week ending 12 SEPTEMBER 2014
PHYSICAL REVIEW LETTERS
PRL 113, 112002 (2014)
Drell-Yan Production at Threshold to Third Order in QCD Taushif Ahmed,1,* Maguni Mahakhud,1,† Narayan Rana,1,‡ and V. Ravindran2,§ 1
RECAPP, Harish-Chandra Research Institute, Allahabad, India 2 The Institute of Mathematical Sciences, Chennai, India (Received 6 May 2014; published 11 September 2014)
The recent computation on the full threshold contributions to Higgs boson production at next to next to next to leading order (N3 LO) in QCD contains valuable information on the soft gluons resulting from virtual and real emission partonic subprocesses. We use those from the real emissions to obtain the corresponding soft gluon contributions to Drell-Yan production and determine the missing δð1 − zÞ part of the N3 LO. The numerical impact of threshold effects demonstrates the importance of our results in the precision study with the Drell-Yan process at the LHC. DOI: 10.1103/PhysRevLett.113.112002
PACS numbers: 12.38.Bx
Discovery of the Higgs boson and the exclusion limits on its mass strongly depend on the precise knowledge of its production mechanism in the standard model (SM) and its extensions. The fixed as well as resummed next to next to leading order (NNLO) [1] and leading log (NNLL) [2] quantum chromodynamics (QCD) corrections supplemented with two-loop electroweak effects [3] played an important role in the discovery of the Higgs boson by ATLAS and CMS collaborations [4] at the Large Hadron Collider. Also for Drell-Yan (DY) production, NNLO [5] and NNLL [6] QCD results are known. There have been several attempts to go beyond the NNLO level in QCD. The gluon and quark form factors [7–9], the mass factorization kernels [10], and the renormalization constant [11] for the effective operator describing the coupling of the Higgs boson with the SM fields in the infinite top quark mass limit up to the three-loop level in dimensional regularization, with space-time dimension d ¼ 4 þ ϵ, enabled one to obtain the next to next to next to leading order (N3 LO) threshold effects [12,13], often called soft plus virtual (SV) contributions, to the inclusive Higgs boson and DY productions at the LHC, excluding the term proportional to δð1 − zÞ where the scaling parameter z ¼ m2H =ˆs for the Higgs boson and z ¼ m2lþ l− =ˆs for DY. Here mH , mlþ l− , and sˆ are the mass of the Higgs boson, invariant mass of the dileptons, and center of mass energy of the partonic reaction responsible for production mechanism, respectively. Note that the finite mass factorized threshold contribution to the inclusive production cross section is expanded in terms of δð1 − zÞ and Di ðzÞ, where i ln ð1 − zÞ Di ðzÞ ¼ : ð1 − zÞ þ
ð1Þ
The δð1 − zÞ part of N3 LO threshold contribution was not known until recently because of the lack of information on the complete soft contributions coming from real radiation processes, while the exact two- [14] and three-loop [9] 0031-9007=14=113(11)=112002(5)
quark and gluon form factors and NNLO soft contributions [15] to all orders in ϵ are already known. The full threshold N3 LO result for Higgs boson production has now become reality due the recent computation by Anastasiou et al. [16] who have computed all these soft effects resulting from gluon radiations, which constitute the missing part. In this Letter, we investigate the impact of these soft gluon contributions on the δð1 − zÞ part of the N3 LO to DY production. This completes the full N3 LO threshold contribution to DY production. The production cross section of a heavy particle, namely, a Higgs boson or a pair of leptons at the hadron colliders can be computed using 2
σ ðs; q Þ ¼ I
XZ
dx1 dx2 f a ðx1 ; μ2F Þf b ðx2 ; μ2F Þ
ab
× σˆ Iab ðˆs; q2 ; μ2F Þ;
ð2Þ
where sˆ ¼ x1 x2 s, s is the hadronic center of mass energy, and σˆ Iab is the partonic cross section with initial state partons a and b. I ¼ g for Higgs boson production with q2 ¼ m2H and I ¼ q for DY production with invariant mass of the dileptons being q2 . μF is the factorization scale. The threshold contribution at the partonic level, denoted by 2 2 2 ΔSV I ðz; q ; μR ; μF Þ, normalized by the Born cross section I;ð0Þ σˆ ab times the Wilson coefficient CIW ðμ2R Þ, is given by 2 2 2 I 2 2 2 ΔSV I ðz; q ; μR ; μF Þ ¼ C expðΨ ðz; q ; μR ; μF ; ϵÞÞjϵ¼0 ; ð3Þ
