Bootstrapping Rapidity Anomalous Dimensions
for Transverse-Momentum Resummation
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Li, Ye, and Hua Xing Zhu. “Bootstrapping Rapidity Anomalous
Dimensions for Transverse-Momentum Resummation.” Physical
Review Letters 118.2 (2017): n. pag. © 2017 American Physical
Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.118.022004
Publisher
American Physical Society
Version
Final published version
Citable link
http://hdl.handle.net/1721.1/106877
Terms of Use
Article is made available in accordance with the publisher's
policy and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Bootstrapping Rapidity Anomalous Dimensions
for Transverse-Momentum Resummation
Ye Li1,* and Hua Xing Zhu2,†
1
Fermilab, P.O. Box 500, Batavia, Illinois 60510, USA
2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 11 April 2016; revised manuscript received 12 October 2016; published 11 January 2017)
A soft function relevant for transverse-momentum resummation for Drell-Yan or Higgs production at hadron colliders is computed through to three loops in the expansion of strong coupling, with the help of the bootstrap technique and supersymmetric decomposition. The corresponding rapidity anomalous dimension is extracted. An intriguing relation between anomalous dimensions for transverse-momentum resummation and threshold resummation is found.
DOI:10.1103/PhysRevLett.118.022004
Introduction.—The transverse-momentum (qT) distribu-tion of generic high-mass color-neutral systems (Drell-Yan lepton pair, Higgs, EW vector boson pair, etc.) produced in hadron collisions has been of great interest since the early days of quantum chromodynamics (QCD)[1–17]. It pro-vides a testing ground for examination and improvement of our understanding of QCD, both perturbatively and non-perturbatively. When qT is small compared with the
invari-ant mass Q of the system, fixed-order perturbation theory breaks down due to the appearance of large logarithms of the form lnkðq2
T=Q2Þ=q2T, with k≥ 0 at each order in strong
couplingαS. These large logarithms originate from
incom-plete cancellation of soft and collinear divergences between real and virtual diagrams. Fortunately, Collins, Soper, and Sterman (CSS) have shown that they can be systematically resummed to all orders in perturbation theory[5], thanks to QCD factorization.
In recent years, there have been increasing interests in applying soft-collinear effective theory (SCET)[18–22] to resum large logarithms in perturbative QCD using the renormalization group (RG) method. For qT resummation this has been done by a number of authors[23–29]. For the transverse-momentum observable, the relevant momentum modes in the light-cone coordinate for fields in the effective theory are soft, ps∼ Qðλ; λ; λÞ, collinear, pc∼ Qðλ2;1; λÞ,
and anticollinear, p¯c∼ Qð1; λ2;λÞ. Here λ ∼ qT=Q is a
power counting parameter. The corresponding effective theory is SCETII. An important feature of SCETII is that
soft and collinear modes live on the same hyperbola of virtuality, p2s∼ p2c∼ p2¯c∼ λ2Q2. Besides the usual large logarithms of the ratio between hard scale Q and soft scale λQ, there are also large rapidity separations between soft, collinear, and anticollinear modes that need to be resummed. In this Letter we adopt the rapidity RG formalism of Chiu, Jain, Neill, and Rothstein[27,28]. According to the rapidity RG formalism, the cross section at small qT factorizes into a hard function H, transverse-momentum-dependent (TMD) beam functions B, and a TMD soft function S⊥. Schematically the factorization formula reads
1 σ d3σðres:Þ d2q~TdYdQ2 ∼ HðμÞ Z d2b~⊥ ð2πÞ2ei~b⊥· ~QT ·½B ⊗ Bð~b⊥;μ; νÞS⊥ð~b⊥;μ; νÞ: ð1Þ Large logarithms in virtuality are resummed by running in the renormalization scale μ, while large logarithms in rapidity are resummed by running in the rapidity scale ν. The μ evolution of the hard function can be derived from the quark or gluon form factor and is well known[30–32]. Since the physical cross section is independent ofμ and ν order by order in the perturbation theory, it follows that the μ and ν evolution of ½B ⊗ B is fixed once the correspond-ing evolution for the soft function is known. The knowl-edge of μ and ν evolution of the hard, beam, and soft function, together with the boundary conditions of these functions at initial scales, determine the all order structure of large logarithms of qT.
