PRL 113, 212003 (2014)
week ending 21 NOVEMBER 2014
PHYSICAL REVIEW LETTERS
Rapidity Distributions in Drell-Yan and Higgs Productions at Threshold to Third Order in QCD 1
Taushif Ahmed,1,* M. K. Mandal,1,† Narayan Rana,1,‡ and V. Ravindran2,§
Regional Centre for Accelerator-based Particle Physics, Harish-Chandra Research Institute, Allahabad, India 2 The Institute of Mathematical Sciences, Chennai, India (Received 6 May 2014; published 20 November 2014) We present the threshold N3 LO perturbative QCD corrections to the rapidity distributions of dileptons in the Drell-Yan process and Higgs boson in gluon fusion. Sudakov resummation of QCD amplitudes, renormalization group invariance, and the mass factorization theorem provide useful guidelines to obtain them in an elegant manner. We use various state of the art three loop results that have been recently available to obtain these distributions. For the Higgs boson, we demonstrate numerically the importance of these corrections at the LHC. DOI: 10.1103/PhysRevLett.113.212003
PACS numbers: 12.38.Bx
Drell-Yan (DY) production [1] of a pair of leptons at the Large Hadron Collider (LHC) is one of the cleanest processes that can be studied not only to test the standard model (SM) to an unprecedented accuracy but also to probe physics beyond the SM (BSM) scenarios in a very clear environment. Rapidity distributions of Z boson [2] and charge asymmetries of leptons in W boson decays [3] constrain various parton densities and, in addition, possible excess events can provide hints to BSM physics, namely, R parity violating supersymmetric models, models with Z0 or with contact interactions and large extra-dimension models. One of the production mechanisms responsible for discovering the Higgs boson of the SM at the LHC [4] is the gluon-gluon fusion through top quark loop. Being a dominant one, it will continue to play a major role in studying the properties of the Higgs boson and its coupling to other SM particles. Distributions of transverse momentum and rapidity of the Higgs boson are going to be very useful tools to achieve this task. Like the inclusive rates [5,6], the rapidity distribution of dileptons in DY process and of the Higgs boson in gluon-gluon fusion are also known to next to next to leading order (NNLO) level in perturbative QCD due to seminal works by Anastasiou et al. [7]. The quark and gluon form factors [8–10], the mass factorization kernels [11], and the renormalization constant [12] for the effective operator describing the coupling of the Higgs boson with the SM fields in the infinite top quark mass limit up to three loop level in dimensional regularization with space-time dimensions n ¼ 4 þ ϵ were found to be useful to obtain the next to next to next to leading order (N3 LO) threshold effects [13] to the inclusive Higgs boson and DY productions at the LHC, excluding δð1 − zÞ terms, where the scaling parameter is z ¼ m2lþ l− =ˆs for the DY process and z ¼ m2H =ˆs for the Higgs boson. Here, mlþ l− , mH and sˆ are the invariant mass of the dileptons, the mass of the Higgs boson, and center of mass energy of the partonic reaction responsible 0031-9007=14=113(21)=212003(5)
for the production mechanism, respectively. Recently, Anastasiou et al. [14] made an important contribution in computing the total rate for the Higgs boson production at N3 LO resulting from the threshold region including the δð1 − zÞ term. Their result, along with three loop quark form factors and mass factorization kernels, was used to compute the DY cross section at N3 LO at threshold in [15]. In this Letter, we will apply the formalism developed in [16] to obtain rapidity distributions of the dilepton pair and of the Higgs boson at N3 LO in the threshold region using the available information that led to the computation of the N3 LO threshold corrections to the inclusive Higgs boson and DY productions. The rapidity distribution can be written as dσ I ¼ σ IBorn ðx01 ; x02 ; q2 ÞW I ðx01 ; x02 ; q2 Þ; dy
I ¼ q; g;
ð1Þ
normalized by W IBorn ðx01 ; x02 ; q2 Þ ¼ δð1 − x01 Þδð1 − x02 Þ. Rapidity y ¼ 12 logðp2 · q=p1 · qÞ ¼ 12 logðx01 =x02 Þ and τ ¼ q2 =S ¼ x01 x02 , q being the momentum of the dilepton pair in the DY process and of the Higgs boson in the Higgs boson production, S ¼ ðp1 þ p2 Þ2 , where pi are the momenta of incoming hadrons Pi ði ¼ 1; 2Þ. For the DY process, I ¼ q and σ I ¼ dσ q ðτ; q2 ; yÞ=dq2 with q2 the invariant mass of the final state dilepton pair, i.e., q2 ¼ m2lþ l− and for the Higgs boson production through gluon fusion, I ¼ g and σ I ¼ σ g ðτ; q2 ; yÞ. The function W I can be expressed in terms of the parton distribution functions f a ðx1 ; μ2F Þ and f b ðx2 ; μ2F Þ renormalized at the factorization scale μF ,
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WI ¼
X Z ab¼q;q;g ¯
1
x01
dz1 z1
Z
1
x02
dz2 I x01 x02 2 H ; ;μ z2 ab z1 z2 F
× ΔId;ab ðz1 ; z2 ; q2 ; μ2F ; μ2R Þ;
ð2Þ
© 2014 American Physical Society
PRL 113, 212003 (2014) with
