Next-to-leading-logarithmic power corrections for $N$-jettiness subtraction in color-singlet production

We present a detailed derivation of the power corrections to the factorization theorem for the 0-jettiness event shape variable $\mathcal{T}$. Our calculation is performed directly in QCD without using the formalism of effective field theory. We analytically calculate the next-to-leading logarithmic power corrections for small $\mathcal{T}$ at next-to-leading order in the strong coupling constant, extending previous computations which obtained only the leading-logarithmic power corrections. We address a discrepancy in the literature between results for the leading-logarithmic power corrections to a particular definition of 0-jettiness. We present a numerical study of the power corrections in the context of their application to the $N$-jettiness subtraction method for higher-order calculations, using gluon-fusion Higgs production as an example. The inclusion of the next-to-leading-logarithmic power corrections further improves the numerical efficiency of the approach beyond the improvement obtained from the leading-logarithmic power corrections.


I. INTRODUCTION
The increasing precision of data from the Large Hadron Collider (LHC) and the anticipated precision of potential future experiments require calculations to increasingly high orders in the perturbative expansion of QCD. The calculation of higher-order corrections is complicated by the fact that the real-emission and virtual contributions to the cross section exhibit infrared singularities that cancel only after they are combined. Currently, the fully differential predictions needed to model the final-state cuts imposed in all experimental analyses can be obtained at next-to-next-to-leading order (NNLO) in the strong coupling constant for 2 → 2 scatter-ing processes. These have become available only recently due to rapid progress in developing new schemes for the efficient cancellation of infrared singularities [1][2][3][4][5][6][7][8]. With these advances a new standard in the comparison of theoretical predictions with data from the LHC has been achieved.
We discuss here one of these successful new approaches to higher-order QCD calculations, the N -jettiness subtraction scheme [7,8]. This method uses the N -jettiness event shape variable T N [9] as a resolution parameter to isolate and cancel the double-unresolved singular limits, where two partons become soft and/or collinear, that complicate the calculation of NNLO cross sections. It has led to some of the first calculations for vector boson production in association with a jet [7,[10][11][12][13] and Higgs production in association with a jet [14] at the LHC through NNLO. It has also led to first predictions for inclusive jet production at NNLO in electronnucleon collisions [15]. The N -jettiness subtraction scheme relies upon the introduction of a cutoff T cut N that separates the N + 1 jet region of an N -jet production process from the doublyunresolved limit. The below-cut region is expanded in T cut N /Q, where Q denotes the hard momentum transfer in the process, in order to allow for an effective field theory calculation using soft-collinear effective theory (SCET) [16][17][18][19][20]. A small cutoff is needed so that the power corrections in T cut N /Q are negligible. However, the below-cut and above-cut contributions separately depend on logarithms of T cut N /Q that only cancel after combining the two regions. For small T cut N these logarithms introduce numerical noise that must be controlled. Although the numerics can already be controlled sufficiently for phenomenological applications, it is desirable for computational efficiency to reduce the sensitivity of the method to the power corrections. One approach is to analytically calculate the power corrections to the SCET factorization theorem used in the below-cut region. The study of the structure of sub-leading power corrections to effective-theory factorization theorems is also of general interest, and has received significant attention recently [21][22][23][24][25][26][27].
The importance of and interest in the N -jettiness subtraction scheme has motivated several calculations of the power corrections to the factorization theorem for the simplest case of colorsinglet production, which uses the 0-jettiness event shape variable. The leading-logarithmic power corrections to the below-cut Drell-Yan cross section, which scale as α s T cut N log T cut N and α 2 s T cut N log 3 T cut N at NLO and NNLO respectively, were derived using an effective field theory approach in Ref. [28]. The leading-logarithmic power corrections for both Drell-Yan and inclusive gluon-fusion Higgs production were derived using both an effective-theory method and a direct QCD calculation in Ref. [29]. The corrections to the differential cross section for gluonfusion Higgs production were subsequently considered using an effective-theory approach in Ref. [30,31].
Motivated by the increasing interest in understanding power corrections in effective field theory, we revisit and extend the calculation of the power corrections to the 0-jettiness factorization theorem. We attempt to fulfill several goals in this work.
• We present a detailed derivation of the leading-power cross section and the sub-leading power corrections. For the leading-power terms we show explicitly how the calculation using the SCET factorization theorem maps onto a direct derivation using QCD. We study gluon-fusion Higgs production as a representative example. Our aim is to provide a detailed, pedagogical discussion to allow interested readers with differing expertise to follow the derivations in detail.
• We present in detail an explicit calculation of the NLO power corrections for the inclusive cross section for gluon-fusion Higgs production directly in QCD, assuming an arbitrary hardness measure used to define T 0 .
• In addition to rederiving in detail the leading-logarithmic sub-leading power corrections, we also derive analytically for the first time the next-to-leading-logarithmic (NLL) O (T 0 ) power corrections.
• We study in detail the numerical impact of the LL and NLL power corrections for two different choices of hardness measure in the T definition on the NLO gluon-fusion Higgs inclusive cross section computed using N -jettiness subtraction. The inclusion of the full power corrections further improves the numerical efficiency of the method beyond the leading-logarithmic improvements observed in previous work [28][29][30]. Deviations from the NLO correction obtained using dipole subtraction remain below 1%, and they can be greatly reduced by changing the normalization factor in the T definition.
Our manuscript is organized as follows. We present our notation and define the versions of 0-jettiness we consider in Section II. For pedagocial purposes, and as a check on our direct derivation of the T distribution including power corrections, we derive the leading-power result using the SCET factorization theorem in Section III. Our direct QCD derivation of the leadingpower result, as well as the full O(T ) power corrections, is shown in detail in Section IV. We present numerical results for two different choices of hardness measure in the T definition in Section V. Finally, we conclude in Section VI. In an Appendix we list the formulae for the NLL power corrections for the qg and qq contributions to Higgs production. The expressions for the dominant gg channel are given in the main text.

