A fast and accurate method for perturbative resummation of transverse momentumdependent observables
HJE
A fast and accurate method for perturbative resummation of transverse momentumdependent observables
Daekyoung Kang 0 1 2 3
Christopher Lee 0 1 2 3
Varun Vaidya 0 1 2 3
0 Shanghai , 200433 , China
1 and Institute of Modern Physics, Fudan University
2 Los Alamos , NM 87545 , U.S.A
3 Theoretical Division , MS B283 , Los Alamos National Laboratory
We propose a novel strategy for the perturbative resummation of transverse momentumdependent (TMD) observables, using the qT spectra of gauge bosons ( , Higgs) in pp collisions in the regime of low (but perturbative) transverse momentum qT as a speci c example. First we introduce a scheme to choose the factorization scale for virtuality in momentum space instead of in impact parameter space, allowing us to avoid integrating over (or cutting o ) a Landau pole in the inverse Fourier transform of the latter to the former. The factorization scale for rapidity is still chosen as a function of impact parameter b, but in such a way designed to obtain a Gaussian form (in ln b) for the exponentiated rapidity evolution kernel, guaranteeing convergence of the b integral. We then apply this scheme to obtain the qT spectra for DrellYan and Higgs production at NNLL accuracy. In addition, using this scheme we are able to obtain a fast semianalytic formula for the perturbative resummed cross sections in momentum space: analytic in its dependence on all physical variables at each order of logarithmic accuracy, up to a numerical expansion for the pure mathematical Bessel function in the inverse Fourier transform that needs to be performed just once for all observables and kinematics, to any desired accuracy.
E ective Field Theories; Perturbative QCD; Resummation

2.1.1
2.1.2
2.1.3
2.1.4
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
3.2.1
3.2.2
3.3.1
3.3.2
3.3.3
Hard function
Soft function and TMDPDFs
RG evolved cross section
How to choose the scales?
2.2
2.3
Scale choice in impact parameter space
Resummation in momentum space
Leading logs
Nexttoleading logs
NNLL and beyond
3.1
3.2
Representing the Bessel function
Expansion in Hermite polynomials
3.3
Fixed order terms
Truncation and resummed accuracy
Stable power counting in momentum space and the scale L
3
Explicit formula for resummed transverse momentum spectrum
Generating function method to integrate against Hermite polynomials 37
Explicit result of integration
Fixedorder prefactors in resummed expression
The large qT limit
Matching to the xed order cross section
1 Introduction
1.1
1.2
1.3
RG and RRG evolution in impact parameter vs. momentum space
A hybrid set of scale choices for convergence of the b integral
A semianalytic result for the b integral with full analytic dependence on
momentumspace parameters
2
Resummed cross section
Final resummed cross section in momentum space
4
5
6
Remarks on nonperturbative region of qT
4.1
4.2
Remarks on nonperturbative e ects
Remarks on perturbative low qT limit
Comparison with previous formalisms Conclusions
{ i {
D.2 A tailored basis for expanding [1 t]
(ix)
[ t]
D Alternative techniques for obtaining analytic resummed result
E
Mathematical proofs
E.1
A proof of the MellinBarnes identity for the Bessel function
E.2 Integral of complex Gaussian
E.3 Proof of Gaussian integral of Hermite polynomials
E.4 Recursion relation for Hn derivative
E.4.1
E.4.2
First proof
Second proof
Introduction
The transverse momentum spectra of gauge bosons is well trodden territory. They are
important for measurements of, e.g. Higgs production, as well as the dynamics of QCD
in DrellYan (DY) processes. There are calculations available at NNLL+NNLO accuracy
using a variety of resummation schemes both using the framework of soft collinear e
ective theory (SCET) [1{5], e.g. [6{9], and CollinsSoperSterman (CSS) [10] formalisms,
e.g. [11{15], and even N3LL+NNLO [16] (see also calculations in, e.g. [17{23]). Joint
resummation of threshold and tranversemomentum logs is even possible to NNLL and
beyond (e.g. [17, 24{26]).Why, then, do we wish to visit this subject anew?
This has mostly to do with the peculiar structure of the factorized cross section which
makes the resummation of large logarithms an interesting problem. The cross section can
be factorized in terms of a hard function, which lives at a virtuality Q, the invariant mass
of the gauge boson, and soft and the beam functions (or TMDPDFs) which describe the
IR physics and live at the virtuality qT
Q, which is the transverse momentum of the
gauge boson. The soft and collinear emissions are the ones providing the recoil for the
transverse momentum of the gauge boson. This automatically means that these functions
{ 1 {
are convolved with each other in transverse momentum space so that the qT of the gauge
boson is a sum of the qT contribution from each emission:
at s = (P1 + P2)2, with colliding protons of momenta P1;2, and gauge boson invariant mass
Q2 and rapidity y. For the case of the Higgs, we have a Wilson coe cient Ct after
integrating out the top quark. (For DY we just set Ct = 1 in eq. (1.1), and consider explicitly only
soft radiation to ~qT , f1?;2 are the TMDPDFs (or beam functions) accounting for the
contribution of radiation collinear to the incoming protons to ~qT , and they depend on kinematic
= Qe y = x1;2ps. The peculiarity of the factorization is that even though
the TMDPDFs form a part of the IR physics, they depend on the hard scale Q (cf. [27]),
which, as we shall see later, will play an important role in our resummation formalism.
The hard function H encodes virtual corrections to the hard scattering process, computed
by a matching calculation from QCD to SCET. The scale
is the renormalization scale
normally encountered in the MS scheme and plays the role of separating hard modes
(integrated out of SCET) from the soft and collinear modes, by their virtuality. The additional
rapidity renormalization scale , introduced in [28, 29], arises from the need to separate
soft and collinear modes, which share the same virtuality
, in their rapidity ( gure 1).
The cross section itself is independent of these arbitrary virtuality and rapidity boundaries,
but the renormalization group (RG) evolution of factorized functions from their natural
scales, where they have no large logs, to arbitrary ; can be used to resum the large logs
in the cross section.
1.1
RG and RRG evolution in impact parameter vs. momentum space
These functions obey the renormalization group (RG) equations in
where Fi can be Ct2(Mt2; ), H(Q2; ), S(~qT s; ; ) or fi?(~qT i; Q; xi; ; ). The RG equations
in
have a more complicated convolution structure:
d
d
Gi(~qT ; ) = i (~qT )
Gi(q~T ; )
where Gi can be soft functions or TMDPDFs. The symbol
here indicates convolution
de ned as
(q~T )
G(q~T ) =
(q~T
p~T )G(p~T )
Apart from the complicated structure of the RG equations, the anomalous dimensions
themselves are not simple functions but are usually plus distributions [29] which makes it
d
d
Fi = i Fi
Z d2pT
(1.4)
cross sections we consider, the small parameter can be taken to be
qT =Q or
Qb0 in impact
parameter b space, where b0 = be E =2. Right: RG and rapidity RG evolution.
runs between the
hard and soft hyperbolas of virtuality shown in the lefthand
gure, while
runs between the soft
and collinear modes which are separated only by rapidity. The evolution is path independent, one
convenient path is shown here.
even harder to solve these equations directly in momentum space. A typical strategy to
get around this is to Fourier transform to position (i.e. impact parameter) space, de ning
Gb(~b)
the latter de nitions accounting for the fact that all the distributions we encounter will
have azimuthal symmetry in ~qT or ~b. This then gives ordinary multiplicative di erential
equations (instead of convolutions), and a closed form solution to the RG equations can
be easily obtained. Moreover the cross section now takes the simpler structure,
d
where J0 is the n = 0 Bessel function of the rst kind. Note we have changed variables
from qT in eq. (1.1) to qT2 in eq. (1.6). The bspace soft and beam functions Se and fei? now
obey multiplicative rapidity RGEs in ,
d
d
Gei = i Gei ;
whose anomalous dimensions and solutions we shall give below. Only the b integration in
eq. (1.6) stands in the way of a having a simple product factorization of the
momentumspace cross section. Finding a way to carry it out will be one the main focuses of this
paper. For perturbative values of qT , the TMDPDF's can be matched onto the PDF's.
