Sudakov Shoulder Resummation for Thrust and Heavy Jet Mass

When the allowed range of an observable grows order-by-order in perturbation theory, its perturbative expansion can have discontinuities (as in the $C$ parameter) or discontinuities in its derivatives (as in thrust or heavy jet mass) called Sudakov shoulders. We explore the origin of these logarithms using both perturbation theory and effective field theory. We show that for thrust and heavy jet mass, the logarithms arise from kinematic configurations with narrow jets and deduce the next-to-leading logarithmic series. The left-shoulder logarithms in heavy jet mass ($\rho$) of the form $r\ \alpha_s^n \ln^{2n}r $ with $r=\frac{1}{3}-\rho$ are particularly dangerous, because they invalidate fixed order perturbation theory in regions traditionally used to extract $\alpha_s$. Although the factorization formula shows there are no non-global logarithms, we find Landau-pole like singularities in the resummed distribution associated with the cusp anomalous dimension, and that power corrections are exceptionally important.


I. INTRODUCTION
It is not uncommon for an observable to have a range that grows order-by-order in perturbation theory. Traditional e + e − event shapes, such as thrust, the C parameter, and heavy jet mass [1] have this property as do some hadron-collider observables like the jet shape [2,3]. Similar behavior can also be seen in the soft-drop jet mass [4]. As observed by Catani and Webber [1], when the range grows order-by-order, there can be incomplete cancellations between the virtual contributions, which are confined to the lower-order range, To understand Sudakov shoulders, consider first the thrust observable [5]. Thrust is defined in the center-of-mass frame of an e + e − collision as where the sum is over all particles in the event and the maximum is over 3-vectors n of unit norm. It is common to use τ = 1 − T in place of T . The vector n that maximizes thrust is known as the thrust axis. When there are only 2 particles, they must be back-to-back, and then τ = 0 exactly. If there are 3 massless particles, then the phase space is 2 dimensional and can be parameterized with s ij = (p i + p j ) 2 /Q 2 constrained by s 12 + s 23 + s 13 = 1 with Q the center-of-mass energy. Then τ = min(s 12 , s 13 , s 23 The phase space point that saturates this bound has s 12 = s 13 = s 23 = 1 3 and comprises the symmetric trijet configuration: 3 particles of equal energy and angular separation, as shown where |M 0 | 2 is the γ * → qq matrix-element-squared. Because the phase space goes to zero at τ = 1 3 , the differential cross section must vanish there. The result is that with the factor of 3 coming from the 3 choices of thrust axis all of which contribute equally near τ = 1 3 . Already here we can see the Sudakov shoulder: there is a discontinuity in the first derivative of the distribution from −144C F αs 4π for τ < 1 3 to 0 for τ > 1 3 . Given the thrust axis from the maximization in Eq. (1), the event is divided into two hemispheres. We can compute the invariant masses m 1 and m 2 of all the partons in hemisphere 1 and 2 and then heavy jet mass is defined as At order α s one hemisphere must be massless and τ = ρ, and thus dσ dρ has a discontinuity in its first derivative at leading order, just like τ .
Now, consider what happens at higher order in perturbation theory. The parton in the light hemisphere will radiate gluons, making the light hemisphere massive. Since the cross section for the light jet having mass less than m after one emission scales like σ ∼ α s ln 2 m 2 there is a Sudakov enhancement to the cross section at small m 2 . As the light hemisphere jet grows, energy must be drawn away from the heavy hemisphere, making it lighter. Roughly speaking, setting Q = 1 for simplicity, ρ 1 3 − m 2 (as we will derive). As a consequence, the cross section at ρ = 1 3 − m 2 will be enhanced by factors of ln 2 m 2 = ln 2 ( 1 3 − ρ). Thus large Sudakov logs associated with radiation into the light hemisphere translate into Sudakov shoulder logs. This is the physical mechanism for the production of large logs in the left shoulder for heavy jet mass.
To properly and systematically resum the Sudakov shoulder logarithms, we must understand this mechanism, as well as the consequences of radiation from the heavy-hemisphere partons. At first glance, the mechanism, which transfers large logs from the light to the heavy hemisphere using energy conservation may seem difficult to reconcile with factorization. Indeed, previous work has noted the recoil sensitivity of Sudakov shoulder logarithms starting at the next-to-leading logarithmic level [3]. Nevertheless, as we will see it is still possible to factorize the matrix elements and phase space near ρ = 1 3 to isolate and extract the large logarithms, at least at the next-to-leading logarithmic level.
One may ask whether Sudakov shoulder resummation is important. For observables with only a right shoulder, such as thrust, one might argue that it is not so important, since there is not much data for τ > 1 3 . However, for heavy jet mass one should generically expect that logs of the form α s ln 2 ( 1 3 − ρ) are as important away from shoulder region as logs α s ln 2 ρ are away from the threshold ρ = 0. This leaves a rather narrow range of intermediate values of ρ where fixed-order perturbation theory might be trusted. Moreover, looking at Fig. 2 it seems that the Sudakov shoulder effects on the left shoulder of heavy jet mass curve tend to pull it down (and away from thrust), so that resumming the left Sudakov shoulder might bring the curves closer together. This difference of the left shoulder in thrust and heavy jet mass could help explain long standing discrepancies between fits for α s using the two event shapes [6,7].
In order to resum the Sudakov logs we first explore the regions of phase space that can contribute logarithms near the shoulder. We do this for the left shoulder of heavy jet mass in Section II. We find that the phase space near ρ 1 3 splits up into regions some of which generate large logarithms of 1 3 − ρ and some of which do not. We find that all the logarithms come from regions with narrow jets in the light and heavy hemispheres. This is in contrast to threshold region, for which every allowed point of phase space near ρ ∼ 0 can contribute logarithms of ρ. It is also in contrast to non-global logarithms, such as for the light jet mass.
There, logarithms of the light jet mass come from regions where the heavy jet side does not have to contain only narrow jets.