Hqab ðx1 ; x2 ; μ2F Þ ¼ f Pa 1 ðx1 ; μ2F Þf Pb 2 ðx2 ; μ2F Þ; Hgab ðx1 ; x2 ; μ2F Þ ¼ x1 f Pa 1 ðx1 ; μ2F Þx2 f Pb 2 ðx2 ; μ2F Þ;
ð3Þ
where xi ði ¼ 1; 2Þ are the momentum fractions of the partons in the incoming hadrons. The threshold contribution to the rapidity distribution denoted by 2 2 2 ΔSV d;I ðz1 ; z2 ; q ; μR ; μF Þ is found to be I 2 2 2 ΔSV ¯ 1 ; z¯ 2 ; ϵÞ)jϵ¼0 ; d;I ¼ C exp (Ψd ðq ; μR ; μF ; z
ð4Þ
where ΨId are finite distributions computed in 4 þ ϵ spacetime dimensions with z¯ 1 ¼ 1 − z1 and z¯ 2 ¼ 1 − z2 : ΨId ¼ ( lnðZI ðaˆ s ; μ2R ; μ2 ; ϵÞ)2 þ ln jFˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞj2 Þδð¯z1 Þδð¯z2 Þ þ 2ΦId ðaˆ s ; q2 ; μ2 ; z¯ 1 ; z¯ 2 ; ϵÞ − C( ln ΓII ðaˆ s ; μ2 ; μ2F ; z¯ 1 ; ϵÞδð¯z2 Þ þ ð¯z1 ↔ z¯ 2 Þ): ð5Þ The definition of double Mellin convolution C can be found in [16]. We drop all the regular functions that result from various convolutions. The bare form factors are denoted by Fˆ I with Q2 ¼ −q2 . The overall operator renormalization constant for the DY process, Zq ¼ 1 and for the Higgs boson, Zg is known up to the three loop level [12] in QCD. ΦId are called the soft distribution functions and ΓII are the mass factorization kernels. μ is the scale introduced to define the dimensionless strong coupling constant aˆ s ¼ gˆ 2s =16π 2 in dimensional regularization and as ðμ2R Þ is the renormalized strong coupling constant which is related to aˆ s through the renormalization constant Z½as ðμ2R Þ, i.e., 2 aˆ s ¼ ðμ=μR Þϵ Zðμ2R ÞS−1 ϵ as ðμR Þ, Sϵ ¼ exp½ðγ E − ln 4πÞϵ=2. SV The fact that Δd;I are finite in the limit ϵ → 0 implies that the pole structure of the soft distribution functions should be similar to that of Fˆ I and ΓII . We find that they must satisfy Sudakov type differential equations which the form factors Fˆ I also satisfy: q2
d I 1 ¯I ¯ I ; Φ ¼ ½K þ G d dq2 d 2 d
where the constants K¯ Id ðaˆ s ; ðμ2R =μ2 Þ; z¯ 1 ; z¯ 2 ; ϵÞ are propor¯ I ðaˆ s ; ðq2 =μ2R Þ; tional to the singular terms in ϵ and the G d 2 2 ðμR =μ Þ; z¯ 1 ; z¯ 2 ; ϵÞ are finite functions of ϵ. It is straightforward to solve the above differential equations yielding ΦId
¼
∞ X i¼1
aˆ is
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PHYSICAL REVIEW LETTERS
2 q z¯ 1 z¯ 2 iðϵ=2Þ i ðiϵÞ2 ˆ I;ðiÞ ϕ ðϵÞ; Sϵ 4¯z1 z¯ 2 d μ2
I;ðiÞ The constants K¯ d ðϵÞ are determined by expanding K¯ Id in powers of the bare coupling constant aˆ s , i.e., 2 iðϵ=2Þ ∞ X μ I;ðiÞ I i ¯ ˆ Siϵ K¯ d ðϵÞ; ð8Þ as R2 K d ¼ δð¯z1 Þδð¯z2 Þ μ i¼1 I;ðiÞ and solving the RG equation for K¯ Id. We find that K¯ d ðϵÞ I;ðiÞ are identical to K¯ ðϵÞ given in [17]. The constants I;ðiÞ ¯ G d ðϵÞ are related to the finite boundary functions I ˆ ¯ Gd ðas ; 1; ðq2 =μ2 Þ; z¯ 1 ; z¯ 2 ; ϵÞ. Defining the G¯ Id;i ðϵÞ through the relation 2 ∞ ∞ X X q z¯ 1 z¯ 2 iðϵ=2Þ i ¯ I;ðiÞ S ðϵÞ ¼ ais ðq2 z¯ 1 z¯ 2 ÞG¯ Id;i ðϵÞ aˆ is G ϵ d 2 μ i¼1 i¼1
and demanding the finiteness of ΔSV d;I given in Eq. (4), we I ¯ find that the structure of Gd;i ðϵÞ is similar to that of the corresponding GI ðϵÞ in the form factors [17], that is ¯ Ii þ G¯ Id;i ðϵÞ ¼ −f Ii þ C
ϵk G¯ I;k d;i ;
ð9Þ
k¼1
¯ I ¼ −2β0 G¯ I;1 , ¯ I ¼ −2β1 G¯ I;1 − ¯ I ¼ 0, C C where C 1 2 3 d;1 d;1 I;2 I ¯ 2β0 ðG¯ I;1 þ 2β Þ, f are given in [8] and β are the G 0 d;1 i i d;2 2 2 2 coefficients of the QCD β function of as ðμR Þ, μR das ðμR Þ= P iþ2 2 ¯ I;k dμ2R ¼ ϵas ðμ2R Þ=2 − ∞ i¼0 β i as ðμR Þ. The constants Gd;i can be expressed in terms of G¯ I;k using the following i relation Z 1 Z 1 Z 1 I 0 0 0 0 N−1 dσ dx1 dx2 ðx1 x2 Þ dττN−1 σ I ; ð10Þ ¼ dY 0 0 0 where the σ I are now known for both DY and the Higgs boson production up to the N3 LO level in the threshold limit [14,15,18]. In the threshold limit, N → ∞, we find the following relation exact to all orders in ϵ, Γð1 þ iϵÞ ˆ I;ðiÞ I;ðiÞ ϕˆ d ðϵÞ ¼ 2 ϕ ðϵÞ; Γ ð1 þ i 2ϵÞ
ð11Þ
where ϕˆ I;ðiÞ ðϵÞ can be found in [16]. Substituting ZI , Fˆ I , and ΦId and ΓII in Eq. (5), and using Eq. (4), we obtain ΔSV d;I in powers of as ðμ2R Þ as ΔSV d;I ðzÞ ¼
∞ X
2 ais ðμ2R ÞΔSV d;I;i ðz; μR Þ;
where
i¼0
ð6Þ
SV ΔSV z1 Þδð¯z2 Þ þ d;I;i ¼ Δd;I;i jδδ δð¯
2i−1 X
ΔSV z2 ÞDj d;I;i jδDj δð¯
j¼0
where 1 I;ðiÞ I;ðiÞ ¯ I;ðiÞ ðϵÞ: ϕˆ d ðϵÞ ¼ ½K¯ d ðϵÞ þ G d iϵ
∞ X
þ ð7Þ 212003-2
2i−1 X j¼0
¯jþ ΔSV z1 Þ D ¯ j δð¯ d;I;i jδD
X ¯ k; ΔSV ¯ k Dj D d;I;i jDj D jⓈk
ð12Þ
PRL 113, 212003 (2014) i ln ð1 − z1 Þ with Di ¼ ; ð1 − z1 Þ þ
PHYSICAL REVIEW LETTERS i ln ð1 − z Þ 2 ¯i ¼ D : ð1 − z2 Þ þ ð13Þ
The symbol jⓈk implies j; k ≥ 0 and j þ k ≤ ð2i − 2Þ. ¯ in Eq. (12) were obtained Terms proportional to D and/or D in [16] and the first term is possible to calculate as the
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results for the threshold N3 LO QCD corrections are now available for DY [15] and the Higgs boson [14] productions. Setting μ2R ¼ μ2F ¼ q2 , in the following, we present this contribution along with the constants G¯ I;k d;i that are needed to determine the soft distribution function ΦId up to N3 LO level using CI ¼ CF ; CA for I ¼ q; g, respectively.