where μR is the renormalization scale, the dimensionless variable z ¼ q2 =ˆs, and ΨI ðz; q2 ; μ2R ; μ2F ; ϵÞ is a finite distribution. The symbol C implies convolution with the following expansion: CefðzÞ ¼ δð1 − zÞ þ
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1 1 fðzÞ þ fðzÞ ⊗ fðzÞ þ . ð4Þ 1! 2! © 2014 American Physical Society
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PHYSICAL REVIEW LETTERS
Here ⊗ means Mellin convolution and fðzÞ is a distribution of the kind δð1 − zÞ and Di . In d ¼ 4 þ ϵ dimensions, ΨI ðz; q2 ; μ2R ; μ2F ; ϵÞ ¼ fln½ZI ðaˆ s ; μ2R ; μ2 ; ϵÞ2 þ ln jFˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞj2 gδð1 − zÞ
can be obtained from the form factors. GI1 ðϵÞ and GI2 ðϵÞ are known to all orders in ϵ and GI3 ðϵÞ is known to Oðϵ3 Þ [9]. The mass factorization kernel Γðz; μ2F ; ϵÞ removes the collinear singularities which arise due to massless partons and it satisfies the following RG equation:
þ 2ΦI ðaˆ s ; q2 ; μ2 ; z; ϵÞ 2
− 2C ln ΓII ðaˆ s ; μ
μ2F
; μ2F ; z; ϵÞ;
where μ is the scale introduced to define the dimensionless coupling constant aˆ s ¼ gˆ 2s =16π 2 in dimensional regularization, Q2 ¼ −q2 , ZI ðaˆ s ; μ2R ; μ2 ; ϵÞ is the overall operator renormalization constant, which satisfies μ2R
∞ X d I ˆ 2 2 ln Z ð a ; μ ; μ ; ϵÞ ¼ ais ðμ2R Þγ Ii−1 ; s R dμ2R i¼1
where as ðμ2R Þ is the renormalized coupling constant that is related to aˆ s through strong coupling constant renormal2 ization Z(as ðμ2R Þ), that is aˆ s ¼ ðμ=μR Þϵ Zðμ2R ÞS−1 ϵ as ðμR Þ, Sϵ ¼ exp½ðγ E − ln 4πÞϵ=2. Because of the gauge and renormalization group invariance, the bare form factors Fˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞ satisfy the following differential equation [17]: d 1 I μ2R Q2 μ2R I I ˆ Q ln F ¼ K aˆ s ; 2 ; ϵ þ G aˆ s ; 2 ; 2 ; ϵ ; 2 dQ2 μ μR μ 2
AIi ’s are the cusp anomalous dimensions. Expanding the μ2R independent part of the solution of the renormalization group (RG) equation for GI , GI (as ðQ2 Þ; 1; ϵ) ¼ P ∞ i 2 I I i¼1 as ðQ ÞGi ðϵÞ, one finds that Gi can be decomposed I I in terms of collinear Bi and soft f i anomalous dimensions as follows [18]: ∞ X
ϵk gI;k i ;
ð5Þ
k¼1 I;1 I;1 I where CI1 ¼ 0, CI2 ¼ −2β0 gI;1 1 , C3 ¼ −2β 1 g1 − 2β0 ðg2 þ
d I 1 ¯I μ2R q Φ ¼ K aˆ s ; 2 ; z; ϵ 2 dq2 μ 2 2 ¯ I aˆ s ; q ; μR ; z; ϵ : þG μ2R μ2 2
¯ I take the forms similar to those of K I and GI of K¯ I and G the form factors in such a way that ΨI is finite as ϵ → 0. The solution to the above equation is found to be Φ ¼ I
þ
4β20 gI;3 1 Þ,
∞ X i¼1
∞ X i¼1
aˆ is
2 ϵ q ð1 − zÞ2 i2 i iϵ ϕˆ I;ðiÞ ðϵÞ; ð7Þ Sϵ 1−z μ2
and βi are the coefficients of the β
function of strong coupling constant as ðμ2R Þ, μ2R das ðμ2R Þ= P I;k iþ2 2 dμ2R ¼ ϵas ðμ2R Þ=2 − ∞ i¼0 β i as ðμR Þ. The coefficients gi
aˆ is
2 iϵ ∞ X qz 2 i ¯ I;ðiÞ S ðϵÞ ¼ ais ðq2z ÞG¯ Ii ðϵÞ; G ϵ μ2 i¼1
ð8Þ
where q2z ¼ q2 ð1 − zÞ2 . Using the fact that ΔSV I is finite as I ¯ ϵ → 0, we can express Gi ðϵÞ in the form similar to that of GIi ðϵÞ by setting γ Ii ¼ 0; BIi ¼ 0 and replacing f Ii → −f Ii ¯ I;j ¯ I;j and gI;j i → Gi . The unknown constants Gi can be extracted from the soft part of the partonic reactions. Since ΦI results from the soft radiations, the constants G¯ Ii ðϵÞ are found to be maximally non-Abelian [13] satisfying C G¯ qi ðϵÞ ¼ F G¯ gi ðϵÞ; CA
I;1 I;1 I;2 I;1 I 2β0 gI;2 1 Þ, C4 ¼ −2β 2 g1 − 2β1 ðg2 þ 4β0 g1 Þ − 2β0 ðg3 þ
2β0 gI;2 2
ð6Þ
¯ I;ðiÞ ðϵÞ=iϵ and μ2R dK¯ I =dμ2R ¼ where ϕˆ I;ðiÞ ðϵÞ¼½K¯ I;ðiÞ ðϵÞþ G 2 I 2 −δð1−zÞμR dK =dμR . This implies that K¯ I;ðiÞ ðϵÞ can be written in terms of AIi and βi . We define G¯ Ii ðϵÞ through
∞ X d I 2 d I K ¼ −μ G ¼ − ais ðμ2R ÞAIi : R dμ2R dμ2R i¼1
GIi ðϵÞ ¼ 2ðBIi − γ Ii Þ þ f Ii þ CIi þ
d 1 Γðz; μ2F ; ϵÞ ¼ Pðz; μ2F Þ ⊗ Γðz; μ2F ; ϵÞ: 2 2 dμF
