a r X i v :h e p -p h /0307215v 1 16 J u l 2003
McGILL-03-02UA/NPPS-02-03
Comparison of approximate to exact next-to-next-to leading order corrections for Higgs and pseudoscalar Higgs boson
production
A.P.Contogouris a,b,?and P.K.Papachristou b,?(a)Department of Physics,McGill University Montreal,Quebec,H3A 2T8,CANADA
(b)Nuclear and Particle Physics,University of Athens
Panepistimiopolis,Athens 15771,GREECE
Abstract
Recently obtained NNLO exact corrections for Higgs and Pseudoscalar Higgs boson production in hadron colliders are compared with approximate ones.As shown before,it is found that there is a range of a proper variable where these corrections di?er little.
Some time ago it was argued that for processes involving structure functions and/or fragmentation functions,over a range of a proper kinematic variable w,there is a part that dominates the next-to leading order(NLO)correction and that this part contains the distributionsδ(1?w)and[ln n(1?w)/1?w)]+n=0,1[1].Subsequently this argument was extended to the then existing next-to-next-to leading order(NNLO)calculations, namely Drell-Yan(D-Y)production of lepton pairs(q+
S is the total c.m.energy of the initial hadrons.Our approach requires inclusion of the regionτlarge(inclusion ofτnear1).Since experiment excludes values of m H(or
√
m A)≤100GeV,we have to consider
S≥2TeV would require m H(or m A)well exceeding1TeV,which would
render questionable the?eld-theoretic approach.We then have considered a nominal en-ergy of
√
p→H+X)mediated via the subprocess gg→H?and we consider the cross-section
σh
1+h2→H+X (m2H,S)=
1
dx1dx2f g/p(x2)σgg→H(m2H,x1x2S)(3)
where h1,h2denote p,p(or p,f g/p(x)is the standard distribution of gluons inside the p(or
x1x2S =
τ
MS scheme)we end up with
the following expression[3]
σh
1+h2→H+X
(τ,S)=τf g/p?f g/p?(σgg(z)/z)(τ)(5) where?denotes the standard convolution de?ned as
[f1?f2](τ)=
1
dx1dx2f1(x1)f2(x2)δ(τ?x1x2).(6)
The partonic cross-sectionσgg(z)is given by the following perturbation expansion
σgg(z)=σ0 η(0)gg(z)+αsπ 2η(2)gg(z)+O(α3s) (7) where the functionsη(k)gg,k=0,1,2,are given in Eqs.(44),(45),(47),(48)and(49)of[3],
and
σ0=π
π 2(8)
withυ≈246GeV the Higgs vacum expectation value.
Subsequently we proceed as in[2]?.We write,for simplicity,σh
1+h2→H+X (τ,S)≡σH(τ,S)
and denote byσ(k)H(τ,S),k=0,1,2,the O(αk s)part ofσH(τ,S),byσ(k)Hs the part ofσ(k)H
arising from distributionsδ(1?z)and[ln n(1?z)/1?z)]+,here n=0,1,2,3(virtual, collinear and soft gluons)and byσ(k)Hh the rest.We also de?ne
L(k)H(τ,S)=
σ(k)Hh(τ,S)
MS CTEQ5M1set of[10].
Fig.1,upper part,shows L(k)H,k=1,2,as functions of
√
τis signi?cant,for
√τ≥.43is smaller than20%.Moreover,both L(k)H decrease fast as
√
τ?. Again,both ratios decrease fast as
√
p→A+X)mediated via the subprocess gg→A.As before,the partonic cross-sectionsσgg(z)have an expansion similar to(7)
σgg(z)=σ0 φ(0)gg(z)+αsπ 2φ(2)gg(z)+O(α3s) (10)
where z is given by(4)withτ=m2A/S,φ(k)gg(z)are given in Eqs.(8)-(11)of[4](together with the expressions ofη(k)gg(z)),and here
σ0=π
π 2.(11)
Now we write tan2βσh
1+h2→A+X≡σA(τ,S)and,as before,denote byσ(k)A(τ,S)the O(αk s) part ofσA(τ,S),byσ(k)As the part ofσ(k)A arising from distributions and byσ(k)Ah the rest.We de?ne
L(k)A(τ,S)=
σ(k)Ah(τ,S)
τ.All the results are similar as for L(k)H.Similar are also the results for the ratiosσ(1)Ah/ σ(0)A+σ(1)A and σ(2)Ah/ σ(0)A+σ(1)A+σ(2)A .
