End-point resummation in squark decays

We study soft and collinear gluon emission in squark decays to quark--neutralino pair, at next-to-next-to-leading logarithmic (NNLL) accuracy in the end-point region, using Soft Collinear Effective Theory (SCET), and at next-to-leading (NLO) fixed order in the rest of the phase space. As a phenomenological case study we discuss the impact of radiative corrections on the simultaneous measurements of squark and neutralino masses at a linear $e^{+}e^{-}$ collider based on $\sqrt{s} = 3$ TeV Compact Linear Collider (CLIC). Since the majority of mass measurement techniques are based on edges in kinematic distributions, and these change appreciably when there is additional QCD radiation in the final state, the knowledge of higher-order QCD effects is required for precise mass determinations.


I. INTRODUCTION
The discovery of Dark Matter (DM) in a collider experiment crucially depends on the ability to measure precisely its properties -its mass and couplings to visible matter.These are the necessary ingredients to test the hypothesis of a "WIMP" miracle [1][2][3][4].Given the importance of such a discovery a number of methods to measure DM mass have been developed [5][6][7][8][9][10][11][12][13][14][15][16][17][18].In this paper we are interested in understanding how QCD radiations modifies the precise determination of DM mass.Many of the methods for DM mass measurements were developed with low energy supersymmetry (SUSY) in mind [19].We will thus also use SUSY as an example, though our results do apply more generally.
A significant effort was devoted in measuring DM mass at hadronic colliders.An ingenious method was put forward in [10,11], where it was applied to g → q qχ decays in gluino pair production.The mass of g and χ can both be measured simultaneously from m T 2 , by computing for each event the value of m T 2 as a function of an assumed χ mass, m T 2 (m trial ).
The envelope of m T 2 (m trial ) curves exhibits a kink at m trial = m χ , where m T 2 = m g.Measuring the kink determines both masses (for the effect of radiative corrections see [20]).For two body decays, e.g., for squark decays, q → qχ, the kinks in the distributions appear only once initial state radiation is included [7].This underscores the importance of radiative corrections for DM mass measurement using kinematical distributions.
In this paper we explore a somewhat simpler case -the squark pair production in e + e − collisions.We focus on a two body decay, q → qχ, with q a light quark and χ a neutralino (for earlier work see [21][22][23]).Emission of a hard gluon converts this to a three body decay, q → qgχ, qualitatively changing the kinematical distributions.Hard gluon emissions, on the other hand, are relatively rare, suppressed by small coupling constant, α s 0.09 for m q 1 TeV.Most commonly the radiated gluons are either soft or collinear with the outgoing quark, affecting the kinematical distributions in the end-point region where the decay is almost two-body.Parameterizing the neutralino energy in the squark rest frame as the end-point region is given by z ∼ 1.Here M (m χ ) is the squark (neutralino) mass, while the dimensionless variable z takes values, z ∈ [m χ /M, 1].Near the end-point the neutralino is maximally boosted and z becomes close to 1.
The collinear and soft singularities of QCD contributions in the end-point regions lead to large logarithms, L ∼ ln(1 − z), in the calculation of the differential decay width, dΓ/dz.
Working to next-to-leading order (NLO) in α s , i.e., to O(α s ), the Sudakov effects result in large double logarithmic contributions of the form α s L 2 .In order to obtain reliable predictions, these logarithms need to be resummed to all orders in α s .At next-to-next-to-leading logarithmic (NNLL) accuracy the resummed decay width is given by with f i (. ..) dimensionless functions that are O(1), counting the large logarithms as L ∼ 1/α s .This shows explicitly the dominance of the end-point region, where the first term on the right-hand side (r.h.s.) is the leading contribution.Keeping just the first term would give the result for decay width at leading logarithmic (LL) accuracy, obtained by resumming the double logarithms in the perturbative expansion of the form exp(Lf 0 (α s L)) = k=0 a k (α s L 2 ) k .The second and the third terms on the r.h.s. in Eq. ( 2), of NLL and NNLL accuracy, then resum terms that are additionally suppressed by α s and α 2 s , respectively.To resum the end-point logarithms we employ soft-collinear effective theory (SCET) [24][25][26], which properly describes collinear and soft gluon radiation in the end-point region.The squark decay near the end-point is governed by three distinct scales: hard (µ H ), jet (µ J ), and soft (µ S ) scales.For large mass splittings, M − m χ ∼ O(M ), the hard scale µ H can be identified with µ H ∼ M .The light quark together with radiated collinear gluons forms a collimated jet, controlled by a typical scale µ J ∼ M √ 1 − z.Finally, the soft gluon radiations arise at the scale µ S ∼ M (1 − z).Note that the kinematics of this problem is very similar to the decay B → X s γ in the end-point region, i.e., in the part of the phase space where the final state photon is close to maximally boosted.The effects of strong interactions are in this case described by collinear and soft gluon radiations.The factorization formalism for B → X s γ near the end-point was established in Refs.[24,26,27].
Similarly to B → X s γ, the differential decay width for q → qχ can be schematically factorized as where H, J, and S are the hard, jet, and collinear functions, respectively.The '⊗' denotes the appropriate convolution over 1 − z, while µ F is the factorization scale.The decay width is independent of the factorization scale, which means that µ F can be chosen arbitrarily.In general there will be large hierarchies between µ F and µ H,J,S , so that one needs to perform renormalization group (RG) evolution for each of the H, J and S functions.These RG evolutions in SCET automatically resum the large end-point logarithms.
Away from the end-point region, where 1 − z ∼ O(1), the differential rate is dominated by hard gluon emissions from squark and quark lines, giving the event rate that is O(α s ).This is of the same order as the NNLL corrections in the end-point region and thus needs to be kept in our expressions.We compute these contributions using fixed order calculation at NLO in α s .We smoothly connect the two expressions, valid in the end-point region and away from the end-point regions, giving our final result for the decay width distribution at NNLL+NLO accuracy.We use the obtained expressions to perform a numerical study of the impact of QCD corrections in e + e − → q q * events, using a weighted Monte-Carlo simulation.
To compare directly with the experiment our results for the decay widths will still need to be supplemented with a resummation of soft and Coulomb gluon radiation contributions connecting the two squarks, see Refs.[28,29] for LHC.These are especially important for slowly moving squarks, i.e., at threshold productions, and can even lead to squark bound states [28,[30][31][32].
The paper is structured as follows.In Section II, we introduce the necessary ingredients of the effective field theory (EFT) approach to the problem, that includes SCET and heavy Scalar Effective Theory (HSET).The HSET describes soft fluctuations of the heavy squark arising from soft gluon radiations.The HSET and SCET are then used to derive the factorization theorem for the squark decay rate near end-point in Section III.The NLO calculation of the decay width in the full kinematical range of z is obtained in Section IV.
Using our results that combine the resummed and fixed calculations, giving the NNLL+NLO accuracy, we perform in Section V a phenomenological study of squark pair production in e + e − annihilation, and then conclude in Section VI.Appendix A contains technical details on ∆-distribution which has been used to regularize infrared (IR) divergences in the fixed NLO calculation.