Pðz; μ2F Þ are Altarelli-Parisi splitting functions. In perturP i 2 ði−1Þ bative QCD, Pðz; μ2F Þ ¼ ∞ ðzÞ. We find i¼1 as ðμF ÞP that only diagonal elements of the kernel, ΓII ðaˆ s ; μ2F ; μ2 ; z; ϵÞ contribute to threshold corrections because they contain δð1 − zÞ and D0 at every order perturbation theory while the nondiagonal ones are regular functions in z, that ðiÞ ðiÞ is, PII ðzÞ ¼ 2½BIiþ1 δð1 − zÞ þ AIiþ1 D0 þ Preg;II ðzÞ. The finiteness of ΔSV I demands that the soft distribution I ˆ 2 2 function Φ ðas ; q ; μ ; z; ϵÞ also satisfies Sudakov-type differential equations [13], namely,
where K I contains all the poles in ϵ and GI contains the terms finite in ϵ. Renormalization group invariance of Fˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞ gives μ2R
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ð9Þ
with CA ¼ N, CF ¼ ðN 2 − 1Þ=2N, N is the number of colors. Equation (9) implies that the entire soft distribution function for the DY production can be obtained from that
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PRL 113, 112002 (2014)
of Higgs boson production. Substituting ZI , the solutions for both Fˆ I and ΦI , and ΓII in Eq. (3), we obtain ΔSV I in powers of as ðμ2R Þ as ΔSV I ðzÞ ¼
∞ X
2 ais ðμ2R ÞΔSV I;i ðz; μR Þ;
where
i¼0 SV 2 ΔSV I;i ¼ ΔI;i ðμR Þjδ δð1 − zÞ þ
2i−1 X
2 ΔSV I;i ðμR ÞjDj Dj :
ð10Þ
j¼0
We have set μ2R ¼ μ2F ¼ q2 and their dependence can be retrieved using the appropriate renormalization group 2 equation. ΔSV I;i ðQ Þ are finite and they depend on the
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anomalous dimensions AIi , BIi , f Ii and γ Ii , the β function coefficients βi and ϵ expansion coefficients of GI ðϵÞ, gI;i j ’s I;i I ¯ ¯ and of the corresponding G ðϵÞ, Gj ’s. Up to the two-loop level, all these terms are known to sufficient accuracy to SV obtain ΔSV I;1 and ΔI;2 exactly. At next to next to next to leading order level, only ΔSV I;3 jDi ’s were known [12,13] as the term ¯Gg;1 needed for ΔSV j has not been available. Recently in 3 I;3 δ Ref. [16], Anastasiou et al. have obtained ΔSV g;3 jδ , using this g;1 ¯ we extract G3 . This along with Eq. (9) can be used to SV determine the corresponding G¯ q;1 3 and hence Δq;3 jδ . This completes the evaluation of full DY soft plus virtual contributions at N3 LO. The result for G¯ I;1 3 is given by
¯GI;1 ¼ CI CA 2 152 ζ 2 3 þ 1964 ζ 2 2 þ 11 000 ζ 2 ζ 3 − 765 127 ζ2 þ 536 ζ3 2 − 59 648 ζ 3 − 1430 ζ 5 þ 7 135 981 3 63 9 9 486 3 27 3 8748 532 2 1208 105 059 45 956 148 716 509 þ CA nf − ζ − ζ ζ þ ζ þ ζ þ ζ − 9 2 9 2 3 243 2 81 3 3 5 4374 152 2 605 2536 112 42 727 þ CF nf ζ 2 − 88ζ2 ζ3 þ ζ2 þ ζ3 þ ζ5 − 15 6 27 3 324 32 2 1996 2720 11 584 þ nf 2 ζ2 − ζ2 − ζ3 þ ; 9 81 81 2187
ð11Þ
with nf being the number of light flavors and CI ≡ CA ; CF for I ¼ g; q, respectively. The ΔSV q;3 jδ is given by ΔSV q;3 jδ
13 264 3 14 611 2 884 400 2 82 385 1 505 881 ζ þ ζ − ζ ζ þ 843ζ 2 − ζ þ ζ − 204ζ5 − ¼ 315 2 135 2 3 2 3 3 3 81 3 972 20 816 3 1664 2 28 736 13 186 3280 2 20 156 39 304 74 321 þ CA C2F − ζ2 − ζ2 þ ζ2 ζ3 − ζ2 þ ζ3 − ζ3 − ζ5 þ 315 135 9 27 3 9 9 36 5756 2 208 28 132 6016 110 651 þ CA CF nf − ζ2 þ ζ2 ζ3 − ζ2 − ζ3 − 8ζ 5 þ 135 3 81 81 243 184 736 3 412 2 130 10 336 2 5599 3 þ CF − ζ þ ζ þ 80ζ2 ζ3 − ζ þ ζ3 − 460ζ 3 þ 1328ζ 5 − 315 2 5 2 3 2 3 6 272 2 5504 2632 3512 5536 421 þ C2F nf ζ − ζ ζ þ ζ þ ζ þ ζ − 135 2 9 2 3 27 2 9 3 9 5 3 2 N −4 4 28 160 128 2 2416 1264 7081 − ζ2 2 þ 20ζ2 þ ζ 3 − ζ5 þ 8 þ CF n2f ζ2 þ ζ2 − ζ3 − ; þ CF nf;v N 5 3 3 27 81 81 243 C2A CF
ð12Þ
where, nf;v is proportional to the charge weighted sum of the quark flavors [9]. 3 We present the contribution from ΔSV q;3 jδ to pure N LOSV SV as δN3 LO and the contributions from Δq;3 jDi s to pure N3 LOSV as DN3 LO in the Table I for different invariant pffiffiffi masses (mlþ l− ≡ Q) of the dileptons. We have used s ¼ 14 TeV for the LHC, number of light quark flavors nf ¼ 5, Fermi constant GF ¼ 4541.68 pb, the Z boson mass
mZ ¼ 91.1876 GeV, and top quark mass mt ¼173.4GeV throughout. The strong coupling constant αs ðμ2R Þ is evolved using the four-loop renormalization group equations with 3 αsN LO ðmZ Þ ¼ 0.117 and for parton density sets we use Martin-Stirling-Thorne-Watt (MSTW) 2008NNLO [19]. We find that the δ contribution is almost equal and opposite in sign to the sum of the contributions from the Di ’s. Hence, adding the δ part reduces the pure N3 LOSV
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TABLE I.