¯GI;1 ¼ CI ð−ζ2 Þ; ¯GI;3 ¼ CI 1 ζ 2 ; ¯GI;2 ¼ CI 1 ζ3 ; d;1 d;1 d;1 3 80 2 ¯GI;1 ¼ CI CA 2428 − 67 ζ2 − 4ζ2 − 44 ζ3 þ CI nf − 328 þ 10 ζ2 þ 8 ζ3 ; 2 d;2 81 3 3 81 3 3 ¯GI;2 ¼ CI CA − 319 ζ 2 2 − 71 ζ2 ζ3 þ 202 ζ2 þ 469 ζ 3 þ 43ζ 5 − 7288 þ CI nf 29 ζ 2 2 − 28 ζ2 − 70 ζ3 þ 976 d;2 120 3 9 27 243 60 9 27 243 2 17392 ζ 3 þ 1538 ζ 2 þ 4136 ζ ζ − 379417 ζ þ 536 ζ 2 − 936ζ − 1430 ζ þ 7135981 G¯ I;1 3 d;3 ¼ CI CA 315 2 45 2 9 2 3 486 2 3 3 3 5 8748 1372 2 392 51053 12356 148 716509 152 2 þ CI CA nf − ζ2 − ζ2 ζ3 þ ζ2 þ ζ3 þ ζ5 − þ CI CF nf ζ − 40ζ 2 ζ 3 45 9 243 81 3 4374 15 2 275 1672 112 42727 152 2 316 320 11584 2 ζ þ ζ þ ζ − þ CI nf ζ − ζ − ζ þ : ð14Þ þ 6 2 27 3 3 5 324 45 2 27 2 81 3 2187 With CA ¼ N, CF ¼ ðN 2 − 1Þ=2N, nf ¼ no:of flavors and nf;v given in [9], the δδ parts of Eq. (12) for I ¼ q; g are 24352 3 2921 2 99289 400 2 125105 1505881 2C ΔSV ζ ζ ζ ζ ζ j ¼ C − − 588ζ ζ þ − þ − 204ζ − A F 2 3 3 5 d;q;3 δδ 315 2 135 2 81 2 3 3 81 972 78272 3 137968 2 10736 39865 1264 2 5972 7624 74321 2 þ CA CF − ζ þ ζ þ ζ2 ζ3 − ζ þ ζ − ζ − ζ þ 315 2 135 2 9 27 2 3 3 3 3 9 5 36 2828 2 272 12112 19888 110651 þ CA CF nf − ζ þ ζ ζ − ζ − ζ − 8ζ5 þ 135 2 3 2 3 27 2 81 3 243 1403 736 2 5599 3 90016 3 3164 2 þ CF ζ − ζ − 160ζ2 ζ3 þ ζ þ ζ − 460ζ 3 þ 1328ζ5 − 315 2 5 2 3 2 3 3 6 19408 2 1472 5848 224 421 þ CF 2 nf − ζ − ζ ζ þ ζ þ 360ζ3 − ζ − 135 2 9 2 3 27 2 9 5 3 2 592 2 2816 304 7081 N −4 4 2 28 160 2 þ CF nf ζ þ ζ − ζ − þ CF nf;v − ζ 2 þ 20ζ2 þ ζ 3 − ζ þ 8 ; ð15Þ 135 2 81 2 81 3 243 N 5 3 3 5 12032 3 40432 2 41914 1600 2 54820 1364 215131 ζ2 þ ζ 2 − 88ζ2 ζ3 þ ζ2 þ ζ3 − ζ3 þ ζ5 þ 105 135 27 3 27 9 81 1240 2 7108 2536 1192 98059 þ CA 2 nf ζ 2 − 272ζ2 ζ3 − ζ2 þ ζ3 þ ζ5 − 27 27 27 9 81 176 2 2270 63991 þ CA CF nf ζ2 þ 288ζ 2 ζ 3 − ζ 2 þ 400ζ 3 þ 160ζ 5 − 45 9 81 208 2 64 112 2515 592 608 2 2 þ CA nf − ζ − ζ2 þ ζ þ þ CF nf ζ − 320ζ 5 þ 15 2 3 3 3 27 3 3 9 32 184 224 8962 þ CF nf 2 − ζ2 2 − ζ − ζ þ : 45 9 2 3 3 81
3 ΔSV d;g;3 jδδ ¼ CA
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ð16Þ
TABLE I.
%
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PHYSICAL REVIEW LETTERS
PRL 113, 212003 (2014)
Relative contributions of pure N3 LO terms.