The bspace cross section, de ned as the following product of factors in the integrand of
eq. (1.6):
e
(b; x1; x2; ; ) = H(Q2; )Se(b; ; )fe1?(b; x1; p ; ; )fe2?(b; x2; p+; ; ) ;
(1.8)
{ 3 {
computed in xedorder QCD perturbation theory then contains logs of Qb0 where b0 =
be E =2 (see eq. (A.37)). Schematically, the expansion takes the form
(2 )3e(b) = fi(x1)fi(x2) exp
evolution for the moment (we include them in eq. (A.37) and in all our analysis below).
This takes the typical form of a series of Sudakov logs. The number of coe cients Gnm
that need to be known is determined by the desired order of resummed accuracy. Using the
1= s in the region of large logs needing resummation,
the leading log (LL) series includes the O(1= s) terms m = n + 1, the nexttoleading log
(NLL) series the O(1) terms up to m = n, at NNLL the O( s) terms up to m = n
1, etc.
When we later talk about resummation in momentum space, we will de ne our accuracy by
the corresponding terms in the bspace integrand that we have successfully inverse Fourier
transformed (cf. [30]).
For a TMD cross section, the logs in the full QCD expansion eq. (1.9) are factored into
logs from the hard and soft functions and TMDPDFs of ratios of the arbitrary virtuality
and rapidity factorization scales ; and the physical virtuality and rapidity scales de ning
each mode. Each function contains logs:
Ct2 = Ct2 ln
2
Mt2
2
H = H ln
Q2 ; Se = Se ln b0; ln
; fe? = fe? ln b0; ln
These logs re ect the natural virtuality and rapidity scales where each function \lives" and
where logs in each are minimized. For example, at one loop, the logs in the QCD result
eq. (A.37) split up into individual hard, soft, and collinear logs from eqs. (A.13), (A.24),
and (A.34),
ZH 2
where the individual anomalous dimension coe cients satisfy the constraints ZH + ZS +
2Zf = 0 and
H0 +
C2 + 2 f0 = 0. (For DY, Ct2 = 0.) RG evolution of each factor 
0
t
hard, soft, and collinear  in both virtuality and rapidity space from scales where the
logs are minimized, namely, H
Q; T
Mt and, naively, S;f
1=b0 for the virtuality
scales, while S
S and f
Q for the rapidity scales, to the common scales ; achieve
resummation of the large logs, to an order of accuracy determined by the order to which the
anomalous dimensions and boundary conditions for each function are known and included.
This will be reviewed in further detail in section 2.
This, at least, is the procedure one would follow to resum logs in impact parameter
space. It corresponds, in SCET language, to how to obtain the result of the standard CSS
resummation through traditional [10] or modern techniques [31], as well as recent EFT
p
:
(1.10)
treatments like [32]. Then the resummed bspace cross section is Fourier transformed back
to momentum space via eq. (1.6). The main issue with this procedure is that the strong
coupling
scale
S;f
s( ) in the soft function and TMDPDFs is then evaluated at a bdependent
1=b0, which enters the nonperturbative regime at su ciently large b in the
integral in eq. (1.6). So the integrand must be cut o
before reaching the Landau pole in
s. There are quite a few procedures in the literature to implement precisely such a cuto
by introducing models for nonperturbative physics, see e.g. [33{37].
Motivated by these observations, in this paper we explore the following main questions:
Even though the natural scale for minimizing the logarithms in the soft function and
TMDPDFs is a function of the impact parameter b, can we actually set scales directly
in momentum space, after performing the b integration? (without an arbitrary cuto
of the b integration?)
If that is possible, can we obtain a closedform expression for the cross section which
will be accurate to any resummation order and ultimately save computation time?
d
In section 2 we shall propose a way to answer the rst question, and in section 3 we
shall develop a method to answer the second. To aid the reader in quickly grasping the
main points of our paper, we o er a more detailedthanusual summary of these sections
here, which is somewhat selfcontained and can be used as a substitute for the rest of the
paper upon a rst reading. Readers interested in the details of our arguments can then
delve into the main body of the paper. Except for a brief discussion near the end, we
emphasize we address only the perturbative computation of the cross section in this paper.
1.2
A hybrid set of scale choices for convergence of the b integral
Regarding the rst question, the issue with leaving the ; scales for the soft function and
TMDPDFs un xed before integrating over b in eq. (1.6) is that the integral, while avoiding
the Landau pole from longdistance/smallenergy scales, is then plagued by a spurious
divergence from largeenergy/short distance emissions [29], e.g. at NLL accuracy:
= 0 (2 )2Ct2(Mt2; T )U NLL( H ; T ; L)H(Q2; H )
db bJ0(bqT )Se(b; L; L)
fe1?(b; x1; p ; L; H )fe2?(b; x2; p+; L; H ) exp
Z
0
( L) ln
H
L
ln( Lb0) ;
(1.12)
although H; Se; fe? are truncated to tree level at NLL while Ct =
s. The hard scale
H
is usually set to iQ to implement what is called
2 resummation to improve perturbative
convergence [38, 39]. This integral, as we will see below in eq. (2.33), is still divergent. At
this point, L and
H;L are bindependent and cannot help with regulating the integral.
What we need in eq. (1.12) is a factor that damps away the integrand for both large and
small b. In this paper, we adopt the approach that there are already terms in the physical
cross section itself that can play the role of this damping factor and that we should use them.
Namely, at NLL0 order and beyond, the soft function evaluated at the low scales L; L in
{ 5 {
the integrand of eq. (1.12) contains logs of Lb0 that we can use to regulate the integral,
see eq. (A.24). Since Se(b; L; L) no longer contains large logs (if L; L are chosen near
the natural soft scales), it is typically truncated to xed order (see table 1). However, we
know that the logs themselves still exponentiate, being predicted by the solution eq. (A.21)
to the RG and
RG equations. If we could keep the exponentiated oneloop double log in
Se in eq. (A.24) in the integrand of eq. (1.12), exp
s( L) ZS 0 ln2
8
Lb0 , where ZS =
4,
it would play precisely the role that we desire. Now, as we argue below, if we are going to
keep this term exponentiated, we should also include a piece of the 2loop rapidity evolution
kernel
s2 ln2
Lb0 ln( H = L) given by eqs. (2.26) and (2.27) in the exponent, as it is of
the same form and same power counting, so that the terms we wish to promote to the
exponent of eq. (1.12) at least at NLL order are:
Seexp = exp
s( L) ZS 2
4
they are subleading, we are in fact free to choose to include them (and not other subleading
terms that are formally of the same order). While this is admittedly a bit ad hoc, we take
the view that it is no more arbitrary than any regulator or cuto
we might choose to
introduce to eq. (1.12), and these are terms that actually exist in the expansion of the
physical cross section. We can rephrase this choice of subleading terms in eq. (1.13) to
include in eq. (1.12) as part of our freedom to choose the precise scale
L in eq. (1.12)
(the variation of which anyway probes theoretical uncertainty due to missing subleading
terms). Namely, if one were otherwise to choose L
then shifting that choice to:
L in Se in eq. (1.12), we propose
L !
L = L( Lb0) 1+p ;
p =
1
1
2
2
s( L) 0 ln
H
L
;
(1.14)
which we derive in eqs. (2.36) and (2.38).
This achieves the shifting of the terms in
the exponent of eq. (1.13) that would otherwise be truncated away into the integrand of
eq. (1.12) where they appear explicitly, and can be used to regulate the b integral. This
particular choice of regulator factor in eq. (1.14) is motivated, furthermore, by the fact that
it will allow us actually to evaluate the b integral eq. (1.12) (semi)analytically, as we show
in section 3. Maintaining a Gaussian form for the exponent in ln b inside the b integral will
be crucial to this strategy.