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PHYSICAL REVIEW LETTERS
PRL 113, 112002 (2014)
Contributions of δN3 LO ; DN3 LO , NNLO (SV), exact NNLO, N3 LO (SV) and N3 LOSV
Q (GeV)
30
50
70
90
100
200
3
400 −2
600
800 −4
1000
3.153 × 10 6.473 × 10 2.006 × 10 7.755 × 10−5 10 δN3 LO (nb) 11.386 2.561 1.724 140.114 5.410 4.567 × 10 103 DN3 LO (nb) −8.397 −2.053 −1.466 −124.493 −4.865 −4.421 × 10−2 −3.368 × 10−3 −7.455 × 10−4 −2.456 × 10−4 −9.959 × 10−5 NNLO (SV) 0.497 0.147 0.117 10.749 0.436 4.917 × 10−3 4.364 × 10−4 1.032 × 10−4 3.538 × 10−5 1.480 × 10−5 NNLO 0.543 0.158 0.124 11.296 0.458 5.233 × 10−3 4.694 × 10−4 1.116 × 10−4 3.836 × 10−5 1.607 × 10−5 3 N LO (SV) 0.500 0.148 0.118 10.765 0.436 4.918 × 10−3 4.362 × 10−4 1.032 × 10−4 3.534 × 10−5 1.478 × 10−5 N3 LOSV 0.546 0.158 0.124 11.311 0.459 5.234 × 10−3 4.692 × 10−4 1.116 × 10−4 3.832 × 10−5 1.605 × 10−5
term to 1 order in magnitude, establishing the dominance of the δ term. We have studied the effect of threshold corrections resulting from distributions such as δð1 − zÞ and Di both at NLO as well as NNLO levels. In the following, we report our findings based on the numerical analysis presented in the table for two different ranges of Q, namely, Q ¼ 200–1000 GeV (above mZ ) and 30–100 GeV (below mZ ). At NLO, if we keep only the distributions and drop contributions from hard radiations ¯ initiated processes, we find coming from qq¯ and qðqÞg that the resulting NLO corrected cross section is about 95% of the exact result at NLO level. Similarly, if we keep the distributions and drop all the hard radiations both in NLO as well as in NNLO terms, we find that the resulting NNLO corrected result [NNLO(SV)] is about 95% of the exact one at NNLO level. Hence, it is expected that the sum (N3 LOSV ) of threshold contributions of N3 LO terms and the exact NNLO corrected result would constitute the dominant contribution at N3 LO level. Like NNLO terms, the threshold contributions in N3 LO terms are also moderate and hence the perturbation theory behaves well. In Fig. 1, we have shown the dependence of our result on the renormalization scale at various orders in perturbation theory. We have plotted RðiÞ ¼ σ iðμR Þ =σ iðQÞ where i ¼ NLO, NNLO, N3 LOSV versus μR =Q and the reduction in the scale dependence is evident as we increase the order of perturbation.
−3
−4
To summarize, we present a systematic way of computing the threshold corrections to inclusive Higgs boson and DY productions in perturbative QCD. We have used several properties of QCD amplitudes, namely, factorization of soft and collinear divergences, renormalization group invariance, and resummation of threshold contributions. For the first time we show that the recent N3 LO soft plus virtual contribution to the Higgs boson production cross section can be used to obtain the corresponding δð1 − zÞ part of DY production at N3 LO. We also present numerical results to establish the importance of the δ term. We find that the impact of the δ contribution is quite large to the pure N3 LOSV correction. We have also demonstrated the dominance of threshold corrections at every order in perturbation theory. We expect that the results presented in this Letter will not only be a benchmark for a full N3 LO order contribution but also an important step in the precision study with the Drell-Yan process at the LHC. T. A., M. M., and N. R. are thankful for the hospitality provided by the Institute of Mathematical Sciences (IMSc) where the work was carried out. We thank the staff of IMSc computer center for their help. We thank M. K. Mandal for his generous help. We also thank the referees for useful suggestions. The work of T. A., M. M., and N. R. has been partially supported by funding from RECAPP, Department of Atomic Energy, Government of India.
1.15 NLO
1.1
NNLO
1 0.95 0.9 1.06 Q = 200 GeV
R
(i)
1.04 1.02 1 0.98 0.2
0.4
0.6
0.8
1 1.2 µR/Q
FIG. 1 (color online).
1.4
1.6
1.8
Scale variation.
