masses of quarks. But on the hadronic level we have to remove these collinear singularities
to avoid double counting, because they are already included in PDF. Therefore we compared
analytical results obtained in our treating of collinear singularities with analogous results
obtained in the MS scheme calculated in the n-dimensional phase space. In this way we
extracted the subtraction terms needed to remove the collinear singularities in our massive
quarks case. We show here also comparison with the corresponding results of the MCFM [11]
package.
One-loop electroweak radiative corrections were computed for the DY processes by SANC
in Refs. [12, 13, 14] and extensively compared with results of other groups, see e.g. Refs. [15,
16, 17].
The paper is organized as follows. In the second section we calculated the hard gluon
bremsstrahlung contributions both to charged and neutral current Drell-Yan processes on a
quark-parton level in the massless MS scheme and compared with our massive quarks results.
Weused FORM3.1 [18]foranalyticalcalculations. Inthethirdsectionwecalculatedthequark–
gluon and gluon-antiquark DY processes with charged and neutral currents in our massive
quarks treating and draw parallel with calculation in the massless MS scheme. In Conclusion
we discuss the results and give some illustrations of DY distributions at the hadronic level.
2 Quark–antiquark Drell–Yan processes
2.1 Massive quarks treatment of the hard gluon contribution
Working with massive quarks we showed in the previous paper [2] the NLO QCD corrections
¯
due to the hard gluon bremsstrahlung of charged current (CC) d(p ) + u(p ) W
1 2
¯ ¯ → →
ν (p )+ℓ(p )+g(p ) and neutral current (NC) q¯(p )+q(p ) [A,Z] ℓ(p )+ℓ(p )+g(p )
ℓ 3 4 5 1 2 3 4 5
→ →
Drell–Yan processes were obtained in the form
α zmax sˆ µ2 µ2
σˆCC = S dz σˆCC(zsˆ) P (z) 2ln +ln +ln 2 , (1)
Hard 2π Zzmin 0 qq " µ2! m2u! m2d!− #
α zmax sˆ µ2
σˆNC = S dzσˆNC(zsˆ)P (z) ln +ln 1 , (2)
Hard π Zzmin 0 qq " µ2! m2q!− #
where µ is the factorization scale. The energy of the emitted gluon is
√sˆ
p0 = (1 z), (3)
5 2 −
2
′
s
′
where z = , sˆ = 2(p .p ), p and p are momenta of the incoming quarks and s =
1 2 1 2
sˆ −
(p +p )2 is the invariant mass of the outgoing leptons1;
3 4
−
1+z2
P (z) = C (4)
qq F
1 z
−
is the leading order (LO) quark-quark splitting function.
2ω¯
Inthechargedcurrentcasewehavethelimitsofintegrationovervariablez: z = 1 ,
max
−√sˆ
m2
whereω¯ √sˆandz = ℓ,m beingthemassofchargedleptons. Theauxiliaryparameter
min ℓ
≪ sˆ
ω¯ is the maximal energy of a soft gluon in the c.m.s. of the incoming partons. In the neutral
4m2
current case we have correspondingly the same z and z = ℓ.
max min
sˆ
The cross sections in the Born approximation read
G2 M4 sˆ 3m2 m6
σˆCC(sˆ) = V 2 F W 1 ℓ + ℓ , (5)
0 | ud | 18π sˆ M2 2 − 2sˆ 2sˆ2!