II. CONSTRUCTION OF EFFECTIVE THEORY LAGRANGIAN
We are interested in the squark decay, q → qχ, where χ is the dark matter (DM) particle, and how this is affected by QCD radiation.Near the end-point, χ and a collimated jet are almost back-to-back in the squark rest frame.DM, χ, escapes detection and manifests itself in the detector as missing energy.The quark interacts strongly -it radiates collinear gluons and quark-antiquark pairs, which form a collimated jet.In addition, there is soft gluon radiation in the event, which does not have a preferred direction.
As explained in the Introduction, the decay is governed by three distinct scales, µ H , µ J , and µ S .We use EFTs to deal with the hierarchies between the three scales and the associated large logarithms.We first integrate out the hard interactions, where the relevant hard scale, µ H , is comparable to the squark mass M .At energy scales below µ H we then have only collinear and soft degrees of freedom.The light quark and the collinear gluon describe collinear interactions for the collimated jet.Also the soft mode decoupled from the collinear quark and the heavy squark describes soft gluon radiations near the end-point.
SCET is the appropriate EFT that describes collinear and soft modes and their interactions.
It provides a systematic way to decouple soft modes from the collinear field.This is very useful when proving factorization in the end-point region.The interactions of heavy squark are described by the HSET, which is obtained by integrating out the hard gluon modes and the squark mass M .In the rest of this section we show how SCET and HSET are constructed.The decay rate of the heavy squark is calculated in the subsequent section.

A. Decay Lagrangian at the hard scale
We take χ to be a Majorana fermion.This is the case in the MSSM where χ is the lightest supersymmetric particle (LSP) -assumed to be the lightest neutralino.The most general Lagrangian describing a two-body decay of a color triplet scalar, q, to a quark, q, and a Majorana fermion χ, is given by1 where we are using the four-component notation with P L,R = (1 ∓ γ 5 )/2.The dimensionless Wilson coefficients B L,R encode the new physics as well as strong interactions above the hard scale µ H ∼ M .Our analysis applies to MSSM, but is also more general and applies to any decays of the form q → qχ, where q is a color triplet scalar.
In the MSSM for each quark flavor there are two squarks, q1,2 , so that the above Lagrangian modifies to The tree level expressions for the Wilson coefficients are, neglecting flavor violating effects, with with g, g the weak and hypercharge gauge couplings, Q q the electric charge of quark q, and T q 3 the weak isospin, while R q1 = L * q2 = cos(θ q), and L q1 = −R * q2 = sin(θ q), with θ q the mixing angle rotating the squark gauge eigenstates qR,L to mass eigenstates q1,2 .The qR -q L mixing is usually important only for the third generation squarks, while for the first two generations gauge and mass eigenstates coincide, θ q = 0.The neutralino mixing matrix is denoted by N ij .If LSP is mostly gaugino then N 11,12 N 13,14 and thus qL → q L χ and qR → q R χ for the first two generations.For well-tempered neutralino, on the other hand, all terms in (7),( 8) may be important.

B. EFTs for the end-point region
We restrict ourselves to the case where quark mass can be neglected compared to M .We will work in the squark rest frame, so that its four-velocity v µ is given by v µ = (1, 0).We orient the coordinate system such that jet goes in the z direction, i.e., that, neglecting its mass, it is on the light cone n µ = (1, 0, 0, 1).We also introduce the opposite light cone fourvector nµ = (1, 0, 0, −1), so that n 2 = n 2 = 0, n • n = 2 and p µ q = M v µ = M (n µ + n µ )/2.We will use light-cone coordinates, in which a four-momentum p µ is given by p µ = (n•p, p ⊥ , n•p).
The effective field theory to reproduce low energy physics in full QCD is obtained by integrating hard degrees of freedom.For instance, the hard gluon exchanges between the heavy squark and the light quark are integrated out.The Wilson coefficients B L,R in Eq. ( 4) thus get modified to C L,R (µ) (see Eq. ( 13) below).The resultant EFT is valid at the scale µ < µ H ∼ M .And the remaining degrees of freedom in EFT are collinear and soft fields scaling as p c = M (1, λ, λ 2 ) and p s = M (λ 2 , λ 2 , λ 2 ) respectively.Here λ is a small expansion parameter in EFT.For the squark decay near end-point, λ is given as ∼ √ 1 − z.
In the heavy squark sector, after integrating out hard fluctuations as well as heavy squark mass M , the heavy squark only interacts with soft gluons.Then full QCD Lagrangian for the heavy squark can be matched onto HSET Lagrangian, where φ v is the squark field in HSET, The covariant derivative D µ s = ∂ µ − igA µ,a s T a includes only the soft gluon field.The second term in (9) is O(1/M ) with a coefficient that is fixed by reparametrization invariance.We work at leading order in 1/M expansion, and thus only keep the first term in (9).
The light quark field matches onto n-collinear field in SCET so that where and thus n /ξ n,p = n /ξ n,p = 0.The summation is over large label momenta given by pµ = n • pn µ /2 + p µ ⊥ that differ by soft fluctuations.The field ξ n is suppressed by λ, and is thus not present as an external quark field in our analysis of squark decays since we work to leading order (LO) in the 1/M expansion.Integrating out ξ n the collinear interactions can be expressed entirely in terms of ξ n .The resulting LO SCET Lagrangian for collinear fields can be found in, e.g., Ref. [25].
The decay Lagrangian (4) matches onto the HSET+SCET effective decay Lagrangian, appropriate for describing the squark decays in the end-point region, In the sum only the p that satisfy momentum conservation are selected.The hard gluon exchanges are encoded in Wilson coefficients C L,R (obtained from B L,R in ( 4)), while collinear gluons emitted from the heavy squark yield the collinear Wilson line Here A µ n is n-collinear gluon field and 'P' indicates the path-ordered integral.To show the factorization of soft and collinear interactions it is useful to perform field The path of integration over s ∈ [−∞, x] indicates that the dressed collinear or squark field is incoming.For the outgoing particles the integration path is over s ∈ [x, +∞], giving for the soft Wilson lines [33], In the LO SCET and HSET Lagrangian the interactions between soft gluons and the redefined collinear fields, ξ n , A µ n , and between the soft gluons and the heavy squark field φ v , drop out (that is, at LO there are no interactions between collinear and soft fields, and no interactions between redefined heavy squark and soft fields).The effects of soft gluons are thus moved into the effective decay Lagrangian, where they appear as a product of two soft Wilson lines in n and v directions, with In Eq. ( 17) a summation over Dirac four-component index a is implied.From now on we will use the form of EFT Lagrangian given in Eq. ( 17), i.e., with ξ n , A µ n and φ v denoting the redefined fields that do not couple to soft gluons and quarks at LO.