δδ
¯0 δD
¯1 δD
¯2 δD
¯3 δD
¯4 δD
¯5 δD
¯0 D0 D
¯1 D0 D
¯2 D0 D
¯3 D0 D
¯4 D0 D
¯1 D1 D
¯2 D1 D
¯3 D1 D
¯2 D2 D
73.3
16.0
9.1
31.4
1.0
−9.9
−23.1
−13.7
−10.7
−0.3
3.1
7.3
−0.2
3.8
8.6
4.2
We present the relative contributions in percentage of the pure N3 LO terms in Eq. (12) with respect to ΔSV d;g;3 , for ¯ rapidity Y ¼ 0 in Table I. The notation Di Dj corresponds to ¯ j and Dj D ¯ i. the sum of the contributions coming from Di D pffiffiffi We have used s ¼ 14 TeV for the LHC, GF ¼ 4541.68 pb, the Z boson mass mZ ¼ 91.1876 GeV, top quark mass mt ¼ 173.4 GeV and the Higgs boson mass mH ¼ 125.5 GeV throughout. For the Higgs boson production, we use the effective theory where top quark is integrated out in the large mt limit. The strong coupling constant αs ðμ2R Þ is evolved using the 4-loop RG equations 3 with αsN LO ðmZ Þ ¼ 0.117 and for parton density sets we use MSTW 2008NNLO [19], as N3 LO evolution kernels are not yet available. In [20], Forte et al. pointed out that the Higgs boson cross sections will remain unaffected with this shortcoming. However, for the DY process, it is not clear whether the same will be true. We find that the contribution from the δð¯z1 Þδð¯z2 Þ part is the largest. The dependence on the renormalization and factorization scales can by studied by varying them in the range ðmH =2Þ < μR ; μF < 2mH . We find that the inclusion of the threshold correction at N3 LO further reduces their dependence. For the inclusive Higgs boson production, we find that about 50% of exact NNLO contribution comes from threshold NLO and NNLO terms. It increases to 80% if we use exact NLO and threshold NNLO terms. Hence, it is expected that the rapidity distribution of the Higgs boson will receive a significant contribution from the threshold region compared to inclusive rate due to the soft emission over the entire range of Y. Our numerical study with threshold enhanced NNLO rapidity distribution confirms our expectation. Comparing our threshold NNLO results against exact NNLO distribution using the FEHiP [21] code, we find that about 90% of exact NNLO distribution comes from the threshold region as can be seen from Table II, in accordance with [22], where it was shown that for low τðm2H =s ≈ 10−5 Þ values the threshold terms are dominant, thanks to the inherent property of the matrix element, which receives the
TABLE II. Y NNLO NNLOSV NNLOSV ðAÞ N3 LOSV N3 LOSV ðAÞ K3
largest radiative corrections from the phase-space points corresponding to Born kinematics. Here we have used the exact results up to the NLO level. Because of an inherent ambiguity in the definition of the partonic cross section at threshold one can multiply a factor zgðzÞ, where z ¼ τ=x1 x2 and limz→1 gðzÞ ¼ 1, with the partonic flux and divide the same in the partonic cross section for an inclusive rate. In [23,24] this was exploited to take into account the subleading collinear logs also, thereby making the threshold approximation a better one. Recently, Anastasiou et al. used this in [14] to modify the partonic flux keeping the partonic cross section unaltered to improve the threshold effects. Following [14,25], we introduce Gðz1 ; z2 Þ such that limz1 ;z2 →1 G ¼ 1 in (2): X Z
1
x01
dz1 z1
Z
1
dz2 I H Gðz1 ; z2 Þ z2 ab ¯ ab¼q;q;g I Δd;ab ðz1 ; z2 Þ lim : z1 ;z2 →1 Gðz1 ; z2 Þ
WI ¼
x02
ð17Þ
We also find that with the choice Gðz1 ; z2 Þ ¼ z21 z22 , the threshold NNLO results are remarkably close to the exact ones for the entire range of Y [see Table II, denoted by ðAÞ]. This clearly demonstrates the dominance of threshold contributions to rapidity distribution of the Higgs boson production at the NNLO level. Assuming that the trend will not change drastically beyond NNLO, we present numerical values for N3 LO distributions for Gðz1 ; z2 Þ ¼ 1; z21 z22, respectively, as N3 LOSV and N3 LOSV ðAÞ in Table II. The threshold N3 LO terms give 6%ðY ¼ 0Þ to 12%ðY ¼ 3.6Þ additional correction over the NNLO contribution to the inclusive DY production. Finally, in Table II, we have presented K3 ¼ N3 LOSV =LO as a function of Y in order to demonstrate the sensitivity of higher order effects to the rapidity Y. To summarize, we present full threshold enhanced N3 LO QCD corrections to rapidity distributions of the dilepton
Contributions of exact NNLO, NNLOSV , N3 LOSV , and K3. 0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
11.21 9.81 10.67 11.62 11.88 2.31
10.96 9.61 10.46 11.36 11.62 2.29
10.70 8.99 9.84 11.07 11.33 2.36
9.13 8.00 8.82 9.44 9.70 2.21
7.80 6.71 7.48 8.04 8.30 2.17
6.10 5.21 5.90 6.27 6.51 2.07
4.23 3.66 4.24 4.33 4.54 1.89
2.66 2.25 2.69 2.70 2.88 1.70
1.40 1.14 1.42 1.40 1.53 1.63
0.54 0.42 0.56 0.53 0.60 1.51
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pair in the DY process and of the Higgs boson in gluongluon fusion at the LHC. We show that the infrared structure of QCD amplitudes, in particular, their factorization properties, along with Sudakov resummation of soft gluons and renormalization group invariance provide an elegant framework to compute these threshold corrections systematically for rapidity distributions order by order in QCD perturbation theory. The recent N3 LO results for inclusive DY and Higgs boson production cross sections at the threshold provide crucial ingredients to obtain δð¯z1 Þδð¯z2 Þ contribution of their rapidity distributions for the first time. We find that this contribution numerically dominates over the rest of the terms in ΔSV d;g;3 at the LHC. Inclusion of N3 LO contributions reduces the scale dependence further. We also demonstrate the dominance of the threshold contribution to rapidity distributions by comparing it against the exact NNLO for two different choices of Gðz1 ; z2 Þ. Finally, we find that threshold N3 LO rapidity distribution with Gðz1 ; z2 Þ ¼ 1; z21 z22 shows a moderate effect over NNLO distribution. We thank F. Petriello for providing the FEHiP code and fruitful discussions. T. A., M. K. M., and N. R. thank IMSc for providing hospitality. We thank M. Mahakhud for discussion. The work of T. A., M. K. M., and N. R. has been partially supported by funding from RECAPP, DAE, Government of India.
*
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M. Spira, A. Djouadi, D. Graudenz, and P. M. Zerwas, Nucl. Phys. B453, 17 (1995); R. V. Harlander and W. B. Kilgore, Phys. Rev. D 64, 013015 (2001); S. Catani, D. de Florian, and M. Grazzini, J. High Energy Phys. 05 (2001) 025; 07 (2003) 028; R. V. Harlander and W. B. Kilgore, Phys. Rev. Lett. 88, 201801 (2002); C. Anastasiou and K. Melnikov, Nucl. Phys. B646, 220 (2002); V. Ravindran, J. Smith, and W. L. van Neerven, Nucl. Phys. B665, 325 (2003). C. Anastasiou, L. J. Dixon, K. Melnikov, and F. Petriello, Phys. Rev. Lett. 91, 182002 (2003); C. Anastasiou, K. Melnikov, and F. Petriello, Phys. Rev. Lett. 93, 262002 (2004). S. Moch, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 08 (2005) 049; Phys. Lett. B 625, 245 (2005). P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Phys. Rev. Lett. 102, 212002 (2009). T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, J. High Energy Phys. 06 (2010) 094; 11 (2010) 102. S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B688, 101 (2004); B691, 129 (2004). T. Inami, T. Kubota, and Y. Okada, Z. Phys. C 18, 69 (1983); K. G. Chetyrkin, B. A. Kniehl, and M. Steinhauser, Phys. Rev. Lett. 79, 353 (1997); Nucl. Phys. B510, 61 (1998). S. Moch and A. Vogt, Phys. Lett. B 631, 48 (2005); E. Laenen and L. Magnea, Phys. Lett. B 632, 270 (2006); A. Idilbi, X.-d. Ji, J.-P. Ma, and F. Yuan, Phys. Rev. D 73, 077501 (2006); V. Ravindran, Nucl. Phys. B746, 58 (2006); B752, 173 (2006). C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, and B. Mistlberger, Phys. Lett. B 737, 325 (2014). T. Ahmed, M. Mahakhud, N. Rana, and V. Ravindran, Phys. Rev. Lett. 113, 112002 (2014). V. Ravindran, J. Smith, and W. L. van Neerven, Nucl. Phys. B767, 100 (2007); V. Ravindran and J. Smith, Phys. Rev. D 76, 114004 (2007). V. Ravindran, Nucl. Phys. B752, 173 (2006). Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, Phys. Rev. D 90, 053006 (2014). A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C 63, 189 (2009). S. Forte, A. Isgr, and G. Vita, Phys. Lett. B 731, 136 (2014). C. Anastasiou, K. Melnikov, and F. Petriello, Nucl. Phys. B724, 197 (2005). T. Becher, M. Neubert, and G. Xu, J. High Energy Phys. 07 (2008), 030. S. Catani, D. de Florian, M. Grazzini, and P. Nason, J. High Energy Phys. 07 (2003) 028. M. Kramer, E. Laenen, and M. Spira, Nucl. Phys. B511, 523 (1998). F. Herzog and B. Mistlberger, arXiv:1405.5685.