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PRL 113, 112002 (2014)
Drell-Yan Production at Threshold to Third Order in QCD Taushif Ahmed,1,* Maguni Mahakhud,1,† Narayan Rana,1,‡ and V. Ravindran2,§ 1
RECAPP, Harish-Chandra Research Institute, Allahabad, India 2 The Institute of Mathematical Sciences, Chennai, India (Received 6 May 2014; published 11 September 2014)
The recent computation on the full threshold contributions to Higgs boson production at next to next to next to leading order (N3 LO) in QCD contains valuable information on the soft gluons resulting from virtual and real emission partonic subprocesses. We use those from the real emissions to obtain the corresponding soft gluon contributions to Drell-Yan production and determine the missing δð1 − zÞ part of the N3 LO. The numerical impact of threshold effects demonstrates the importance of our results in the precision study with the Drell-Yan process at the LHC. DOI: 10.1103/PhysRevLett.113.112002
PACS numbers: 12.38.Bx
Discovery of the Higgs boson and the exclusion limits on its mass strongly depend on the precise knowledge of its production mechanism in the standard model (SM) and its extensions. The fixed as well as resummed next to next to leading order (NNLO) [1] and leading log (NNLL) [2] quantum chromodynamics (QCD) corrections supplemented with two-loop electroweak effects [3] played an important role in the discovery of the Higgs boson by ATLAS and CMS collaborations [4] at the Large Hadron Collider. Also for Drell-Yan (DY) production, NNLO [5] and NNLL [6] QCD results are known. There have been several attempts to go beyond the NNLO level in QCD. The gluon and quark form factors [7–9], the mass factorization kernels [10], and the renormalization constant [11] for the effective operator describing the coupling of the Higgs boson with the SM fields in the infinite top quark mass limit up to the three-loop level in dimensional regularization, with space-time dimension d ¼ 4 þ ϵ, enabled one to obtain the next to next to next to leading order (N3 LO) threshold effects [12,13], often called soft plus virtual (SV) contributions, to the inclusive Higgs boson and DY productions at the LHC, excluding the term proportional to δð1 − zÞ where the scaling parameter z ¼ m2H =ˆs for the Higgs boson and z ¼ m2lþ l− =ˆs for DY. Here mH , mlþ l− , and sˆ are the mass of the Higgs boson, invariant mass of the dileptons, and center of mass energy of the partonic reaction responsible for production mechanism, respectively. Note that the finite mass factorized threshold contribution to the inclusive production cross section is expanded in terms of δð1 − zÞ and Di ðzÞ, where i ln ð1 − zÞ Di ðzÞ ¼ : ð1 − zÞ þ
ð1Þ
The δð1 − zÞ part of N3 LO threshold contribution was not known until recently because of the lack of information on the complete soft contributions coming from real radiation processes, while the exact two- [14] and three-loop [9] 0031-9007=14=113(11)=112002(5)
quark and gluon form factors and NNLO soft contributions [15] to all orders in ϵ are already known. The full threshold N3 LO result for Higgs boson production has now become reality due the recent computation by Anastasiou et al. [16] who have computed all these soft effects resulting from gluon radiations, which constitute the missing part. In this Letter, we investigate the impact of these soft gluon contributions on the δð1 − zÞ part of the N3 LO to DY production. This completes the full N3 LO threshold contribution to DY production. The production cross section of a heavy particle, namely, a Higgs boson or a pair of leptons at the hadron colliders can be computed using 2
σ ðs; q Þ ¼ I
XZ
dx1 dx2 f a ðx1 ; μ2F Þf b ðx2 ; μ2F Þ
ab
× σˆ Iab ðˆs; q2 ; μ2F Þ;
ð2Þ
where sˆ ¼ x1 x2 s, s is the hadronic center of mass energy, and σˆ Iab is the partonic cross section with initial state partons a and b. I ¼ g for Higgs boson production with q2 ¼ m2H and I ¼ q for DY production with invariant mass of the dileptons being q2 . μF is the factorization scale. The threshold contribution at the partonic level, denoted by 2 2 2 ΔSV I ðz; q ; μR ; μF Þ, normalized by the Born cross section I;ð0Þ σˆ ab times the Wilson coefficient CIW ðμ2R Þ, is given by 2 2 2 I 2 2 2 ΔSV I ðz; q ; μR ; μF Þ ¼ C expðΨ ðz; q ; μR ; μF ; ϵÞÞjϵ¼0 ; ð3Þ