| − W |
4πα2 1 m2 m2
σˆNC(sˆ) = β(sˆ,m2) f 1 ℓ V (sˆ)+ ℓV (sˆ) , (6)
0 3sˆ ℓ "3 − sˆ ! 0 sˆ a #
where sˆ= (p +p )2, p and p are 4-momenta of the initial quarks; M2 = M2 iM Γ ;
− 1 2 1 2 W W − W W
4m2
β(sˆ,m2) = 1 ℓ. Here we denoted f
ℓ s − sˆ
V (sˆ) = Q2Q2 +2Q Q χ (sˆ) v v + χ (sˆ) 2 v2 +I(3)2 v2 +I(3)2 ,
0 q ℓ q ℓ | Z | q ℓ | Z | q q ℓ ℓ
(cid:16) (cid:17)(cid:18) (cid:19)
V (sˆ) = V (sˆ) 2 χ (sˆ) 2 v2 +I(3)2 I(3) 2,
a 0 − | Z | q q ℓ
v = I(3) 2Q sin2(cid:16)θ , (cid:17)(cid:16)v =(cid:17)I(3) 2Q sin2θ . (7)
q q − q W ℓ ℓ − ℓ W
The Z/γ propagator ratio χ (sˆ) with sˆ–dependent or constant Z-width is
Z
sˆ 1
χ (sˆ) = . (8)
Z Γ 4sin2θ cos2θ
s M2 +isˆ Z W W
− Z M
Z
We see in Eqs. (1) and (2) that the collinear singularities appear as quark mass singularities.
2.2 Hard gluon contribution in MS scheme with massless quarks
Because the collinear singularities calculated in the MS scheme with massless quarks are
already included in PDF, we have to find which terms in our massive quarks treatment of the
cross sections have to be subtracted to avoid the double counting. Therefore we calculate the
1We use the ( ,+,+,+) metrics, p=(p0,p~).
−
3
same cross sections in the MS scheme following the well known [19] manner of working with
massless quarks and compare with our results.
Calculating the three particle phase space element of the hard gluon emission in the n-
dimensional phase space we used a cascade in two steps:
′
ds
dΦ(3) = dΦ(2)dΦ(2). (9)
2π 1 2
¯ ∗ ′
For the first step of the charged current process d(p )+u(p ) W (Q)+g(p ) we obtained
1 2 5
→
the same formula as in [19]:
1 4πµ2 ε 1 1
Φ(2) = zε(1 z)1−2ε dy y−ε(1 y)−ε, (10)
1 8π s′ ! Γ(1−ε) − Z0 −
where we introduced in addition to z the variable y
1+cos(θ )
g
y = . (11)
2
Here θ is an angle between vectors p~ and ~p , the angle of the emitted gluon.
g 1 5
The phase space element of the second step W∗(Q′) ν (p )+ℓ+(p ) of the cascade is:
ℓ 3 4
→
1 4πµ2 ε 1 m2 1−2ε 1 2π
Φ(2) = 1 ℓ dy y −ε(1 y )−ε dϕ , (12)
2 16π2 s′ ! Γ(1−ε) − s′ ! Z0 R R − R Z0 R
where
1+cos(θR)
yR = , (13)
2
and θR is an angle between the charged lepton and gluon in the rest frame of the outgoing
leptons.
HavinginmindthatthecrosssectionofthechargedcurrentprocessinBornapproximation
has the form
G2M4sˆ 4πµ2 ε 1 m2 2−2ε 1
σˆCC(sˆ,ε) = V 2 F W 1 ℓ dy y −ε(1 y )−ε
0 | ud | 6π| sˆ−MW2 |2 sˆ ! Γ(1−ε) − sˆ ! Z0 0 0 − 0
m2 1
y 1 ℓ fy (1 y ) ε , (14)
0 0 0
" − − sˆ ! − − 2 #
we obtained a factorized expression of the hard gluon NLO correction to the charged current
process.
σˆCC (ε) = αS C zmax dz σˆCC(zsˆ,ε) 4πµ2 ε 1 zε(1 z)−2ε 1dy y−ε(1 y)−ε
Hard 2π F Zzmin 0 zsˆ ! Γ(1−ε) − Z0 −
1 1 1 ε
1+ (1 z) +(1 z) 1 , (15)
"y(1−y) (cid:18)1−z − 2 − (cid:19) − − − 2y(1−y)!#
4
Integration over y gives
σˆCC (ε) = αS zmaxdzσˆCC(zsˆ,ε) 4πµ2 ε Γ(1−ε) zε(1 z)−2εP (z) 2 . (16)
Hard 2π Zzmin 0 zsˆ ! Γ(1−2ε) − qq (cid:18)−ε(cid:19)
1
Onecansee thatthecollinear divergence appearshereasapole . To comparethisexpression
ε
with the analogous expression (1) where the collinear divergence manifests itself in the form
sˆ sˆ
of logarithms ln , ln one have to take the limit ε 0 . Then one obtains an
m2u! m2d! →
expression
α zmax 2 sˆ
σˆCC = S dz σˆCC(zsˆ) P (z) +2ln +4ln(1 z) (17)
Hard 2π Zzmin 0 qq −ε¯ µ2! − !