Beyond NLL, we will choose to keep the same shifted scale choice eq. (1.14), but
to ensure that we do not introduce higher powers of logs of
Lb0 than quadratic into
the exponent of the integrand in eq. (1.12), we make one additional modi cation to how
we treat the rapidity evolution kernel. Namely, in the all orders form of the rapidity
evolution kernel:
{ 6 {
given in eq. (2.26), where the rapidity anomalous dimension takes the form eq. (2.27),
we divide the anomalous dimension into a purely \conformal" part containing only the
diagonal pure cusp terms with a single log of
Lb0 and the same for the noncusp part
RS . We divide the rapidity evolution kernel eq. (1.15) into corresponding \conformal"
and \nonconformal" parts:
where V contains pure anomalous dimension coe cients,
V ( L; H ; L) = exp ln
1
H X
L n=0
and V contains all the terms with beta function coe cients, whose expansion is shown
in eq. (2.44). We will keep V
exponentiated as in eq. (1.18), and the shift L !
L in
eq. (1.14) will turn it into a Gaussian in ln Lb0 and thus allow us to carry out the b integral
in eq. (1.12) (updated beyond NLL). However, to keep this Gaussian form of the exponent,
we will then choose to truncate V at xed order. The logs of Lb0 in V
will give integrals
in eq. (1.12) that we can carry out by di erentiating the basic result we obtain in section 3.
Admittedly, this expansion and truncation of V is not part of any usual scheme for
NkLL resummation, but is our addition. In particular, V still contains large logs of H = L
as seen in eq. (2.44).
This means that starting at NNLL order, we will not actually
exponentiate all the logs that appear at this accuracy, as usual log counting schemes in
the exponent require. This is the price we choose to pay for the (semi)analytic solution
we obtain in section 3, which requires a Gaussian exponent in ln b in the b integrand.
This is essentially an implementation of Laplace's method for evaluating the b integral.
As we argue in section 2.3.3, our truncation of V in
xed order is not as bad as failing
to exponentiate large logs involving
(which we do exponentiate) would be. There is
never more than a single large log of H = L appearing in the exponent of the rapidity
evolution. Thus, the series of terms in the xedorder expansion of the exponentiated V
are suppressed at every order by another power of s
.
1 Our expansion of V
should be
viewed as asymptotic expansion, which, indeed, we nd truncating at a
nite order yields
a good numerical approximation to the resummed cross section (within the theoretical
uncertainties otherwise present in the resummed cross section at NkLL accuracy) in the
1In the
evolution kernels, the exponents, e.g. eqs. (A.7a) and (A.8a), themselves contain higher and
higher powers of large logs of H = L, and truncating any part of it to
xed order would not be sensible.
Truncating V in eq. (2.44) to the same order as other corresponding genuinely
xedorder terms at NkLL
accuracy makes more sense. Loosely speaking, we maintain counting of logs in the exponent for most of the
cross section, except for V , in which we revert to older log counting in the xedorder expansion (NkLLE
vs. NkLLF in [27]).
{ 7 {
(1.16)
(1.17)
(1.18)
perturbative region.2 Note, furthermore, that in the conformal limit, V
= 1, and the
exponentiated part V of the rapidity evolution would be exact.
We should point out that, through the shift eq. (1.14), we do introduce b
dependence into our choice of scale
L, so we would not call our resummation scheme entirely
a momentumspace scheme. (See [40, 41] for such proposed methods.) We do, however,
leave the
L scale un xed until after the b integration, and this still allows us to avoid
integrating over a Landau pole in
s( L) in eq. (1.6).
In section 2 we also use our freedom to determine exactly where L
qT should be in
order to improve the convergence of the resummed perturbative series. We argue it should
be set at a value such that other unresummed xedorder logs make a minimal contribution
to the nal momentumspace cross section. For small values of qT , this scale turns out to
be shifted to slightly higher values L
qT +
qT . Without making such a shift we nd
instabilities in the evaluation of the cross section. This is similar in spirit to the shift
L ! qT + qT proposed in [42, 43], though not identical in motivation, implementation, or
interpretation in terms of nonperturbative screening.
1.3
A semianalytic result for the b integral with full analytic dependence on
momentumspace parameters
If we stopped there, our choice of scale eq. (1.14) might be no more than just another in a
long series of proposed schemes to avoid the Landau pole in eq. (1.6), and, in addition, our
division of the rapidity evolution kernel eq. (1.15) into an exponentiated and a xedorder
truncated part in eq. (1.17) would be quite unnecessary and inexplicable. However, what
we nd in section 3 is that all of these scheme choices together yield a form of the bspace
integrand eq. (1.8) that is Gaussian in ln b so that we can integrate it analytically into
a fairly simple form, modulo a numerical approximation for the pure Bessel function in
eq. (1.6). The dependence on all physical parameters and scales such as qT ; H;L; H;L, is
obtained analytically. We now brie y summarize our procedure and results.
With the division eq. (1.15) of the rapidity evolution kernel and the scale choice L in
eq. (1.14), the momentumspace cross section eq. (1.6) can be written in the form, given
in eq. (3.1),
0 Ct2(Mt2; T )H(Q2; H )U ( L; H ; T )Ib(qT ; Q; L; L; H ) ;
(1.19)
d
where we isolated the b integral,
Ib(qT ; Q; L; L; H )
db bJ0(bqT )Fe(b; x1; x2; Q; L; L; H )V ( L; H ; L) ; (1.20)
Fe contains the xedorder terms, including powers of logs of Lb0, contained in the soft
function, TMDPDFs, and the part V in eq. (1.17) of the rapidity evolution kernel that we
choose to truncate at xed order. As we will show in section 3, the exponentiated part of
2We expect that the way in which it breaks down for small qT will yield clues to the behavior of the
nonperturbative contributions to the cross section, which however are not the subject of this paper.
{ 8 {
the rapidity evolution kernel V in eq. (2.43), with the scale choice L in eq. (1.14), can be
written in the form of a pure Gaussian in ln b,
V = Ce A ln2( Lb0 ) ;
where C; A; are functions of the scales L; L; H and the rapidity anomalous dimension,
given explicitly in eq. (3.6). In particular A
[ s( L)]. If we could
gure out how to
integrate this Gaussian against the Bessel function in eq. (3.2), we would be done. Now,
the presence of terms in Fe in eq. (3.2) with nonzero powers of ln Lb0 can be obtained from
the basic result by di erentiation, as we will derive in section 3.3, so we really only need
to gure out how to evaluate the basic integral,
Now, our mathematical achievements in this paper do not reach so far as to
evaluate eq. (1.22) analytically in its precise form. We will, however, develop a procedure to
evaluate it in a closed form, with analytic dependence on qT ; A;
(and thus all scales and
anomalous dimensions), to arbitrary numerical accuracy determined by the goodness of an
approximation we use for the Bessel function. We nd a basis in which to expand the pure
Bessel function, in which just a few terms are su cient to reach a precision better than
needed for NNLL accuracy in the resummed cross section, and which can be systematically
improved as needed. The details of this derivation are in section 3, but we summarize the
key steps here.
The rst step is to use a MellinBarnes representation for the Bessel function,
where the contour lies to the left the poles of the gamma function ( t), so c < 0. The
choice c =
1 turns out to be well behaved, and useful as it is closely related to the
xedorder limit of eq. (1.20) (see section 3.3.2). This trades the b integral in eq. (1.22) for the
t integral, and we obtain
Ib0 =
2 e AL2 Z 1
q
T
2 p
A
1
dx ( c
ix)2 sin[ (c + ix)]e 21 [x i(c t0)]2 ;
(1.24)
(1.21)
(1.22)
(1.23)
(1.25)
Ib0 =
0
db bJ0(bqT ) e A ln2( b) ;
f (t)
( c
ix)2 ;
{ 9 {
Gaussian with a width
the rest of the integrand, in particular
p
where we parametrized the contour in eq. (1.23) as t = c + ix, and where t0 =
1 + AL,
where L = ln(2 =qT ). We also used the re ection formula ( t) (1 + t) =
csc( t).
It may appear that we are no farther along than when we started with eq. (1.22)we
still have to do the x integral. However, we now observe that thanks to the form of the
A, which vanishes in the limit s ! 0, we only need to know
in a fairly small region of x. In fact we shall not need it out to more than jxj
1:5 for any
of our applications. Thus if we can
nd a good basis in which to expand f where every
term gives an analytically evaluable integral in eq. (1.24), we shall be in good shape.
