Hemisphere jet mass distribution at finite $N_c$

We perform the leading logarithmic resummation of nonglobal logarithms for the single-hemisphere jet mass distribution in $e^+e^-$ annihilation including the finite-$N_c$ corrections. The result is compared with the previous all-order result in the large-$N_c$ limit as well as fixed-order perturbative calculations.


Introduction
The hemisphere jet mass distribution is an event shape variable in e + e − -annihilation defined as the distribution of invariant mass in a single hemisphere whose axis coincides with the thrust axis. As is usually the case with all event shapes, it receives logarithmically enhanced perturbative corrections when the shape variable becomes small. However, unlike other event shapes for which systematic resummation methods are available (see [1,2] and references therein), the resummation of logarithms for the hemisphere jet mass distribution has turned out to be thorny and so far remained unsatisfactory even at the leading logarithmic level. This is due to the presence of the so-called nonglobal logarithms [3] which arise from the energy-ordered radiation of soft gluons in a restricted region of phase space.
The difficulty of resumming nonglobal logarithms stems from the fact that one has to keep track of the distribution of an arbitrary number of secondary soft gluons emitted at large angle. (For a recent review, see [4].) The original work by Dasgupta and Salam [3] employed a Monte Carlo algorithm, valid to leading logarithmic accuracy and in the large-N c approximation, to actually generate soft gluon cascades on a computer. Later, Banfi, Marchesini and Smye (BMS) [5] reduced the problem, still at large-N c , to solving a nonlinear integro-differential equation. This latter approach paved the way for the inclusion of the finite-N c corrections in the resummation [6] which has been recently put on a firmer ground [7], and the first quantitative finite-N c result can be found in [8]. However, so far only one particular observable ('interjet energy flow') has been computed and the full impact of the finite-N c resummation is yet to be uncovered.
In this work, we apply the method developed in [8] to the hemisphere jet mass distribution and numerically carry out the resummation of nonglobal logarithms at finite-N c , thereby achieving the full leading logarithmic accuracy for this observable. 1 In the next section we define the observable and introduce the BMS equation which resums the nonglobal logarithms in the large-N c limit. In Section 3, we discuss the resummation strategy at finite-N c . It turns out that a naive application of the previous method is plagued by large numerical errors, and we shall propose a refined method which cures this problem. In Section 4, we present the numerical result and compare it with the previous all-order result at large-N c [3] as well as the recent fixed-order calculations [9,10].

Hemisphere jet mass distribution
Consider a two-jet event in e + e − -annihilation with the center-of-mass energy Q. Without loss of generality, we assume that the quark jet is right-moving with momentum p µ = Q 2 (1, 0, 0, 1) ≡ Q 2 (1, n R ) and the antiquark jet is left-moving withp µ = Q 2 (1, 0, 0, −1) ≡ Q 2 (1, n L ). Suppose soft gluons with momentum k µ i = ω i (1, n i ) (n = (sin θ cos φ, sin θ sin φ, cos θ)) are emitted in the right hemisphere 0 ≤ θ i ≤ π 2 . The invariant mass in the right hemisphere is We shall be interested in the probability that the rescaled invariant mass is less than some value ρ < 1. When ρ ≪ 1, one has to resum large logarithms ln n 1/ρ in the perturbative calculation of P LR (ρ). As observed in [3], this resummation consists of two parts. One is the Sudakov double logarithms (α s ln 2 1/ρ) n which can be resummed via exponentiation. The other is the nonglobal logarithms (α s ln 1/ρ) n which arise from the fact that measurement is done only in a part of phase space (i.e., in the right hemisphere). The latter resummation affects various single-hemisphere event shapes in e + e − annihilation and DIS [3]. It is also relevant to the so-called soft function in the dijet mass distribution d 2 σ/dM R dM L in the asymmetric limit M L ≫ M R [13,14,9]. So far, the resummation of nonglobal logarithms for P LR has been carried out only in the large-N c limit [3], or to finite orders of perturbation theory at large-N c [9] and finite-N c [10]. As stated in the introduction, we shall perform the all-order leading logarithmic resummation at finite-N c along the lines of [6,8]. To explain our approach, it is best to start with the BMS equation which resums both the Sudakov and nonglobal logarithms in the large-N c limit [5]. Adapted to the hemisphere jet mass distribution [9], the equation reads where is the soft gluon emission kernel and we defined Θ L/R (γ) is the 'step function' which restricts the angular integral dΩ γ = d cos θ γ dφ γ to the left/right hemisphere. [Below we also use a shorthand notation L/R dΩ to represent this.] In (4), P αβ = P (Ω α , Ω β ) is the generalization of P LR defined for arbitrary pairs of solid angle directions. Taken at its face value, Eq. (4) is ill-defined. When α or β is in the right hemisphere, the dΩ γ integral in the second term on the right-hand-side (the virtual term) is divergent, and this is precisely the situation (αβ) = (LR) we are eventually interested in. Physically, this collinear divergence should be cut off by the kinematical effect, yielding the Sudakov factor e −O(αs) ln 2 1/ρ . However, since the Sudakov factor is well understood anyway, one can leave it out of consideration by defining 2Nc and 2C F ≈ N c in the large-N c limit.) Unlike (4), the equation satisfied by g αβ is well-defined and amenable to analytical and numerical approaches. In particular, Ref. [9] analytically calculated g LR to five loops using the hidden SL(2,R) symmetry of the BMS equation [15].