where μR is the renormalization scale, the dimensionless variable z ¼ q2 =ˆs, and ΨI ðz; q2 ; μ2R ; μ2F ; ϵÞ is a finite distribution. The symbol C implies convolution with the following expansion: CefðzÞ ¼ δð1 − zÞ þ
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1 1 fðzÞ þ fðzÞ ⊗ fðzÞ þ . ð4Þ 1! 2! © 2014 American Physical Society
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PHYSICAL REVIEW LETTERS
Here ⊗ means Mellin convolution and fðzÞ is a distribution of the kind δð1 − zÞ and Di . In d ¼ 4 þ ϵ dimensions, ΨI ðz; q2 ; μ2R ; μ2F ; ϵÞ ¼ fln½ZI ðaˆ s ; μ2R ; μ2 ; ϵÞ2 þ ln jFˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞj2 gδð1 − zÞ
can be obtained from the form factors. GI1 ðϵÞ and GI2 ðϵÞ are known to all orders in ϵ and GI3 ðϵÞ is known to Oðϵ3 Þ [9]. The mass factorization kernel Γðz; μ2F ; ϵÞ removes the collinear singularities which arise due to massless partons and it satisfies the following RG equation:
þ 2ΦI ðaˆ s ; q2 ; μ2 ; z; ϵÞ 2
− 2C ln ΓII ðaˆ s ; μ
μ2F
; μ2F ; z; ϵÞ;
where μ is the scale introduced to define the dimensionless coupling constant aˆ s ¼ gˆ 2s =16π 2 in dimensional regularization, Q2 ¼ −q2 , ZI ðaˆ s ; μ2R ; μ2 ; ϵÞ is the overall operator renormalization constant, which satisfies μ2R
∞ X d I ˆ 2 2 ln Z ð a ; μ ; μ ; ϵÞ ¼ ais ðμ2R Þγ Ii−1 ; s R dμ2R i¼1
where as ðμ2R Þ is the renormalized coupling constant that is related to aˆ s through strong coupling constant renormal2 ization Z(as ðμ2R Þ), that is aˆ s ¼ ðμ=μR Þϵ Zðμ2R ÞS−1 ϵ as ðμR Þ, Sϵ ¼ exp½ðγ E − ln 4πÞϵ=2. Because of the gauge and renormalization group invariance, the bare form factors Fˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞ satisfy the following differential equation [17]: d 1 I μ2R Q2 μ2R I I ˆ Q ln F ¼ K aˆ s ; 2 ; ϵ þ G aˆ s ; 2 ; 2 ; ϵ ; 2 dQ2 μ μR μ 2
AIi ’s are the cusp anomalous dimensions. Expanding the μ2R independent part of the solution of the renormalization group (RG) equation for GI , GI (as ðQ2 Þ; 1; ϵ) ¼ P ∞ i 2 I I i¼1 as ðQ ÞGi ðϵÞ, one finds that Gi can be decomposed I I in terms of collinear Bi and soft f i anomalous dimensions as follows [18]: ∞ X
ϵk gI;k i ;
ð5Þ
k¼1 I;1 I;1 I where CI1 ¼ 0, CI2 ¼ −2β0 gI;1 1 , C3 ¼ −2β 1 g1 − 2β0 ðg2 þ
d I 1 ¯I μ2R q Φ ¼ K aˆ s ; 2 ; z; ϵ 2 dq2 μ 2 2 ¯ I aˆ s ; q ; μR ; z; ϵ : þG μ2R μ2 2
¯ I take the forms similar to those of K I and GI of K¯ I and G the form factors in such a way that ΨI is finite as ϵ → 0. The solution to the above equation is found to be Φ ¼ I
þ
4β20 gI;3 1 Þ,
∞ X i¼1
∞ X i¼1
aˆ is
2 ϵ q ð1 − zÞ2 i2 i iϵ ϕˆ I;ðiÞ ðϵÞ; ð7Þ Sϵ 1−z μ2
and βi are the coefficients of the β
function of strong coupling constant as ðμ2R Þ, μ2R das ðμ2R Þ= P I;k iþ2 2 dμ2R ¼ ϵas ðμ2R Þ=2 − ∞ i¼0 β i as ðμR Þ. The coefficients gi
aˆ is
2 iϵ ∞ X qz 2 i ¯ I;ðiÞ S ðϵÞ ¼ ais ðq2z ÞG¯ Ii ðϵÞ; G ϵ μ2 i¼1
ð8Þ
where q2z ¼ q2 ð1 − zÞ2 . Using the fact that ΔSV I is finite as I ¯ ϵ → 0, we can express Gi ðϵÞ in the form similar to that of GIi ðϵÞ by setting γ Ii ¼ 0; BIi ¼ 0 and replacing f Ii → −f Ii ¯ I;j ¯ I;j and gI;j i → Gi . The unknown constants Gi can be extracted from the soft part of the partonic reactions. Since ΦI results from the soft radiations, the constants G¯ Ii ðϵÞ are found to be maximally non-Abelian [13] satisfying C G¯ qi ðϵÞ ¼ F G¯ gi ðϵÞ; CA
I;1 I;1 I;2 I;1 I 2β0 gI;2 1 Þ, C4 ¼ −2β 2 g1 − 2β1 ðg2 þ 4β0 g1 Þ − 2β0 ðg3 þ
2β0 gI;2 2
ð6Þ
¯ I;ðiÞ ðϵÞ=iϵ and μ2R dK¯ I =dμ2R ¼ where ϕˆ I;ðiÞ ðϵÞ¼½K¯ I;ðiÞ ðϵÞþ G 2 I 2 −δð1−zÞμR dK =dμR . This implies that K¯ I;ðiÞ ðϵÞ can be written in terms of AIi and βi . We define G¯ Ii ðϵÞ through
∞ X d I 2 d I K ¼ −μ G ¼ − ais ðμ2R ÞAIi : R dμ2R dμ2R i¼1
GIi ðϵÞ ¼ 2ðBIi − γ Ii Þ þ f Ii þ CIi þ
d 1 Γðz; μ2F ; ϵÞ ¼ Pðz; μ2F Þ ⊗ Γðz; μ2F ; ϵÞ: 2 2 dμF