to be compared with the corresponding expression (1). We see which terms in the expression
1
(1) correspond to the collinear divergent term which in MS scheme has to be subtracted
−ε¯
from the hard gluon contribution to the considered process because it is already included into
PDF.
So, on the quark-parton level we have subtract from σˆCC (1), the following expression:
Hard
α zmax µ2 µ2
σˆCC (µ2) = S dz σˆCC(zsˆ) P (z) ln +ln 2 4ln(1 z) . (18)
HSubtr 2π Zzmin 0 qq " m2u! m2d!− − − #
Factorization properties and general relations between amplitudes with massive and massless
partons can be found in Ref. [20].
For the neutral current process calculating the three particle phase space element (9) of
the hard gluon emission in the n-dimensional phase space we obtain the same result (10) for
∗ ′
the first step q¯(p )+q(p ) γ,Z (Q)+g(p ). The phase space element of the second step
1 2 5
γ,Z ∗(Q′) ℓ−(p )+ℓ+→(p {) of t}he cascade is similar to (12):
3 4
{ } →
1 4πµ2 ε 1 1 2π
Φ(2) = β1−2ε(s′,m ) dy y −ε(1 y )−ε dϕ . (19)
2 16π2 s′ ! Γ(1−ε) ℓ Z0 R R − R Z0 R
Analogously, we obtained a factorized expression for the hard gluon NLO correction to the
neutral current process. It has exactly the same structure as Eq. (15), but the cross section
in Born approximation σˆNC(zsˆ,ε) of the neutral current process has a different form:
0
4πα2 4πµ2 ε 1 1
σˆNC(sˆ,ε) = β1−2ε(sˆ,m2) dy y −ε(1 y )−ε
0 3sˆ ℓ sˆ ! Γ(1−ε) Z0 0 0 − 0
1 m2
V (sˆ) β2(sˆ,m2)y (1 y )+(1 ε) ℓ
" 0 − ℓ 0 − 0 − 2 − sˆ !!
m2 1
+V (sˆ)(1 ε) ℓ +A (sˆ)(1 ε)(1 2ε)β(sˆ,m2)( y ) , (20)
a − sˆ 0 − − ℓ 2 − 0 #
where
A (sˆ) = 2Q Q χ (sˆ) I(3)I(3) + χ (sˆ) 24v v I(3)I(3). (21)
0 q ℓ | Z | q ℓ | Z | q ℓ q ℓ
5
Integration over y gives for σˆNC (sˆ,ε) the same result as (16). So, in the limit ε 0 we come
Hard →
to the expression almost the same as (17),
α zmax 2 sˆ
σˆNC = S dz σˆNC(zsˆ) P (z) +2ln +4ln(1 z) . (22)
Hard 2π Zzmin 0 qq −ε¯ µ2! − !
1
Comparison of this expression where the collinear divergence appears as a pole , with
ε
(2) where the collinear divergence manifests itself in the form of logarithms permits us to find
what expression one has to subtract from σˆNC (2), namely
Hard
α zmax µ2
σˆNC (µ2) = S dz σˆNC(zsˆ) P (z) ln 1 2ln(1 z) , (23)
HSubtr π Zzmin 0 qq " m2q!− − − #
where m is the mass of the pair quark and antiquark coming from the both protons.