In Section III we discuss the factorization of ρ and τ near 1 3 . We find that near the That is, the LO is the αs 2π times the "A" function and the NLO curves are ( αs 2π ) 2 times the "B" functions, in the notation of [8]. Right is a zoom-in of Sudakov shoulder region near 1 3 . The NLO computation is performed with the program event2 [9,10]. All distributions are normalized to Born cross section σ 0 .
shoulder region, the phase space and matrix elements both neatly factorize. This allows us to define a soft function, which along with the inclusive jet function, can be used to reproduce all the logarithms at NLO, and more generally the next-to-leading logarithmic series. In Section IV we analyze the resummed expression. We show that there are no nonglobal logarithms for the Sudakov shoulder; only regions related to the trijet configuration by soft or collinear radiation can generate the shoulder logs. We also find an unusual pole in the the resummed distribution, qualitatively similar to the Landau pole in the running coupling. Unlike the QCD Landau pole however, the singularity in the resummed heavy jet mass shoulder distribution is determined by the cusp anomalous dimension. Thus it is a kind of Sudakov Landau pole. Similar poles can be found in other observables, such as the Drell-Yan spectrum at small p T [11][12][13]. We show that for the Sudakov shoulder case, the large Sudakov anomalous dimension contributing to this pole also enhances subleading power effects, making them comparable to the leading power result allowing the pole to be cancelled in the full distribution. We conclude in Section VI.

II. NEXT-TO-LEADING ORDER ANALYSIS
As a first step towards understanding Sudakov shoulder logarithms, we analyze the matrix elements and phase space near the shoulder region in full QCD. We concentrate here on the heavy jet mass for concreteness, but the same analysis works for thrust.
At next-to leading order in QCD, there is the virtual contribution with 3 partons in the final state and a real emission contribution with 4 partons. The virtual contribution is proportional to the LO cross section and serves to regularize infrared and collinear divergences.
Thus we focus on the real emission contributions to extract the logarithms.
To have ρ 1 3 we can have configurations which differ from the trijet configuration by soft and collinear emissions, or configurations which do not. For example, one could take a non-planar 4-parton configuration with 4 well-separated partons and ρ ∼ 0.4, then adjust their momenta to lower ρ. Staring from such a configuration, one would not expect anything unusual to happen as ρ is lowered through 1 3 . Indeed, ρ = 1 3 is only special because it is a kinematic limit for 3-body phase space. Thus we expect that the only 4-parton configurations which will contribute Sudakov shoulder logarithms are those close to the trijet configuration.
We will find that this is in fact the case.

A. Kinematics
Let us define the momenta of the 4 particles in the final state as p µ 1 , p µ 2 , p µ 3 and p µ 4 . After momentum conservation, on-shell conditions and a frame choice, there are 5 independent degrees of freedom of these four momenta. Although we will not restrict the momenta to be soft or collinear, it is helpful to choose variables so that the soft and collinear limits are transparent. To impose the on-shell constraints, it is helpful to parameterize the momenta initially in lightcone coordinates: where n µ = (1, 0, 0, 1) andn µ = (1, 0, 0, −1) are back-to-back lightlike directions. Imposing momentum conservation and defining φ as the azimuthal angle between the 1-2 and 3-4 planes we can then express all the momenta in terms of We conventionally define φ by p µ ⊥ = (0, p T sin φ, p T cos φ, 0). The variables ω = 1 2n · (p 3 + p 4 ) and s 234 are hard variables, approaching 1 3 at the trijet configuration. s 34 is the invariant mass of one of the jets in the collinear limit which approaches zero in the trijet limit. The collinear momentum fraction z and the azimuthal angle are order 1 in the collinear limit, but z → 0 in the limit that p 4 is soft. We also find it sometimes convenient to trade cos φ for s 23 using When using s 23 the physical constraint −1 ≤ cos φ ≤ 1 must be imposed on the region of integration. Another useful exact relation is We can use this relation to trade ω for ρ when ρ = s 12 .
To compute thrust or heavy jet mass, we need to determine the thrust axis from the formula in Eq. (1). With 4 partons, the two possibilities are that 3 partons are in one hemisphere and 1 parton in the other, or 2 partons can be in each hemisphere. If we know that partons p 1 . . . p m are to be clustered in the same hemisphere then This dot product will be maximized if n = | p j | −1 p j so that the thrust axis will always align with the sum of momenta in each hemisphere. So there are 7 possibilities for the thrust axis. For each axis choice Thus to determine the thrust axis, we need to find which set of partons has the largest value of 2| p j | or equivalently In terms of our variables in Eq. (8), the T j with one parton in one hemisphere are relatively simple We also have and All of these T j values are exact. Now we would like to consider the region ρ < 1 3 . The heavy hemisphere can have either 2 partons or 3 partons. We can therefore choose it to be ρ = s 234 with T 1 is maximal, or s 12 with T 12 is maximal. The other cases are given by permutation of the indices. Figure 3 shows examples of the phase space regions labeled by which T j is greatest. All regions in these plots contribute to some value of ρ. However, to avoid overcounting we only need to consider the green region on the left plot and the blue region in the right plot.

B. Matrix Elements
Let us define As we have discussed, we expect contributions to the NLO heavy jet mass cross section with factors of ln r or ln 2 r to come from soft or collinear regions of phase space close to the trijet configuration. We can therefore power-expand the matrix elements and phase space constraints in soft and collinear limits. This dramatically simplifies the calculation. There are two ways to confirm that only soft and collinear limits are relevant. First, we can extend the integration limits to the full phase space and verify that no additional logarithms can be generated. Second we can compare the logarithms we extract with a numerical computation of the heavy jet mass distribution at NLO.
For power counting we take r ∼ λ 1. In the collinear limit where p 4 || p 3 , the phase space variables scale as In the soft limit, where p 3 is soft, the scaling is the same except that z ∼ λ instead of z ∼ λ 0 .
First we compute the matrix elements-squared at leading power. We do this by summing all the relevant Feynman diagrams, squaring the amplitudes and summing over spins, after which we take the leading power expansion. We cross check the results against the expectation for soft and collinear limits from factorization.