III. DIFFERENTIAL DECAY RATE AT THE END-POINT
The total decay rate for q → χ 0 q L averaged over the squark color is where T L (E χ , m χ , M ) is related to the matrix elements squared for squark decays into lefthanded quarks Explicitly, the matrix elements squared are with the summation over color indices, α, β, and Lorentz indices, a, b implied.We do not show color index of squark external state: we always consider color-averaged initial states, Note that in Eq. ( 21) we already used the fact that neutralinos are not charged under QCD and thus only contribute as spin u b ūa = (p / χ − m χ ) ba .For q → χ 0 q R decays the same results apply, but with L ↔ R. The interference terms between the two decays are m q /M suppressed and can be safely neglected.
At tree level the decay rate is given by with A. Factorization theorem near the end-point We start by reviewing the decay kinematics near the end-point, where in the final states we have an energetic neutralino and a collimated jet as well as soft gluons.Following Eq. ( 1), we define the kinematic variable in terms of which the neutralino and other final states momenta are given by Here z can take values z ∈ [ √ r χ , 1] (we oriented the coordinate system such that p χ is always along negative z-axis, while p X is along positive z-axis).The missing energy is and therefore one can use z and E χ interchangeably.The invariant mass of the collinear and soft final states is p 2 The limit of a very collimated jet, p 2 X → 0, is thus obtained in the limit z → 1.In the end-point region, 1 − z 1, we can apply the HSET and SCET formalism, introduced in the previous section.The differential rate is with The EFT operators O L,R are given in Eq. ( 18), and |q = √ 2M |φ v up to 1/M corrections.
The Wilson coefficients C L,R can be decomposed into where B L,R are the unknown new physics Wilson coefficients in Eq. ( 4) and C L,R are the Wilson coefficients as a result of integrating out the squark mass M and the hard gluon exchanges between squark and quark.The NLO matching onto HSET+SCET has been performed in Section III B, with the result for C L,R given in Eq. (37).
In the remainder of this section we use the operator production expansion (OPE) to arrive at a more practically useful expression for T L .We first use the completeness relation, The large label momentum pc is the n µ -component of p µ X , given in Eq. (25).The phase in (29) is therefore equal to so that where we used the shorthand notation y − = n • y (y + = n • y) and that φ α v |φ β v = δ αβ .The collinear field ξ n describes an inclusive jet in n-direction.The corresponding jet function is defined as At LO in α s the jet function is simply The product of Wilson lines in Eq. ( 31) forms the soft function S(l + ), defined as so that its Fourier transform is The expression for T L in Eq. ( 31) can therefore be written as the convolution of the soft function, Eq. ( 34), and the jet function, Eq. ( 32), Collecting all the terms, the differential decay rate dΓ/dz in the z → 1 limit can thus be written as where the large momentum, M (1 − r χ /z), in the jet function can be further simplified to The factorization formula (36) is similar to the one for B → X s γ [24,26,27].The main difference are the hard interactions.Another difference is that the soft function in the q → χq decay would be treated perturbatively.For the squark mass M O(1TeV), the typical soft scale µ S ∼ M (1 − z) would be of a few tens of GeV, and it is much larger than the hadronic scale Λ QCD 1 GeV.Unlike B → X s γ, where the predictions in the end-point region are given in terms of the nonperturbative B meson shape functions, the nonperturbative physics here affects the region of phase space less than 1 − z ∼ O(Λ QCD /M ) ∼ 10 −3 , which does not significantly change our phenomenological conclusions in Sec.V B. (a) x 9 a w W o 8 h N Z 5 2 9 h N / q D w 2 h e e a s l g q c 3 S P Q O d i q W O E I U o e e o u J 6 w U + P X A 5 B y m u i M S 6 V U 3 U + b 6 L Q i 8 W K k L U G v A P W l x + I B 1 j b O j w 9 j Z n 4 4 H J + + b x 2 0 X P U c v 0 S s U o z f o B H 1 E I 3 S K M J q j n + g X + t 1 7 1 4 P e v P d l h W 5 v N W u e o d b o 6 X + n i Q y K < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x 9 a w W o 8 h N Z 5 2 9 h N / q D w 2 h e e a s l g q c 3 S P Q O d i q W O E I U o e e o u J 6 w U + P X A 5 B y m u i M S 6 V U 3 U + b 6 L Q i 8 W K k L U G v A P W l x + I B 1 j b O j w 9 j Z n 4 4 H J + + b x 2 0 X P U c v 0 S s U o z f o B H 1 E I 3 S K M J q j n + g X + t 1 7 1 4 P e v P d l h W 5 v N W u e o d b o 6 X + n i Q y K < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x 9 a w W o 8 h N Z 5 2 9 h N / q D w 2 h e e a s l g q c 3 S P Q O d i q W O E I U o e e o u J 6 w U + P X A 5 B y m u i M S 6 V U 3 U + b 6 L Q i 8 W K k L U G v A P W l x + I B 1 j b O j w 9 j Z n 4 4 H J + + b x 2 0 X P U c v 0 S s U o z f o B H 1 E I 3 S K M J q j n + g X + t 1 7 1 4 P e v P d l h W 5 v N W u e o d b o 6 X + n i Q y K < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = "    z Z p n q D V 6 5 T + v X Q 2 A < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K 9 q n V v 0 1 Z u j 0 g 1 1 3 E P e p E g 3 < l a t e x i t s h a 1 _ b a s e 6 4 = " K D U i q z 9 t 8 P 3 e O S 6 / H P k s 1 o P u t t 8 = " >         Here

