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arXiv:hep-ph/0511302v2 17 Jan 2006
Revisiting parton evolution and the large-x limit
Yu. L. Dokshitzer1∗, G. Marchesini2,1and G. P. Salam1
1LPTHE, Universities of Paris-VI and VII and CNRS, Paris, France
2University of Milano–Bicocca and INFN Sezione di Milano, Milan, Italy
Bicocca–FT–05–31
LPTHE–05–31
hep-ph/0511302 v2
Abstract
This remark is part of an ongoing project to simplify the structure of the multi-loop anoma-
lous dimensions for parton distributions and fragmentation functions. It answers the call for
a “structural explanation” of a “very suggestive” relation found by Moch, Vermaseren and
Vogt in the context of the x→1 behaviour of three-loop DIS anomalous dimensions. It also
highlights further structure that remains to be fully explained.
1 Introduction
This letter stems from a project to better understand the structure of multi-loop anomalous
dimensions both for parton distributions and fragmentation functions [1]. These distributions,
which we shall generically denote as D (DN(Q2) in moment space, D(x,Q2) in x-space) satisfy
a renormalisation group equation
dDN(Q2)
dlnQ2
≡ ∂tD = γ(N,αs(Q2))DN(Q2), (1)
where ∂tis a compact notation for the derivative with respect to t = lnQ2and γ(N,αs) are
elements of an anomalous dimension matrix. The latter have been calculated in terms of an
expansion in the coupling αsup to three (two) loops in the space-like (time-like) cases [2–5].
They become increasingly cumbersome beyond leading order.
Conventionally one defines parton splitting functions, P(x), as the inverse Mellin transform
of the corresponding anomalous dimensions, giving evolution equation in x space in terms of
a direct convolution
∂tD(x,Q2) =
?1
0
dz
zP(z,αs(Q2))D
?x
z,Q2?
, (2)
where D(x,Q2) has the physical support x ≤ 1. We have reason to suspect that there might
exist a reformulation of the evolution equations (2) in which, by generalising the structure on
the right-hand-side, one is able to simplify the splitting functions. This is equivalent to stating
∗On leave of absence: St. Petersburg Nuclear Physics Institute, 188350, Gatchina, Russia
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that the higher-loop structure of the anomalous dimensions in (1) can in part be understood
as inherited from non-linear combinations of lower loops.
In this picture, the new splitting functions would not only be more compact, but they
would also exhibit some important physical properties: beyond first loop they should vanish
at large x, and they should be identical for space-like and time-like evolution, thus restoring
Gribov-Lipatov reciprocity [6] (broken beyond first loop in the standard formulation, see [4]).
A possible reformulation of (2) is1
∂tD(x,Q2) =
?1
0
dz
zP(z,αs(z−1Q2))D
?x
z,zσQ2?
, (3)
where σ = −1 (+1) for the space-like (time-like) case. The choice of zσQ2as the logarithmic
ordering parameter (parton evolution ‘time’) corresponds to ordering parton splittings in the
fluctuation lifetimes of successive virtual parton states. Assuming the new P(x) splitting
functions to be identical in the space-like and time-like cases one obtains that the ‘traditional’
two-loop splitting functions in the two cases, P(2),S(x) and P(2),T(x) differ by
1
2(P(2),T
ns
− P(2),S
ns
) =
?1
0
dz
?1
0
dy δ(x − yz) P(1)
qq(z)lnz ·
?
P(1)
qq(y)
?
+. (4)
This is precisely the relation that was noted by Curci, Furmanski and Petronzio in [4] for
non-singlet quark evolution. In the singlet case (both for quarks and gluons) there are also
interesting patterns, see appendix. Furthermore, if one writes P(z,αs) as a series in the
physical coupling, αPh= αMS+K
of P(x,αs) vanishes as (1−x) for x → 1. This corresponds to the wisdom of Low, Barnett and
Kroll [9] according to which the classical nature of soft radiation reveals itself at the level of
the 1/(1 − x) and constant terms. This classical nature of soft radiation allows one to absorb
all soft singularities into the first loop and to look upon higher-loop splitting functions as due
to true multi-parton (quantum) fluctuations.
Finally, one notes also the z-dependence of the argument of the coupling in (3). Its trace
is visible in the explicit structure of all the diagonal two-loop anomalous dimensions. More-
over, this argument naturally emerges when using dispersive reasoning to carry out a careful
treatment of the appearance of the running coupling in inclusive processes [10].
We are still far from a good understanding of how to simplify the structure of multi-
loop anomalous dimensions, notably because of complications that arise from off-diagonal
transitions (see appendix). Nevertheless the belief that the x → 1 limit is under control
has led us to investigate the implications of eq. (3) for the large-N structure of three loop
anomalous dimensions.