The naive definition of the TMD soft function is a vacuum expectation value of lightlike Wilson loops with a transverse separation, which suffers from light-cone or rapidity divergence [3]. A proper definition of the TMD soft function requires the introduction of the appropriate regulator for the rapidity divergence. Proposals to regular-ize the rapidity divergence include the nonlightlike axial gauge without Wilson lines[5], tilting Wilson lines off the light cone [33], nearly lightlike Wilson lines with sub-traction of the soft factor[34], modifying the phase space measure [26,27,35], modifying the iε prescription of the eikonal propagator[36], etc. In this Letter, we follow the recent proposal [37] by Neill and the current authors of implementing an infinitesimal shift in the time direction to the Wilson loop correlator. Specifically, the TMD soft function with the rapidity regulator of Ref.[37]reads S⊥ð~b⊥;μ;νÞ¼ lim ν→þ∞SF:D:ð~b⊥;μ;νÞ ≡ lim ν→þ∞ 1 da h0jT½S† ¯nð−∞;0ÞSnð0;−∞Þ · ¯T½S†n(−∞;yνð~b⊥Þ)S¯n(yνð~b⊥Þ;−∞)j0i; ð2Þ
where the two Wilson loops are separated by the distance yνð~b⊥Þ ¼ ðib0=ν; ib0=ν; ~b⊥Þ, with b0¼ 2e−γE. S
nð¯nÞ are
path-ordered Wilson lines on the light cone. They carry fundamental or adjoint color indices, depending on whether the color-neutral system is produced in q¯q annihilation (da¼ Nc) or gg fusion (da¼ N2c− 1). T is the time-ordered
operator. The soft function S⊥ in Eq.(2)is closely related to the so-called fully differential soft function[25]SF:D:. The
limitν → þ∞ means that only the nonvanishing terms of SF:D:are kept in that limit. The important role of SF:D:in our calculation will be explained in the next section. Note that our definition for the TMD soft function does not rely on perturbation theory. However, we restrict to the perturba-tively calculable part of the soft function in this Letter.
After minimal subtraction of the dimensional regulariza-tion pole 1=ϵn in the MS scheme, the soft function S
⊥
depends on both the renormalization scaleμ and the rapidity scaleν. The μ evolution of the TMD soft function is specified by the RG equation:
d ln S⊥ð~b⊥;μ; νÞ
d lnμ2 ¼ Γcusp½αSðμÞ ln μ2
ν2− γs½αSðμÞ; ð3Þ
where Γcusp is the well-known lightlike cusp anomalous
dimension[38,39], which is known to three loops in QCD
[40]. γs is the soft anomalous dimension governing the
single logarithmic evolution, which can be extracted through to three loops from the QCD splitting function [40] and quark and gluon form factor[30–32], as is confirmed by the explicit three-loop calculation[41]. The rapidity evolution equation for the TMD soft function reads
d ln S⊥ð~b⊥;μ; νÞ d lnν2 ¼ Z b20=~b2⊥ μ2 d¯μ2 ¯μ2 Γcusp½αSð¯μÞ þ γr½αSðb0=j~b⊥jÞ; ð4Þ
where the rapidity anomalous dimension γr is introduced
for the single logarithmic evolution of rapidity logarithms. Thanks to the non-Abelian exponentiation theorem[42–44], which our regularization procedure [37] preserves, the perturbative soft function can be written as an exponential: S⊥ð~b⊥;μ; νÞ ¼ exp½aSS⊥1 þ a2SS⊥2 þ a3SS⊥3 þ Oða4SÞ; ð5Þ where we have defined aS¼ αSðμÞ=ð4πÞ as our perturbative expansion parameter throughout this Letter. The one- and two-loop coefficients S⊥1;2can be found in Ref.[37]. In the next section we outline the procedure we used to calculate the three-loop coefficient S⊥3, from which the rapidity anomalous dimensions can be extracted to the same order. Method.—To obtain the TMD soft function S⊥ through to three loops, we first calculate the fully differential soft function to the same order. SF:D: obeys a RG equation identical to Eq.(3) [25]:
d ln SF:D:ð~b⊥;μ; νÞ
d lnμ2 ¼ Γcusp½αSðμÞ ln μ2
ν2− γs½αSðμÞ: ð6Þ
In SF:D:, ν is a parameter of the theory, not a regulator.
Therefore, theν dependence of SF:D:is in general
compli-cated. The perturbative solution to SF:D:is then determined
by Eq.(6)and the boundary condition at the initial scale, SF:D:ð~b⊥;μ ¼ ν; νÞ. Similar to S⊥, SF:D:can also be written
as an exponential, as in Eq. (5). The one- and two-loop coefficients SF:D:
1;2 were first computed in Ref. [45], and
reproduced in Ref.[37].