< l a t e x i t s h a 1 _ b a s e 6 4 = " 3 W 0 d F R 4 A v y T M H o r x p 3 N v e t 1 e W j 0 = " >
In diagrams (b), the first (second) diagram is for collinear (soft) gluon exchange.Self energy diagrams are also needed for the matching process.

B. Radiative Corrections
We now return to the calculation of C L,R in Eq. ( 28).At scale µ ∼ M , we need to integrate out the heavy squark mass M and the hard gluon fluctuations of order M , matching onto HSET+SCET.For this we match QCD calculations in α s for the effective operators in full QCD (Eq.( 4)) and ones in SCET+HSET (Eq.( 13)).And we obtain the Wilson coefficients C L,R at the higher order in α s .At tree level the matching is trivial, For NLO results of C L,R we compute Feynman diagrams shown in Fig. 1 and self energy diagrams at one loop.Throughout this paper, we employ dimensional regularization in the MS scheme with D = 4 − 2 in order to handle ultraviolet (UV) singularity.As a result we obtain Here Li 2 (x) is the dilogarithm function.The anomalous dimensions for C L,R are given Here, are the wave function renormalizations for the heavy squark and the collinear quark fields respectively.Computing the one loop diagrams in Fig. 1-(b) gives where n • p is the large momentum component for the quark and equals n in the z → 1 limit, while n • v = 1 in the squark rest frame.From Eq. ( 40) we obtain the anomalous dimension for 38) and ( 41) the anomalous dimensions for the unknown B L,R are obtained as In order to employ the standard plus distribution for the radiative corrections in Eq. (36), it is convenient to introduce dimensionless jet and soft functions, where y is related to l + through The two new variables are defined in the interval z ≤ x, y ≤ 1.The limit of soft momenta in the soft function corresponds to y → 1.In terms of the dimensionless jet and soft functions, the differential decay rate in Eq. ( 36) can be rewritten as We computed the jet and the soft functions at next-to-leading order (NLO) in α s , and the results read [27,34] Jn (x, µ) =δ where We can check that the obtained differential decay rate does not depend on the scale choice, µ, to the order we are working.Differentiation with respect to d(log µ) gives, where γ J and γ S are the anomalous dimensions for Jn and S, At the lowest order in α s they are given by From ( 41), (52), and (53) it then follows immediately that Eq. ( 49) vanishes at O(α s ).

C. Resummed result for the differential decay rate near the end-point
The factorized result in Eq. ( 46) still contains large logarithms.We resum these by RG evolving |C L | 2 , Jn , and S from the factorization scale µ F down to the respective "typical scales" for each of the three quantities.The RG evolution then automatically resums the large logarithms and exponentiates them.Here "the typical scale" denotes the scale at which the logarithms in the expressions for |C L | 2 , Jn , and S are minimized.The typical hard scale for |C L | 2 can be chosen as µ H ∼ M .On the other hand, Eqs. ( 47) and ( 48) imply that we for the jet and soft functions, respectively.We perform the resummation to NNLL accuracy, counting large logarithms to be of O(1/α s ).For resummation to NNLL accuracy, we express the anomalous dimension of each factorized part as follows: From Eqs. ( 41), ( 52) and ( 53) [35,36], which can be expanded as k=0 Γ k (α s /4π) k+1 .The first two coefficients in the expansion are, where The noncusp anomalous dimensions in Eqs. ( 55) and ( 56) can be expanded as γf=J,S = k=0 γf,k (α s /(4π)) k .From Eqs. ( 52) and ( 53), the leading coefficients are given as The two loop coefficients required for NNLL accuracy are given by [38] γJ, where β 0 is the first coefficient of QCD beta function.The γC in Eq. ( 54) can be written as γC = −γ J − γS from the fact that the the differential decay width is scale independent.
Performing the RG evolutions using Laplace transform [39,40] leads to the resummed result near the end-point at NNLL accuracy as For integrated quantities such as the total decay width the NNLL accuracy requires jet and soft functions, J and S, to be calculated at O(α s ) (see, e.g., Table 1 in Ref. [41]).For decay width distributions, such as dΓ/dz, the log enhanced O(α 2 s ) terms in Jn and S need to be included as well [42] (see, e.g., Table 6 in Ref. [42]).To the required order the two functions are where the ellipses denote the O(α 2 s ) terms, which are given in Appendix B. For the NLL result, we only keep the first two terms in Eq.( 63) and (64).
The exponantiation factor in Eq. ( 62) is given by with the Sudakov factor S Γ and the evolution function a[f ] defined as Here α and α 0,1 denote α s (µ) and α s (µ 0,1 ), while b(α s ) = dα s /d ln µ is the QCD beta function.To NNLL accuracy S Γ and a[Γ C ] are given by [41] wehere r = α s (µ 0 )/α s (µ 1 ).Finally, the evolution parameter η in Eq. ( 62) is defined as For the NLL result, we only keep the first line of Eq. ( 67), and the first term in Eq. (68).