The γ → qqg matrix element depends on whether the gluon is polarized in the plane of scattering or out of the plane. We find spins M in when the gluon polarization in = (0, 0, 1, 0) in the conventions of Fig. 1, where p g = when the gluon polarization is out = (0, 1, 0, 0). The sum of these agrees with Eq. (3).
The matrix elements depend on which partons are gluons and which are quarks. If p 2 is a quark and p 4 is a gluon, then to leading power in collinear scaling Here the blob represents all the diagrams that can contribute. We derive this by squaring the full matrix element for γ → qgqg using Qgraf [14] and Form [15] or FeynCalc [16][17][18], summing over spins, and then and power expanding in the small s 34 limit. The splitting function naturally appears.
When p 3 and p 4 are gluons, then we find Note the azimuthal angle dependence is due to the polarization of the gluons. Indeed, the leading order γ → qqg matrix element is polarized, and we must therefore use polarized splitting functions (see [19] for example). We have checked that summing the polarized leading order matrix elements in Eqs. (21) and (22) with the polarized splitting functions (see [19] for example) reproduces Eq. (24).
And finally when p 3 and p 4 are quarks (or antiquarks), the leading power result is the same whether they are identical or not.
M collinear This expression also depends on the azimuthal angle, and like the gluon case, is consistent with using the polarized 3 parton matrix elements and polarization-dependent splitting functions.
For the soft limits, we can power expand the full matrix elements in the soft limit. When z is soft we cannot drop s 34 with respect to z, or vice-versa. When p 3 and p 4 are both gluons, the result can be written as This is consistent with the Eikonal approximation.
To avoid double-counting we also need the soft collinear matrix elements which come from taking the soft limit (small z) of the collinear matrix elements or equivalently the collinear limit (s 34 z) of the soft matrix elements. These are therefore the same as the soft matrix elements but keeping only the final term in Eqs. (28)- (30).

C. Phase space
For the phase space limits, we will first examine the soft-collinear limit where z ∼ λ. To leading power in the soft-collinear limit For the case where T 1 is maximal, ρ = s 234 and y = r. We can then impose the constraints T 1 > T 2 , T 1 > T 3 , and so on. Since we are using the variable s 23 instead of cos φ we also have to impose −1 ≤ cos φ ≤ 1. Reducing these constraints leads to five integration regions where The Jacobian scales like J ∼ λ 0 . For the leading double log we need to compute Analyzing the integrals we find that none of them generate ln r terms; the limit r → 0 in each of the integrals is smooth. Thus the region with T 1 max does not contribute to the Sudakov shoulder at NLO. The logs must therefore come from regions with two partons in each hemisphere.
Next, we consider configurations where T 12 is maximal. As before, we expand first assuming collinear scaling. In this case, we no longer have r = 1 3 − ρ = y but instead So that r = 1 2 s 34 +2x. To hold r fixed we then can use r, s 34 , z, y, s 23 as independent variables (instead of r, s 34 , z, x, s 23 in the T 1 max case). Now we find 40 relevant integration regions.
In most of these r can be set to zero without consequence. Only four can possibly generate logs of r: where For the C 2 F color structure, using the power-expanded matrix elements in the collinear limit, Eq. (23), only the first two integrals in Eq. (41) contribute. We find Similarly, integrating against the soft matrix element and the soft-collinear overlap region, we find The constants in the integrals come from the permutations of final state particles and we have accounted the symmetry factor for identical gluons. The total is This is compared to the exact (numerical) NLO calculation in the shoulder region in Fig. 6.
For the C F C A color structure, there can be single logarithms coming from both the the z ∼ 0 and z ∼ 1 regions. Moreover, the splitting functions in this case depend on the polarization of the gluon that splits. However, because the only integration regions that contribute logarithms are uniform in φ (the first two in Eq. (41)), one can simply azimuthally-average the splitting functions, reducing them to the unpolarized case. The final resumts we find are we find which gives Again, this is compared to NLO in the shoulder region in Fig. 6.
The n f T F C F color structure only contains a single logarithm since there is no soft region.
Integrating the collinear matrix element Eq. (26) over power expanded phase space gives No overlap subtraction is needed. This is also shown in Fig. 6.
One can perform a similar leading-power computation for the right shoulder for thrust and heavy jet mass. For these cases, we find it is only the phase space regions with 1 parton in one hemisphere and 3 partons in the other hemisphere that contribute. Since the equivalent calculation is significantly easier using Soft-Collinear Effective Theory, we skip the details of the right-shoulder cases using the full theory and turn instead to the effective theory approach.

III. FACTORIZATION AND RESUMMATION
In Section II, we computed the Sudakov shoulder logs for heavy jet mass and thrust at NLO using full QCD expanded to leading power. We now want to generalize the analysis to all orders leading to a factorization formula. To do so, we first review the approach of [1] and and discuss recoil sensitivity. We then demonstrate a different approach inspired by the NLO calculation that leads to a systematically improvable factorization formula.

A. Recoil sensitivity
One approach to resummation of Sudakov shoulders [1] is that emissions from one of the hard partons will cause an additive shift in heavy jet mass (or thrust) from ρ → ρ + m 2 .
Then one could write the resummed distribution as a convolution. Heuristically, With J(m 2 ) representing some sort of jet function and σ LO (ρ) the leading order cross section.
Unfortunately, when one tries to make this formula more precise it produces ambiguities beyond the leading logarithmic order. To see this, consider how ρ changes due to emissions in the light hemisphere making the light hemisphere have a mass m 2 . With 3 massless partons taking p 1 and p 2 in the heavy hemisphere and p 3 in the light hemisphere for concreteness, the heavy jet mass is with E 3 the energy of the light-hemisphere parton. Now say the p 3 parton becomes massive (i.e. turns into a jet) with p 2 3 = m 2 . Then we have the exact relation So it seems ρ → ρ + m 2 , as in Eq. (50). However, this was a little too quick. For suppose instead of expressing ρ in terms of E 3 we expressed it in terms of | p 3 |. Then when p 3 is massless, However after the emissions, Thus the way ρ shifts depends on whether we hold the energy or the momentum of the jet fixed after the emission. This recoil-sensitivity seems to violate factorization. Moreover, if ρ → ρ − 2m 2 one cannot write down a convolution for the distribution as in (50), since the shift implies that emissions only decrease the value of the heavy jet mass. Thus it becomes clear that while one might use the emission picture for the double-logarithmic analysis of [1], it is inadequate for NLL resummation.