Pðz; μ2F Þ are Altarelli-Parisi splitting functions. In perturP i 2 ði−1Þ bative QCD, Pðz; μ2F Þ ¼ ∞ ðzÞ. We find i¼1 as ðμF ÞP that only diagonal elements of the kernel, ΓII ðaˆ s ; μ2F ; μ2 ; z; ϵÞ contribute to threshold corrections because they contain δð1 − zÞ and D0 at every order perturbation theory while the nondiagonal ones are regular functions in z, that ðiÞ ðiÞ is, PII ðzÞ ¼ 2½BIiþ1 δð1 − zÞ þ AIiþ1 D0 þ Preg;II ðzÞ. The finiteness of ΔSV I demands that the soft distribution I ˆ 2 2 function Φ ðas ; q ; μ ; z; ϵÞ also satisfies Sudakov-type differential equations [13], namely,
where K I contains all the poles in ϵ and GI contains the terms finite in ϵ. Renormalization group invariance of Fˆ I ðaˆ s ; Q2 ; μ2 ; ϵÞ gives μ2R
week ending 12 SEPTEMBER 2014
ð9Þ
with CA ¼ N, CF ¼ ðN 2 − 1Þ=2N, N is the number of colors. Equation (9) implies that the entire soft distribution function for the DY production can be obtained from that
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of Higgs boson production. Substituting ZI , the solutions for both Fˆ I and ΦI , and ΓII in Eq. (3), we obtain ΔSV I in powers of as ðμ2R Þ as ΔSV I ðzÞ ¼
∞ X
2 ais ðμ2R ÞΔSV I;i ðz; μR Þ;
where
i¼0 SV 2 ΔSV I;i ¼ ΔI;i ðμR Þjδ δð1 − zÞ þ
2i−1 X
2 ΔSV I;i ðμR ÞjDj Dj :
ð10Þ
j¼0
We have set μ2R ¼ μ2F ¼ q2 and their dependence can be retrieved using the appropriate renormalization group 2 equation. ΔSV I;i ðQ Þ are finite and they depend on the
week ending 12 SEPTEMBER 2014
anomalous dimensions AIi , BIi , f Ii and γ Ii , the β function coefficients βi and ϵ expansion coefficients of GI ðϵÞ, gI;i j ’s I;i I ¯ ¯ and of the corresponding G ðϵÞ, Gj ’s. Up to the two-loop level, all these terms are known to sufficient accuracy to SV obtain ΔSV I;1 and ΔI;2 exactly. At next to next to next to leading order level, only ΔSV I;3 jDi ’s were known [12,13] as the term ¯Gg;1 needed for ΔSV j has not been available. Recently in 3 I;3 δ Ref. [16], Anastasiou et al. have obtained ΔSV g;3 jδ , using this g;1 ¯ we extract G3 . This along with Eq. (9) can be used to SV determine the corresponding G¯ q;1 3 and hence Δq;3 jδ . This completes the evaluation of full DY soft plus virtual contributions at N3 LO. The result for G¯ I;1 3 is given by
¯GI;1 ¼ CI CA 2 152 ζ 2 3 þ 1964 ζ 2 2 þ 11 000 ζ 2 ζ 3 − 765 127 ζ2 þ 536 ζ3 2 − 59 648 ζ 3 − 1430 ζ 5 þ 7 135 981 3 63 9 9 486 3 27 3 8748 532 2 1208 105 059 45 956 148 716 509 þ CA nf − ζ − ζ ζ þ ζ þ ζ þ ζ − 9 2 9 2 3 243 2 81 3 3 5 4374 152 2 605 2536 112 42 727 þ CF nf ζ 2 − 88ζ2 ζ3 þ ζ2 þ ζ3 þ ζ5 − 15 6 27 3 324 32 2 1996 2720 11 584 þ nf 2 ζ2 − ζ2 − ζ3 þ ; 9 81 81 2187
ð11Þ
with nf being the number of light flavors and CI ≡ CA ; CF for I ¼ g; q, respectively. The ΔSV q;3 jδ is given by ΔSV q;3 jδ
13 264 3 14 611 2 884 400 2 82 385 1 505 881 ζ þ ζ − ζ ζ þ 843ζ 2 − ζ þ ζ − 204ζ5 − ¼ 315 2 135 2 3 2 3 3 3 81 3 972 20 816 3 1664 2 28 736 13 186 3280 2 20 156 39 304 74 321 þ CA C2F − ζ2 − ζ2 þ ζ2 ζ3 − ζ2 þ ζ3 − ζ3 − ζ5 þ 315 135 9 27 3 9 9 36 5756 2 208 28 132 6016 110 651 þ CA CF nf − ζ2 þ ζ2 ζ3 − ζ2 − ζ3 − 8ζ 5 þ 135 3 81 81 243 184 736 3 412 2 130 10 336 2 5599 3 þ CF − ζ þ ζ þ 80ζ2 ζ3 − ζ þ ζ3 − 460ζ 3 þ 1328ζ 5 − 315 2 5 2 3 2 3 6 272 2 5504 2632 3512 5536 421 þ C2F nf ζ − ζ ζ þ ζ þ ζ þ ζ − 135 2 9 2 3 27 2 9 3 9 5 3 2 N −4 4 28 160 128 2 2416 1264 7081 − ζ2 2 þ 20ζ2 þ ζ 3 − ζ5 þ 8 þ CF n2f ζ2 þ ζ2 − ζ3 − ; þ CF nf;v N 5 3 3 27 81 81 243 C2A CF
ð12Þ
where, nf;v is proportional to the charge weighted sum of the quark flavors [9]. 3 We present the contribution from ΔSV q;3 jδ to pure N LOSV SV as δN3 LO and the contributions from Δq;3 jDi s to pure N3 LOSV as DN3 LO in the Table I for different invariant pffiffiffi masses (mlþ l− ≡ Q) of the dileptons. We have used s ¼ 14 TeV for the LHC, number of light quark flavors nf ¼ 5, Fermi constant GF ¼ 4541.68 pb, the Z boson mass
mZ ¼ 91.1876 GeV, and top quark mass mt ¼173.4GeV throughout. The strong coupling constant αs ðμ2R Þ is evolved using the four-loop renormalization group equations with 3 αsN LO ðmZ Þ ¼ 0.117 and for parton density sets we use Martin-Stirling-Thorne-Watt (MSTW) 2008NNLO [19]. We find that the δ contribution is almost equal and opposite in sign to the sum of the contributions from the Di ’s. Hence, adding the δ part reduces the pure N3 LOSV
112002-3
TABLE I.