IV. DECAY DISTRIBUTION IN THE FULL RANGE
Even though the decay distribution dΓ/dz in the region z → 1 is the dominant contribution to the total decay width, it is useful for phenomenological analyses to obtain the decay distribution in the full range of z, while retaining the O(1 − z) corrections.The expression for dΓ/dz away from z → 1 should be obtained using full QCD.Away from the end-point region the gluon emissions are hard so that the total invariant mass of final state jets can be comparable to M .
To perform the calculation of T L in Eq. ( 20) in full QCD we introduce the structure The W L is thus given by where p χ is the momentum of the neutralino (the expression for it in terms of z and r χ is given in Eq. ( 25)).The differential decay rate is then At tree level we have simply To obtain the NLO expression for W L we computed the Feynman diagrams shown in Fig. 2 as well as self energy diagrams for the squark and the light quark.As a result, one loop corrections to W L in MS scheme are given as  Here we introduced the so called 'delta distribution', [. ..] ∆ , in order to deal with the infrared (IR) singularity as z → 1.The definition and some useful properties of the ∆ distribution are given in Appendix A. For a region of integration that is, as in our case, over z ∈ [ √ r χ , 1] (rather than over the interval [0, 1]) the introduction of a ∆ distribution shortens the expressions compared to the standard plus distribution.In Eq. (72) the functions H 1,2 (r χ ) and g(z, r χ ) are given by, The anomalous dimension γ W (z), controlling the RG evolution of structure functions, is given by Using the fact that dΓ/dz is scale-invariant then gives the anomalous dimension for B L (µ) as γ B = −3α s C F /(2π) + O(α 2 s ), which, as expected, is the same result as given in Eq. (42).
Finally, we combine our results for the differential decay rates in the full z range and near the end-point, z → 1, to obtain the decay distribution at NNLL+NLO, 2 The first and the second terms on the right-hand side are the resummed result near the end-point, Eq. ( 62), and the NLO result for the full z range, Eq. ( 71), respectively.The double counting of contributions between the two terms is removed by the third term on the right-hand side of Eq. (78), i.e., the dΓ f E /dz.The expression for dΓ f E /dz follows from dΓ res /dz by identifying the multiple scales as µ H = µ J = µ S = M .

V. PHENOMENOLOGICAL STUDY
In this section, we present a detailed study of NLO, NLL+NLO, and NNLL+NLO predictions for the q → qχ decay.In section V A we show our results for the normalized differential width distributions as well as the total decay widths, and discuss the impact of soft gluon resummations on the NLL+NLO and NNLL+NLO results.In section V B, we perform a numerical analysis at NLL+NLO and NNLL+NLO accuracies for the decays of pair-produced squarks in a linear e + e − collider based on √ s = 3 TeV Compact Linear Collider (CLIC).Since the decay topology of a squark can be significantly altered by higherorder corrections, it is necessary to scrutinize these effects for the precise measurements of a squark and neutralino masses, which is an important part of the CLIC physics program.

A. Differential width distributions and total widths
The resummed results that we calculated at NNLL+NLO and NLL+NLO accuracies in Eq. ( 62) depend on the choices of scales, µ H , µ J , and µ S .To illustrate the scale dependences, we independently vary the µ i , i = H, J, S, between 2µ 0 i and µ 0 i /2, where µ 0 i are the default choices of the hard, jet, and soft scales.We take µ 0 H = M for the default hard scale.The default jet and soft scales are chosen as the running scales µ 0 For z → 1 we would have µ 0 i → 0 for these choices of running scales and therefore the IR Landau poles in the running of the strong coupling 2 One may wish to consider other matching schemes that turn off smoothly the resummation effects in the region far away from the end-point.In our case the role of the smooth matching is performed by the running of jet and soft scales which are respectively given as µ 0 While the prescription in Eq. (78) does not recover exactly the NLO result anywhere in the physical region z ≥ √ r χ , this treatment does suffice for our purposes.First of all, alternative prescriptions that have µ H = µ J = µ S at z = 0 similarly do not lead to perfectly smooth matchings anywhere in the physical region for z.More importantly, the two decay distributions with and without resummations, dΓ res /dz and dΓ f NLO /dz, are completely dominated by the end-point region and have only negligible contributions from the rest of the z range.We therefore expect only numerically subleading corrections from alternative matching prescriptions relative to the results using Eq.(78  constants.In order to avoid this problem we adopt the following profile function for the soft scale, Therefore we make the soft scale frozen as µ min as z → 1.The parameters a and z 0 , where 1 − z 0 1, are determined by z 0 = 1 − 2µ min /M and a = M/(4µ min ) to ensure that µ pf S is smoothly continuous at z 0 .We use µ min = 0.5 GeV so that M (1 − z 0 ) = 1 GeV.The impact of nonperturbative physics grows as z becomes larger than z 0 , and therefore µ 0 S goes from 1 GeV to 0.5 GeV.The precise choice of µ min and the estimation of its uncertainty would be possible from a nonperturbative model or from a fit to experimental data, if these become available.This is beyond the scope of this paper, where we focus on perturbative resummation effects near the end-point.
We also modify the jet scale using the following profile function3 Table I shows the total decay widths of a squark with mass M = 1.45 TeV obtained at LO, NLO, NLL+NLO, and NNLL+NLO accuracies, with the Wilson coefficient B L normalized by B L (M ) = 1 at the scale M .Benchmark neutralino masses are chosen in the interval 200 < M χ < 1000 GeV.The scale uncertainty of NLL+NLO prediction turns out to be ∼ 18%, which is improved to ∼ 14% at NNLL+NLO.These are obtained by varying the hard, jet and soft scales in the range from µ 0 H,J,S /2 to 2µ 0 H,J,S each independently.Figure 3 shows the ratios of total decay widths of NLL+NLO and NNLL+NLO predictions with Concerning the scale variations of the NLL+NLO and NNLL+NLO distributions, we find that the dominant uncertainties come from when we vary the soft scale down to µ 0 S /2, while other scale variations give quite small uncertainties.Note that µ 0 S varies from 5.8 GeV to 1.45 GeV when z is changed from z = 0.996 to z = 0.999, and reaches µ min = 0.5 GeV at the very right end of panels in Fig. 4. The large variation still present at NNLL for z > 0.999 can thus be traced to this very low soft scale that is reached at the very end-point of the spectrum.