By dimensional analysis, SF:D:ð~b⊥;ν; νÞ is a function of x¼ −~b2⊥ν2=b20. A strategy based on the bootstrap program for scattering amplitudes[46] is proposed in Ref.[37] to compute SF:D:ð~b⊥;ν; νÞ, which we briefly recall below. In
Ref. [45], the one- and two-loop coefficients SF:D: 1;2 are
written in terms of classical and Nielsen’s polylogarithms with argument x. A crucial observation made in Ref.[37]is that the same results can be written in terms of harmonic polylogarithms (HPLs) Hw~ðxÞ [47], with weight indices
drawn from the set f0; 1g. Furthermore, for the available one- and two-loop data, the leftmost and the rightmost index of the weight vectors were found to be 0 and 1, respectively. The rightmost index has to be 1, because the two cusp points of the Wilson loops are separated by Euclidean distance for x <0, and no branch cut is expected. On the other hand, the condition on the leftmost index comes empirically from the observation of the one-and two-loop results; as we will show below; this condition breaks down at three loops in QCD. Nevertheless, for now we proceed with the empirical ansatz for the L-loop fully differential soft function proposed in Ref.[37], which is a linear combination of HPLs with undetermined rational coefficients, and whose weight vectors obey the leftmost-and rightmost-index conditions. The undetermined coef-ficients of the HPLs can then be fixed by performing an expansion around x∼ 0, together with the constraint that rapidity divergence is only a single logarithmic divergence at each order for the expansion coefficients in Eq.(5). It turns out that the x→ 0 limit of SF:D: is smooth, and the expansion is simply a Taylor series in x. As explained in Ref.[37], the leading x0term of the expansion reproduces the threshold soft function[41], while the coefficient of xn
can be obtained by inserting a numeratorðlþl−− l2Þn into
the integrand of the threshold soft function, where l is the total momentum of real radiation from the time-ordered Wilson loop. Furthermore, using integration-by-part iden-tities[48,49], integrals with high rank numerator insertion can be reduced to a small number of master integrals, which have been computed for other purpose recently[50–55].
Although the strategy outlined above is straightforward, it has two caveats. First, the maximal weight of HPLs at three loops for massless perturbation theory is 6. It follows that the number of coefficients that need to be fixed is P4
i¼02i¼ 31. In other words, one needs to insert a
high-rank numerator ðlþl−− l2Þ31 into the integrand of the threshold soft function in order to have enough data to
fix the coefficients, which is unfortunately beyond the ability of the tools for integration-by-part reduction [56–59]. Second, it is not clear whether the conjectured sets of functions in Ref. [37]are sufficient to describe the three-loop soft function. To circumvent the above difficulties, we first perform the calculation for soft Wilson loops whose matter content[41,51,53]resembles those ofN ¼ 4 super-symmetric Yang-Mills theory (SYM). This has a number of advantages. (1) It has been observed that for soft Wilson loops in SCET[41], the results inN ¼ 4 SYM have uniform degrees of transcendentality with transcendental weight 2L at L loops. Furthermore, the N ¼ 4 results match the maximal-weight part of the corresponding QCD results. A similar phenomenon was first observed for the anomalous dimension of the twist-2 operator for Wilson lines[60]. It also holds for some other quantities, e.g., the perturbative form factor[30,61,62]. Assuming that this is also true in our current calculation, by calculating SF:D:inN ¼ 4 SYM first,
we should automatically obtain the maximal-weight part of SF:D: in QCD. (2) Since the N ¼ 4 SYM results have uniform degrees of transcendentality, there are only 16 coefficients to be fixed at three loops, which can be achieved
within the current computation power. (3) The remaining parts of the QCD result have a transcendental weight lower than 6 and, therefore, only require 15 coefficients to be fixed. Alternatively, since the Feynman diagrams corresponding to the lower-weight part have a less complicated analytical structure, they can be computed by brute force. Direct calculation can also test the completeness of the ansatz. And it turns out that although the ansatz remain complete for the three-loop N ¼ 4 SYM result, it fails for the three-loop QCD one. Fortunately, for the QCD result, a brute-force calculation for the terms proportional to nfis possible using the method of Ref.[54]. More importantly, the result for the nfterms indicate which set of functions we should add to the existing ansatz. The full results, for bothN ¼ 4 SYM and QCD, are presented in the next section.
