A simplified expression for the one-loop soft-gluon current with massive fermions
aa r X i v : . [ h e p - ph ] A p r A simplified expression for the one-loop soft-gluon current with massivefermions
Micha l Czakon a , Alexander Mitov b a Institut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany b Cavendish Laboratory, University of Cambridge, CB3 0HE Cambridge, UK
Abstract
We present a much simpler analytic expression for the UV un-renormalized one-loop soft-gluon current whichcontrols the infra-red behavior of one-loop QCD amplitudes with massive fermions when an external gluonbecomes soft. The result is given entirely in terms of standard polylogarithms, is of uniform transcendentalityand is well-suited for computer implementation.In ref. [1] the expression for the one-loop correction to the so-called soft-gluon current [2] with massivefermions was derived. This process-independent result controls the singular behavior of one-loop QCDamplitudes with massive fermions when one of the external gluons becomes soft. It is also an essentialingredient to NNLO QCD calculations for processes with massive fermions.For one-loop amplitudes with a single massive fermion leg the analytic expression for that current isquite compact. It was re-derived in ref. [3]. For one-loop amplitudes with two massive fermion legs thesoft-gluon current through order O ( ǫ ), where ǫ = (4 − d ) / O ( ǫ ), however, is quitelarge in size and contains a function which is not written through polylogarithms.In this work we rewrite the result of ref. [1] in an equivalent but much more compact form which containsonly standard polylogarithms of up to weight three. The result is given in a form that is particularly wellsuited for computer implementation.Following ref. [1], the one-loop correction to the UV un-renormalized soft-current is expressed throughthe function g (1) ij = − a bS p i · p j ) µ p i · q )2( p j · q ) ! ǫ " ǫ + X n = − ǫ n (cid:16) R ( n ) ij + iπI ( n ) ij (cid:17) , (1)where the fixed indices i, j label the two massive legs with momenta p i , p j such that p i = m i and p j = m j .We note that the function g (1) ij in eq. (1) corresponds precisely to the function g (1) ij in ref. [1].We give the result for the function g (1) ij for both time-like ( T L ) and space-like ( SL ) kinematics. In thetime-like case both momenta p i and p j are outgoing. This configuration corresponds to Case 3 of ref. [1].The case of both p i and p j incoming is obtained by changing the sign of the imaginary part. The space-likecase corresponds to p i outgoing and p j incoming. Its real part can be expressed through the real part of thetime-like case as follows R ( − SL ] ij = R ( − T L ] ij ,R (0)[ SL ] ij = R (0)[ T L ] ij − ζ v − v ,R (1)[ SL ] ij = R (1)[ T L ] ij + 12 ζ v " d i + d j ) (cid:0) α j v + − α i v − (cid:1) ln (cid:16) α j v + (cid:17) + (cid:0) v + ln( v + ) − ln( v ) (cid:1) , (2) ∗ Preprint numbers: Cavendish-HEP-18/08, TTK-18-13
Preprint submitted to Elsevier April 9, 2018 hile its imaginary part can be written explicitly as I ( − SL ] ij = 1 ,I (0)[ SL ] ij = 2 v ( d i + d j ) (cid:16)(cid:0) α i − v − (cid:1) ln (cid:16) α i v + (cid:17) − (cid:0) d i v − + α j v (cid:1) ln (cid:16) α j v + (cid:17)(cid:17) + ln( v + ) ,I (1)[ SL ] ij = 1( d i + d j ) ((cid:0) − ( d i + d j ) (cid:1)" (cid:16) α i v + (cid:17) ln (cid:16) − α i v + (cid:17) − ln (cid:16) α j v + (cid:17)(cid:16) (cid:16) − α j v + (cid:17) + ln( v + ) (cid:17) − (cid:16) α j v + (cid:17) + 2Li (cid:16) α i v + (cid:17) + 1 v "(cid:0) α i − v − (cid:1) ln (cid:16) α i v + (cid:17) + (cid:0) d i v − + α j v (cid:1) ln (cid:16) α j v + (cid:17) + 2 ln (cid:16) α i v + (cid:17)(cid:16)(cid:0) v + − α i (cid:1) ln( v + ) − d i ln( v ) (cid:17) + d i ln (cid:16) α j v + (cid:17)(cid:16) ln( v + ) − v ) (cid:17) − d i Li ( x ) − ζ d j + 12 ln ( v + ) − ζ (cid:16) − v (cid:17) . (3)The corresponding time-like results read: R ( − T L ] ij = ln( v + ) − v − v (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) ,R (0)[ T L ] ij = 1 v " d i + d j ) (cid:16) ( α i v + − α j v − ) ln (cid:16) α i v + (cid:17) + (cid:0) α j v + − α i v − (cid:1) ln (cid:16) α j v + (cid:17)(cid:17) + (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17)(cid:0) v + ln( v + ) − ln( v ) (cid:1) − Li ( x ) + 12 ln ( v + ) + ζ (cid:16) v − (cid:17) ,R (1)[ T L ] ij = 1( d i + d j ) ((cid:0) − ( d i + d j ) (cid:1)" ln (cid:16) − α i v + (cid:17) ln (cid:16) α i v + (cid:17) + ln (cid:16) − α j v + (cid:17) ln (cid:16) α j v + (cid:17) + 2 (cid:16) ln (cid:16) α i v + (cid:17) Li (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17) Li (cid:16) α j v + (cid:17)(cid:17) − Li ( x ) (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) + 2 (cid:16) Li ( x ) − Li (cid:16) α i v + (cid:17) − Li (cid:16) α j v + (cid:17) + ζ (cid:17) − ζ (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) + 1 v "(cid:16)(cid:0) α j v + − α i v − (cid:1) ln (cid:16) α i v + (cid:17) + (cid:0) α i v + − α j v − (cid:1) ln (cid:16) α j v + (cid:17)(cid:17) ln( v + )+ (cid:0) α i − α j (cid:1)(cid:16) ln (cid:16) α i v + (cid:17) − ln (cid:16) α j v + (cid:17)(cid:17) ln( v ) − (cid:16) d i ln (cid:16) α i v + (cid:17) + d j ln (cid:16) α j v + (cid:17)(cid:17)(cid:0) Li ( x ) − ζ (cid:1) + 1 v (" ln( v + ) (cid:16) v v + ) − ln( v ) (cid:17) − v − ζ ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) − v − (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) + 2Li (1 − x ) + Li ( x ) − " Li ( x ) + ζ (cid:16) v (cid:17) ln( v + )+ 12 ζ ln( v ) ) + 16 ln ( v + ) − (cid:16)
73 + 1 v (cid:17) ζ , (4)2nd I ( − T L ] ij = 2 − v ,I (0)[ T L ] ij = 2 v " d i + d j ) (cid:16)(cid:0) α i − v − (cid:1) ln (cid:16) α i v + (cid:17) + (cid:0) α j − v − (cid:1) ln (cid:16) α j v + (cid:17)(cid:17) + (cid:16)
12 + v (cid:17) ln( v + ) − ln( v ) ,I (1)[ T L ] ij = 1( d i + d j ) ( (cid:0) − ( d i + d j ) (cid:1)(cid:16) ln (cid:16) α i v + (cid:17) ln (cid:16) − α i v + (cid:17) + ln (cid:16) α j v + (cid:17) ln (cid:16) − α j v + (cid:17) + Li (cid:16) α i v + (cid:17) + Li (cid:16) α j v + (cid:17)(cid:17) + (cid:16) ln (cid:16) α i v + (cid:17) + ln (cid:16) α j v + (cid:17)(cid:17) ln( v + ) − (cid:0) Li ( x ) + ζ (cid:1) + 1 v "(cid:16)(cid:0) α i − v − (cid:1) ln (cid:16) α i v + (cid:17) + (cid:0) α j − v − (cid:1) ln (cid:16) α j v + (cid:17)(cid:17) + (cid:16) d i ln (cid:16) α i v + (cid:17) + d j ln (cid:16) α j v + (cid:17)(cid:17)(cid:0) ln( v + ) − v ) (cid:1) − v (cid:16) v − Li ( x ) + 12 (cid:0) ln( v + ) − v ) (cid:1) (cid:17) + ln ( v + ) − ζ (cid:16) − v (cid:17) . (5)The above results have been expressed through the following variables α i ≡ m i p j · q )2( p i · p j )2( p i · q ) , α j ≡ m j p i · q )2( p i · p j )2( p j · q ) , d i ≡ − α i , d j ≡ − α j ,v ≡ p − α i α j , v ± ≡ ± v , x ≡ v − v + , (6)and ζ n is the Riemann zeta function. For ease of comparison with ref. [1] we have retained as much aspossible the notation introduced there. We only note that the variables α i and α j used here differ from theirdefinition in ref. [1] by a factor of 1 /
2, the normalization of the functions R ij and I ij differs from the one inref. [1] by the overall factor of − / x used here corresponds to x from ref. [1].The above result for the UV un-renormalized soft-gluon current is of uniform transcendentality. It alsohas the nice property that the limits m i → m j → m i = 0, the correct result isobtained by setting ln( α i /v + ) = 0 followed by α i = 0. Similarly for m j = 0. In either case v = v + = 1 and v − = x = 0.The UV renormalization of the results presented here is trivial; see ref. [1] for details. Acknowledgments
A.M. thanks the Department of Physics at Princeton University for hospitality during the completion ofthis work. The work of A.M. is supported by the UK STFC grants ST/L002760/1 and ST/K004883/1 andby the European Research Council Consolidator Grant NNLOforLHC2.
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