B. Precision studies of squarks and neutralinos at CLIC
The squark decay, q → qχ, results in a two-body final state at LO, which at higher orders becomes a multi-body final state due to additional hard or soft QCD radiation.This can potentially affect the methods for precise measurements of squark and neutralino masses.As an illustration we take the impact of QCD corrections on such measurements at CLIC [43][44][45][46], a future linear e + e − collider designed to provide collision energies up to 3 TeV.If supersymmetric particles are light enough to be produced at such machine, CLIC will provide a platform for precision studies where their properties could be determined with considerable accuracy [47][48][49][50][51].In the phenomenological analysis we focus exclusively on the impact of QCD radiations in the squark decay.For realistic study other important effects, in particular the initial state QED radiation, that results in the reduced effective e + e − collision energy, need to be included.
For pair-produced squarks that decay into light quarks and neutralinos, e + e − → q q * → q χ q χ , an interesting technique to simultaneously measure squark and neutralino masses, is to search for the edges in the event distributions.We will discuss two such methods, i) based on edges in energy distribution of the light quark jets, E 1 + E 2 , and ii) a method based on the kinematic variable M C [52].The numerical analysis is based on LO version of Mad-Graph5 aMC@NLO [53,54] with PYTHIA 6 [55] showering, but no hadronization, which was used to generate the event chain in (81), utilizing the Minimal Supersymmetric Standard Model (MSSM) implementation from [56,57].The PYTHIA events were clustered with the FastJet [58] implementation of the anti-k T algorithm [59], taking r = 0.4 for the cone size.For events to pass the selection cuts we require at least two jets p T > 50 GeV.To obtain the NLL+NLO and NNLO+NLO (including two-loop-log terms) samples we reweight PYTHIA events, on an event-by-event basis, according to the d log Γ/dz normalized differential distributions in Figure 4 for each of the decay chains.We first rewrite the variable z in Lorentz-invariant form where p q and p χ are four-momenta of a squark and a neutralino respectively 4 .We plug two z values (from two decay chains) into the normalized NLL+NLO and NNLO+NLO distributions in Figure 4 using µ min = 0.5 GeV to obtain probabilities.Then we multiply the two probabilities to obtain the weight for each event.In this way, the simulated events acquire the correct NLL+NLO and NNLO+NLO distributions in z variable, but are only approximately NLL+NLO and NNLO+NLO in the other phase space variables.The NLO samples, on the other hand, cannot be obtained using the same reweighting method.Since the NLO distributions in Figure 4 diverge at z = 1, the probabilistic interpretation of the differential width distributions is not well-defined.As a result we do not include reweighted NLO distributions.We derive results for two benchmarks, setting squark mass to M = 1.45 TeV, while taking the lightest neutralino mass to be M χ = 1 TeV or 0.5 TeV, and assume a negligible squark decay width.In this study, the beamstrahlung, initial state radiation, and detector effects are not included.
For two body squark decays, (81), the minimal and maximal light quark energy are directly related to M and M χ [60, 61] and thus in our case E 1 + E 2 ∈ [2E q,min , 2E q,max ], neglecting the small squark boosts in the lab frame.At LO the E 1 + E 2 distributions start at 0.59 TeV and 0.98 TeV, for M χ = 0.5 TeV and M χ = 1 TeV, respectively.In Figure 5, on the other hand, the NLL+NLO and NNLL+NLO E 1 + E 2 distributions extend well below these boundaries (see the green and blue lines).This behavior is easy to understand -the collinear radiation leads to nonzero jet masses, or equivalently, to a d log dΓ/dz squark decay distribution with most of the events having z < 1 (two-body decays have z = 1), see Figure 4.This in turn means that the jet energy is smaller than in the two body decay, cf. ( 25), softening the E 1 + E 2 spectrum.The effect is present, but less pronounced, also at the upper edge of the E 1 + E 2 distribution.
The original PYTHIA distributions (before reweighting) are shown with gray lines.
The extraction of M , and M χ from the E 1 + E 2 distribution is still possible, as indicated by the fact that the E 1 + E 2 distributions shifts significantly between the M χ = 0.5 TeV and M χ = 1 TeV benchmarks.However, one would need to use the full matrix element and not just the edges, in this way controlling the shift of the edges due to the soft and collinear radiations.In addition to the NLL + NLO and NNLL+NLO decay width distributions that we have calculated in the present manuscript, one would also control other systematics and theoretical uncertainties.The method, for instance, requires precise knowledge of the center of mass energy, which can be potentially distorted by beamstrahlung [62,63] and initial state radiations (ISR), causing sizable uncertainties in the measurements of the edges, see, e.g., Ref. [61] .
An alternative mass measurement method exploits the kinematic variable M C , invariant under contra-linear boosts of equal magnitude, Here E q,1 , p q,1 and E q,2 , p q,2 are the energies and three-momenta of the two final state quarks, respectively.The maximal value of M C is reached when the two jets are co-linear.It is given by showing that M C is sensitive to both M and M χ .The virtue of the M C variable is that it does not depend on the center of mass energy, and is therefore less susceptible to beamstrahlung distortions [61].Similarly to the E 1 + E 2 distribution, the collinear and soft radiations cause the M C spectrum to soften.However, as can be seen in Figure 6, the effect is more pronounced at the maximal value of M C , which is exactly the quantity that enters the determination of M and M χ .Comparing the shift in the distributions for M χ = 0.5 TeV and M χ = 1 TeV one sees that the LO sensitivity to M, M χ , Eq. (86), still applies to a good extent also to the resummed distribution with M C constructed using the two hardest jets.
For instance, for the numerical examples in Fig. 6 there are still appreciable numbers of events within O(5%) of M max C , with the peak of the distribution shifted by O(10 − 20%) at NNLL+NLO compared to Pythia.This gives a rough sense of associated errors on M max C due to the softening of distributions in the case of limited statistics available in an experiment.However, once CLIC collects enough statistics a precise determination of M max C (M, M χ ) using a matrix element method based on resummed distributions can be attempted.
Finally, we show in Figure 7 the PYTHIA (gray), NLL+NLO (green), and NNLL+NLO (blue) missing energy E miss distributions.Here the E miss is due to the two neutralinos in the final state, and we do not include any detector effect.Unlike the other two observables, E 1 + E 2 and M C , the effect of resummations is negligible for the E miss distribution.This is because the neutralino mass is too heavy for the effect of recoiling against the soft gluon radiations from the quark-sector to be significant.