Now, this would not have been a good strategy in eq. (1.20) for the Bessel function
itself, as it is highly oscillatory out to fairly large b, and the Gaussian does not damp the
integrand away quickly  its width only grows as s (i.e. A) goes to zero. However, inside
eq. (1.24), we nd an expansion of f (eq. (1.25)) in terms of Hermite polynomials Hn to
work very well:
(1
ix)2 = e a0x2 X1 c2nH2n( x) +
i E
e b0x2 X1 c2n+1H2n+1( x) ;
n=0
n=0
(1.26)
1 and factor out Gaussians with widths set by a0; b0 which
closely (but not exactly) resemble the real and imaginary parts of (1
ix)2 itself, near
x = 0. Their departures from an exact Gaussian are accounted for by the remaining series
of Hermite polynomials. It would be natural to choose the scaling factors ;
for the
Hermite polynomials to be
2 = a0 and
2 = b0, but instead we leave them free, to be
determined empirically to optimize fast convergence of the series.
We
nd we can get
su cient numerical accuracy acceptable for NNLL accuracy in the nal cross section with
just a few (3 or 4) terms in each series, real and imaginary. The coe cients cn in eq. (1.26)
still have to be determined by the numerical integrals eq. (3.25), which unfortunately
prevents us from having a fully analytic result for the momentumspace cross section.
However, the series eq. (1.26) with these numerical coe cients depends only on properties
of the pure mathematical function
(1
ix)2 itself  not on any physical parameters.
The dependence on these we keep analytically. All that is left is to evaluate analytically
the integral of each Hermite polynomial against the Gaussian in eq. (1.24), leading to the
result we derive in eq. (3.37),
2
q
1
2
T n=0
where each term Hn is de ned by the integral,
Hn( ; a0) = p
1
A
e A(L i =2)2 Z 1
1
dx Hn( x)e a0x2 A1 (x+z0)2 ;
each of which has the closed form result,
Hn( ; a0) =
( 1)nn! e
A(L i =2)2 bn=2c 1
1+a0A
X
(1+a0A)n+ 12
m=0
m! (n 2m)!
1
n[A( 2 a0) 1](1+a0A)om
(2 z0)n 2m;
(1.27)
(1.28)
(1.29)
the rst several of which are written out explicitly in eq. (E.14). In eqs. (1.28) and (1.29),
z0 = A( =2 + iL), in terms of which the integral eq. (1.24) can be written, the shifted
exponents arising from absorbing the sine function in eq. (1.24).
The results eq. (1.29) for the integrals eq. (1.28) are the primary mathematical result
of our paper. The nal and primary physics result of our paper, eq. (3.77), the resummed
cross section in momentum space, is given in terms of the analytic result eq. (1.27) for
Ib0 above.
While a rst glance at these formulas may not be particuarly illuminating, we would
like to emphasize that the results eq. (1.29) of the integrals eq. (1.28) in terms of which
the
nal result is written contain within them explicit dependence on all the physical
parameters such as qT and the scales
L;H ; L;H that one would want to vary not only
to evaluate the cross section but estimate its theoretical uncertainties. This is made very
fast to compute by our explicit analytic formula, modulo only the numerically computed
coe cients in eq. (1.26), but that can be done once and for all, for any TMD observable
or kinematics.
It is important to emphasize that the result eq. (3.77) we give for the resummed
momentumspace cross section represents, then, a triple expansion:
Perturbative expansion: usual expansions in
s of matching coe cients and
resummed exponents in eq. (1.19), counting
s ln( H = L)
1 or s ln( H = L)
1,
and xedorder tails (not shown in eq. (1.19)).
V expansion: the additional truncation of the V part of the rapidity evolution kernel
in eqs. (1.17) and (2.54) to a xed order in
s, according to table 1, makes possible
the integration of a rapidity exponential eq. (1.21) Gaussian in ln b, and behaves as an
asymptotic expansion. This expansion becomes exact in the conformal limit of QCD.
Hermite expansion: the integral of the Gaussian in eq. (1.21) against the Bessel
function in eq. (1.22) is performed in terms of analytic integrals, by expanding J0
through the representation eq. (1.23) and the series of Gaussianweighted Hermite
polynomials eq. (1.26), truncated to a
nite number of terms, as needed to achieve
a numerical accuracy in the cross section better than the perturbative uncertainty
already present.
These are the expansions we nd necessary to obtain the analytical (up to the numerical
Hermite coe cients) result for the cross section in eq. (3.77). Each expansion is
systematically and straightforwardly improvable. The last two expansions could be avoided if one
is satis ed with a fully numerical evaluation of the b integral in eq. (1.20). We
nd the
expansions worthwhile as they yield the faster and similarly accurate formula eq. (3.77).3
In the rest of the paper, we will do our best to make clear which expansion(s) are being
used at each stage.
The remainder of our paper is organized as follows. Before concluding section 3 we
match our resummed result onto xedorder perturbation theory in section 3.3.3 and obtain
and illustrate our results resummed to NNLL accuracy and matched to O( s) xed order.
In section 4 we o er some comments about expected nonperturbative corrections to our
perturbative predictions, and in section 5 we survey other methods to resum TMD cross
sections in the literature as compared to ours. We conclude in section 6. In the appendices
3In our calculations, we found a factor of 5 improvement in speed with our formula for the qT distribution
vs. numerically integrating eq. (1.20) at every qT .
we o er an array of technical results we need to evaluate the integrals and cross sections
in the rest of the paper, as well as some alternatives to particular choices of schemes or
methods we made in the main body of the paper.
2
Resummed cross section
In this section, we rst review RG and rapidity RG methods to resum logs of separated
hard and soft/collinear virtuality scales and collinear and soft rapidity scales in TMD cross
sections. We review a standard procedure to set scales in impact parameter space, and
then inverse Fourier transforming to momentum space. Then we propose a hybrid scale
setting scheme where the soft rapidity scale is chosen to depend on b, but the virtuality
scales are chosen only after we transform back to momentum space, allowing evaluation of
the b integral without encountering a Landau pole. We also organize the rapidity evolution
kernel in a way that anticipates making use of it to perform the b integral semianalytically
in section 3. We also address the choice of the soft virtuality scale itself in momentum
space to ensure stable power counting of logs.
2.1
RGE and
RGE solutions
We de ned the bspace cross section in eq. (1.8). The cross section is independent of the
virtuality and rapidity factorization scales ; , but each factor H; Se; fe? does depend on
them, and contains logs of ratios of the scales ; to their \natural" virtuality or rapidity
scales
H , ( L; L) and ( L; H ), at which no large logarithms exist. Thus we would like to
evaluate each factor at these separate scales, and then use RG and
RG evolution to take them to the common scales ( ; ) at which the cross section is evaluated. The solutions to these evolution equations are in a form where the large logs of ratios of separated scales are resummed or exponentiated.
2.1.1
Hard function
HJEP04(218)9
d
d
1
X
i=0
The hard function H = jCj2 depends only on the virtuality scale , and obeys the RGE,
d
d
C(Q2; ) = C ( )C(Q2; ) )
H(Q2; ) =
H ( )H(Q2; ) ;
where the anomalous dimension takes the form,
C ( ) =
ZH
2
cusp[ s( )] ln
+ C [ s( )] ;
Q
H = C + C ;
where cusp is known as the cusp anomalous dimension, the proportionality constant ZH =
4, and
C [ s] is the noncusp part of the anomalous dimension. The cusp anomalous
dimension can be written as an expansion in the strong coupling s( ).
cusp[ s( )] =
Explicit expressions for these kernels up to NNLL accuracy are given in appendix A.1.
For the case of the Higgs production, we have another Wilson coe cient (Ct2) obtained
from integrating out the top quark. So in addition to the hard function, we also have a
running for this coe cient.