Resummation at finite N c
We now turn to the physically relevant case N c = 3. Temporarily forgetting about the issue of the collinear divergence, we recapitulate the resummation strategy developed in [16,6,8]. First we make the formal identification where U α is the Wilson line in the fundamental representation of SU(N c ) from the origin to infinity in the Ω α direction. The product in (8) represents the propagation of the qq jets ('dipole') in the eikonal approximation. As τ is increased, more and more soft gluons are emitted from the dipole and also from the secondary gluons. This can be simulated as a stochastic process in which the Wilson lines receive random kicks in the color SU(N c ) space, and is described by the where and ξ (1) , ξ (2) are the Gaussian noises This is equivalent to the following 'Fokker-Planck' equation to be compared with (4) 2 We write the evolution (9) in a slightly different, but equivalent form compared to Ref. [8]. It should be understood that various exponentials are meaningful only to O(ǫ) [8], although in practice we keep all orders in √ ǫ in order to preserve the unitarity of U α .
where ... ξ denotes averaging over the noises. In principle, P LR (τ ) at finite-N c can be evaluated by computing 1 Nc trU L U † R for a given random walk trajectory with the initial condition U α (τ = 0) = 1, and then averaging over many trajectories. In this calculation, it suffices to define U α in the left hemisphere and at a single point α = R in the right hemisphere.
However, this strategy does not apply straightforwardly to our present problem. P LR ξ quickly goes to zero due to the collinear divergence in the Sudakov factor. 3 One may try to regularize the divergence by introducing a cutoff δ and extract the finite part Unfortunately, this does not work in practice because P δ LR ξ becomes very small and the exponential factor becomes very large as δ → 0. It is difficult to numerically achieve the precise cancelation between the two factors.
As a matter of fact, the same problem was already noticed in the original Monte Carlo simulation at large-N c [3]. There the authors subtracted the Sudakov contribution step-by-step, by modifying the emission probability as M αβ (γ) → M αβ (γ) − Θ R (γ)M LR (γ). Here we shall implement a similar subtraction directly in the evolution of U α . The origin of the collinear divergence can be traced to the factors e iS (i) α in (9). They give, after averaging over the noise, which is indeed the Sudakov factor (7) generated in a single step. (16) can be checked by using the identity [8] M αβ (γ) = 2K αβ (γ) − K αα (γ) − K ββ (γ) , K αβ (γ) ≡ (n α − n γ ) · (n γ − n β ) 2(1 − n α · n γ )(1 − n γ · n β ) . (17) In the special case (αβ) = (LR), we have that K LR (γ) ≡ 0 and We see that the singularity entirely comes from the second order term in the expansion of e iS (i) R .
It is thus tempting to remove the factor e iS (i) α altogether and use a modified evolution equation U α (τ + ε) = e iAα U α (τ )e iBα . However, this also removes an essential part of the nonglobal logarithms. The reason is that the linear term in the expansion of e iSα(ξ) = 1 + iS α (ξ) + · · · can give finite contributions when the Gaussian noise ξ is contracted with that implicit in U γ t a U † γ in (11). Physically, the factor U γ t a U † γ = U ab A t b (with U A being the Wilson line in the adjoint representation) represents the emission of real gluons which is restricted to the left hemisphere. These gluons together with the original qq pair form a QCD antenna which coherently emits the softest gluon in the right hemisphere, thereby producing nonglobal logarithms.
To make the last statement more concrete, we follow the evolution (9) analytically up to τ = 2ε (two steps) and collect the non-Sudakov contributions. We find in agreement with the lowest order (two-loop) result [3,9]. 4 In the first line of (19), the factors K Lγ and K γR come from the linear terms in e iS L ≈ 1 + iS L and e iS R ≈ 1 + iS R , respectively. They both seem to be essential for obtaining the correct result. Importantly, however, the term K γR (λ) vanishes when integrating over the azimuthal angle where we used cos θ λ > 0 > cos θ γ . Moreover, by following the evolution (9) a few more steps, it is easy to convince oneself that the linear term iS R does not produce nonglobal logarithms to all orders because this term always reduces to factors like K γ (n) R (λ) (after contracting with the n-th gluon emitted in the left hemisphere) and vanishes when integrating over φ λ in the right hemisphere. This observation brings in a major simplification in our resummation strategy. We can neglect the factors e iS (1,2) R in (9) for α = R and use the modified Langevin equation As for U α in the left hemisphere, we may continue to use the same evolution (9). Actually, we can make a slight improvement which speeds up the numerical simulation. The two independent noises ξ (1,2) defined in the right hemisphere always give identical contributions for the observable at hand. Therefore, we can eliminate one of them and use a modified equation Note the factor of √ 2. One can check that (22) leads to the same equation (14) for the product of two Wilson lines. 5 Using these Langevin equations, we finally compute the average The multiplicative factor in front subtracts the finite part of the Sudakov factor (18) which is included in the evolution of U L .