B. Factorization
To proceed, recall from Section II which configurations contributed to the NLO logs.
With 4 partons, we can have either 2 in each hemisphere or 1 in one light hemisphere and 3 in the heavy hemisphere. For the left shoulder of heavy jet mass at NLO we found that only the case with 2 partons in each hemisphere contributed. Moreover, the two partons in the heavy hemisphere were hard, with invariant mass ρ ∼ 1 3 , while the two partons in the light hemisphere formed a jet of small invariant mass, s 34 ∼ 1 3 − ρ 1. In contrast, for the right shoulder of heavy jet mass or thrust, only the region with 1 parton the light hemisphere contributed. Moreover, the configuration in the heavy hemipshere had two hard partons and one parton which was soft or collinear to one of the hard partons.
In the r = 1 3 − ρ 1 region, we found integrals like The integrals over s 34  with ρ close to 1 3 that are not close to the trijet configuration. However, such configurations contribute to the cross section both for r < 0 and r > 0 and will be smooth across r = 0.
Hence they cannot produce large logarithms (in Section IV B we use this same argument to show there are no non-global logs in the Sudakov shoulders).
So let use consider a generic configuration with 3 jets pointing in the n 1 , n 2 and n 3 directions. Such a configuration can have particles collinear to the 3 directions as well as soft partons scattered throughout phase space. At leading power, we can treat the collinear radiation as generating masses m 1 , m 2 and m 3 for the three jets. Thus we can approximate the state as having three hard, massive particles with momenta p 1 , p 2 and p 3 and soft radiation.
To compute heavy jet mass and thrust, we need to know which direction the thrust axis points for a given amount of collinear and soft radiation. To determine this, we first observe that as in Eq. (11), the thrust axis is determined by the set of momenta in a given hemisphere that maximize Then τ and ρ can be computed from the set {p i }.
Let us begin with the case where there is only collinear momenta, so we only have the 3 massive momenta to consider. In this case, phase space is described by s 12 , s 13 and s 23 Then and similarly for T 2 2 and T 2 3 by permutation. Let us take the case where T 1 sets the thrust axis, so that r = 1 3 − s 23 . Then at leading power (assuming So the conditions T 1 > T 2 and T 1 > T 3 imply These limits on s 12 pinch off when r = m 2 2 + m 2 3 − m 2 1 . At m = 0 the linear scaling with r of the s 12 integration region is what generates the linear fall off of the thrust or heavy jet mass cross section as in Eq. (4). For the integration region to be nonzero we therefore have In other words, at fixed m 2 , m 3 and r, there is an upper limit on the light-hemisphere jet mass. The probability of finding a light jet of mass at most m 1 at leading power is proportional to ln 2 m 1 , so for m 2 = m 3 = 0 the integral over m 1 up to r will give the ln 2 r left Sudakov shoulder logarithms. Combined with the factor of r from the s 12 integration gives an overall r ln 2 r behavior. If m 2 and m 3 are parametrically larger than r then we can drop r in Eq. (61). In that case, no logs are generated. Thus the shoulder logs are determined by the region of small m 1 , m 2 and m 3 , consistent with a global observable.
The right shoulder for heavy jet mass is constrained by Eq. (61), but with r < 0. For the right shoulder we define s = ρ − 1 3 . Then replaces Eq. (61).
For the thrust case, we define t = τ − 1 3 . When T 1 determines the thrust axis, then at leading power and Eq. (61) becomes Thus the right shoulder for thrust is defined by integrals over any of the masses with a lower limit of t. Since the inclusive integral, without this constraint, has no t dependence, one can equivalently get the right Sudakov shoulder logarithms by integrating over the masses constrained by m 2 1 + m 2 2 + m 2 3 < t. For the soft radiation, we first need to determine when it affects the thrust axis. Let's start with the configuration with 3 massive partons and suppose some soft radiation k enters hemisphere 1. We want to know whether the thrust axis should shift so that hemisphere 1 excludes k or if it should stay fixed, to include k. To find out, we need to compare T 1k , the thrust value with p 1 and k included in the hemisphere, to T 1 where k is not the 1-hemisphere, but is still included overall. A quick calculation shows that Defining pj as p j with its 3-momentum reversed, so we can write When k is in the 1 hemisphere it must be closer to p 1 than p1. In that case p1 · k > p 1 · k. We conclude that thrust is maximized when all the soft radiation in the hemisphere centered on p 1 is included. In other words, if radiation is slightly on the opposite side of the hemisphere boundary, the thrust axis should not shift to cluster k with p 1 . (backwards to the 1-jet) is characterized byn 2 · k > n 2 · k andn 3 · k > n 3 · k. (backwards to the 1-jet) is characterized byn 2 · k > n 2 · k andn 3 · k > n 3 · k.
Now suppose there is a lot of soft radiation with momenta with {k µ i }. Since the thrust value goes up when radiation is included in a given hemisphere, to find the thrust axis we only have to consider 3 sets of momenta: for each j the set includes a hard jet's momentum p j and all the soft radiation k hemi j in the jet's hemisphere. That is, the maximal value of thrust for a hemisphere containing p j will be given by Since the jet hemispheres overlap, there will be some soft radiation included in both k hemi 1 and k hemi 2 , for example. To avoid overcounting, let us decompose the soft momenta into 6 regions, as shown in Fig. 4. So and so on. Here k j is the soft radiation in the sextant centered on p j and kj is the soft radiation in the sextant opposite to p j .