Results.—We first present the results for SF:D:inN ¼ 4 SYM. We only give the results at the initial scale,μ ¼ ν. The full scale dependence can be inferred from Eq.(6). The one- and two-loop coefficients can be found in Ref.[37]. The three-loop coefficient in the four-dimensional-helicity scheme[63]reads SF:D: 3;N ¼4jμ¼ν¼ cs3;N ¼4þ Nc3ð16ζ2H4þ 48ζ2H2;2þ 64ζ2H3;1þ 96ζ2H2;1;1þ 120ζ4H2þ 48H6þ 24H2;4þ 40H3;3 þ 72H4;2þ 128H5;1þ 16H2;1;3þ 56H2;2;2þ 80H2;3;1þ 80H3;1;2þ 144H3;2;1þ 224H4;1;1þ 64H2;1;1;2 þ 96H2;1;2;1þ 160H2;2;1;1þ 256H3;1;1;1þ 192H2;1;1;1;1Þ; ð7Þ where cs
3;N ¼4¼ 492.609N3cis the three-loop constant for the
threshold soft function inN ¼ 4 SYM[41]. We have used the shorthand notation for the HPLs[47]and neglected the argument x. It is interesting to note that each term in Eq.(7)
has a uniform sign and integer coefficient. Furthermore, the
overall sign is alternating at each order inαS[37]. A similar behavior of alternating uniform signs in perturbative ex-pansion with increasing loop order for a certain observable was known before, see Ref.[64]. The corresponding results for QCD in the’t Hooft–Veltman scheme reads
SF:D:3 jμ¼ν ¼ cs3þCaC 2 A N3c ðSF:D: 3;N ¼4ðxÞjμ¼ν− cs3;N ¼4Þ þ CaC2A −1072 9 ζ2H2− 176ζ3H2− 88 3 ζ2H3þ 88ζ2H2;1 þ 30790 81 H2þ 712027 H3− 104 9 H4− 440 3 H5− 8 3 H1;1−H1;1 x −712027 H2;1−1072 9 H2;2− 88 3 H2;3 −3112 9 H3;1− 88H3;2− 352 3 H4;1− 392 3 H2;1;1þ 883 H2;1;2þ 3523 H2;2;1þ 3523 H3;1;1þ 352H2;1;1;1 þ CaCAnf 160 9 ζ2H2þ 163 ζ2H3− 16ζ2H2;1− 7988 81 H2− 2312 27 H3− 64 3 H4þ 803 H5þ 83 H1;1−H1;1 x þ 2312 27 H2;1þ 1609 H2;2þ 163 H2;3þ 2243 H3;1þ 16H3;2þ 643 H4;1− 32 9 H2;1;1− 16 3 H2;1;2− 64 3 H2;2;1 −64 3 H3;1;1− 64H2;1;1;1 þ Can2f 400 81 H2þ 16027 H3þ 329 H4− 160 27 H2;1− 32 9 H3;1þ 329 H2;1;1 þ CaCFnf 32ζ3H2−1103 H2− 8H3þ 8H2;1 ; ð8Þ
where Ca¼ CF for the Drell-Yan process, and Ca¼ CA
for Higgs production. cs3 is the three-loop scale independent part of the threshold soft function in QCD, c3s¼ Sthr:3 ðτ; μ ¼ τ−1Þ, see, for example, Refs.[37,41,65]. It
can be found in Eq. (3.2) of Ref. [41] multiplying by a Casimir rescaling factor Ca=CA. We note that the only term that goes beyond the empirical ansatz [37] is ðH1;1− H1;1=xÞ, which can be inferred from the direct
calculation of the nf-dependent part using the Feynman
diagram method. (This term cancels out in the N ¼ 4 combination, as is clear from Eq.(7). It also cancels out in the pureN ¼ 1 SYM with an adjoint gluino, in which one simply sets nf→ CA and CF→ CA We thank Mingxing
Luo and Lance Dixon for pointing out this.) Specifically, if all the relevant integrals are known, the result forN ¼ 4 SYM in Eq.(7) can also be obtained using the Feynman diagram method, in a gauge theory with nf¼ 4 adjoint
fermions, ns¼ 6 adjoint real scalars, and proper Yukawa
interaction between the fermions and scalars. While the integrals for the pure gluon contribution are challenging, we manage to compute the nf- and ns-dependent terms by a
brute-force Feynman diagram calculation. We observe that for both the fermion and scalar contributions, the only addition needed to correct the empirical ansatz at three loops is the combination ðH1;1− H1;1=xÞ. From there we can readily extract the gluon contribution, which is the same inN ¼ 4 SYM and QCD, by subtracting from Eq.(7)
the corresponding fermion and scalar contributions. We can also conclude that the only addition to the ansatz of the gluon contribution is the combinationðH1;1− H1;1=xÞ.