II. NOTATION AND T DEFINITIONS
In this section we present the notation and conventions used in this paper, and define the N -jettiness event-shape variable. To simplify comparison with other results in the literature we adopt whenever possible the notation of Ref. [30].
The most general definition of the N -jettiness event-shape variable is where the index i runs over the two beam directions and over the N final-state jets. The index k runs over all final-state partons. The Q i are arbitrary hardness measures that lead to different definitions of N -jettiness. For our gluon-fusion Higgs production example we will be interested in 0-jettiness in the situation where there is only a single parton in the final state. In this case the relevent event shape variable is where p 3 is the momentum of the emitted final-state parton. The two beam momenta q a and q b are where x a and x b are the momentum fraction carried by the two partons, and As we will be considering gluon-fusion Higgs production as a pedogogical example, we will have Q = m H . From now on the subscript 0 for jettiness will be implicit.
The hardness measures Q a and Q b in Eq. (2) are arbitrary. The leading-power cross section is not affected by this choice, but the sub-leading power corrections do depend on these normalization factors. While our derivation will be valid for arbitrary Q i , we will call particular attention to two possible choices: hadronic T and fixed T , the former following the terminology of Ref. [30]. Fixed T is defined by while hadronic T is defined by III. LEADING-POWER DERIVATION USING SCET As a check on our direct QCD derivation of the 0-jettiness power corrections in Section IV we will compare the leading-power expression obtained there with the result from the SCET factorization theorem [9,32]. According to this result the differential cross section in T to leading order in the T /Q expansion can be expressed in terms of universal objects in the effective theory known as hard, soft and beam functions that respectively describe hard radiation, soft radiation, and radiation collinear to a beam direction. We can write the leading-power (LP) result as [9,32] Here, Φ H denotes the Born phase space for Higgs production, x a and x b denote the Bjorken- x variables for each beam, and H, B, and S respectively denote the hard, beam and soft functions. The variables t a and t b parameterize the contributions of the beam sectors to the total 0-jettiness T . For gluon-fusion Higgs production the Born-level phase space takes the The factorization formula of Eq. (7) is true to all orders in the strong coupling constant α s . In our study we are interested in the leading-power result at NLO, so we can expand each function to the NLO level to obtain the fixed-order result for the differential cross section. Using a convolution symbol to abbreviate the integrals appearing in Eq. (7), we find the following four convolutions contributing to the NLO cross section: We have introduced a superscript on each function to denote the order of α s contributing to the convolution (for example, S (0) denotes the leading-order soft function, while S (1) denotes the O(α s ) coefficient of the soft function).
We will now use the known results from the literature to separately derive the contributions above. We begin by compiling the various functions required in the factorization theorem. For simplicitly we focus in this section on those terms that involve the gluon distribution function.
Contributions involving the quark distribution functions can be obtained in an identical fashion.
• The beam function is a non-perturbative object that can be written as a convolution of perturbative matching coefficients with the usual parton distribution functions: The matching coefficients have an expansion in α s : A similar expansion for the beam function can be obtained upon substituting the matching coefficients into Eq. (10): The matching coefficients through NNLO can be found in [33]. For our purposes we need only the LO and NLO results. Simplifying the relevant expressions from this work and keeping only those terms containing the gluon PDF for simplicity, we find tree-level result is while the one loop correction can be parameterized as In our case, the indices i and j can only take the values 1, 2. We have Furthermore, we only have the hemisphere contribution to the soft function in the language of Ref. [34]: The constantŝ ab isŝ where theq i are defined asq We have assumed for simplicity that the normalization constants are chosen so that , as is the case for the leptonic and hadronic definitions of T . There is an • After renormalization the NLO hard function contains the finite contributions to the virtual corrections. We can obtain the leading-order hard function for Higgs production from [35] and [36]. The NLO result can be taken from Ref. [37]. The tree-level hard function is simply the tree level amplitude squared, which in the effective theory with m H m top takes the form whereŝ ≡ x a x b s. At tree level, the d = 4 dimensional version of the Born matrix element is sufficient. At NLO, we have, in the MS scheme, ing out the top quark. As this term is treated identically in SCET and in the direct QCD derivation of the differential cross section we do not discuss it explicitly here. It is included in all numerical results.
We must now use these expressions in the expanded factorization formula of Eq. (9). At the NLO level there is no non-trivial convolution to perform, since in each case all but one function takes on its simple tree-level form, and all integrals can be done straightforwardly.
We separately present the individual contributions to the differential cross section in T for the gluon-gluon partonic channel below: There is a fourth contribution B