Assuming p 1 is the thrust axis, then heavy jet mass For ρ < 1 3 we want to express constraints in terms of r = 1 3 − ρ. For the hard kinematic variable, we can use anything equal to s 12 at leading power. A convenient choice is The variable ξ is defined so that T max We have fixed the signs of the vj so that they all have positive energy. Since v1 = p1, and k1 is close to p1, we will have v1 · k1 ≥ 0 for all k1. For the other directions, v2 · p 2 = 0 and v3 · p 3 = 0, and they will also have v2 · k2 ≥ 0 and v3 · k3 ≥ 0.
For the integration range over ξ to be nonzero we therefore need which is the same as W (r, m j , k i ) > 0, with W in Eq.(72). Every term in this expression is a positive quantity. This inequality applies to both the left and right shoulder for heavy jet mass (for the right shoulder we prefer to use So that t < m 2 1 + m 2 2 + m 2 3 + 2p 1 k 1 + 2p 2 k 2 + 2p 3 k 3 + 2v1k1 + 2v 2 k2 + 2v 3 k3 For thrust, as for heavy jet mass, every term in this inequality is positive. As observed in Section II, we can set m j = k i = 0 to zero in the hard matrix elements at leading power. Then the integral over hard phase space simply gives the maximum value of ξ from Eqs. (72) or (77). That is, each channel of the LO integral in Eq. (4) gets modified as 48C F α s 4π with θ(x) the Heaviside step function.
The rate for producing collinear radiation is given by splitting functions, and the cross section for producing collinear radiation of mass m is given by the inclusive jet function J(m 2 ). The rate for soft radiation is given by a soft function, defined as an integral over emissions from Wilson lines using a measurement function (see Section III C where The arguments of the 6-parameter soft function S 6 (q i ) are the projections q i = n i · k i and qī = vī · kī. In terms of the q i , Eq. (72) becomes We can simplify the factorized expression by defining a 2-parameter trijet hemisphere soft where q and q h represent the soft radiation in the light and heavy hemispheres. This soft function contributes to the doubly-differential distribution of the hemisphere masses as And then One also must sum over channels, corresponding to which jet is the quark jet, which is antiquark and which is gluon.
The factorization formula for thrust is similar: The soft function for thrust is the same as for heavy jet mass after changing v → v . As we will show in Appendix A changing v → v has no effect on the parts of the soft function relevant to NLL resummation, so we will treat the HJM and thrust trijet hemisphere soft functions as being the same.

C. Soft function
According to the analysis in the previous section, the factorization formula requires a soft function giving the rate for producing gluons k i entering one of 6 sextants, as in Fig. (4). In each sextant we need the projection p i · k i for i = 1, 2, 3 (sextants containing a jet) or v i · · · k i for i =1,2,3 (sextants between jets). For NLL resummation, we only need the anomalous dimension of the soft function at 1-loop. This can be determined by RG invariance. However, as a cross check on the factorization formula, it is important to compute the soft function explicitly.
It is convenient to introduce the scaleless vectors for the six direction that appear in the measurment function where the n i can be read off from Fig. 1 The matrix element for Eikonal emission of one gluon off of 3 Wilson lines is the same as for direct photon production [20,21] or hard W/Z production [22,23]. There are 3 Wilson lines in the trijet configuration, pointing in the n 1 , n 2 and n 3 directions (see Fig. 1). When the jet in the 1 direction is a gluon, the 1-loop soft function is The soft function with a quark Wilson line in the 1 direction has the color structures interchanged: Despite the preponderance of directions, the integrals required are all of the same general form. By rotational invariance, we can always take the Wilson lines to be the n 1 and n 2 directions. Then all the required integrals are special cases of the general form a single θ-function, appear in the iterative solution of the BMS equation [24] for non-global logarithms of the light-jet mass distribution. There, a larger SL(2, R) symmetry constrains the functional form even more [25]. Here, the SL(2, R) is broken by the second θ-function, so the integration region is a cats-eye shaped wedge inside the Poincare disk. However the conformal coordinates proposed in [25] can still provide a useful change of variables which we used to understand and simplify the integrals.
In the regions without a Wilson line, the anomalous dimension of the soft function is insensitive to the projection vectors N i ; it only depends on the location of the measurement region relative to the Wilson lines. Thus for NLL resummation there are only 4 independent integrals, as illustrated in Fig. 5. A detailed calculation of the soft integrals can be found in Appendix A. Here we just summarize the results. We find for the 4 integrals I 1 (q) = I n 2 ,n 3 ,n 1 (q) = 1 q 1+2 1 − 7 2 ln 2 + ln 3 − 3κ 2π (99) I 2 (q) = I n 1 ,n 2 ,n 3 (q) = 1 q 1+2 − ln 2 + 3κ π (100) where κ = Im Li 2 e Then, when we add in the colors structures, the soft function is and so on for the other four q i sectors and for I 6q (q i ). For the trijet hemisphere soft function in Eq. (85), we can set all the q i in each hemisphere equal For the channel with a gluon jet in the light hemisphere we find with Γ 0 = 4 and γ sqq = −4C F ln 6, γ sg = −2C A ln 3 + 4C F ln 2 (106) Notation for the distributions can be found in [20,22,[26][27][28].
In the channel where the light hemisphere has quark jet, the trijet hemisphere soft function has terms of the form irrational. In this context, transecendentality-2 refers to the representation of κ as a 2-fold iterated polylogarithic integral.
The resummed quark and gluon jet functions have the form [20]: where the Laplace transform of the 1-loop jet functions arẽ and the Casimirs and 1-loop anomalous dimensions are The Sudakov RG kernel is where To NLL order Finally, where The hard function can be extracted from [34], or using the general forms for hard functions in [35] or from the hard function for n-jettiness [32]. It is with The trijet hemisphere soft functions can be resummed in exactly the same manner as the hemisphere soft function [7,[27][28][29]36]. At NLL level they factorize into the product of soft functions for each hemisphere: The single-variable soft functions all have the same form where and The only difference is the anomalous dimensions. The coefficient of the Sudakov logs are determined by Casimir scaling as the sum of the color factors for each parton in the hemi- The anomalous dimensions γ sg , γ sq , γ sqq and γ sqg are in Eqs. For thrust, with t = τ − 1 3 > 0, the core measurement function integral following from Eq. (80) is For the left shoulder of heavy jet mass, the integral is similar, but the sign flip in Eq. (76) as compared to Eq. (80) gives an important change.