q
2.3 Virtual and soft gluon contribution
Working in our massive quark treatment we obtained for the sum of virtual and soft gluon
contributions an expression free from infrared divergences. In the case of charged current
processes we have
α 3 4ω¯2 sˆ sˆ π2
σˆCC +σˆCC = S C σˆCC(sˆ) +ln ln +ln 2 1 . (24)
Virt Soft 2π F 0 ( 2 sˆ !!" m2u! m2d!− #− − 3 )
Correspondingly, in the case of neutral current processes we have
α 3 4ω¯2 sˆ 1 π2
σˆNC +σˆNC = S C σˆNC(sˆ) +ln ln 1 . (25)
Virt Soft π F 0 ( 2 sˆ !! " m2q!− #− 2 − 6 )
One can find the collinear divergent expressions to be subtracted from these virtual and
soft gluon contributions because they are already included into PDF, taking the correspond-
ing expressions which one has subtract from the hard gluon contributions to the considered
processes. But one has to take them with opposite sign, to substitute the argument of Born
cross sections taken z = 1, namely
σˆCC(zsˆ) = σˆCC(sˆ) and σˆNC(zsˆ) = σˆNC(sˆ) (26)
0 ⇒ 0 0 ⇒ 0
and to integrate over z from 0 to z . In this way we obtained the expressions to be
max
subtracted from virtual and soft gluon contributions.
For charged current processes collinear divergent subtraction is:
α zmax µ2 µ2
σˆCC (µ2) = S σˆCC(sˆ) dz P (z) ln +ln 2 4ln(1 z) . (27)
SVSubtr −2π 0 Z0 qq " m2u! m2d!− − − #
And for neutral current processes it is:
α zmax µ2
σˆNC (µ2) = S σˆNC(sˆ) dz P (z) ln 1 2ln(1 z) , (28)
SVSubtr − π 0 Z0 qq " m2q!− − − #
6
Integration over z gives the expressions to be found, namely, first - a subtraction for the
charged current contribution:
α 3 4ω¯2 µ2 µ2
σˆCC (µ2) = S C σˆCC(sˆ) +ln ln +ln 2
SVSubtr 2π F 0 ( 2 sˆ !!" m2u! m2d!− #
4ω¯2
+7 ln2 (29)
− sˆ !)
and second - a subtraction for the neutral current contribution:
α 3 4ω¯2 µ2 7 1 4ω¯2
σˆNC (µ2) = S C σˆNC(sˆ) +ln ln 1 + ln2 . (30)
SVSubtr π F 0 ( 2 sˆ !!" m2q!− # 2 − 2 sˆ !)
One can see that subtractions (29) and (30) really subtract the collinear divergent terms in
the expressions of virtual and soft gluon contributions (24) and (25), correspondingly.
3 Quark–gluon Drell–Yan processes
3.1 Charged current quark–gluon processes
In the framework of the Drell–Yan process pp W ℓν we have to take into account
ℓ
→ →
the presence of gluons in the protons. Therefore we consider on the quark-parton level the
attendant processes with incoming gluon u(p ) + g(p ) d(p ) + ν (p ) + ℓ+(p ), see the
2 5 1 ℓ 3 4
FeynmandiagramsontheFig.1,and d¯(p )+g(p ) u¯(p )→+ν (p )+ℓ+(p )(similardiagrams).
1 5 2 ℓ 3 4
→
d(p ) ℓ+(p ) d(p ) ℓ+(p )
1 4 1 4
g(p )
5
g(p )
5
(W+,φ+) (W+,φ+)
u(p ) ν (p ) u(p ) ν (p )
2 ℓ 3 2 ℓ 3
Figure 1: Charged current diagrams with coming gluon.
The contributions of these processes do not contain infrared divergences but have quark
mass singularities.
The three particle phase space element of the process u g d ℓ+ ν can be treated as a
ℓ
→
cascade analogously to (9) but here gluon g is incoming and d - quark is outgoing. The phase
∗ ′
space element of the first step g(p )+u(p ) W (Q)+d(p ) of the cascade is the same as
5 2 1
→ ′
s
in Eq. (10). Variable z has the same meaning, z = , and s′ = (p +p )2 is the invariant
3 4
sˆ −
mass of outgoing leptons, but here sˆ= 2(p .p ). Variable y has the same form (11) and θ
2 5 g
−
7
is the angle between vectors p~ and p~ , but now p~ is the momentum of the incoming gluon
1 5 5
and ~p is the momentum of the outgoing quark.