We briefly describe the available checks on our results in Eqs. (7) and (8). First, as mentioned above, due to the relative simplicity in the resulting integrals, we have been able to compute all the nf-dependent part in Eq. (8) by directly calculating the Feynman diagrams. We find that our ansatz, even including the ð1 − 1=xÞH1;1 term, is insufficient to express the result in the intermediate step of the direct calculation. The additional terms needed are ð1 − 1=xÞH1, H2=x, ζ2H1− H1;2. Interestingly, they all
cancel out in the sum of real and virtual contributions. Second, our ansatz can be uniquely fixed at three loops using the data from a Taylor expansion over x through to x10. However, we have obtained the expansion data through to x17, leading to an over constrained system of equations. We found that the solution exists and is unique for the system, thus providing a strong check of our calculation. See, e.g., Ref.[66]for a similar discussion on using an over constrained system of equations to fix an ansatz.
With the fully differential soft function at hand, it is straightforward to obtain S⊥ by taking the limitν → þ∞ using the packageHPL[67]. The soft anomalous dimension
γsthrough to three loops can be found, e.g., in Eqs. (A.4)–
(A6) of Ref.[41]by a rescaling factor Ca=CA. The rapidity
anomalous dimensions are given by γr 0¼ 0; γr 1¼ CaCA 28ζ3−80827 þ112Canf 27 ; γr 2¼ CaC2A −176 3 ζ3ζ2þ 6392ζ81 2þ 12328ζ27 3þ 154ζ3 4− 192ζ5− 297029 729 þ CaCAnf −824ζ2 81 − 904ζ3 27 þ 20ζ34þ 62626729 þ Can2f −32ζ3 9 − 1856 729 þ CaCFnf −304ζ3 9 − 16ζ4þ 171127 : ð9Þ Note that γr
0 and γr1 can be obtained from the QCD anomalous dimension known a long time ago [68–70]. They
have also been reproduced in SCET recently [37,71–73]. The three-loop coefficientγr
2 is new and is one of the main
results of this Letter. It is also straightforward to obtain the boundary condition of S⊥ at the initial scale c⊥3 ≡ S⊥3ð~b⊥;μ ¼ b0=j~b⊥j; ν ¼ b0=j~b⊥jÞ: c⊥3 ¼ CaC2A 928ζ2 3 9 þ 11009 ζ2ζ3− 151132ζ3 243 − 297481ζ2 729 þ 3649ζ27 4þ 1804ζ9 5− 3086ζ6 27 þ 521194913122 þ CaCAnf 40 9 ζ3ζ2þ 74530ζ729 2þ 8152ζ81 3− 416ζ4 27 − 184ζ5 3 − 412765 6561 þ CaCFnf −80 3 ζ3ζ2þ 275ζ9 2þ 3488ζ81 3þ 152ζ9 4þ 224ζ9 5− 42727 486 þ Can2f −136ζ2 27 − 560ζ3 243 − 44ζ4 27 − 256 6561 : ð10Þ 022004-4
Discussion.—The explicit results for the rapidity anoma-lous dimension in Eq.(9)can be rewritten in a remarkable form: γr 0¼ γs0; γr 1¼ γs1− β0cs1; γr 2¼ γs2− 2β0cs2− β1cs1þ 2CaCAβ0ζ4: ð11Þ
Equation(11)is interesting because it connects between very different objects: the rapidity anomalous dimensionγr, the
soft anomalous dimensionγs, the threshold constant cs, and
the QCD beta function. A similar relation also holds in N ¼ 4 SYM by dropping the beta function terms in Eq.(11). In the CSS formalism, the resummation of large qT logarithms is controlled by two anomalous dimensions, A½αSðμÞ¼Pi¼1ai
SAi and B½αSðμÞ¼
P
i¼1aiSBi. It is
straightforward to express these anomalous dimensions in terms of the anomalous dimension in SCET, see, e.g., Ref. [26,74]. In particular, we obtain the B anomalous dimension in the original CSS scheme through to three loops:
B1¼ γV 0 − γr0; B2¼ γV 1 − γr1þ β0cV1; B3¼ γV 2 − γr2þ β1cV1 þ 2β0 cV 2 −12ðcV1Þ2 ; ð12Þ whereγV is the anomalous dimension of the hard function
results from matching QCD onto SCET, and cV is the scale
independent term of the hard matching[75]. For Drell-Yan production they can be extracted from the quark form factor
[30–32], while for Higgs production they can be extracted from the gluon form factor[30–32], and additionally from effective coupling of the Higgs boson to gluons [76]. Equation(12)partially explains the close connection between γrand γs, because the combinationγV− γsis given by the
δð1 − xÞ part of the single pole in the QCD splitting function
[40]. Substituting the actual numbers in Eq.(12), we find BDY
1 ¼ −8; BDY2 ¼ 13.3447 þ 3.4138nf;
BDY
3 ¼ 7358.86 − 721.516nfþ 20.5951n2f ð13Þ
for Drell-Yan production. For Higgs production, the results are BH 1 ¼ −22 þ 1.33333nf; BH2 ¼ 658.881 − 45.9712nf; BH 3 ¼ 35134.6 − 7311.10nfþ 293.017n2f − ð836 þ 184nf− 14.2222n2fÞ ln m2t m2H: ð14Þ The one- and two-loop results have been known for a long time[68–70]. The three-loop results are new. We note that numerically BDY3 is quite large for nf¼ 5.