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PHYSICAL REVIEW LETTERS
Rapidity Distributions in Drell-Yan and Higgs Productions at Threshold to Third Order in QCD 1
Taushif Ahmed,1,* M. K. Mandal,1,† Narayan Rana,1,‡ and V. Ravindran2,§
Regional Centre for Accelerator-based Particle Physics, Harish-Chandra Research Institute, Allahabad, India 2 The Institute of Mathematical Sciences, Chennai, India (Received 6 May 2014; published 20 November 2014) We present the threshold N3 LO perturbative QCD corrections to the rapidity distributions of dileptons in the Drell-Yan process and Higgs boson in gluon fusion. Sudakov resummation of QCD amplitudes, renormalization group invariance, and the mass factorization theorem provide useful guidelines to obtain them in an elegant manner. We use various state of the art three loop results that have been recently available to obtain these distributions. For the Higgs boson, we demonstrate numerically the importance of these corrections at the LHC. DOI: 10.1103/PhysRevLett.113.212003
PACS numbers: 12.38.Bx
Drell-Yan (DY) production [1] of a pair of leptons at the Large Hadron Collider (LHC) is one of the cleanest processes that can be studied not only to test the standard model (SM) to an unprecedented accuracy but also to probe physics beyond the SM (BSM) scenarios in a very clear environment. Rapidity distributions of Z boson [2] and charge asymmetries of leptons in W boson decays [3] constrain various parton densities and, in addition, possible excess events can provide hints to BSM physics, namely, R parity violating supersymmetric models, models with Z0 or with contact interactions and large extra-dimension models. One of the production mechanisms responsible for discovering the Higgs boson of the SM at the LHC [4] is the gluon-gluon fusion through top quark loop. Being a dominant one, it will continue to play a major role in studying the properties of the Higgs boson and its coupling to other SM particles. Distributions of transverse momentum and rapidity of the Higgs boson are going to be very useful tools to achieve this task. Like the inclusive rates [5,6], the rapidity distribution of dileptons in DY process and of the Higgs boson in gluon-gluon fusion are also known to next to next to leading order (NNLO) level in perturbative QCD due to seminal works by Anastasiou et al. [7]. The quark and gluon form factors [8–10], the mass factorization kernels [11], and the renormalization constant [12] for the effective operator describing the coupling of the Higgs boson with the SM fields in the infinite top quark mass limit up to three loop level in dimensional regularization with space-time dimensions n ¼ 4 þ ϵ were found to be useful to obtain the next to next to next to leading order (N3 LO) threshold effects [13] to the inclusive Higgs boson and DY productions at the LHC, excluding δð1 − zÞ terms, where the scaling parameter is z ¼ m2lþ l− =ˆs for the DY process and z ¼ m2H =ˆs for the Higgs boson. Here, mlþ l− , mH and sˆ are the invariant mass of the dileptons, the mass of the Higgs boson, and center of mass energy of the partonic reaction responsible 0031-9007=14=113(21)=212003(5)
for the production mechanism, respectively. Recently, Anastasiou et al. [14] made an important contribution in computing the total rate for the Higgs boson production at N3 LO resulting from the threshold region including the δð1 − zÞ term. Their result, along with three loop quark form factors and mass factorization kernels, was used to compute the DY cross section at N3 LO at threshold in [15]. In this Letter, we will apply the formalism developed in [16] to obtain rapidity distributions of the dilepton pair and of the Higgs boson at N3 LO in the threshold region using the available information that led to the computation of the N3 LO threshold corrections to the inclusive Higgs boson and DY productions. The rapidity distribution can be written as dσ I ¼ σ IBorn ðx01 ; x02 ; q2 ÞW I ðx01 ; x02 ; q2 Þ; dy
I ¼ q; g;
ð1Þ
normalized by W IBorn ðx01 ; x02 ; q2 Þ ¼ δð1 − x01 Þδð1 − x02 Þ. Rapidity y ¼ 12 logðp2 · q=p1 · qÞ ¼ 12 logðx01 =x02 Þ and τ ¼ q2 =S ¼ x01 x02 , q being the momentum of the dilepton pair in the DY process and of the Higgs boson in the Higgs boson production, S ¼ ðp1 þ p2 Þ2 , where pi are the momenta of incoming hadrons Pi ði ¼ 1; 2Þ. For the DY process, I ¼ q and σ I ¼ dσ q ðτ; q2 ; yÞ=dq2 with q2 the invariant mass of the final state dilepton pair, i.e., q2 ¼ m2lþ l− and for the Higgs boson production through gluon fusion, I ¼ g and σ I ¼ σ g ðτ; q2 ; yÞ. The function W I can be expressed in terms of the parton distribution functions f a ðx1 ; μ2F Þ and f b ðx2 ; μ2F Þ renormalized at the factorization scale μF ,
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WI ¼
X Z ab¼q;q;g ¯
1
x01
dz1 z1
Z
1
x02
dz2 I x01 x02 2 H ; ;μ z2 ab z1 z2 F
× ΔId;ab ðz1 ; z2 ; q2 ; μ2F ; μ2R Þ;
ð2Þ
© 2014 American Physical Society
PRL 113, 212003 (2014) with