1
The phase space of the second step W∗(Q′) ν (p )+ℓ+(p ) of the cascade, obviously,
ℓ 3 4
→
has the same form (12).
In our massive quarks treatment in the 4-dimensional phase space we obtain the cross
section of the process u g d ν ℓ+ in the form factorized to the Born cross section of the
ℓ
corresponding quark-antiqu→ark process u d¯ ν ℓ+:
ℓ
→
α zmax sˆ 7 9 1
σˆCC = S dz σˆCC(zsˆ) P (z) ln +2 ln(1 z) + z , (31)
ug 2π Zzmin 0 ( qg " m2d! − − 4# 8 − 4 )
where
P (z) = T z2 +(1 z)2 (32)
qg f
−
h i
is the quark-gluon splitting function and T = 1. We neglected the quark masses except
f 2
of in the logarithm, where we have a mass of the outgoing quark, namely we face a mass
2
m
d
singularity. Integration over z here is up to z = 1 1.
max
− √sˆ! ≈
In the MS scheme with massless quarks working in the n-dimensional phase space we
obtain a factorized expression:
σˆCC(ε) = 3αS CF zmaxdzσˆCC(zsˆ,ε) 4πµ2 ε zε(1−z)−2ε 1dy y−ε(1 y)−ε
ug 8π (1−ε) Zzmin 0 zsˆ ! Γ(1−ε) Z0 −
1 1 1
z2 z + (1 ε) +z z2 +(1 z)ε+ y(1 z)2(1 ε) . (33)
× (y (cid:18) − 2 − (cid:19) − − 2 − − )
We divided here by 1 ε because of the number of the gluon spin projections in the
−
n-dimensional phase space is n 2 = 2(1 ε). After integration over y we have
− −
α zmax 4πµ2 ε Γ(1 ε) zε(1 z)−2ε
σˆCC(ε) = S dzσˆCC(zsˆ,ε) − −
ug 2π Zzmin 0 zsˆ ! Γ(1−2ε) 1−ε
1 3 1 9 1 7 3
P (z) ε + z +ε z . (34)
qg
× −ε − 4 − 4 8 − 4 8 − 4
(cid:26) (cid:20) (cid:21) (cid:18) (cid:19)(cid:27)
1
In this expression the collinear divergence appears as a pole . It has to be compared with
ε
the analogous expression (31) where the collinear divergence appears as a mass singularity.
In the limit ε 0 we have
→
1 4πµ2 ε Γ(1 ε) zε(1 z)−2ε 1 sˆ
− − = 1+ln +2ln(1 z)+ (ε). (35)
− ε zsˆ ! Γ(1 2ε) 1 ε −ε¯− µ2! − O
− −
In this way we obtain
α zmax 1 sˆ 7 9 1
σˆCC(ε) = S dzσˆCC(zsˆ) P (z) +ln +2ln(1 z) + z . (36)
ug 2π Zzmin 0 (cid:26) qg "−ε¯ µ2! − − 4# 8 − 4 (cid:27)
8
Comparing with (31) we see that the collinear divergent term which in MS scheme has to be
subtracted from the cross section of the considered process (because it is already included
into PDF of gluons) in our treatment has the form
α 1 µ2
σˆCC = S dz σˆCC(zsˆ) P (z) ln . (37)
ugSubtr 2π Zzmin 0 qg " m2d!#
Here m is the mass of the outgoing quark.
d
We have the same situation with the process g d¯ u¯ ν ℓ+. Analogously we obtain that
ℓ
→
from the cross section of this process we have to subtract the corresponding expression:
α 1 µ2
σˆCC = S dz σˆCC(zsˆ) P (z) ln . (38)
dgSubtr 2π Zzmin 0 qg " m2u!#
In this process anti-up quark is outgoing and it can move collinearly with the gluon, so the
logarithm is from the mass m of the u-quark.
u
3.2 Neutral current quark-gluon processes
In the case of the DY processes pp [A,Z] ℓ−ℓ+ we consider on the quark-parton
level the additional process q(p )+g(p→) q(p )→+ℓ−(p )+ℓ+(p ) (corresponding Feynman
2 5 1 3 4
diagrams are similar to the Fig.1) and proc→ess q¯(p )+g(p ) q¯(p )+ℓ−(p )+ℓ+(p ) with an
1 5 2 3 4
→
incoming gluon. The contributions of these processes also do not contain infrared divergences
but have quark mass singularities. We considered these processes analogously as the charged
current quark–gluon processes. Difference is only in the Born cross section to which they are
factorized.