In summary, we have presented the first calculation of the soft function for transverse-momentum resummation in rapidity RG formalism through to three loops, using the rapidity regulator recently introduced in Ref. [37]. As a
by-product, we have also obtained the fully differential soft function to the same order. Our calculation combines the use of the bootstrap technique and supersymmetric decomposition in transcendental weight. We found a sur-prising relation between the anomalous dimensions for the transverse-momentum resummation and the threshold resummation, whose explanation calls for further investiga-tion. Our three-loop results pave the way for transverse-momentum resummation for the production of a color neutral system at hadron colliders at N3LLþ NNLO accuracy. The method and results of our calculation also make generalizing the qT-subtraction method[77]to N3LO promising.
We are grateful for a useful conversation with Duff Neill, and helpful comments on the Letter by Iain Stewart. This work was supported by the Office of Nuclear Physics of the U.S. Department of Energy under Contract No. DE-SC0011090. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy.
*
yli32@fnal.gov
†zhuhx@mit.edu
[1] Y. L. Dokshitzer, D. Diakonov, and S. I. Troian, On the transverse momentum distribution of massive lepton pairs,
Phys. Lett. B 79, 269 (1978).
[2] G. Parisi and R. Petronzio, Small transverse momentum distributions in hard processes, Nucl. Phys. B154, 427 (1979).
[3] J. C. Collins and D. E. Soper, Back-to-back jets in QCD,
Nucl. Phys. B193, 381 (1981);B213, 545 (1983). [4] J. C. Collins and D. E. Soper, Back-to-back jets: Fourier
transform from B to K-transverse,Nucl. Phys. B197, 446 (1982).
[5] J. C. Collins, D. E. Soper, and G. F. Sterman, Transverse momentum distribution in Drell-Yan pair and W and Z boson production,Nucl. Phys. B250, 199 (1985). [6] P. B. Arnold and R. P. Kauffman, W and Z production at
next-to-leading order: From large q(t) to small,Nucl. Phys. B349, 381 (1991).
[7] G. A. Ladinsky and C. P. Yuan, The nonperturbative regime in QCD resummation for gauge boson production at hadron colliders,Phys. Rev. D 50, R4239 (1994).
[8] R. K. Ellis and S. Veseli, W and Z transverse momentum distributions: Resummation in qTspace,Nucl. Phys. B511,
649 (1998).
[9] C. Balazs and C. P. Yuan, Soft gluon effects on lepton pairs at hadron colliders, Phys. Rev. D 56, 5558 (1997). [10] J. w. Qiu and X. f. Zhang, QCD Prediction for Heavy Boson
Transverse Momentum Distributions, Phys. Rev. Lett. 86, 2724 (2001).
[11] S. Catani, D. de Florian, and M. Grazzini, Universality of nonleading logarithmic contributions in transverse momen-tum distributions,Nucl. Phys. B596, 299 (2001).
[12] E. L. Berger and J. Qiu, Differential cross-section for Higgs boson production including all orders soft gluon resumma-tion,Phys. Rev. D 67, 034026 (2003).
[13] G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC, Nucl. Phys. B737, 73 (2006).
[14] T. C. Rogers and P. J. Mulders, No generalized TMD-factorization in hadro-production of high transverse mo-mentum hadrons,Phys. Rev. D 81, 094006 (2010). [15] S. M. Aybat and T. C. Rogers, TMD parton distribution and
fragmentation functions with QCD evolution,Phys. Rev. D 83, 114042 (2011).