Hqab ðx1 ; x2 ; μ2F Þ ¼ f Pa 1 ðx1 ; μ2F Þf Pb 2 ðx2 ; μ2F Þ; Hgab ðx1 ; x2 ; μ2F Þ ¼ x1 f Pa 1 ðx1 ; μ2F Þx2 f Pb 2 ðx2 ; μ2F Þ;
ð3Þ
where xi ði ¼ 1; 2Þ are the momentum fractions of the partons in the incoming hadrons. The threshold contribution to the rapidity distribution denoted by 2 2 2 ΔSV d;I ðz1 ; z2 ; q ; μR ; μF Þ is found to be I 2 2 2 ΔSV ¯ 1 ; z¯ 2 ; ϵÞ)jϵ¼0 ; d;I ¼ C exp (Ψd ðq ; μR ; μF ; z
ð4Þ
where ΨId are finite distributions computed in 4 þ ϵ spacetime dimensions with z¯ 1 ¼ 1 − z1 and z¯ 2 ¼ 1 − z2 : ΨId ¼ ( lnðZI ðaˆ s ; μ2R ; μ2 ; ϵÞ)2 þ ln jFˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞj2 Þδð¯z1 Þδð¯z2 Þ þ 2ΦId ðaˆ s ; q2 ; μ2 ; z¯ 1 ; z¯ 2 ; ϵÞ − C( ln ΓII ðaˆ s ; μ2 ; μ2F ; z¯ 1 ; ϵÞδð¯z2 Þ þ ð¯z1 ↔ z¯ 2 Þ): ð5Þ The definition of double Mellin convolution C can be found in [16]. We drop all the regular functions that result from various convolutions. The bare form factors are denoted by Fˆ I with Q2 ¼ −q2 . The overall operator renormalization constant for the DY process, Zq ¼ 1 and for the Higgs boson, Zg is known up to the three loop level [12] in QCD. ΦId are called the soft distribution functions and ΓII are the mass factorization kernels. μ is the scale introduced to define the dimensionless strong coupling constant aˆ s ¼ gˆ 2s =16π 2 in dimensional regularization and as ðμ2R Þ is the renormalized strong coupling constant which is related to aˆ s through the renormalization constant Z½as ðμ2R Þ, i.e., 2 aˆ s ¼ ðμ=μR Þϵ Zðμ2R ÞS−1 ϵ as ðμR Þ, Sϵ ¼ exp½ðγ E − ln 4πÞϵ=2. SV The fact that Δd;I are finite in the limit ϵ → 0 implies that the pole structure of the soft distribution functions should be similar to that of Fˆ I and ΓII . We find that they must satisfy Sudakov type differential equations which the form factors Fˆ I also satisfy: q2
d I 1 ¯I ¯ I ; Φ ¼ ½K þ G d dq2 d 2 d
where the constants K¯ Id ðaˆ s ; ðμ2R =μ2 Þ; z¯ 1 ; z¯ 2 ; ϵÞ are propor¯ I ðaˆ s ; ðq2 =μ2R Þ; tional to the singular terms in ϵ and the G d 2 2 ðμR =μ Þ; z¯ 1 ; z¯ 2 ; ϵÞ are finite functions of ϵ. It is straightforward to solve the above differential equations yielding ΦId
¼
∞ X i¼1
aˆ is
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PHYSICAL REVIEW LETTERS
2 q z¯ 1 z¯ 2 iðϵ=2Þ i ðiϵÞ2 ˆ I;ðiÞ ϕ ðϵÞ; Sϵ 4¯z1 z¯ 2 d μ2
I;ðiÞ The constants K¯ d ðϵÞ are determined by expanding K¯ Id in powers of the bare coupling constant aˆ s , i.e., 2 iðϵ=2Þ ∞ X μ I;ðiÞ I i ¯ ˆ Siϵ K¯ d ðϵÞ; ð8Þ as R2 K d ¼ δð¯z1 Þδð¯z2 Þ μ i¼1 I;ðiÞ and solving the RG equation for K¯ Id. We find that K¯ d ðϵÞ I;ðiÞ are identical to K¯ ðϵÞ given in [17]. The constants I;ðiÞ ¯ G d ðϵÞ are related to the finite boundary functions I ˆ ¯ Gd ðas ; 1; ðq2 =μ2 Þ; z¯ 1 ; z¯ 2 ; ϵÞ. Defining the G¯ Id;i ðϵÞ through the relation 2 ∞ ∞ X X q z¯ 1 z¯ 2 iðϵ=2Þ i ¯ I;ðiÞ S ðϵÞ ¼ ais ðq2 z¯ 1 z¯ 2 ÞG¯ Id;i ðϵÞ aˆ is G ϵ d 2 μ i¼1 i¼1
and demanding the finiteness of ΔSV d;I given in Eq. (4), we I ¯ find that the structure of Gd;i ðϵÞ is similar to that of the corresponding GI ðϵÞ in the form factors [17], that is ¯ Ii þ G¯ Id;i ðϵÞ ¼ −f Ii þ C
ϵk G¯ I;k d;i ;
ð9Þ
k¼1
¯ I ¼ −2β0 G¯ I;1 , ¯ I ¼ −2β1 G¯ I;1 − ¯ I ¼ 0, C C where C 1 2 3 d;1 d;1 I;2 I ¯ 2β0 ðG¯ I;1 þ 2β Þ, f are given in [8] and β are the G 0 d;1 i i d;2 2 2 2 coefficients of the QCD β function of as ðμR Þ, μR das ðμR Þ= P iþ2 2 ¯ I;k dμ2R ¼ ϵas ðμ2R Þ=2 − ∞ i¼0 β i as ðμR Þ. The constants Gd;i can be expressed in terms of G¯ I;k using the following i relation Z 1 Z 1 Z 1 I 0 0 0 0 N−1 dσ dx1 dx2 ðx1 x2 Þ dττN−1 σ I ; ð10Þ ¼ dY 0 0 0 where the σ I are now known for both DY and the Higgs boson production up to the N3 LO level in the threshold limit [14,15,18]. In the threshold limit, N → ∞, we find the following relation exact to all orders in ϵ, Γð1 þ iϵÞ ˆ I;ðiÞ I;ðiÞ ϕˆ d ðϵÞ ¼ 2 ϕ ðϵÞ; Γ ð1 þ i 2ϵÞ
ð11Þ
where ϕˆ I;ðiÞ ðϵÞ can be found in [16]. Substituting ZI , Fˆ I , and ΦId and ΓII in Eq. (5), and using Eq. (4), we obtain ΔSV d;I in powers of as ðμ2R Þ as ΔSV d;I ðzÞ ¼
∞ X
2 ais ðμ2R ÞΔSV d;I;i ðz; μR Þ;
where
i¼0
ð6Þ
SV ΔSV z1 Þδð¯z2 Þ þ d;I;i ¼ Δd;I;i jδδ δð¯
2i−1 X
ΔSV z2 ÞDj d;I;i jδDj δð¯
j¼0
where 1 I;ðiÞ I;ðiÞ ¯ I;ðiÞ ðϵÞ: ϕˆ d ðϵÞ ¼ ½K¯ d ðϵÞ þ G d iϵ
∞ X
þ ð7Þ 212003-2
2i−1 X j¼0
¯jþ ΔSV z1 Þ D ¯ j δð¯ d;I;i jδD
X ¯ k; ΔSV ¯ k Dj D d;I;i jDj D jⓈk
ð12Þ
PRL 113, 212003 (2014) i ln ð1 − z1 Þ with Di ¼ ; ð1 − z1 Þ þ
PHYSICAL REVIEW LETTERS i ln ð1 − z Þ 2 ¯i ¼ D : ð1 − z2 Þ þ ð13Þ
The symbol jⓈk implies j; k ≥ 0 and j þ k ≤ ð2i − 2Þ. ¯ in Eq. (12) were obtained Terms proportional to D and/or D in [16] and the first term is possible to calculate as the
week ending 21 NOVEMBER 2014
results for the threshold N3 LO QCD corrections are now available for DY [15] and the Higgs boson [14] productions. Setting μ2R ¼ μ2F ¼ q2 , in the following, we present this contribution along with the constants G¯ I;k d;i that are needed to determine the soft distribution function ΦId up to N3 LO level using CI ¼ CF ; CA for I ¼ q; g, respectively.