For the right shoulder of heavy jet mass we define s = −r = ρ − 1 3 > 0. Then the core integral is These integrals are all UV and IR divergent, and so analytic continuation has been used to complete them. We discuss the integrals in more detail in Sections IV B.
Putting everything together and applying algebraic simplifications as in [20,28], we find that all 3 observables can be written in terms of the same RG evolution kernel. For the gluon channels We have chosen the same jet scales for the light and heavy hemispheres although one could also choose them to be different. Similarly, we have taken the same soft scales for the left and right hemispheres.
One can read off from Eq. (133) that the large logs will be resummed for the left-shoulder of heavy jet mass with the canonical scale choices For thrust or the right shoulder of heavy jet mass, the canonical scale choices are the same with r replaced by t or s respectively. We have verified that the expansion of the resummed distribution is independent of the matching scales µ h , µ j and µ s at order α s .
The quark channels have the same form as Eqs. (132) to (134) but with The the final resummed distribution for thrust is and similarly for heavy jet mass.

IV. ANALYSIS
In Section III derived a factorization formula for the left and right Sudakov shoulders for heavy jet mass as well as the right Sudakov shoulder for thrust (thrust has no left shoulder).
We will now perform some cross checks on those results. We first perform the fixed order expansion and compare to a numerical computation of the exact NLO expression to verify the singular behavior. Then we demonstrate that there are no non-global logarithms and discuss power corrections.

A. Fixed-order expansions
First of all, we observe that the full resummed distributions are renormalization-group invariant. This invariance has let us write the evolution kernels in Eqs. (135) and (139) in a form that depends only on the hard, jet and soft matching scales µ i , and not on µ. The cancellation of the µ-dependence is non-trivial and requires the Casimirs associated with the Sudakov double logs to cancel and the anomalous dimensions to satisfy These relations can be checked explicitly using Eqs. (122), (111), (106) and (108).
Expanding the resummed distributions to order α s we find for some B 1 .The linear terms 3t and B 1 t are not predicted with NLL resummation. So to be consistent we should remove all the terms linear in t. This can be done to all orders by subtracting from the full resummed distribution σ(t) the boundary condition t σ(1). That is, we consider 1 which has only terms of the form t ln n t to all orders in α s . We use an analogous definition with t replaced by r or s for the subtracted form of the heavy jet mass distribution. Plugging in the anomalous dimensions This is shown in comparison to the NLO calculation in Fig. 6.
For the left shoulder of heavy jet mass, the expansion gives This agrees with our fixed-order computation in Section II and with the leading shoulder logarithms at NLO as can be seen in Fig. 6.
Breaking down the expression in Eq. (147) the anomalous dimensions which appear are γ jg + 2γ sg from the gluon channel and γ jq + 2γ sq from the quark and antiquark channels. So in each channel only anomalous dimensions associated with light-hemisphere side are contributing logarithms as order α s . This is a somewhat remarkable feature of the factorization formula: although both sides contribute 1-loop anomalous dimensions, as is required for renormalization-group invariance, Eq. (143) only one side contributes logarithms. Mechanically, what happens that So the sin(πη ) sin(π(η +η h )) factor replaces the full anomalous dimension γ + γ h with just γ . For the right shoulder of heavy jet mass In this case, only the anomalous dimensions in the heavy hemisphere contribute at NLO.
This distribution is also shown in Fig. 6 and compared to the exact NLO calculation. to the cross section at order dσ/dρ ∼ α 2 s ln 2 ρ so it is the same order as terms in NLL resummation. The leading logarithmic series of non-global logs for the light jet mass and related observables is understood and can be resummed [24,25]. Progress has also been made on systematic higher-order resummation of non-global logarithms [38? ? ? , 39]. Thus, if there were non-global logs in the Sudakov shoulders it would not pose an insurmountable obstacle. Nevertheless, we will show that for the Sudakov shoulders of thrust and heavy jet mass, non-global logs are absent.
For the right shoulder of thrust, the constraint in Eq. (80) is of the form t < x+y where x and y represent contributions to the mass of the heavy or light hemispheres respectively from soft and collinear radiation near the trijet region (i.e. x = m 2 2 + m 2 3 + 2p 2 k 2 + 2p 3 k 3 + 2v1k1 and y = m 2 1 +2p 1 k 1 +v 2 k2 +v 3 k3 when the light jet is in the 1 direction). Since the constraint imposes a lower bound on t, demanding t 1 does not force x and y to be small, suggesting that the Sudakov shoulder for thrust might be non-global. However, we can rewrite the core convolution integral in Eq. (129) as The first integral in brackets in Eq. (152) is divergent but either independent of t or linear in t, so it is smooth across t = 0 and does not generate Sudakov shoulder logarithms. The remaining integral has z = x + y < t, so taking t 1 does force x, y 1. We conclude that the right shoulder of thrust should be free of non-global logarithms. The actual divergence is an artifact of expanding the phase space limits to leading power. In the full theory, the divergences would cut off by the hard scale Q, but still would not generate logarithms of t.
It is also worth noting that the scaleless integral in Eq. (152) does generate a divergent term proportional to t. This would be the same order as terms in the NNLL resummation of the Sudakov shoulder. The presence of such a term does not imply that the factorization formula is valid only to NLL. Indeed, this divergent contribution is smooth across t = 0, suggesting that it contributes similarly to the left and right sides of τ = 1 3 and therefore does not give a discontinuity or a kink at τ = 1 3 . In any case, since we are only working to NLL in this paper, we can safely ignore it.