[16] T. Becher, M. Neubert, and D. Wilhelm, Electroweak gauge-boson production at small qT: Infrared safety from the collinear anomaly, J. High Energy Phys. 02 (2012) 124.
[17] P. Sun, J. Isaacson, C.-P. Yuan, and F. Yuan, Universal non-perturbative functions for SIDIS and Drell-Yan processes,
arXiv:1406.3073.
[18] C. W. Bauer, S. Fleming, and M. E. Luke, Summing Sudakov logarithms in B− X in effective field theory,
Phys. Rev. D 63, 014006 (2000).
[19] C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, An effective field theory for collinear and soft gluons: Heavy to light decays,Phys. Rev. D 63, 114020 (2001).
[20] C. W. Bauer, D. Pirjol, and I. W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65, 054022 (2002).
[21] C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Hard scattering factorization from effective field theory,Phys. Rev. D 66, 014017 (2002).
[22] M. Beneke, A. P. Chapovsky, M. Diehl, and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power,Nucl. Phys. B643, 431 (2002). [23] Y. Gao, C. S. Li, and J. J. Liu, Transverse momentum
resummation for Higgs production in soft-collinear effective theory,Phys. Rev. D 72, 114020 (2005).
[24] A. Idilbi, X. d. Ji, and F. Yuan, Transverse momentum distribution through soft-gluon resummation in effective field theory,Phys. Lett. B 625, 253 (2005).
[25] S. Mantry and F. Petriello, Factorization and resummation of Higgs boson differential distributions in soft-collinear effective theory,Phys. Rev. D 81, 093007 (2010). [26] T. Becher and M. Neubert, Drell-Yan production at small
qT, transverse parton distributions and the collinear anomaly,Eur. Phys. J. C 71, 1665 (2011).
[27] J. y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett. 108, 151601 (2012).
[28] J. Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, A formalism for the systematic treatment of rapidity logarithms in quan-tum field theory,J. High Energy Phys. 05 (2012) 084.
[29] M. G. Echevarria, A. Idilbi, and I. Scimemi, Factorization theorem for Drell-Yan at low qTand transverse momentum distributions on-the-light-cone, J. High Energy Phys. 07 (2012) 002.
[30] P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnov, and M. Steinhauser, Quark and Gluon Form Factors to Three Loops,Phys. Rev. Lett. 102, 212002 (2009).
[31] R. N. Lee, A. V. Smirnov, and V. A. Smirnov, Analytic results for massless three-loop form factors,J. High Energy Phys. 04 (2010) 020.
[32] T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD,J. High Energy Phys. 06 (2010) 094.
[33] X. Ji, J. P. Ma, and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse mo-mentum,Phys. Rev. D 71, 034005 (2005).
[34] J. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 32 (Cambridge University Press, Cambridge, 2013).
[35] T. Becher and G. Bell, Analytic regularization in soft-collinear effective theory,Phys. Lett. B 713, 41 (2012). [36] M. G. Echevarria, A. Idilbi, A. Schäfer, and I. Scimemi,
Model-independent evolution of transverse momentum dependent distribution functions (TMDs) at NNLL, Eur. Phys. J. C 73, 2636 (2013).
[37] Y. Li, D. Neill, and H. X. Zhu, An exponential regulator for rapidity divergences,arXiv:1604.00392.
[38] A. M. Polyakov, Gauge fields as rings of glue,Nucl. Phys. B164, 171 (1980).
[39] G. P. Korchemsky and A. V. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD,Phys. Lett. B 171, 459 (1986). [40] S. Moch, J. A. M. Vermaseren, and A. Vogt, The three loop
splitting functions in QCD: The nonsinglet case,Nucl. Phys. B688, 101 (2004).
[41] Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, Soft-virtual corrections to Higgs production at N3LO,Phys. Rev. D 91, 036008 (2015).
[42] G. F. Sterman, Infrared divergences in perturbative QCD,
AIP Conf. Proc. 74, 22 (1981).
[43] J. G. M. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories,Phys. Lett. B 133B, 90 (1983). [44] J. Frenkel and J. C. Taylor, Nonabelian eikonal
exponen-tiation, Nucl. Phys. B246, 231 (1984).
[45] Y. Li, S. Mantry, and F. Petriello, An exclusive soft function for Drell-Yan at next-to-next-to-leading order,Phys. Rev. D 84, 094014 (2011).
[46] L. J. Dixon, J. M. Drummond, and J. M. Henn, Bootstrap-ping the three-loop hexagon, J. High Energy Phys. 11 (2011) 023.