¯GI;1 ¼ CI ð−ζ2 Þ; ¯GI;3 ¼ CI 1 ζ 2 ; ¯GI;2 ¼ CI 1 ζ3 ; d;1 d;1 d;1 3 80 2 ¯GI;1 ¼ CI CA 2428 − 67 ζ2 − 4ζ2 − 44 ζ3 þ CI nf − 328 þ 10 ζ2 þ 8 ζ3 ; 2 d;2 81 3 3 81 3 3 ¯GI;2 ¼ CI CA − 319 ζ 2 2 − 71 ζ2 ζ3 þ 202 ζ2 þ 469 ζ 3 þ 43ζ 5 − 7288 þ CI nf 29 ζ 2 2 − 28 ζ2 − 70 ζ3 þ 976 d;2 120 3 9 27 243 60 9 27 243 2 17392 ζ 3 þ 1538 ζ 2 þ 4136 ζ ζ − 379417 ζ þ 536 ζ 2 − 936ζ − 1430 ζ þ 7135981 G¯ I;1 3 d;3 ¼ CI CA 315 2 45 2 9 2 3 486 2 3 3 3 5 8748 1372 2 392 51053 12356 148 716509 152 2 þ CI CA nf − ζ2 − ζ2 ζ3 þ ζ2 þ ζ3 þ ζ5 − þ CI CF nf ζ − 40ζ 2 ζ 3 45 9 243 81 3 4374 15 2 275 1672 112 42727 152 2 316 320 11584 2 ζ þ ζ þ ζ − þ CI nf ζ − ζ − ζ þ : ð14Þ þ 6 2 27 3 3 5 324 45 2 27 2 81 3 2187 With CA ¼ N, CF ¼ ðN 2 − 1Þ=2N, nf ¼ no:of flavors and nf;v given in [9], the δδ parts of Eq. (12) for I ¼ q; g are 24352 3 2921 2 99289 400 2 125105 1505881 2C ΔSV ζ ζ ζ ζ ζ j ¼ C − − 588ζ ζ þ − þ − 204ζ − A F 2 3 3 5 d;q;3 δδ 315 2 135 2 81 2 3 3 81 972 78272 3 137968 2 10736 39865 1264 2 5972 7624 74321 2 þ CA CF − ζ þ ζ þ ζ2 ζ3 − ζ þ ζ − ζ − ζ þ 315 2 135 2 9 27 2 3 3 3 3 9 5 36 2828 2 272 12112 19888 110651 þ CA CF nf − ζ þ ζ ζ − ζ − ζ − 8ζ5 þ 135 2 3 2 3 27 2 81 3 243 1403 736 2 5599 3 90016 3 3164 2 þ CF ζ − ζ − 160ζ2 ζ3 þ ζ þ ζ − 460ζ 3 þ 1328ζ5 − 315 2 5 2 3 2 3 3 6 19408 2 1472 5848 224 421 þ CF 2 nf − ζ − ζ ζ þ ζ þ 360ζ3 − ζ − 135 2 9 2 3 27 2 9 5 3 2 592 2 2816 304 7081 N −4 4 2 28 160 2 þ CF nf ζ þ ζ − ζ − þ CF nf;v − ζ 2 þ 20ζ2 þ ζ 3 − ζ þ 8 ; ð15Þ 135 2 81 2 81 3 243 N 5 3 3 5 12032 3 40432 2 41914 1600 2 54820 1364 215131 ζ2 þ ζ 2 − 88ζ2 ζ3 þ ζ2 þ ζ3 − ζ3 þ ζ5 þ 105 135 27 3 27 9 81 1240 2 7108 2536 1192 98059 þ CA 2 nf ζ 2 − 272ζ2 ζ3 − ζ2 þ ζ3 þ ζ5 − 27 27 27 9 81 176 2 2270 63991 þ CA CF nf ζ2 þ 288ζ 2 ζ 3 − ζ 2 þ 400ζ 3 þ 160ζ 5 − 45 9 81 208 2 64 112 2515 592 608 2 2 þ CA nf − ζ − ζ2 þ ζ þ þ CF nf ζ − 320ζ 5 þ 15 2 3 3 3 27 3 3 9 32 184 224 8962 þ CF nf 2 − ζ2 2 − ζ − ζ þ : 45 9 2 3 3 81
3 ΔSV d;g;3 jδδ ¼ CA
212003-3
ð16Þ
TABLE I.
%
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PHYSICAL REVIEW LETTERS
PRL 113, 212003 (2014)
Relative contributions of pure N3 LO terms.
