NNLO predictions for event shapes and jet rates in electron-positron annihilation
aa r X i v : . [ h e p - ph ] J a n NNLO predictions for event shapes and jet rates inelectron-positron annihilation
Stefan Weinzierl ∗ University of MainzE-mail: [email protected]
The strong coupling constant is a fundamental parameter of nature. It can be extracted from ex-periments measuring three-jet events in electron-positron annihilation. For this extraction precisetheoretical calculations for jet rates and event shapes are needed. In this talk I will discuss theNNLO calculation for these observables.
RADCOR 2009 - 9th International Symposium on Radiative Corrections (Applications of Quantum FieldTheory to Phenomenology)October 25-30 2009Ascona, Switzerland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NLO predictions for event shapes and jet rates in electron-positron annihilation
Stefan Weinzierl
1. The calculation
The process e + e − → a s . Three-jet events are well suited for this task because the leading term in a perturbative calcula-tion of three-jet observables is already proportional to the strong coupling. For a precise extractionof the strong coupling one needs in addition to a precise measurement of three-jet observables inthe experiment a precise prediction for this process from theory. This implies the calculation ofhigher order corrections. The process e + e − → h O i = s (cid:229) n ≥ Z d f n O n ( p , ..., p n , q , q ) (cid:229) helicity | A n | , (1.1)where q and q are the momenta of the initial-state particles and 1 / ( s ) corresponds to the fluxfactor and the average over the spins of the initial state particles. The observable has to be infraredsafe, in particular this implies that in single and double unresolved limits we must have O ( p , ..., p , q , q ) → O ( p ′ , ..., p ′ , q , q ) for single unresolved limits , O ( p , ..., p , q , q ) → O ( p ′ , ..., p ′ , q , q ) for double unresolved limits . (1.2) A n is the amplitude with n final-state partons. At NNLO we need the following perturbative ex-pansions of the amplitudes: | A | = (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) + (cid:16) A ( ) ∗ A ( ) (cid:17) + (cid:16) A ( ) ∗ A ( ) (cid:17) + (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) , | A | = (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) + (cid:16) A ( ) ∗ A ( ) (cid:17) , | A | = (cid:12)(cid:12)(cid:12) A ( ) (cid:12)(cid:12)(cid:12) . (1.3)Here A ( l ) n denotes an amplitude with n final-state partons and l loops. We can rewrite symbolicallythe LO, NLO and NNLO contribution as h O i LO = Z O d s ( ) , h O i NLO = Z O d s ( ) + Z O d s ( ) , h O i NNLO = Z O d s ( ) + Z O d s ( ) + Z O d s ( ) . (1.4)The computation of the NNLO correction for the process e + e − → e + e − → ¯ qqg up to two-loops [3, 4], the amplitudesof the four-parton final states e + e − → ¯ qqgg and e + e − → ¯ qq ¯ q ′ q ′ up to one-loop [5 – 8] and the five-parton final states e + e − → ¯ qqggg and e + e − → ¯ qq ¯ q ′ q ′ g at tree level [9 – 11]. The most complicatedamplitude is of course the two-loop amplitude. For the calculation of the two-loop amplitudespecial integration techniques have been invented [12 – 15]. The analytic result can be expressed interms of multiple polylogarithms, which in turn requires routines for the numerical evaluation ofthese functions [16, 17]. 2 NLO predictions for event shapes and jet rates in electron-positron annihilation
Stefan Weinzierl
2. Subtraction and slicing
Is is well known that the individual pieces in the NLO and in the NNLO contribution ofeq. (1.4) are infrared divergent. To render them finite, a mixture of subtraction and slicing isemployed. The NNLO contribution is written as [18] h O i NNLO = Z (cid:16) O d s ( ) − O ◦ d a single − O ◦ d a ( , ) (cid:17) + Z (cid:16) O d s ( ) + O ◦ d a single − O ◦ d a ( , ) (cid:17) + Z (cid:16) O d s ( ) + O ◦ d a ( , ) + O ◦ d a ( , ) (cid:17) . (2.1) d a single is the NLO subtraction term for 4-parton configurations, d a ( , ) and d a ( , ) are genericNNLO subtraction terms, which can be further decomposed into d a ( , ) = d a double + d a almost + d a so ft − d a iterated , d a ( , ) = d a loop + d a product − d a almost − d a so ft + d a iterated . (2.2)In a hybrid scheme of subtraction and slicing the subtraction terms have to satisfy weaker condi-tions as compared to a strict subtraction scheme. It is just required that(a) the explicit poles in the dimensional regularisation parameter e in the second line of eq. (2.1)cancel after integration over unresolved phase spaces for each point of the resolved phasespace.(b) the phase space singularities in the first and in the second line of eq. (2.1) cancel after azimuthalaveraging has been performed.Point (b) allows the determination of the subtraction terms from spin-averaged matrix elements.The subtraction terms can be found in [19 – 21]. The subtraction term d a ( , ) without d a so ft would approximate all singularities except a soft single unresolved singularity. The subtractionterm d a so ft takes care of this last piece [2, 22]. The azimuthal average is not performed in theMonte Carlo integration. Instead a slicing parameter h is introduced to regulate the phase spacesingularities related to spin-dependent terms. It is important to note that there are no numericallylarge contributions proportional to a power of ln h which cancel between the 5-, 4- or 3-partoncontributions. Each contribution itself is independent of h in the limit h →