For heavy jet mass, the analogous constraint is in Eq. (76) which corresponds to x < r +y for the left shoulder or x + s < y for the right shoulder, as in Eqs. (130) and (131). We can rewrite Eq. (130) as This integral is both UV and IR divergent (for a, b > 0) and gets contributions from all scales, suggesting, again, that it may generate non-global logarithms. To separate out the UV and IR divergences, we can take two derivatives with respect to r, leaving an integral which is UV finite for a, b > 0. We also introduce a new scale R to separate small r from To complete the computation, as far as the Sudakov shoulder logs are concerned, we have where we have taken R r to simplify the second integral. Integrating twice with respect to r then gives At small a and b (these are proportional to α s ), the second term on the right-hand side is suppressed by a factor of r R compared to the first term, so it only gives power corrections and no Sudakov shoulder logs, as anticipated. We should fix the integration constants c 1 and c 2 so that the expansion of f (r) at small a and b only has terms of the form r ln n r with n > 0. The constant term we can simply discard, c 0 = 0. To fix c 1 we should set c 1 = −f (1).
This corresponds to integrating f (r) from 1 to r. These integration constants were used in Eq. (145).
In summary, the heavy jet mass distribution at a value of ρ ≈ 1 3 does get contributions from phase space regions with jets whose masses are not small. In this sense it is similar to light jet mass near ρ = 0 which gets contributions from phase space regions where ρ is not small. However, the contributions corresponding to heavy jets for the Sudakov shoulder do not generate large logarithms. This is because the phase space regions with heavy jets can contribute to both ρ 1 3 and ρ 1 3 and are smooth across ρ = 1 3 . All the contributions to the distribution that are not smooth across ρ = 1 3 come from the regions with one nearlymassless jet in the light hemisphere and two nearly massless jets in the heavy hemisphere.
There is no analog of this continuity argument for light jet mass, which cannot have ρ < 0.
Thus, the Sudakov shoulders of heavy jet mass (and thrust) are free of non-global logarithms. sin(π(a+b)) with a = b = −6 αs π ln r and α s = 0.1. The pole is at a + b = 1. The dashed curve shows this same function once the power-suppressed term − r 2 2 R a+b−1 1 1−a−b is added with R = 1. The power suppression is apparent at small r, where the difference between the two curves is negligible. The pole at a + b = 2 is not canceled and appears as the spike at r ≈ 0.0003. nears 1 a pole from the sin −1 (π(a + b)) factor in Eq. (159) is approached. However, when a + b ≈ 1, power-suppressed term is no-longer power suppressed. Indeed, it has precisely the behavior needed to remove the singular behavior from the leading power term. We show this in Fig. 7.
To see what is happening analytically, noting that the leading power expression scales like r 1+a+b we can write the subleading power expression as For a + b 1 there is a r R linear power suppression. However for a + b ∼ 1 there is no power suppression at all; for a + b = 1 this expressions reduces to r 2 as does r 1+a+b . In effect, the scaling dimensions of the leading power and subleading power pick up such large anomalous contributions that their relative scaling changes.
Taking R → ∞ gives the leading contribution. The first subleading power contribution in this limit cancels the pole at a + b = 1. To cancel subsequent poles, one can use the exact with η h and η in Eqs. (141) and (140). This expression has singularities whenever η +η h ∈ Z.
Choosing canonical scales as in Eq. (138) at leading logarithmic level gives The singularity η + η h = 0 occurs when µ j = µ s , which happens at r = 1. At r = 1 there are no logarithms, so this singularity is entirely removed by the subtraction in Eq. (145).
That is, is regular at r = 1. Note, however, if the soft and jet scales meet at some lower scale, this singularity may be reintroduced.
The singularity at η + η h = 1 is more troublesome. Similar singularities have been seen in other processes, such as Drell-Yan or Higgs production at small p T [11-13, 41, 42] or the jet shape [2,43]. Writing L = ln 1 r , resummation at order NLL is meant to get right all terms of order α n s L j with j ≥ 2n − 1 in R(r) or equivalently all terms of order α n s L j with j ≥ n in the exponent, i.e. in ln R(r). In the notation of [44], we can write ln dσ sub dr = Lg 1 (α s L) + g 2 (α s L) + · · · =∼ · · · − ln sin(π(η + η h )) + · · · (165) with g 1 (α s L) and g 2 (α s L) completely fixed by the expansion and reorganization of our resummed expression. Normally, when α s L ∼ 1 then we must go to higher order in RGimproved perturbation theory; at NNLL level, we would have additionally Lg 3 (α s L) which would extend the validity of the theoretical prediction. Here, instead we find a singularity in the exponent: ln R sub is infinite at α s L ∼ 1 due to the η + η h = 1 singularity. Therefore, going beyond NLL would not allow us to make perturbative predictions beyond where the singularity occurs. Instead, the singularity is canceled by including subleading power effects, as discussed in Section IV C and shown in Fig. 7.
The singularity at α ln r ∼ 1 is reminiscent of the Landau pole in QCD. There, already at We emphasize that excluded range is larger than that associated with strong coupling. Because of the Sudakov Landau pole in the resummed distribution it is difficult to make quantitative predictions, particularly at the NLL level, without a better understanding of the power corrections. There are a number of approaches that could be applied to ameliorate the problem. In [11], a similar pole in the Drell-Yan spectrum at small p T (at q = m Z exp(− 3π 8αs ) [45]) was shown to be associated with power-suppressed region of small impact parameter, but could be softened with higher-order resummation. In [13] it is argued that one could also do resummation in momentum space directly with a modified expansion of the Sudakov radiator. Related ideas can be found in [43,46]. It will be important to understand which of these approaches might apply for Sudakov shoulder resummation, but we do not attempt a complete analysis here.
At the LL level, however, because the Sudakov Landau pole is very close to the shoulder, we can at least begin to get a quantitative feel of how important resummation is. Consider the LL distribution using canonical scales in Eq. (138). When the jet in the light hemisphere is a gluon, it has the form as in Eq. (162) with Π g from Eq. (135) becoming Note that we include every term with Γ 0 in it for leading-log resummation, not just the exponential prefactor. Including only the prefactor would give the double-logarithmic approx- imation, as used in previous work on resummation of the C parameter Sudakov shoulder [1].