IV. DIRECT QCD DERIVATION OF LEADING AND SUB-LEADING POWER
In order to study in detail the structure of the power corrections to the 0-jettiness factorization formula it is useful to expand the cross section directly in QCD. It turns out to be possible to obtain the full O(T cut ) power corrections, not just the logarithmically-enhanced terms studied previously. We will derive here the leading-power result as well to compare to the SCET expression of the previous section. This serves as a check on our result, and we hope that it is also useful to the reader interested in understanding the connection between the SCET formalism and standard QCD. For clarity, the T behavior of the coefficients at leadingpower and next-to-leading-power is shown in Table IV. We show the power counting both for the differential cross section in T and for the result integrated up to a cutoff T cut . We begin by discussing the factorization of the phase space and the expansion of the matrix elements in T . Most aspects of our derivation are applicable to arbitrary 0-jettiness processes and not just gluon-fusion Higgs production. We identify in the text which parts of our derivation are generic and which parts must be modified for other processes.
TABLE I. Power counting in T of the NLO cross section differential in T (left) and integrated up to T cut (right), at leading-power, next-to-leading-power and next-to-next-to-leading power. We note that there is no NNLL contribution at NLO in α s .

A. Factorization of the phase space
The SCET result for the below-cut region in Eq. (7) is written in a form that explicitly factors out the Born-level phase space. This is possible directly in QCD with straightforward changes of variables. The strategy used here is to absorb the kinematics of the emitted finalstate gluon into one of the two incoming gluons. The approach adopted here works as long as we are inclusive in the Higgs rapidity. We begin by deriving the Born-level phase space for the leading-order process g(q a )+g(q b ) → H, including also the convolution over parton distribution functions and the flux factor. The kinematics for the initial state was described in Section II, from which we find At NLO we consider the partonic process g(q a ) + g(q b ) → H + g(p 3 ). The same approach is applicable to the other NLO partonic channels, which are q(q a ) + g( We have included µ 0 , the usual dimensional-regularization mass scale, in the phase space.
In the second line we have used momentum conservation to remove the p H integral, and have suppressed the energy theta-function for notational simplicity. We use a Sudakov decomposition for the gluon momentum p 3 : We now incorporate the definition of 0-jettiness into the phase space. The emitted gluon p 3 can be closer to either the n or n direction, leading to different expressions for T according to the definition in Eq. (2). We will assume the first case. The second case can be easily obtained by exchanging a ↔ b. In this region we have We note that T has been defined using the q a and q b defined in Eq. (3). We will see how to define these quantities at NLO shortly. We can now write down the differential phase space for this partition as We rescale n · p 3 so that it ranges from 0 to 1 using the variable change leading to the expression At this point we wish to identify and factor out the Born phase space defined in Eq. (25) from this expression. We can do so with the following variable changes, which also serve the purpose of defining the x a and x b that appear in the 0-jettiness definition: These transformations force the delta function in Eq. (31) into the form which appears in the LO phase space. We note one point that appears when we perform this variable change. In general, Q a is a function of x a (for example, for the hadronic T it is Q a (x a ) = √ sx a ). Similarly, . Therefore derivatives of the Q i appear when changing variables from ξ a to x a .
Upon making these variable changes and expanding the phase-space measure to O(T ), we have The prime denotes a derivative with respect to x a . This has the desired factorized Born-level phase space. We note that the lower limit on the z a integral comes from the requirement ξ a ≤ 1, while the upper limit comes from the 0-jettiness requirement We have checked that the z a integral can be extended down to x a , since the region between x a and x a + QaT m 2 H does not contribute to the LL nor to the NLL power corrections. We will express our results in the usual MS scheme by replacing We note that this factorization of the phase space is valid for any process using 0-jettiness as a resolution parameter.