In our massive quarks treatment in the 4-dimensional phase space the cross section of
both kind of neutral current quark–gluon processes have equal form:
α zmax sˆ 3 9 1
σˆNC = S dzσˆNC(zsˆ) P (z) ln 1+2 ln(1 z) + z , (39)
qg 2π Zzmin 0 ( qg " m2q!− − − 4# 8 − 4 )
where m is the mass of incoming as well as outgoing quark (or anti-quark). The upper limit
q
m2 2
q
of integration over z is the same, namely z = 1 1. Here we also neglected the
max
− √sˆ! ≈
quark masses except of under the logarithm.
IntheMSschemewithmasslessquarksworkinginthen-dimensionalphasespaceweobtain
the same expressions (33), (34), (36). as those of charged current quark–gluon processes.
Analogously comparing gave us that the collinear divergent term which in MS scheme has
to be subtracted from the cross section of the both considered processes, because is already
included into PDF of gluons, in our treatment has a form
α 1 µ2
σˆNC = S dz σˆNC(zsˆ) P (z) ln . (40)
qgSubtr 2π Zzmin 0 qg " m2q!#
9
4 Numerical calculations on hadronic level
4.1 Hadronic level kinematics
In the c.m.s. of the quark-quark or quark-gluon pair (see Fig.2a) we have (neglecting masses
of quarks):
√sˆ
p0 = p0 = ~p = ~p = , p~ = p~ , sˆ= 2p p . (41)
1 2 | 1 | | 2 | 2 2 − 1 − 1 · 2
p~
3 p~
3
~p1 θq p~1N θN
z θ z
θ p~ N p~
q 2 2N
~p
p~ 4
4
a) b)
Figure 2: a) Quark c.m.s. b) Proton c.m.s. (Born approximation).
In the c.m.s. of protons p,p (see Fig.2b) we have
√s
p0 = p0 = p~ = p~ = , p~ = p~ , s = 2p p
1N 2N | 1N | | 2N | 2 2N − 1N − 1N · 2N
p = x p , p = x p , sˆ= 2p p = 2p p x x = sx x . (42)
1 1 1N 2 2 2N − 1 · 2 − 1N · 2N 1 2 1 2
Equations of transition from quarks c.m.s to protons c.m.s. for any momentum have the form
Qˆ0 = γ Q0 βQz , Qˆz = γ Qz βQ0 , (43)
N − N N − N
(cid:16) (cid:17) (cid:16) (cid:17)
where
x +x x x
1 2 1 2
γ = , γβ = − . (44)
2√x x 2√x x
1 2 1 2
Replacing the scalar products
p p = p 0p 0 (1 cosθ ), p p = p 0p 0 (1+cosθ ), (45)
1 3 1 3 q 2 3 2 3 q
· − − · −
where θ is an angle between the 3-momenta of the quark ~p and neutrino ~p , from quarks
q 1 3
c.m.s. to protons c.m.s.
p p = x p p = x p0 p 0 (1 cosθ ), p p = x p p = x p0 p 0 (1+cosθ ),
1· 3 1 1N· 3 − 1 1N 3 − N 2· 3 2 2N· 3 − 2 2N 3 N
(46)
and the angle θ is between 3-momenta of the proton p~ and neutrino ~p , after some algebra
N 1N 3
we come to the relation between angles θ and θ
q N
x x +(x +x )cosθ
2 1 1 2 N
cosθ = − . (47)
q
x +x +(x x )cosθ
1 2 2 1 N
−
10