As we shall see this will provide insight concerning a “very suggestive” relation noted by
Moch, Vermaseren and Vogt (MVV) which, in their words, “seems to call for a structural
explanation”: in both the non-singlet quark [2] and diagonal singlet quark-quark and gluon-
gluon three-loop splitting functions [3], they observed that the third-loop coefficients, Ca
4πα2
MS+··· (with K = (67
9−π2
3)CA−10
9nf), the two-loop part
3, in
1Such a reformulation of the notion of parton splitting functions originally proposed in [7] has been carried out in
detail in the context of heavy-quark fragmentation functions, where it was found to greatly improve the perturbative
series [8].
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the large-N expansion of the n-loop anomalous dimensions,2
γaa(N) = −Aa(lnN + γe) + Ba− CaN−1lnN + O?N−1?,
γaa≡
?
γn,aa
?αMS
4π
?n
,Aa≡
?
Aa
n
?αMS
4π
?n
, etc. (5)
are simply related with Aa
two-loop relation, Ca
2and Aa
1)2[4].
1, i.e. Ca
3= 2Aa
1Aa
2, where a = q,g. This supplements the
2= (Aa
2 MVV relation
For the purpose of studying the x → 1 limit we initially approximate P(x,αs) by the product of
the physical coupling, αPh, and the 1-loop splitting function, P(x,αs) ≃ (αPh/4π)P(1)(x) (for
compactness we will write (αPh/4π) ≡ α). To deal with the correlated z and Q2dependences
in the right-hand side of (3), we rewrite it as
∂tD(x,Q2) =
?1
0
dz
zP(1)(z)
?
elnz β(α)∂αα
? ?
eσ lnz ∂tD
?x
z,Q2??
,(6)
where both α and D are now evaluated at scale Q2and β(α) ≡ −dα/dt = −?
The Mellin transform of this equation results in the formal expression,
n=0βnαn+2.
∂tDN= γ1(N + β(α)∂α+ σ∂t) αDN, (7)
where γ1(N) is the Mellin transform of the first order splitting function P(1)(x). We note
that ∂toperates only on DN and not on α. This gives an all-order model for the anomalous
dimension, γ(N) ≡ D−1
N∂tDN.
Expanding (7) results in
γ ≡ γ[α]= αγ1+ ˙ γ1D−1(β∂α+ σ∂t)(αD) + +1
αγ1+ ˙ γ1(β + σαγ) +1
αγ1+ ˙ γ1(β + σαγ) +1
2¨ γ1D−1(β∂α+ σ∂t)2(αD) + ...
2¨ γ1D−1(β∂α+ σ∂t)(βD + σαDγ) + ...
2¨ γ1
?αγ2+ σ(2βγ + αβ∂αγ) + β∂αβ?+ O?α4?,
=
=
(8)
where dots indicate derivatives with respect to N and γ1≡ γ1(N). Solving this iteratively
produces
γ = αγ1+α2˙ γ1(β0+σγ1)+α3?˙ γ1(β1+ σ˙ γ1(β0+ σγ1)) +1
2¨ γ1(γ2
1+ 3σβ0γ1+ 2β2
0)?+O?α4?.
(9)
For the purpose of understanding the MVV relation it suffices to take γ1= −A1lnN + O(1)
and to keep in (9) only the term ∝ ˙ γ1γ1, giving
γ = −αA1lnN + const. + σα2A2
1
lnN
N
+ O?N−1?.(10)
2We define γn(N) =?1
the sign in front Caas compared to eq. (3.10) of ref. [2], which contains a misprint.
0dzzN−1P(n)(z); this has the opposite sign to the convention of MVV; we have also changed
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Recalling that Aa
all-order relation between Caand Aa,
1α ≡ Aa(with Aq
1= 4CF and Ag
1= 4CA), we can then write the following
Ca= −σ(Aa)2,(11a)
or equivalently, in terms of the expansion coefficients Cn,
C1= 0,C2= −σA2
1,C3= −2σA1A2,C4= −σ(A2
2+ 2A1A3), etc. (11b)
where we have suppressed the index a = q,g. For the space-like case (σ = −1) this explains
the MVV observation. For the time-like case we have only the two-loop result [4] to compare
to, and it agrees.
3Pushing our luck
Motivated by the idea that the universality of soft gluon emission holds both in singular and
constant terms in gluon energy [9], one may attempt to trace further terms of the large-N
expansion generated by eq. (3). We definitely expect this push to fail at the level of 1/N2
(possibly modulo logarithms, see below) because this corresponds to ‘quantum’ terms in the
splitting function, which vanish as 1−x. However we would expect to have control over the
1/N term in the anomalous dimension,
γ(N) = −A(ψ(N+1) + γe) + B − C(ψ(N) + γe)N−1+ DN−1+ O?N−2logpN?.