B. Expansion of the matrix elements
We now consider the matrix elements for the NLO real-emission correction gg → Hg, expanded in T . This partonic channel is numerically the most important contribution to gluonfusion Higgs production. Results for these other channels are given in the Appendix. We note that the structure of the power corrections takes the form of leading-power phase space combined with sub-leading power matrix elements, plus sub-leading power phase space combined with leading-power matrix elements. While the exact expressions for the matrix elements are of course process-dependent, the structure of the power corrections is the same for any 0-jettiness process.
The Born matrix element for the partonic process gg → H in d dimensions is [35] |M(gg → H) For the real emission of a gluon, we have The invariants that appear in the matrix elements are given in our phase-space parameterization by the first correction as

C. Derivation of the leading-power result
We begin by deriving the leading-power expression to compare with our result of Section III.
To obtain this term we take the leading-power expression for both the phase space and the matrix element. We can write the differential cross section for the first phase-space partition To proceed we divide the integral in z a into two regions as follows: We can now establish the connection between the direct QCD calculation performed here and the SCET result of Section III. The first integral will give us the beam function contribution of SCET, while the second integral will give us the the soft function contribution. We expand in where the plus distribution is written as The term inside the braces, together with the overall factor outside of the integral, is exactly the bare gluon beam function [32]. The finite terms are the same that we found in Eq. We now consider the second region of the z a integral in Eq. (41). We can expand the integrand around z a = 1: The first term in this expansion contributes to the leading-power and to the sub-leading power result 1 . The second term contributes only to sub-leading power, and will be needed in the next sub-section. For the first term, the following integral is needed: We compile here for use in the sub-leading power derivation the following result as well: We again assume for simplicity that Q a Q b = m 2 H as we did in Section III, leading to the result: dσ We can again compare the finte part of this result to the SCET soft-function contribution of Eq. (23). This is equal to half of the result there. The remaining factor of two is provided by the other phase-space partition, establishing the exact correspondence between the QCD calculation and the SCET derivation. We have performed this check for the other partonic channels as well.
It only remains to consider the virtual corrections to the cross section and to establish the cancellation of poles. Since the hard function of SCET is exactly the finite part of the virtual corrections in QCD as can be checked by comparing Refs. [37] and [38], the correspondence between the QCD virtual corrections and the SCET hard-function contribution of Eq. (24) Combining these terms we arrive at We note that in writing this expression we have made use of the following relation: This is the expression for the poles of the real-emission corrections. The first two terms cancel against the virtual corrections, whose pole structure at NLO in the MS scheme is given by [38] The second two terms are removed by mass factorization into the initial-state PDFs, leaving a finite cross section. This completely establishes both the cancellation of poles and the equivalence of QCD and SCET at leading power in T . We note that we have not considered the additional contribution in the Higgs effective theory coming from integrating out the top quark, since it is treated identically in both a direct QCD calculation and in the effective theory.
For the soft region we also must include the contribution from the second term in the right hand side of (47) and from the term in the right hand side of (48), leading to the total soft contribution dσ NLP,(a) soft The full NLP correction when the emitted gluon is close to the n µ direction can be found by simply summing the beam and soft contribution. As expected, the apparent pole cancels: The total differential cross section is the sum of the two regions (a) and (b): This is the full O(1) power correction for the gg channel, including both the LL and NLL terms.
Results for the other channels are given in the Appendix.
Let us first focus on the LL log T term. Combining both regions we arrive at the following form valid for arbitrary Q a and Q b , subject to the (easily-removeable) restriction that We can apply this form to obtain the LL power corrections to the different definitions of T introduced in Section II. For the fixed T definition we have and therefore the leading-logarithmic power corrections are For the hadronic T we have The result becomes These results for the NLO corrections hold at the level of the inclusive cross section.