We note the following?:suppose that inσ(k)Hs andσ(k)As,apart from the terms arising from the distributionsδ(1?z)and[ln n(1?z)/1?z)]+we include also the terms ln m(1?z), m=1,2,3.De?ning asσ(k)Hs andσ(k)Ah the rest,we?nd that the ratiosσ(k)Hh/σ(k)H andσ(k)Ah/σ(k)A decrease signi?cantly in magnitude over the entire range of
√
τincreases towards 1both cross-sections approach each other,and forτ≥0.8practically coincide.The same is observed in Fig.2,lower part,which shows the quantitiesσ(0)A+σ(1)A+σ(2)A(dashed)and σ(0)A+σ(1)As+σ(2)As(solid).
In conclusion,under the assumptions discussed at the beginning,we have shown that not only in D-Y production and DIS[2],but also in Higgs and pseudoscalar Higgs boson production(gg→H and gg→A)there is a part containing the distributionsδ(1?z) and[ln n(1?z)/1?z)]+,here n=0,1,2,3(virtual,soft and collinear part)that for
√S or m
A /√
?A similar remark regarding resummations was?rst made by M.Kramer,https://www.doczj.com/doc/6715183095.html,enen and M.Spira, Nucl.Phys.B511(1998)523.
?In the?rst of Ref.[11]an error was found in the calculation of T.Matsuura et al.,Nucl.Phys.B319 (1989)570on D-Y production.We have repeated the relevant calculations of[2]and found no signi?cant change.
Engineering Research Council of Canada,by the Research Committee of the University of Athens and by the Greek State Scholarships foundation(IKY).
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0.10.20.30.40.50.60.70.80.9 1.0
(
)
(2)(0)(1)(2)
/
A h
A
A
A s
s
s
s
++(2)
A
L
(
)
(1)(0)(1)
/
A h
A
A s
s
s
+(1)
A
L
(
)
(2)(0)(1)(2)
/
H h
H
H
H
s
s
s
s
++(2)
H
L
(
)
(1)(0)(1)
/
H h
H
H s
s
s
+(1)
H
L
10
-1
10
-2
10
-2
10
-1
10-110
-2
Figure 1:Upper part:The ratios L (1)H and σ(1)Hh / σ(0)H +σ(1)
H
(dashed lines)and the
ratios L (2)H and σ(2)Hh / σ(0)H +σ(1)H +σ(2)
H (solid lines)versus √S .Lower
part:The quantities L (1)A and σ(1)Ah / σ(0)A +σ(1)A
(dashed)and the quantities L (2)
A and
σ(2)Ah / σ(0)A +σ(1)A +σ(2)
A (solid)versus
√S .
0.10.20.30.40.50.60.70.80.9 1.0
(
0)(1)(2)
H
H s
H s
s
s
s
++(0)(1)(2)
A
A s
A s
s
s
s
++(0
)
(1)
(
2)
A
A
A
s
s
s
++
(0)(1)(2)
H
H
H
s
s
s
++10
10
-1
10
-3
10
-5
10
-7
10
-9
10
-11
10
-13
s
A
(f b )
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
1
10
2
s
H
(f b )
Figure 2:Upper part:The cross-sections σ(0)H +σ(1)H +σ(2)H (dashed line)and σ(0)
H +σ(1)
Hs +σ(2)
Hs
(solid line)versus
√S .Lower part:The quantities σ(0)A +σ(1)A +σ(2)
A (dashed)and σ(0)A +σ(1)As +σ(2)
As (solid)versus √S .