The anomalous dimension takes the general form
where i is a number. The RGE has the solution
ZH
2
( H ; ) ln
Q
H + K C
( H ; ) ;
HJEP04(218)9
Ct2(Mt2; ) = Ct2(Mt2; T )UCt2 ( T ; )
The RGE eq. (2.1) has the solution
C(Q2; ) = C(Q2; H )UC ( H ; ) ) H(Q2; ) = H(Q2; H )UH ( H ; ) ;
where the evolution kernel is
UC ( H ; ) = exp
= exp
Z
and UH = jUC j2. The pieces K ; ; K of the evolution kernel are given by:
where the evolution kernel is
where
UCt2 ( T ; ) = exp
( Z
T
Explicit expressions for these kernels up to NNLL accuracy are given in appendix A.1.
(2.4)
(2.5)
(2.6a)
(2.6b)
(2.6c)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
The soft function in b space obeys the  and RGEs,
d
d Se(b; ; ) = S( ; )Se(b; ; ) ;
d Se(b; ; ) = S( ; )Se(b; ; )
d
while the TMDPDFs/beam functions obey
The
anomalous dimensions take the form:
Solving this equation in , we obtain
i ( ; ) = Zi
Z
1=b0
where the boundary condition of the evolution at 1=b0 determines the noncusp part Ri[ s]
of the
anomalous dimension. The independence of the cross section eq. (1.8) on requires,
again, ZS =
2Zf , and RS =
2 Rf .
The solutions of the and
RGEs for Se and fe? are:
d ln 0 cusp[ s( 0)] + Ri[ s(1=b0)] = Zi (1=b0; ) + Ri[ s(1=b0)] ;
of the
the values
where
S
p
where
and
independence of the cross section require ZH = 2Zf =
ZS = 4, and
[ s]. In f we recall the large rapidity scales are given by p =
Qe y = x1;2 s for the two colliding hard partons. Note p+p
= Q2. As for the form
anomalous dimensions, at oneloop xed order in perturbation theory, they take
the scale at which rapidity logs are minimized in b space. Beyond O( s), the form of the
anomalous dimensions can be deduced from the consistency relation:
d
d
d fei?(b; xi; p ; ; ) = f ( ; )fei?(b; xi; p ; ; ) ;
d fei?(b; xi; p ; ; ) = f ( ; )fei?(b; xi; p ; ; ) :
S( ; ) =
ZS cusp[ s( )] ln
f ( ; ) = Zf cusp[ s( )] ln
+ S
Se(b; ; ) = Se(b; L; L)US( L; ; )VS( L; ; L)
= Se(b; L; L)VS( L; ; )US( L; ; L)
fei?(b; xi; p ; ; ) = fei?(b; xi; p ; L; H )Uf ( L; ; )Vf ( H ; ; L)
= fei?(b; xi; p ; L; H )Vf ( H ; ; )Uf ( L; ; H ) ;
(2.12)
(2.13)
(2.14a)
(2.14b)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19a)
(2.19b)
where each pair of equalities accounts for two, equivalent paths for RG evolution in the
twodimensional ; space (see gure 1). The evolution kernels US;f in the
direction are:
US( L; ; ) = exp
ZSK ( L; )
ZS ( L; ) ln L + K S ( L; )
Uf ( L; ; ) = exp Zf ( L; ) ln
+ K f ( L; ) :
p
Note that the
anomalous dimension for fe? in eq. (2.14b) does not have a log of
its cusp anomalous dimension term, so no K
term appears in its evolution kernel Uf
in eq. (2.20b). Meanwhile, the
evolution kernels VS;f are given by integrals over
in
of
Utot( i; i; ; ) = exp 4K ( L; H ) 4 ( L; H ) ln
K H
( L; H )
K C2 ( L; T )
+
h
4 (1=b0; L) + R S s(1=b0) ln H
;
i
in which we observe that the explicit dependence on the arbitrary scales and has exactly
canceled out, leaving only the dependence on the natural scales H;T;L;b and L;H where
L
t
t
(2.20a)
(2.20b)
(2.21a)
(2.21b)
i
:
(2.23)
H
(2.24)
VS( L; ; ) = exp
ZS (1=b0; ) + RS[ s(1=b0)] ln
Vf ( H ; ; ) = exp
Zf (1=b0; ) + Rf [ s(1=b0)] ln
:
L
H
2.1.3
RG evolved cross section
We can now put these pieces together to express the cross section eq. (1.8) in terms of the
hard, soft, and beam functions evolved from their natural scales where logs in each are
minimized, and thus logs in the whole cross section are resummed:
where
Utot( i; i; ; ) UCt2( T ; )UH ( H ; )US( L; ; )VS( L; ; L) Uf2( L; ; )Vf2( H ; ; L)
= exp
ZH K ( H ; ) ZSK ( L; ) ZH
h
+ ( L; )
h
ZS ln L +2Zf ln Q
+ ZS (1=b0; L)+ RS[ s(1=b0)] ln
i
i
Using the relations ZH =
obtain the simpler expression,
ZS = 2Zf = 4, H + Ct2 =
S
2 f , and RS =
2 Rf , we
( H ; ) ln QH +K H
( H ; )+K C2 ( T ; )
+K S ( L; )+2K f ( L; )
L
+2hZf (1=b0; L)+ Rf [ s(1=b0)]i ln
the Higgs.
the hard, soft, and beam functions live. Note UCt2 ; K C2 are present only in the case of
t
In eq. (2.24), we envision that the rapidity evolution takes place at (or around) the
scale 1=b0 (see gure 1). Then we can actually just expand the evolution factor
(1=b0; L)
and the rapidity anomalous dimension
RS in a
xedorder expansion in
s( L), to the
order required for NkLL accuracy. This is in fact what we will do below. Then it becomes
useful to split up Utot in eq. (2.24) into two factors,
Utot( i; i; ; ) = U ( L; H )V ( L; H ; L) ;
(2.25)
HJEP04(218)9
(2.26)
(2.27)
where
U ( L; H ; T ) = exp 4K ( L; H ) 4 ( L; H ) ln
(2.10), and (2.21a). For brevity in the rest of the paper we will just use U; V in eq. (2.26).
Inside V in eq. (2.26), we use the xedorder expansion of S( L) given in eq. (2.18)
using the expansions eq. (A.8b) for
and eq. (A.22) for RS:
Q
L
K H
( L; H ) K C2
( L; T )
t
S
In practice we truncate this expansion at the appropriate order of logarithmic accuracy. We
will always pick L in such a way that none of these generate large logs (either L
1=b0
in b space, or in momentum space in such a way that they remain small after inverse
transformation  see section 2.3.5), except the factor of ln( H = L) in eq. (2.26). This is
an observation that will become key below, when we split
into two separate parts in
section 2.3.3.
2.1.4
How to choose the scales?
To evaluate the cross section eq. (2.22) (and its inverse Fourier transform back to
momentum space eq. (1.6)) explicitly, we need to make explicit choices for the scales
H;L
and
L;H between which to run in eq. (2.24). Choosing these near the scales at which
the logs in each individual function are minimized in principle achieves resummation of
all large logarithms. However, these natural choices are di erent in impact parameter and
momentum space.
There are various possible ways in which this resummation can be handled. In this
paper, we envision, in eqs. (2.25) and (2.26) and
gure 1, running the hard function in
to the natural low scale of the soft and collinear functions, and the soft function in
the natural rapidity scale of the TMDPDFs. The high scales
H and
H for the running
of the hard and soft functions are unambiguously best chosen near the invariant mass
of the gauge boson. The choices of the low scales L and L are under debate since we can
to
Q
choose those scales either in b space or in momentum space.
The
scales, like in a usual EFT, are a measure of virtuality of the modes that
contribute to that function. For the hard function, this virtuality scale, not surprisingly,
is the hard scale Q, which also happens to be the scale choice for which the logarithms
in the hard function are minimized. The virtuality for soft and beam functions is of the
order of the transverse momentum that the function contributes to the total transverse
momentum. This can be seen in momentum space where the product of these functions
in impact parameter space turns into a convolution over transverse momentum, eq. (1.1).
Since the total transverse momentum is a sum over the transverse momenta contributed by
each function, for a given total qT , the contribution of any one of these functions traverses
a range of scales. While this situation is not unique to this observable, what is di erent
is the dependence of the TMDPDF on the hard scale Q. As we will see, due to this Q
dependence, the conjugate natural scale to b0 in the resummed result is no longer qT but
is shifted away from qT towards Q.