Numerical results and discussions
The numerical procedure is explained in Ref. [8] which we refer to for details. We discretize the solid angle 1 ≥ cos θ ≥ −1 and 2π > φ ≥ 0 into a lattice of 80 × 80 grid points and put a SU(3) matrix U α at each grid point on the left hemisphere cos θ < 0. In addition, we define a SU(3) matrix U R at a single point n R in the right hemisphere. The Gaussian noise ξ is randomly generated at all grid points and at each time step. 6 We then evolve U α and U R according to (22) and (21), respectively, with ε = 5 × 10 −5 and the initial conditions U α = U R = 1. As in the previous work [8], we observe large event-by-event fluctuations. In order to obtain a reasonably smooth curve, we typically have to average over O(10 3 ) random walks. Fig. 1 shows the result from 2600 random walks. 7 In the same figure, we make comparisons with the following results in the literature: The blue line is a parameterization of the all-order Monte Carlo result in the large-N c limit by Dasgupta and Salam (DS) [3] g DS (τ ) = exp −C F N c π 2 τ 2 12 with a = 0.85N c , b = 0.86N c and c = 1.33. Here we set C F ≈ N c /2 = 1.5 which is what was actually used in [3]. The black dashed line is a combination of the fixed-order analytical results by Schwartz and Zhu (SZ) to five-loop at large-N c [9] and Khelifa-Kerfa and Delenda (KD) to four-loop at finite N c [10] g SZ−KD (τ ) = 1 − C F N c π 2 12 where C F = 4/3. Actually, the complete finite-N c result at O(τ 5 ) is not available, and the above formula is a well-motivated guess [10] which reduces to the known result in the large-N c limit. Finally, the blue dash-dotted line is the following 'resummed' expression suggested by KD based on their four-loop result with C F = 4/3. Note, however, that at the moment it is not known whether the nonglobal logarithms actually exponentiate to all orders.
We see that our result agrees very well with the most-advanced fixed-order result (25) up to τ 0.5. Beyond this, the perturbative result quickly deviates and eventually blows up. It has been observed that higher loop contributions alternate in sign and converge rather poorly [17]. In addition, fixed-order results are numerically sensitive to the 1/N c -suppressed corrections when τ ∼ O(1). This can be partly remedied in the resummed formula (26). On the other hand, the all-order large-N c result (24) stays close to our curve up to τ = 1. In fact, the difference can be partly accounted for by choosing C F = 4/3 in (24), which is what was actually suggested by DS as the likely functional form at finite-N c and has been used for phenomenological purposes [18,19]. This is shown by the red line in Fig. 1. To correct the remaining difference, we independently determined a, b, c in (24) with C F = 4/3 and obtained a = 0.62 ± 0.06 , b = 0.06 ± 0.03 , c = 0.37 ± 0.04 .
In conclusion, we have completed the resummation project for the single-hemisphere jet mass distribution initiated in [3] by including the finite-N c corrections to all orders. We find that the finite-N c effect is numerically small, and this is consistent with the previous finding in [8]. However, it should be kept in mind that the observables calculated at finite-N c so far are defined in e + e − annihilation where the two outgoing jets are represented by the product of two Wilson lines tr U α U † β . In hadron-hadron collisions, or in processes including hard gluons, one needs to consider the evolution of more complicated objects such as tr(U α U † β U γ U † δ ) and tr(U α U † β )tr(U γ U † δ ) (cf., [20]). The finite-N c effects in the resummation of nonglobal logarithms for these observables have not been studied so far.