VI. SUMMARY
In this paper we have studied QCD corrections to the squark decay, q → qχ.The large logarithms that arise in the end-point region, z → 1, were resummed using SCET up to the NNLL accuracy.Away from the end-point we computed hard gluon radiations at NLO.Finally, we provided an expression that smoothly interpolates between the NNLL and NLO results, giving the NNLL+NLO prediction for the total decay width and the decay distribution, dΓ/dz.The additional QCD radiation in the decay softens the decay distributions for many observables.As a case study for the phenomenological impact of higher order QCD corrections we explored the methods for simultaneous measurements of squark and neutralino masses at a linear e + e − collider based on √ s = 3 TeV CLIC.A majority of mass measurement techniques are based on edges in kinematic distributions.
Such kinematic edges are modified by having additional QCD radiation in the event.For instance, the distribution of the combined energy of the hardest two jets, E 1 + E 2 , now extends below the lower boundary that is otherwise obtained in the case of two body decays.
Similarly, the distributions in the M C variable get softened near its maximal value, which is precisely the region used for the quark and neutralino mass extractions.With limited available statistics in experiments this softening of the distributions would result in a shift in measured squark and neutralino masses.The induced shift in the masses could be estimated from a matrix element based method using the NNLL+NLO resummed decay distributions that we provided.In a quantitative analysis one would also need to include additional effects such as the beamstrahlung, initial state radiation, and detector effects.
t b 2 j Y + i m 4 g n g u 3 g Z 7 s 0 2 z 3 u I o 1 m j + P 8 / Y 4 1 m n k l F t h s O / W 9 v 3 d n r 3 H + w + 7 D 9 6 / O T p s 7 3 9 5 y e 6 K B U m x 7 h g h T t b 2 j Y + i m 4 g n g u 3 g Z 7 s 0 2 z 3 u I o 1 m j + P 8 / Y 4 1 m n k l F t h s O / W 9 v 3 d n r 3 H + w + 7 D 9 6 / O T p s 7 3 9 5 y e 6 K B U m x 7 h g h T t b 2 j Y + i m 4 g n g u 3 g Z 7 s 0 2 z 3 u I o 1 m j + P 8 / Y 4 1 m n k l F t h s O / W 9 v 3 d n r 3 H + w + 7 D 9 6 / O T p s 7 3 9 5 y e 6 K B U m x 7 h g h T e N n a T P y i 8 9 o W n W j J Y a r N 0 z 0 C n Y q k j R C F K n r r L C S s F f j 0 w O Y e p 7 o h E e t X N l L l + C w I v V u o C 1 B p w T 1 o c P m B d 4 / T o M H b 2 5 z e D 4 w / N 4 7 a L n q O X 6 B W K 0 V t 0 j D 6 h E T p B G F 2 g n + g X + t 1 7 3 8 O 9 i x 5 b o d t b T b b 8 W y n U D L / H T w C f x b / D X a a t Y 0 z X N U 6 3 f f j O / t 8 c S x o p n S / / 3 d j 8 9 Z W 5 / a d 7 b v d e / c f P H y 0 s / v 4 W O W F x G S I c 5 r L 0 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K D U i q z 9 t 8 P 3 e O S 6 / H P k s 1 o P u t t8 = " > A A A F w n i c d Z T f b 9 M w E M e 9 s c I o v z Z 4 5 C V Q T R o v o 5 m Q 4 H E C H n h A o i C 6 T W q q 6 u I 4 X T b b 8 W y n U D L / H T w C f x b / D X a a t Y 0 z X N U 6 3 f f j O / t8 c S x o p n S / / 3 d j 8 9 Z W 5 / a d 7 b v d e / c f P H y 0 s / v 4 W O W F x G S I c 5 r L 0 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K D U i q z 9 t 8 P 3 e O S 6 / H P k s 1 o P u t t8 = " > A A A F w n i c d Z T f b 9 M w E M e 9 s c I o v z Z 4 5 C V Q T R o v o 5 m Q 4 H E C H n h A o i C 6 T W q q 6 u I 4 X T b b 8 W y n U D L / H T w C f x b / D X a a t Y 0 z X N U 6 3 f f j O / t8 c S x o p n S / / 3 d j 8 9 Z W 5 / a d 7 b v d e / c f P H y 0 s / v 4 W O W F x G S I c 5 r L 0 l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K D U i q z 9 t 8 P 3 e O S 6 / H P k s 1 o P u t t8 = " > A A A F w n i c d Z T f b 9 M w E M e 9 s c I o v z Z 4 5 C V Q T R o v o 5 m Q 4 H E C H n h A o i C 6 T W q q 6 u I 4 X T b b 8 W y n U D L / H T w C f x b / D X a a t Y 0 z X N U 6 3 f f j O / t8 c S x o p n S / / 3 d j 8 9 Z W 5 / a d 7 b v d e / c f P H y 0 s / v 4 W O W F x G S I c 5 r L 0 r H e S r m p 7 X d u b E r Z + C a 4 g 8 V y 8 D f Y m T b L e 4 i j W a P 4 / z 9 j j W S e S Z d p 0 u 3 8 3 N u 9 s t e 7 e 2 7 7 f f v D w 0 e M n O 7 t P T 3 R e K I r H N G e 5 O k t A I 8 s E H p v M M D y T C o E n D E + T i 4 9 e P 5 2 i 0 9 8 p i W D m T Y z 9 w w 0 K p Y 4 Q u S i 4 I m 7 n L B S 4 N c D k + c w 0g 0 R p V f d n D H X b 0 H g 6 V y d g l o C 7 k n r h Q 9 Y 0 z g 5 P O g 5 + 8 u b z t G H x e O 2 T Z 6 T l 2 S f 9 M h b c k Q + k T 4 5 J p Q w 8 p P 8 I r 9 b 7 1 v Y Y i 0 x R z c 3 F m u e k d p o X f 0 D d E c N 2 Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 W 0 d F R 4 A v y T M H o r x p 3 N v e t 1 e W j 0 = " > A A A F v X i c d Z T N b h M x E M f d 0 k A J X y 0 c u S x E l c q l J B U S 3 K j g w o F D Q P R D y k b R r H e Sr m p 7 X d u b E r Z + C a 4 g 8 V y 8 D f Y m T b L e 4 i j W a P 4 / z 9 j j W S e S Z d p 0 u 3 8 3 N u 9 s t e 7 e 2 7 7 f f v D w 0 e M n O 7 t P T 3 R e K I r H N G e 5 O k t A I 8 s E H p v M M D y T C o E n D E + T i 4 9 e P 5 2 i 0 9 8 p i W D m T Y z 9 w w 0 K p Y 4 Q u S i 4 I m 7 n L B S 4 N c D k + c w 0g 0 R p V f d n D H X b 0 H g 6 V y d g l o C 7 k n r h Q 9 Y 0 z g 5 P O g 5 + 8 u b z t G H x e O 2 T Z 6 T l 2 S f 9 M h b c k Q + k T 4 5 J p Q w 8 p P 8 I r 9 b 7 1 v Y Y i 0 x R z c 3 F m u e k d p o X f 0 D d E c N 2 Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 W 0 d F R 4 A v y T M H o r x p 3 N v e t 1 e W j 0 = " > A A A F v X i c d Z T N b h M x E M f d 0 k A J X y 0 c u S x E l c q l J B U S 3 K j g w o F D Q P R D y k b R r H e Sr m p 7 X d u b E r Z + C a 4 g 8 V y 8 D f Y m T b L e 4 i j W a P 4 / z 9 j j W S e S Z d p 0 u 3 8 3 N u 9 s t e 7 e 2 7 7 f f v D w 0 e M n O 7 t P T 3 R e K I r H N G e 5 O k t A I 8 s E H p v M M D y T C o E n D E + T i 4 9 e P 5 2 i 0 9 8 p i W D m T Y z 9 w w 0 K p Y 4 Q u S i 4 I m 7 n L B S 4 N c D k + c w 0g 0 R p V f d n D H X b 0 H g 6 V y d g l o C 7 k n r h Q 9 Y 0 z g 5 P O g 5 + 8 u b z t G H x e O 2 T Z 6 T l 2 S f 9 M h b c k Q + k T 4 5 J p Q w 8 p P 8 I r 9 b 7 1 v Y Y i 0 x R z c 3 F m u e k d p o X f 0 D d E c N 2 Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 W 0 d F R 4 A v y T M H o r x p 3 N v e t 1 e W j 0 = " > A A A F v X i c d Z T N b h M x E M f d 0 k A J X y 0 c u S x E l c q l J B U S 3 K j g w o F D Q P R D y k b R r H e Sr m p 7 X d u b E r Z + C a 4 g 8 V y 8 D f Y m T b L e 4 i j W a P 4 / z 9 j j W S e S Z d p 0 u 3 8 3 N u 9 s t e 7 e 2 7 7 f f v D w 0 e M n O 7 t P T 3 R e K I r H N G e 5 O k t A I 8 s E H p v M M D y T C o E n D E + T i 4 9 e P 5 2 i 0