However, the nal aim of any resummation is to have a well behaved perturbative series.
Whenever the xed order logs become too large, the expansion in s does not converge, and
it becomes necessary to reorganize the series in terms of resummed exponents. A successful
resummation is then one in which the xed order terms that are left behind form a rapidly
converging series in
s. Since the large logarithms are, in fact, the terms that spoil the
convergence of the xed order perturbative series, the general strategy is then to minimize
the e ect of these logarithms in the residual xed order series.
Keeping these issues in mind, we explore two possible sets of scale choices for L; L
for resummation: the standard choices in impact parameter space in section 2.2, and a new
proposed set of choices in section 2.3 allowing evaluation of the resummed cross section in
momentum space.
2.2
Scale choice in impact parameter space
To choose scales for resummation, we need some idea about the natural scales at which
each of the three functions (hard, soft and TMDPDF) live. This is easily seen by looking
at their behavior up to one loop. From the results given in appendix A, we nd that each
of these functions are function of the logs,
;
fe? = fe? ln b0; ln
;
(2.28)
p
given in impact parameter space for Se; fe?. In this space, it is perfectly evident from the
xedorder calculation that the natural scale which minimizes all the logs in
for the soft
and beam functions is
=
L
1=b0. Since the nal cross section at a given qT involves
an integral over a range of b, the scale choice is in fact spread over a range of scales. This
is to be expected from the earlier discussion of their being no unique physical scale for the
soft and beam functions. The natural scales for the various functions then are
=
H
for the hard function, ( ; ) = ( L; L) for the soft function and ( ; ) = ( L; H ) for the
beam functions, where L; L
1=b0 and H ; H
Q (recall p+p
= Q2).
All the logs can then be resummed by running the hard function from the scale Q to
1=b0 and the soft function in
from Q to 1=b0. This will produce the result (for the central
values, not counting scale variations) of the CSS formalism. Therefore, this scheme resums
logarithms of the form ln(Qb0). The power counting adopted for this resummation then is
straightforward since there is only one type of log. It is usually chosen as s ln(Qb0)
Leading log (LL) accuracy then resums sn lnn+1(Qb0), with NLL and NNLL down by one
and two powers of the logarithm respectively.
Since the lower scales
L are chosen in b space, the cross section involves an inverse
Fourier transform over arbitrarily large values of b, so eventually we hit the nonperturbative
scale which manifests itself in the form of the Landau pole:
s(1=b0). This corresponds
to the fact that the beam and soft functions can contribute arbitrarily small values of
transverse momentum even when the total transverse momentum is perturbative. This is
usually handled by putting a sharp or smooth cuto in b space which provides a way to
model nonperturbative physics [33{37]. The impact of these nonperturbative e ects will
be discussed in section 4.
The obvious advantage of this scheme is that the power counting is unambiguous and
we can guarantee that with the central values of scale choices in b space, all the logs in
the residual xed order series are set exactly to zero. As far as the choice of central values
is concerned, the terms that are resummed are exactly equal to the CSS resummation
formalism [10]. However, due to the introduction of the new rapidity renormalization scale
, there is much better control over which terms can be included in the exponent and
which terms remain in the xed order [28, 29]. This directly translates into a much better
estimates of error due to missing higher order terms.
Another advantage of having control over what exactly goes in the exponent is when
we match the resummed cross section to the
xed order cross section at large qT . To
maintain accuracy over the full (perturbative) range of qT (Q
qT
QCD), we need to
turn o resummation at the value qT where the nonsingular contribution is the same order
as that of the singular one. Due to the two independent scales ;
available, this can be
done very easily by using pro les in these scales to smoothly turn o the resummation and
simultaneously match onto the full (including nonsingular pieces) xed order crosssection.
This technique was implemented for the Higgs transverse spectrum in [32] to obtain the
cross section to NNLL+NNLO accuracy. In this paper, for the purposes of comparison
with other resummation schemes, we present the results for the cross section at NNLL
accuracy for both the Higgs and DY using this scheme ( gure 2).
After having decided on the central values, we next need to estimate the size of
higherorder perturbative corrections we have missed by using scale variations. This is
accomplished by varying the two renormalization scales
and
independently as detailed in [32].
Since we resummed and chose scales in b space, our nal result involves an inverse Fourier
HJEP04(218)9
their explicit expressions to NNLL accuracy are (suppressing the superscript q on ),
where
these are:
s( 0)
2
0
1
r
ln r +
1 r2 +ln r +
1
0
s( )= s( 0) and the running coupling is given to 3loop order by the expression
s( )
=
X
s( 0)
+
4
1 ln X +
s( 0)
16 2
s( 0)
2
0 ln
0
:
The expressions eq. (A.4) resum to all orders in s terms in the xedorder expansion
associated with expansion of the running coupling eq. (A.5) in xed orders in s. Sometimes
it is useful, though, to look at the explicit
xedorder expansions to see which terms
+
+
s( 0)
4
+
1
2 0 ln2
+
exhibiting the towers of leading logs, nexttoleading logs, and nexttonextto leading
logs from left to right. The LL (O(1= s)), NLL (O(1)), and NNLL (O( s)) terms in
eq. (A.4) automatically sum each in nite tower of logs through functions of the ratios r.
Alternatively, the expansion around s( ) is given by:
+
+
s( )
4
+
1
2 0 ln2
3 0 1 +
2
+ 0 ln
s( 0)
4
s( )
4
+ 0 ln
0
is always multiplied by another large log (e.g. ln(Q= L) or ln( H = L) in eq. (2.24)),
the rst tower is again part of the LL series, the second the NLL series, etc. Alternatively,
expanded in s( ),
+ ( 0 1 + 2 1 0) ln2
0
+ 2 ln
0
+
:
Finally, K is given by the same expansions as eqs. (A.8a) and (A.8b) with i ! i. But
K is not multiplied by any additional large logs, so in this case the rst column of terms
in eqs. (A.8a) and (A.8b) for K
begins at NLL, the second NNLL, etc.
In our numerical analysis we use the full NNLL expressions for K ; ;
in eq. (A.4),
but to be consistent with the value of s( ) used in the NLO PDFs we only use the
twoloop truncation of eq. (A.5), dropping the 2 and 12 terms, to obtain numerical values for
s( ). Up to three loops, the coe cients of the beta function [52, 53] and cusp anomalous
dimension [54, 55] in MS are
+
+
+
where Ci is CF and CA for the quark and gluon, respectively.
0
0
CA2 +
1267 TF2 nf2 ;
CF +
CA
4TF2 nf2 ;
418
27
+
3 TF nf ;
20
3
0 = 4Ci ;
1 = 4Ci
2 = 4Ci
CF2
2
3
55
3
CA + 4CF
TF nf ;
67
9
245
6
+
CA
+
The hard function H is given as the square of the SCET matching coe cient C arising from
matching QCD and SCET amplitudes, H = jCj2. The form of the xedorder expansion
of C can be deduced from eq. (2.4) and the xedorder expansion of UC in eq. (2.5), using
the evolution kernels expanded in powers of s in eqs. (A.7a) and (A.8a). All logs in the
hard coe cient are zero at H = iQ [38, 39], leaving
C(Q2; H = iQ) = 1 + X
1
n=1
s(iQ) n
2
cn :
Then using eqs. (2.4) and (A.10) to express the hard coe cient at an arbitrary scale ,
HJEP04(218)9
C(Q2; ) = C(Q2; iQ)UC (iQ; )
At an arbitrary scale , C then has the expansion,
= C(Q2; iQ) exp
ZH K (iQ; ) + K C (iQ; ) ;
2
1
n=1
C(Q2; ) = 1 + X
The anomalous dimension for the hard function can be written as
In the Higgs production we have the matching coe cient after integrating out the top
quark and it can be written as
Ct2(Mt2; t = Mt) = s(Mt)2 1 + X
"
1
n=1
2
s(Mt) n #
c
tn ;
C(1) =
C(2) =
2
0
ZH 16
0 ln2
Q2 + H ln
4
1
48 ZH 0 0 ln3
2
Q2 +
2
Q2 +
c
1
2
where the oneloop constant for DY is given by
and for Higgs production by
The 2loop constant terms can be found in [56{59].