[47] E. Remiddi and J. A. M. Vermaseren, Harmonic polylogar-ithms,Int. J. Mod. Phys. A 15, 725 (2000).
[48] K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate beta functions in 4 loops,Nucl. Phys. B192, 159 (1981).
[49] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000).
[50] C. Anastasiou, C. Duhr, F. Dulat, and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO,J. High Energy Phys. 07 (2013) 003.
[51] Y. Li and H. X. Zhu, Single soft gluon emission at two loops,J. High Energy Phys. 11 (2013) 080.
[52] C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization,Phys. Lett. B 727, 452 (2013). [53] Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, N3LO Higgs boson and Drell-Yan production at threshold: The one-loop two-emission contribution,Phys. Rev. D 90, 053006 (2014).
[54] H. X. Zhu, On the calculation of soft phase space integral,J. High Energy Phys. 02 (2015) 155.
[55] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, F. Herzog, and B. Mistlberger, Soft expansion of double-real-virtual corrections to Higgs production at N3LO, J. High Energy Phys. 08 (2015) 051.
[56] C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations,J. High Energy Phys. 07 (2004) 046.
[57] A. V. Smirnov, Algorithm FIRE—Feynman integral REduction,J. High Energy Phys. 10 (2008) 107.
[58] A. von Manteuffel and C. Studerus, Reduze 2—distributed Feynman integral reduction,arXiv:1201.4330.
[59] R. N. Lee, Presenting LiteRed: a tool for the loop InTEgrals REDuction,arXiv:1212.2685.
[60] A. V. Kotikov, L. N. Lipatov, and V. N. Velizhanin, Anoma-lous dimensions of Wilson operators in N¼ 4 SYM theory,
Phys. Lett. B 557, 114 (2003).
[61] T. Gehrmann, J. M. Henn, and T. Huber, The three-loop form factor in N¼ 4 super Yang-Mills, J. High Energy Phys. 03 (2012) 101.
[62] A. Brandhuber, G. Travaglini, and G. Yang, Analytic two-loop form factors in N¼ 4 SYM,J. High Energy Phys. 05 (2012) 082.
[63] Z. Bern, A. De Freitas, L. J. Dixon, and H. L. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts,Phys. Rev. D 66, 085002 (2002).
[64] J. M. Henn and T. Huber, Systematics of the cusp anoma-lous dimension,J. High Energy Phys. 11 (2012) 058.
[65] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, and B. Mistlberger, High precision determination of the gluon fusion Higgs boson cross-section at the LHC,J. High Energy Phys. 05 (2016) 058.
[66] J. M. Henn and T. Huber, J. High Energy Phys. 09 (2013) 147.
[67] D. Maitre, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174, 222 (2006).
[68] C. T. H. Davies and W. J. Stirling, Nonleading corrections to the Drell-Yan cross-section at small transverse momentum,
Nucl. Phys. B244, 337 (1984).
[69] C. T. H. Davies, B. R. Webber, and W. J. Stirling, Drell-Yan cross-sections at small transverse momentum, Nucl. Phys. B256, 413 (1985).
[70] D. de Florian and M. Grazzini, Next-to-Next-to-Leading Logarithmic Corrections at Small Transverse Momentum in Hadronic Collisions,Phys. Rev. Lett. 85, 4678 (2000). [71] T. Gehrmann, T. Luebbert, and L. L. Yang, Calculation of
the transverse parton distribution functions at next-to-next-to-leading order, J. High Energy Phys. 06 (2014) 155 (2014);
[72] M. G. Echevarria, I. Scimemi, and A. Vladimirov, The Universal Transverse Momentum Dependent Soft Function at NNLO, Phys. Rev. Lett. 93, 054004 (2016).
[73] T. Luebbert, J. Oredsson, and M. Stahlhofen, Rapidity renormalized TMD soft and beam functions at two loops,
J. High Energy Phys. 03 (2016) 168.
[74] Y. Li, D. Neill, M. Schulze, I. Stewart, and H. X. Zhu (to be published).
[75] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.118.022004for explicit formulas for TMD factorization at one loop, TMD and fully differential soft function through to three loops, and relevant anomalous dimensions and matching coefficients. [76] K. G. Chetyrkin, B. A. Kniehl, and M. Steinhauser,
Decou-pling relations to Oðα3sÞ and their connection to low-energy theorems,Nucl. Phys. B510, 61 (1998).
[77] S. Catani and M. Grazzini, An NNLO Subtraction Formalism in Hadron Collisions and its Application to Higgs Boson Production at the LHC,Phys. Rev. Lett. 98, 222002 (2007).