FIG. 1 .
FIG. 1.One loop diagrams for matching between full QCD (a) and SCET+HSET (b).HereO L(R) = q L(R) P R(L) χ q and O L(R) = ξn,L(R) W n P R(L) χ φ v .In diagrams (b), the first (second) diagram is for collinear (soft) gluon exchange.Self energy diagrams are also needed for the matching process.

)
Note that full Wilson coefficients C L,R = B L,R C L,R involve the unknown new physics Wilson coefficients B L,R .But, since the effective interaction Lagrangian in (13) should be scale invariant, we can compute γ C L,R by considering the renormalization behavior of the effective operators O L,R in SCET+HSET.The relation between the bare and renormalized effective operators can be written as Z and (. ..) + the standard plus distribution.Note that both of the above results are infrared finite.The logarithms are minimized at µ = Q(1 − x) 1/2 and µ = M (1 − y) for the jet and soft functions, respectively.
t b 2 j Y + i m 4 g n g u 3 g Z 7 s 0 2 z 3 u I o 1 m j + P 8 / Y 4 1 m n k l F t h s O / W 9 v 3 d n r 3 H + w + 7 D 9 6 / O T p s 7 3 9 5 y e 6 K B U m x 7 h g h T FIG.2.The NLO Feynman diagrams for q → χX q L .The dashed lines at the center of each diagram denotes the discontinuity cut for forward scattering amplitudes.

FIG. 3 .FIG. 4 .
FIG.3.The ratios of NLL+NLO (green line) and NNLL+NLO (blue) total decay widths normalized to the LO result as a function of mass M .The neutralino masses are fixed to M χ = 1 (0.5) TeV in the left (right) panel.The details on hard, jet, and soft scale variations, giving the corresponding bands, are explained in Eq. (79) and Eq.(80), with µ min = 0.5 GeV.
FIG. 5. Distributions for the sum of energies of the first two hardest jets for M χ = 1 TeV (left) and M χ = 500 GeV with fixed M = 1.45 TeV (right).
is the number of colors and n f is the number of flavors.The three loop ).

TABLE I .
Total decay widths of a 1.45 TeV squark based on the LO, fixed-order (NLO), NLL+NLO, and NNLL+NLO calculations, with the Wilson coefficient B L at the scale M taken to be B L (M ) = 1.Benchmark neutralino masses are varied in the interval 200 < M χ < 1000 GeV.The impact of scale uncertainties for NLL+NLO and NNLL+NLO predictions are shown as well.