The hard function is then given by
c1 =
16 +
CF ;
c1 =
2
3
2
3
CA :
H(Q2; ) = C( Q2; )
2
H (Q2; ) = C ( Q2; ) + c.c.
ZH 16
1 + H0 0 ln2
8
2
Q2 +
1
4
H +
c
1
2 0 ln
Q2 +
2
(A.10)
(A.11)
(A.12)
(A.13a)
c
2
2
;
(A.14a)
(A.14b)
(A.15)
(A.16)
(A.17)
where ct1 = 5CA 3CF and the 2loop constant is given in [58, 59]. Its anomalous dimension
and RGE take the same form as eqs. (2.1) and (2.5) except for the cusp part being zero.
The MS noncusp anomalous dimension H = 2 C for the DY hard function H can
be obtained [56, 57] from the IR divergences of the onshell massless quark form factor
C(q2; ) which are known to three loops [60]. Here we write results up to 2 loops
H 0 = 2 C 0 = 12CF ;
H 1 = 2 C 1 = 2CF
82
9
52 3 CA + 3 4 2 +48 3 CF +
(A.18)
The Higgs production can be obtained from virtual corrections with the virtual
topquark loop providing the e ective ggH vertex.
The matching coe cient is known at
NLO [61, 62] and at NNLO [58, 59]. The noncusp anomalous dimension of the top
matching coe cient Ct2 and gg SCET hard function H are given by
t 0 =
t 1 =
4 0
+
The xedorder expansion of the soft function Se can be deduced from its (R)RG solution
eq. (2.19a) and the xed order expansions of US and VS in eqs. (2.20a) and (2.21a). At the
scales L = L = 1=b0, all the logs in Se vanish,
2 Se(b; L = 1=b0; L = 1=b0) = 1 + X
1
n=1
4
s(1=b0) ncn :
S
e
Evolving it to arbitrary scales ; by eq. (2.19a), we obtain
RS [ s (1=b0)] = X
s (1=b0) n
4
=
1
n=1
+
4
s ( ) R0S +
s ( ) 3
4
n
RS
s ( ) 2
4
Se(b; ; ) = Se(b; 1=b0; 1=b0)VS(1=b0; ; 1=b0)US(1=b0; ; )
= Se(b; 1=b0; 1=b0) exp
ZSK (1=b0; ) + ZS (1=b0; ) ln b0
+ K S (1=b0; ) + RS[ s(1=b0)] ln b0 :
We expand the exponent using eqs. (A.7a) and (A.8a), the constants in front using
eq. (A.20), and the rapidity anomalous dimension in powers of s( ), using
(A.19)
(A.20)
(A.21)
(A.22)
2 R0S 0 ln b0 + R1S
In principle all the higherorder function terms could contribute at the same order as
lowerorder ones, but we will always evaluate
close to 1=b0 in a xedorder soft or beam
function or RRG evolution factor, so the higherorder logs are not large, and those terms
are genuinely suppressed by powers of s relative to lowerorder terms.
Putting together these pieces, we obtain the expansion of the soft function,
2 Se (b; ; ) = 1 + X
1
n=1
;
+
S1 + 2c1Se 0 ln b0 + R1S ln b0 + c2 :
S
e
2
3
Note we used that the oneloop noncusp anomalous dimensions S0 =
0
RS = 0 vanish.
These expansions agree with those given in [21]. The constant terms are given by
+ Ci CA
208
27
The noncusp  and anomalous dimensions for the TMDPDF and soft function for
gluon at 1 loop were calculated in [29]. Their anomalous dimensions are f 0 = 2 0 and
S 0 = 0. By replacing CA by CF we obtain anomalous dimension of the soft function for
the quark from the one for gluon and by using the consistency relation 2 f =
H
S
we can nd anomalous dimension for the TMDPDF although their one loop results for the
quark are separately known: f 0 = 6CF and S 0 = 0. At two loops we only need the total
anomalous dimension
S1 + 2 f1 =
H1 which was given in eq. (A.18).
The anomalous dimensions up to 2 loops are given by (e.g. [17{19, 21, 22])
R S 0 =
R S 1 =
2 R f 0 = 0 ;
2 R f 1 =
2Ci
64
9
where Ci = CF ; CA for the quark and for the gluon, respectively.
A.4
TMDPDF
The TMDPDF matches onto ordinary PDFs via the matching relation
(A.23)
(A.24a)
(A.24b)
HJEP04(218)9
2 f~i?(b; x; ; ) =
j
x z Iij (b; z; ; ) fj
z
:
(A.27)
evolution equation eq. (2.19b) for f~? and from the DGLAP evolution of the PDFs:
The xedorder expansion of the matching coe cients I can be deduced from the (R)RG
where the splitting function Pij have the perturbative expansion:
d
d
fi (x; ) =
x z
Pij (z; ) fj
z
;
;
Pij (z; ) =
1
X
n=0
L = 1=b0 and
H = p , all logs in the TMDPDF vanish, and it has
2 fi?(b; x; L = 1=b0; H = p ) = X Z 1 dz
x z Iij (b; z; L = 1=b0; H = p )fj z
x
; L = 1=b0 ;
At the scales
the form
where
(A.28)
(A.29)
(A.30)
(A.31)
(A.32)
p
;
(A.33)
HJEP04(218)9
Iij (b; z; L = 1=b0; H = p ) = ij (1
z) +
4
s(1=b0) Ii(j1)(z) +
4
s(1=b0) 2Ii(j2)(z) +
contains just nite matching functions. Using eqs. (2.20b) and (2.21b) to evolve fi? to
arbitrary scales ; ,
fei?(b; x; ; ) = fei?(b; x; 1=b0; p )Vf (p ; ; 1=b0)Uf (1=b0; ; )
= fei?(b; x; 1=b0; p ) exp Zf
(1=b0; ) ln
+ K f (1=b0; ) + Rf [ s(1=b0)] ln
p
and using eq. (A.28) to evolve the PDF as well as eq. (A.22) to expand the rapidity
anomalous dimension in powers of s( ), we nd that the beam function matching coe cients in
eq. (A.27) have xedorder expansions taking the form:7
Iij = ij (1
z) + X
1
n=1
7The gluon beam function is decomposed into 2 tensor structures contributing to diagonal part and to
o diagonal parts. Here, we restrict ourselves to the diagonal part because the o diagonal part begins to
contribute at two loops and is not necessary at NNLL accuracy.
Ii(j1) = ij (1
Ii(j2) = ij (1
+ hIi(j1) (z)
2Pi(j0) (z) ln b0
Zf 0 ln b0 ln
2 ln b0
k
X Z 1 dyy Ii(k1) (y) Pk(j0) z
z y
4Pi(j1)(z) ln b0 + Ii(j2)(z) :
+ 4 ln2 b0
The oneloop constant terms Ii(j1)(z) [29, 63] are given by
1
2
z) Zf 0 ln b0 ln
Zf 0 ln b0 ln
p
+ f0 ln b0
2Pi(j0)(z) ln b0 + Ii(j1)(z) ;
+ f0 ln b0
+ Zf 0 0 ln2 b0 ln
(A.34a)
(A.34b)
+ f0 0 ln2 b0 + Zf 1 ln b0 ln
+ f1 ln b0 + R1f ln
p
+ f0 ln b0 + 2 0 ln b0
k
X Z 1 dyy Pi(k0) (y) Pk(j0) z
z y
Iq(q10)(z) = Iq(q1)(z) = Ig(1g)(z) = 0 ;
Iq(q1)(z) = 2CF (1
z) ;
Iq(g1)(z) = Iq(g1)(z) = 4TF z(1
Ig(1q)(z) = Ig(1q)(z) = 2CF z :
functions de ned in eq. (A.29) are given by8
The twoloop terms Ii(j2)(z) can be found from [21] (see also [20]). The 1loop splitting
Pq(i0q)j (z) = 2CF ij L0(1
z)(1 + z2) +
(1
z) ;
Pq(i0g)(z) = Pq(i0g)(z) = 2TF (1
Pg(g0)(z) = 4CA L0(1
Pg(q0i)(z) = Pg(q0i)(z) = 2CF
z)2 + z2 ;
z + z2)2
z
z
1 + (1
z)2
:
The anomalous dimensions Rif that we need are given above in eq. (A.26).