We subtract off from the resummed distribution r times its r → 1 limit as done in Eq. (145).
Note that this subtraction must be done before setting canonical scales. We then match to the fixed order LO+NLO calculation by subtracting from the resummed distribution its expansion to order α s . In this case, the matching subtraction is Finally we include the subtraction of the first subleading power contribution. For the gluon channel this amounts to subtracting: from the resummed distribution. The resulting LL resummed and matched result at Q = m Z and α s (Q) = 0.119 with R = 1 3 for the left shoulder is shown in Fig. 8. To be clear, this is not the complete power correction, but amounts to integrating the leading power soft and collinear matrix elements outside of their formal region of validity upto the kinematic limit of r = 1 3 .

VI. CONCLUSION
Thrust τ and heavy jet mass ρ are two of the most important observables at e + e − colliders.
They have been used for decades for tests of precision QCD and measurements of α s . At leading order in perturbation theory, both ρ and τ are phase-space limited to be less than 1 3 and have a non-vanishing slope as 1 3 is approached. At next-to-leading order, thrust behaves like α 2 s (τ − 1 3 ) ln 2 (τ − 1 3 ) for τ > 1 3 so that the slope diverges as 1 3 is approached from the right. This behavior is called a right Sudakov shoulder. Heavy jet mass has a slope which diverges as ρ nears 1 3 both from the left and the right: it has two Sudakov shoulders. The left shoulder of heavy jet mass is particularly important as the large logarithms can extend well into the region where α s fits are typically done (0.1 ρ 0.24). Thus understanding and resumming its Sudakov shoulders could be very important for improving agreement of theoretical predictions with data and subsequent extractions of α s . We also point out that it has been noted recently in the literature that in the context of other event shape observables such as fractional moments of energy-energy correlation [42] or projected energy correlators [47], one must resort to a joint resummation of the Sudakov shoulders and endpoint peaks.
We derived a factorization formula for both thrust and heavy jet mass in the Sudakov shoulder region. The basic mechanism for generating Sudakov shoulder logs is when a soft or collinear emission goes into one hemisphere a global constraint such as m 2 < ρ − 1 3 transfers large logs from the emissions to the shoulder. Although the constraint seems nonlocal, involving both hemispheres, and therefore might violate factorization, we show that it does not. Moreover regions of large jet mass do not contribute Sudakov shoulder logs, showing that there is no non-global log contribution in the shoulder region. We checked our factorization formula by expanding to NLO and comparing to the exact numerical NLO calculation very close to the shoulder region. As can be seen in Fig. 6 the agreement is excellent.
The calculation involves some unusual ingredients. Since the emissions come off a trijet configuration with 2 quarks and 1 gluons, there is no azimuthal symmetry (unlike the threshold case), and the polarization of the gluon affects the spectrum. At leading order, only a uniform azimuthal angle integral was needed for the resummed expression, but in general polarized splitting function may be necessary. We also saw the appearance of Gieseking's constant, a transcendentally-two number. Although it also drops out of the NLL expression, at higher orders it or related constants may be involved.
The resummed distribution for heavy jet mass has a term of the form sin −1 (πη) with η ∼ α s Γ 0 ln r, where r = 1 3 − ρ. The expansion near α s = 0 (or ln r = 0) produces the leading and next-to-leading logarithmic series: terms like α n ln 2n r. However, there is also a pole at η = 1. This pole in the resummed distribution is not due to the running coupling -it is present even with β(α s ) = 0 -but due to the cusp anomalous dimension. Thus it is a kind of Sudakov Landau pole. Similar behavior has been seen before, in the Drell-Yan process at small p T , for example [11,13,41]. In both cases there is a connection between the pole and subleading power effects (subleading in r for the shoulder, or in impact parameter b for Drell-Yan). We show that subleading power terms can in fact cancel the η = 1 pole but do not affect the NLL series. This implies that a better understanding of power corrections will be necessary to establish proper theoretical uncertainty on the resummed distribution. There are many approaches that may help improve the convergences of the resummed distribution [13,30,46].
Although our results are only valid to NLL level, the factorization formula applies to all orders. In fact, since the anomalous dimensions of the jet and hard functions are known to two loops, and therefore the soft function anomalous dimension as well by renormalizationgroup invariance, NNLL resummation should be possible. At NNLL level, terms linear in ρ or τ are determined. The slope can be discontinuous from the left to right side of the shoulder, as it is already at LO. This discontinuity should be computable. However, because there is also a linear term in the distribution not associated with the shoulder, confirming the predictions at NNLL will be challenging. Nevertheless, pushing the limits of Sudakov shoulder resummation, not just for e + e − event shapes but for collider observables more broadly, provides opportunities to improve our understanding of precision QCD.
With these preliminariles, the integral I 1 (q) = I n 2 ,n 3 ,n 1 (q) takes the form in d = 4 − 2 dimensions The θ-functions impose that To handle the UV divergence as k + → ∞, we can add and subtract the integral I div As a cross check, we can add the three sextants in one hemisphere with the same axis projection and compare to the hemisphere soft function [7,20,29,48,49].
Another interesting fact is that the divergent part of our trijet soft function does not depend on the projection vector N . The soft function integral is If we rotate the projection vector N to another direction N , then the δ-function transforms as Then after rescaling k → N · q N · q k (A31) the integral becomes I na,n b ,nc,n d ,N (q) ∼ N · q N · q 2 q d−5 d d k n a · n b (n a · k)(n b · k) δ(k 2 )δ q − 2 3 N · k × [· · · ] (A32) So the effect of using different N 's only shows up at order in the expansion. This only affects the anomalous dimension for the integrals which have soft-collinear divergences. This is only I 1 (q), since that is the only integral where a Wilson line is in within the integration region. However, for I 1 (q) the projection is on n 1 (the Wilson line direction) in both thrust and heavy jet mass. For the others, using the definitions in Eqs.(A5), (A6) and (A8), we see that at NLL level after rescaling the projection vector N I 5 (q) = I 3 (q), I 6 (q) = I 4 (q), and I 7 (q) = I 6 (q) (A33)