δδ
¯0 δD
¯1 δD
¯2 δD
¯3 δD
¯4 δD
¯5 δD
¯0 D0 D
¯1 D0 D
¯2 D0 D
¯3 D0 D
¯4 D0 D
¯1 D1 D
¯2 D1 D
¯3 D1 D
¯2 D2 D
73.3
16.0
9.1
31.4
1.0
−9.9
−23.1
−13.7
−10.7
−0.3
3.1
7.3
−0.2
3.8
8.6
4.2
We present the relative contributions in percentage of the pure N3 LO terms in Eq. (12) with respect to ΔSV d;g;3 , for ¯ rapidity Y ¼ 0 in Table I. The notation Di Dj corresponds to ¯ j and Dj D ¯ i. the sum of the contributions coming from Di D pffiffiffi We have used s ¼ 14 TeV for the LHC, GF ¼ 4541.68 pb, the Z boson mass mZ ¼ 91.1876 GeV, top quark mass mt ¼ 173.4 GeV and the Higgs boson mass mH ¼ 125.5 GeV throughout. For the Higgs boson production, we use the effective theory where top quark is integrated out in the large mt limit. The strong coupling constant αs ðμ2R Þ is evolved using the 4-loop RG equations 3 with αsN LO ðmZ Þ ¼ 0.117 and for parton density sets we use MSTW 2008NNLO [19], as N3 LO evolution kernels are not yet available. In [20], Forte et al. pointed out that the Higgs boson cross sections will remain unaffected with this shortcoming. However, for the DY process, it is not clear whether the same will be true. We find that the contribution from the δð¯z1 Þδð¯z2 Þ part is the largest. The dependence on the renormalization and factorization scales can by studied by varying them in the range ðmH =2Þ < μR ; μF < 2mH . We find that the inclusion of the threshold correction at N3 LO further reduces their dependence. For the inclusive Higgs boson production, we find that about 50% of exact NNLO contribution comes from threshold NLO and NNLO terms. It increases to 80% if we use exact NLO and threshold NNLO terms. Hence, it is expected that the rapidity distribution of the Higgs boson will receive a significant contribution from the threshold region compared to inclusive rate due to the soft emission over the entire range of Y. Our numerical study with threshold enhanced NNLO rapidity distribution confirms our expectation. Comparing our threshold NNLO results against exact NNLO distribution using the FEHiP [21] code, we find that about 90% of exact NNLO distribution comes from the threshold region as can be seen from Table II, in accordance with [22], where it was shown that for low τðm2H =s ≈ 10−5 Þ values the threshold terms are dominant, thanks to the inherent property of the matrix element, which receives the
TABLE II. Y NNLO NNLOSV NNLOSV ðAÞ N3 LOSV N3 LOSV ðAÞ K3
largest radiative corrections from the phase-space points corresponding to Born kinematics. Here we have used the exact results up to the NLO level. Because of an inherent ambiguity in the definition of the partonic cross section at threshold one can multiply a factor zgðzÞ, where z ¼ τ=x1 x2 and limz→1 gðzÞ ¼ 1, with the partonic flux and divide the same in the partonic cross section for an inclusive rate. In [23,24] this was exploited to take into account the subleading collinear logs also, thereby making the threshold approximation a better one. Recently, Anastasiou et al. used this in [14] to modify the partonic flux keeping the partonic cross section unaltered to improve the threshold effects. Following [14,25], we introduce Gðz1 ; z2 Þ such that limz1 ;z2 →1 G ¼ 1 in (2): X Z
1
x01
dz1 z1
Z
1
dz2 I H Gðz1 ; z2 Þ z2 ab ¯ ab¼q;q;g I Δd;ab ðz1 ; z2 Þ lim : z1 ;z2 →1 Gðz1 ; z2 Þ
WI ¼
x02
ð17Þ
We also find that with the choice Gðz1 ; z2 Þ ¼ z21 z22 , the threshold NNLO results are remarkably close to the exact ones for the entire range of Y [see Table II, denoted by ðAÞ]. This clearly demonstrates the dominance of threshold contributions to rapidity distribution of the Higgs boson production at the NNLO level. Assuming that the trend will not change drastically beyond NNLO, we present numerical values for N3 LO distributions for Gðz1 ; z2 Þ ¼ 1; z21 z22, respectively, as N3 LOSV and N3 LOSV ðAÞ in Table II. The threshold N3 LO terms give 6%ðY ¼ 0Þ to 12%ðY ¼ 3.6Þ additional correction over the NNLO contribution to the inclusive DY production. Finally, in Table II, we have presented K3 ¼ N3 LOSV =LO as a function of Y in order to demonstrate the sensitivity of higher order effects to the rapidity Y. To summarize, we present full threshold enhanced N3 LO QCD corrections to rapidity distributions of the dilepton
Contributions of exact NNLO, NNLOSV , N3 LOSV , and K3. 0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
11.21 9.81 10.67 11.62 11.88 2.31
10.96 9.61 10.46 11.36 11.62 2.29
10.70 8.99 9.84 11.07 11.33 2.36
9.13 8.00 8.82 9.44 9.70 2.21
7.80 6.71 7.48 8.04 8.30 2.17
6.10 5.21 5.90 6.27 6.51 2.07
4.23 3.66 4.24 4.33 4.54 1.89
2.66 2.25 2.69 2.70 2.88 1.70
1.40 1.14 1.42 1.40 1.53 1.63
0.54 0.42 0.56 0.53 0.60 1.51
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PRL 113, 212003 (2014)
PHYSICAL REVIEW LETTERS
pair in the DY process and of the Higgs boson in gluongluon fusion at the LHC. We show that the infrared structure of QCD amplitudes, in particular, their factorization properties, along with Sudakov resummation of soft gluons and renormalization group invariance provide an elegant framework to compute these threshold corrections systematically for rapidity distributions order by order in QCD perturbation theory. The recent N3 LO results for inclusive DY and Higgs boson production cross sections at the threshold provide crucial ingredients to obtain δð¯z1 Þδð¯z2 Þ contribution of their rapidity distributions for the first time. We find that this contribution numerically dominates over the rest of the terms in ΔSV d;g;3 at the LHC. Inclusion of N3 LO contributions reduces the scale dependence further. We also demonstrate the dominance of the threshold contribution to rapidity distributions by comparing it against the exact NNLO for two different choices of Gðz1 ; z2 Þ. Finally, we find that threshold N3 LO rapidity distribution with Gðz1 ; z2 Þ ¼ 1; z21 z22 shows a moderate effect over NNLO distribution. We thank F. Petriello for providing the FEHiP code and fruitful discussions. T. A., M. K. M., and N. R. thank IMSc for providing hospitality. We thank M. Mahakhud for discussion. The work of T. A., M. K. M., and N. R. has been partially supported by funding from RECAPP, DAE, Government of India.
*
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