Compared to (5) we have shifted the argument of the logarithm in the A-term, N → N + 1,
added the constant γein the C-term and then replaced logarithms with ψ functions. These
modifications do not affect the first three functions A, B and C but serve to simplify the next
subleading term ∝ 1/N. Additionally they lead to a compact x-space image of (12),
(12)
P(x) =
Ax
(1−x)+
+ B δ(1−x) + C ln(1−x) + D + O((1−x)logp(1 − x)) . (13)
Here, the presence of x/(1 − x) in the first term (as opposed to 1/(1 − x)) is a consequence of
Low’s theorem [9]. In the calculation of the C coefficients we could safely ignore the O(1) piece
of γ1. This is no longer possible when calculating D because, in our non-linear construction,
this constant (unity in Mellin space) is multiplied by ˙ γ1∼ 1/N thus contributing to D. The
extension of (9) to account for B in all orders is obtained by generalising
αγ1→ α(γ1− B1) + B = −A(ln(N+1) + γe) + B + O?N−2?.
The reason why the structure of the Taylor expansion (9) is unmodified modulo this simple
substitution is that B, being a constant, disappears everywhere but un-dotted factors of γi.
This leads to the following all-order expectation for D,
(14)
Da= Aa
?∂tAa
Aa
− σBa
?
,(15)
where we have rewritten the Aa
definition of β in terms of the physical coupling). The MS expansion for Dais then
1β term that comes from eq. (8) as −(∂tAa) (recalling the
D1= 0,D2= −A1(σB1+ β0),D3= −A1(σB2+ β1) − A2(σB1+ 2 · β0),etc.(16)
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Hard luck.
we find agreement for D2(space and time-like, and quark and gluon channels); however the
result for the space-like D3is as follows (for both quarks and gluons)
Examining the full known results for the two and three-loop splitting functions,
D3= A1(B2− β1) + A2(B1− 1 · β0). (17)
There is one mismatch between eqs. (16) and (17), which we have highlighted in boldface. Had
(15) contained (∂tαMS)/αMSinstead of (∂tAa)/Aawe would have obtained agreement with the
full result for D3, however we see no reason why it should be the MS coupling that appears
there instead of the physical coupling (which is equivalent to putting Aa). The remarkable
simplicity of the mismatch calls for a further structural explanation.
Good luck.
leading D term (which we hope can be understood) we have also investigated the coefficient
of terms that vanish for x → 1 but that are logarithmically enhanced there. We have found
agreement using (9) for the α2
(space-like) non-singlet anomalous dimensions, while α3
that remain to be understood. As for the diagonal singlet anomalous dimensions, at three loops
the coefficient of α3
additional unexpected α3
only be explained once the higher-order structure of the off-diagonal splittings is elucidated.
Despite the disagreement in the comparison with the exactly calculated sub-
s(1 − x)ln(1 − x) and the α3
s(1 − x)ln2(1 − x) terms in the
s(1 − x)ln(1 − x) contains structures
s(1 − x)ln2(1 − x) agrees only in its nf-independent parts and there is an
s(1 − x)ln3(1 − x) contribution proportional to nf, whose origin may
The wealth of structure that is present in higher-order splitting functions is suggestive of
underlying simplicity. Possible sources of such simplicity, as proposed here, are the universal
nature of soft gluon radiation and the reformulation of the notion of parton splitting functions
with the aim of preserving universality between space and time-like parton multiplication
(Gribov-Lipatov reciprocity). Whether this picture can be made fully consistent remains to
be seen. We look forward to future work shedding more light on this question.
Acknowledgements
We wish to thank Andreas Vogt for numerous discussions on the subject of splitting functions
as well as comments on the manuscript. The comparisons performed here would not have been
possible without the explicit code provided in the arXiv versions of [2,3] and fortran [11] and
mathematica [12] packages for dealing with harmonic polylogarithms. We additionally thank
Einan Gardi and Alberto Guffanti for a helpful comment, and Werner Vogelsang for bringing
to our attention ref. [14]. Its detailed study of Gribov-Lipatov reciprocity in the second loop
should help elucidate the structure of non-diagonal transitions in eq. (3).
Appendix
We do not know how to generalise (3) to non-diagonal transitions. These are considerably more
divergent at large x [13] than would be expected based on the non-linear relations that we
propose here, going as αn
logarithms may be that in the MS factorisation scheme, for a → b transitions with a ?= b,
sln2n−2(1−x) at large x. We suspect that the origin of these additional
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the splitting functions could pick up residues from ratios of non-cancelling divergent Sudakov
exponents (as well as from singular integrals of these ratios). This belief is not inconsistent with
the MVV observation that in the supersymmetric case most of these logarithmic enhancements
cancel, since in this case the Sudakov exponents become identical for quarks and gluons and
do cancel.
These problems of non-diagonal terms may be responsible for the following fact: at two
loops, the P that appears in (3) for gluon-gluon splitting is universal (identical for space and
time-like cases) only for two of the colour structures, C2
the remaining colour structure, CFnfrelates gluon-gluon and singlet quark-quark splittings.
On the left-hand side one finds the combinations P(2),T
on the right-hand side one has convolutions involving P(1)
lnzP(1)
gq (z) [7].
Aand CAnf. The analogue of (4) for
gg
− P(2),S
qq
gq (x/z) · lnzP(1)
and P(2),T
qg (z) and P(1)
qq
− P(2),S
gg
, while
qg (x/z) ·
References
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