3. Monte Carlo integration
The integration over the phase space is performed numerically with Monte Carlo techniques.Efficiency of the Monte Carlo integration is an important issue, especially for the first momentsof the event shape observables. Some of these moments receive sizable contributions from theclose-to-two-jet region. In the 5-parton configuration this corresponds to (almost) three unresolvedpartons. The generation of the phase space is done sequentially, starting from a 2-parton config-uration. In each step an additional particle is inserted [21, 23]. In going from n partons to n + NLO predictions for event shapes and jet rates in electron-positron annihilation
Stefan Weinzierl partons, the n + d f n + = d f n d f unresolved i , j , k (3.1)The indices i , j and k indicate that the new particle j is inserted between the hard radiators i and k .For each channel we require that the product of invariants s i j s jk is the smallest among all consideredchannels. For the unresolved phase space measure we have d f unresolved i , j , k = s i jk p Z dx Z dx p Z d j Q ( − x − x ) (3.2)We are not interested in generating invariants smaller than ( h s ) , these configurations will be re-jected by the slicing procedure. Instead we are interested in generating invariants with values largerthan ( h s ) with a distribution which mimics the one of a typical matrix element. We therefore gener-ate the ( n + ) -parton configuration from the n -parton configuration by using three random numbers u , u , u uniformly distributed in [ , ] and by setting x = h u PS , x = h u PS j = p u . (3.3)The phase space parameter h PS is an adjustable parameter of the order of the slicing parameter h .The invariants are defined as s i j = x s i jk , s jk = x s i jk , s ik = ( − x − x ) s i jk . (3.4)From these invariants and the value of j we can reconstruct the four-momenta of the ( n + ) -partonconfiguration [24]. The additional phase space weight due to the insertion of the ( n + ) -th particleis w = p s i j s jk s i jk ln h PS . (3.5)Note that the phase space weight compensates the typical eikonal factor s i jk / ( s i j s jk ) of a singleemission. As mentioned above, the full phase space is constructed iteratively from these singleemissions.
4. Numerical results
Fig. 1 shows the results for the Durham three jet rate and the thrust distribution at the LEPI centre-of-mass energy p Q = m Z with a s ( m Z ) = . m = Q / m = Q . Note that the theory predictions in these plots are the pure perturbativepredictions. Power corrections or soft gluon resummation effects are not included in these results.In a recent calculation the logarithmic terms of the NNLO coefficient of the thrust distributionhave been calculated based on soft-collinear effective theory [26]: dC t d t = t (cid:2) a ln t + a ln t + a ln t + a ln t + a ln t + a + O ( t ) (cid:3) , t = − T . (4.1)4 NLO predictions for event shapes and jet rates in electron-positron annihilation
Stefan Weinzierl
Aleph dataNNLONLOLODurham three-jet rate y cut j e t r a t e − T − T s d s d ( − T ) Figure 1:
The scale variation of the Durham three jet rate and the thrust distribution at p Q = m Z with a s ( m Z ) = . m = m Z / m = m Z . NNLO1 − T ( − T ) d C ( − T ) d ( − T ) − T ( − T ) d C ( − T ) d ( − T ) h = − h = − h = − NNLO1 − T ( − T ) d C ( − T ) d ( − T ) Figure 2:
A comparison of the NNLO coefficient of the thrust distribution as obtained from the numericalprogram with the logarithmic terms obtained from SCET.
The values of the a j ’s are for N f = a = − . , a = − . , a = − . , a = . , a = − . , a = − . . The logarithmic terms give a good description of the thrust distribution in the close-to-two jetregion. They are not expected to give an accurate result in the hard region. Fig. 2 shows the com-parison of the NNLO coefficient of the thrust distribution as obtained from the numerical programwith eq. (4.1). In the left plot of fig. 2 the x-axis shows ( − T ) on a linear scale. This correspondsto the hard region, where the NNLO result from the numerical program is expected to give the cor-rect answer. The middle plot of fig. 2 shows ( − T ) on a logarithmic scale around ( − T ) ≈ . ( − T ) on a logarithmic scalearound ( − T ) ≈ . h = − , h = − and h = − are plotted. Forsmaller values of h the SCET result is approached.5 NLO predictions for event shapes and jet rates in electron-positron annihilation
Stefan Weinzierl
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