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PHYSICAL REVIEW LETTERS
PRL 113, 112002 (2014)
Contributions of δN3 LO ; DN3 LO , NNLO (SV), exact NNLO, N3 LO (SV) and N3 LOSV
Q (GeV)
30
50
70
90
100
200
3
400 −2
600
800 −4
1000
3.153 × 10 6.473 × 10 2.006 × 10 7.755 × 10−5 10 δN3 LO (nb) 11.386 2.561 1.724 140.114 5.410 4.567 × 10 103 DN3 LO (nb) −8.397 −2.053 −1.466 −124.493 −4.865 −4.421 × 10−2 −3.368 × 10−3 −7.455 × 10−4 −2.456 × 10−4 −9.959 × 10−5 NNLO (SV) 0.497 0.147 0.117 10.749 0.436 4.917 × 10−3 4.364 × 10−4 1.032 × 10−4 3.538 × 10−5 1.480 × 10−5 NNLO 0.543 0.158 0.124 11.296 0.458 5.233 × 10−3 4.694 × 10−4 1.116 × 10−4 3.836 × 10−5 1.607 × 10−5 3 N LO (SV) 0.500 0.148 0.118 10.765 0.436 4.918 × 10−3 4.362 × 10−4 1.032 × 10−4 3.534 × 10−5 1.478 × 10−5 N3 LOSV 0.546 0.158 0.124 11.311 0.459 5.234 × 10−3 4.692 × 10−4 1.116 × 10−4 3.832 × 10−5 1.605 × 10−5
term to 1 order in magnitude, establishing the dominance of the δ term. We have studied the effect of threshold corrections resulting from distributions such as δð1 − zÞ and Di both at NLO as well as NNLO levels. In the following, we report our findings based on the numerical analysis presented in the table for two different ranges of Q, namely, Q ¼ 200–1000 GeV (above mZ ) and 30–100 GeV (below mZ ). At NLO, if we keep only the distributions and drop contributions from hard radiations ¯ initiated processes, we find coming from qq¯ and qðqÞg that the resulting NLO corrected cross section is about 95% of the exact result at NLO level. Similarly, if we keep the distributions and drop all the hard radiations both in NLO as well as in NNLO terms, we find that the resulting NNLO corrected result [NNLO(SV)] is about 95% of the exact one at NNLO level. Hence, it is expected that the sum (N3 LOSV ) of threshold contributions of N3 LO terms and the exact NNLO corrected result would constitute the dominant contribution at N3 LO level. Like NNLO terms, the threshold contributions in N3 LO terms are also moderate and hence the perturbation theory behaves well. In Fig. 1, we have shown the dependence of our result on the renormalization scale at various orders in perturbation theory. We have plotted RðiÞ ¼ σ iðμR Þ =σ iðQÞ where i ¼ NLO, NNLO, N3 LOSV versus μR =Q and the reduction in the scale dependence is evident as we increase the order of perturbation.
−3
−4
To summarize, we present a systematic way of computing the threshold corrections to inclusive Higgs boson and DY productions in perturbative QCD. We have used several properties of QCD amplitudes, namely, factorization of soft and collinear divergences, renormalization group invariance, and resummation of threshold contributions. For the first time we show that the recent N3 LO soft plus virtual contribution to the Higgs boson production cross section can be used to obtain the corresponding δð1 − zÞ part of DY production at N3 LO. We also present numerical results to establish the importance of the δ term. We find that the impact of the δ contribution is quite large to the pure N3 LOSV correction. We have also demonstrated the dominance of threshold corrections at every order in perturbation theory. We expect that the results presented in this Letter will not only be a benchmark for a full N3 LO order contribution but also an important step in the precision study with the Drell-Yan process at the LHC. T. A., M. M., and N. R. are thankful for the hospitality provided by the Institute of Mathematical Sciences (IMSc) where the work was carried out. We thank the staff of IMSc computer center for their help. We thank M. K. Mandal for his generous help. We also thank the referees for useful suggestions. The work of T. A., M. M., and N. R. has been partially supported by funding from RECAPP, Department of Atomic Energy, Government of India.
1.15 NLO
1.1
NNLO
1 0.95 0.9 1.06 Q = 200 GeV
R
(i)
1.04 1.02 1 0.98 0.2
0.4
0.6
0.8
1 1.2 µR/Q
FIG. 1 (color online).
1.4
1.6
1.8
Scale variation.
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