V. NUMERICAL RESULTS
We study in this section the numerical impact of the power corrections computed in the previous section. Our intent is to compare the full NLL power corrections with the LL ones and the hadronic T definition versus the fixed definition. We first consider the leading gluongluon partonic channel, before also considering the qg + gq and qq results. The following parameter choices are used in all results: All numerical results are obtained using NNPDF 3.0 NNLO parton distribution functions [39] and MCFM 8.0 [40]. We expand the cross section in the strong coupling constant according to and display results for the NLO coefficient σ NLO below.
Finally, we note that the leading-logarithmic power corrections in Eq. (58) are defined differentially in T . When the below-cut contribution which includes this power correction is integrated up to T cut to obtain the cross section using N -jettiness subtraction, both leadinglogarithmic and sub-leading power corrections in T cut are produced: In our definition of the leading-logarithmic power corrections for the cross section integrated in T we include both terms arising from the integral above.
We begin by comparing the impact of the power corrections on the hadronic T distribution. Fig. 1  T cut = 1 GeV. This is reduced to 3% with the LL improvements, and to less than 1% at NLL.

Shown in
Requiring a deviation from the exact result of less than 1% requires T cut ≤ 0.1 GeV without power corrections. This is reduced to T cut ≤ 0.25 GeV with LL power corrections included and further to T cut ≤ 1 GeV with both LL and NLL corrections included. We now study the impact of power corrections on the fixed T variable. We also compare the deviations from the dipole subtracton result between the hadronic and fixed definitions. We included, as shown in Fig. 3. In particular the fixed definition becomes nearly identical to the dipole subtraction result up to T cut = 1 GeV. This has a significant effect on the numerical efficiency of the method. Each of the above-cut and below-cut contributions depends separately on T cut as log 2 (T cut ), and these terms only cancel after the two pieces are added. We note that with a fixed number of integrand evaluations for our numerical integration, the estimated statistical error is a factor of 2 smaller for T cut = 1 GeV than for T cut = 0.05 GeV, which would have to be used for sub-percent agreement with the dipole subtraction result for the hadronic definition without power corrections.
We now consider the qg + gq initial state. Analytic results for this channel are given in the Appendix. We focus on the hadronic T definition and study the impact of including the full NLL power corrections. We note, however, that our derivation is equally valid for all definitions of T . Results for the inclusive cross section are shown are shown in Fig. 4. It is interesting to note that the LL power corrections worsen the agreement between N -jettiness subtraction and the dipole subtraction result. Only upon including the full NLL power corrections is a better agreement with the dipole subtraction result obtained. We note that for the total cross section the agreement at NLL is excellent, with sub-percent deviations observed all the way up to T cut = 1 GeV. The deviation at LL reaches 20% at this T cut value. This pattern cannot be seen when studying the full NLO cross section since the qg channel is much smaller than the dominant gg scattering process. We note that the deviation between the no power-correction result and the dipole subtraction cross section is non-monotonic as a function of T cut , indicating an accidental cancellation between sub-leading powers that leads to the observed behavior. The full NLL corrections still produce a sub-percent deviation from the dipole subtraction result up to fairly large values T cut ≈ 0.3 GeV.
Finally we consider the qq initial state. Analytic results for this channel are given in the Appendix. This contribution is numerically very subdominant to the other channels. It does not have either a leading-power or LL sub-leading power result, and begins first at the NLL level.
Results for the inclusive cross section are shown in Fig. 5. The agreement between the dipole and N -jettiness subtraction results after including the NLL power corrections is excellent, with sub-percent deviations for all studied values of T cut .

VI. CONCLUSIONS
In this manuscript we have studied in detail the power corrections to the effective-theory factorization theorem for the 0-jettiness event shape variable. In comparison to previous works we have derived the next-to-leading-logarithmic power corrections at next-to-leading order, and not just the leading-logarithmic corrections obtained previously. We have derived our result directly in QCD, without use of the effective field theory formalism, and have discussed the connection between this method and the SCET approach. We hope that our discussion is of interest to the reader who desires to better understand the connection between traditional QCD and the effective-theory approach. We would like to stress that our result can only be applied to the inclusive cross section, and not to the cross section differential in the Higgs rapidity. We have also presented a numerical study of the power corrections for two different T definitions.
Including the NLL power corrections improves the performance of the N -jettiness subtraction method applied to color-singlet production, For the hadronic T , including the power corrections reduces the deviation from the NLO result as computed by dipole subtraction by nearly a factor of three. The fixed T shows nearly no deviation from the dipole subtraction result after the It would be interesting in the future to extend the derivation here to the sub-leading logarithmic level at NNLO as well, which we believe is possible. Applications of N -jettiness subtraction to jet production would also benefit from an understanding of power corrections, which should benefit from the techniques used in our direct QCD derivation of the 0-jettiness power corrections. We look forward to these future investigations.