A.5
TMD cross section
Combining the above xedorder expansions of the hard and soft functions and TMDPDFs
according to the factorization formula eq. (1.8), we obtain for the xedorder expansion up
8This de nition di ers by a factor of 2 from the ones used in [64].
z) ;
3
2
+ 0 (1
z) ;
(A.35)
(A.36)
to O( s2) of the singular pieces of the full QCD cross section (for the qq channel in DY):
3 ZH 0 0 ln3 Qb0
H0 ln Qb0
2
+
ZH 21 + H0 0 ln2 Qb0 +(
H1 + R1S +2c1Se 0) ln Qb0 +c2H +c2S
e
ZH 2
;
0 y
+ s( ) X Z 1 dy Iq(j1)(y) 2Pq(j0)(y) ln b0
fq(z1; )fj y
;
z2
+fj y
;
z1
fq(z2; )
1+ s( )
4
ZH 2
y
j
0 y
s( ) 2 X Z 1 dy Iq(j2)(y) 4Pq(j1)(y) ln b0
fq(z1; )fj y
;
z2
+fj y
;
z1
fq(z2; ) :
The third line of the O( s2) pieces cancels out the running of s( ) in the O( s) piece on
the very rst line. All the Pi(j0;1) pieces cancel out the evolution of the PDFs fq(zi; ). The
remaining pieces on the rst three lines that contain logs are all xed by the RG evolution
of the hard and soft functions and TMDPDFs in eq. (1.8).
A.6
Gaussian rapidity exponent
The pieces is in the exponentiated rapidity evolution kernel eq. (3.5) are given to all orders
by eq. (3.6), and to NLL accuracy by:
A =
ZS 0 4
s( L) 1 +
2
4
s( L) ln H
L
= exp
ln H = L
1 + s( L) ln H = L
2
;
C = eA ln2
;
and to NNLL accuracy by:
A =
ZS
0
2
= exp 4
C = exp A ln2
s( L)
4
2
+ 1 1s6( L2)
1
2
4
s( L) ln
ln H = L
1 + s( L) ln H = L
2
2ZS
(1) s( L)
RS 4
0 + 1 s4( L) 5
H
L
3
4
s( L) 2 R(1S) ln
H
L
(A.38)
(A.39)
A general scheme for soft resummation
The default prescription for running in our scheme uses the choice (eq. (2.36))
L = L( Lb0) 1+p
(B.1)
which automatically resums the leading double logarithms of ln( Lb0). This scheme also
partially resums single and double logarithms of the argument
Lb0 at higher orders in s
.
This is the simplest scheme that provides a stable b space kernel that respects the power
counting that l = ln( Lb0) is small (see section 2.3). This scale choice is by no means
unique. There is still a lot of space to play around with the choice of this scale, where
each choice would di er from the other in exactly which set of small logs l get included
in the exponent. All of these di erent schemes, therefore, would only di er from each
other in subleading terms, and hence we would expect that each of these would lead to an
overlapping error band at any given order in resummation.
In this section, we give a general prescription for the scale choice
L, that covers all
of these schemes, while still allowing us to obtain an analytical expression. So we still
obey the constraint of putting terms at most of quadratic power in l in the exponent. The
generalization that we propose is
(B.2)
(B.3)
(B.4)
where we now expand out both r and s as a power series in s
. 0 is a constant
~L = L( Lb0)r 0s
r =
1
X ri si
;
i=0
s =
1
X si si
:
i=0
The soft exponent eq. (2.43) now looks like
V ( L = ~L; H ; L) = exp ln
(
= exp
ln
H
L
ln( Lb0) X ri si
ln 0
1
X
n=0
1
H X
L n=0
1
i=0
(ZS n ln Lb0 + RnS) :
In practice, while resumming to a particular order, we truncate the series in r and s to that
order in accuracy. The ri parameters will control the coe cient of both l and l2 at each
order in
s, while the si parameters will control the coe cient of l. The si parameters also
induces constant terms in the exponent, which ideally, one would not nd in an exponent,
however, their e ect at each order can be cancelled out by including the corresponding
constant terms induced in the
xed order by this scale choice.
We have checked that
the e ect of several di erent choices for the parameters r and s produce variations in the
resummed cross sections smaller than the inherent perturbative uncertainty already present
at each order seen in gure 6.
u^ = Q
2
x1mT pse y
(C.2)
For Higgs production, the treelevel and partonic cross sections are given by
(C.1)
(C.3)
(C.4)
mT =
xmin = p
1
qqT2 + Q2 ;
mT ey
s
Q
mT e y
s^ = x1x2s ;
For DY, they are
x2 =
t^ = Q
2
mT e y x1ps
s
x1ps
x2mT psey ;
Q2=mT ey
mT ey
;
0 =
64s
Ggg = CA
Ggq = CF
Gqq = 2CF2 t^2 + u^2
;
Q8 + s^4 + t^4 + u^4
Gqg = Ggqjt^$u^ ;
0 =
Ggg = 0 ;
Ggq =
2s^
Gqq = CF s^
Q2 (t^
4
2
2
3sQ4NC ei ;
Q2 s^2 + u^2 + 2Q2t^2
u^s^
Q2)2 + (u^
;
t^u^
Gqg = Ggqjt^$u^ ;
Q2)2
Perturbative QCD results at NLO
The QCD results of qT spectrum for Higgs and for DY are known up to NNLO [65{72].
Here we give NLO expression [66, 73{75] which is used to obtain the nonsingular part
de ned in eq. (3.72)
d pert
where Gij is a reduced partonic cross section depending on partonic Mandelstam variables.
All variable above are de ned as
HJEP04(218)9
where ei is quark charge 2=3 and
1=3 for the up and downtype quarks and summation
over i and j is implicitly implied in eq. (C.1).
D
Alternative techniques for obtaining analytic resummed result
In this section, we present two other ways in which we can obtain an analytic expression
for our resummed soft exponent. One of them involves using again a weighted Hermite
basis, now applied to the function f (t) =
(1+t)
( t) . The other uses a more generalized basis,
however, is less systematic in terms of determining the expansion coe cients. In the last
subsection we consider an alternative expansion applicable to the low qT regime of the
perturbative distribution.
Note that the values of a~0 and ~b0 di er from those for a0; b0 in eq. (3.24) by 2=6.
Then, the series expansion can be written as
= ix e a~0x2 X1 c~2nH2n( ~x)
2 Ex2 e ~b0x2 X1 d~2nH2n( ~x) :
So the coe cients in eq. (D.2) are given by
n=0
Z 1
1
22n(2n)!
E
22n+1(2n)!
dx Re x 1 (1
Z 1
1
dx Im
H2n( x)e ( 2 a0)x2 ;
ix)
2
H2n( x)e ( 2 b0)x2 :
c~2n = p
d2n =
((1+tt)) .
(1
(D.1)
(D.2)
(D.3)
(D.6)
Empirical tests imply the series converges well for ~
Figure 7 shows the exact result and series expansion up to 4th order for ~
~b0 = 4. Note that the deviations from the exact results above x = 1:5 is suppressed
by the Gaussian kernel in eq. (3.12) and resulting error in the integral should be smaller
2
a~0 = 4 and
a~0 and ~2
~b0 around 3
than that appearing in gure 13.
The coe cients c~2n and d~2n are given by
c~0 = 1:02257 ;
d~0 = 1:00941 ;
c~2 = 0:02162 ;
d~2 = 0:00818 ;
c~4 = 0:00168 ;
d~4 = 0:00042 ;
c~6 = 3:33
d~6 =
1:43
10 6
;
10 5
:
(D.4)
(D.5)
Now it is straightforward to rewrite the integration in eq. (3.12) in terms of the
basis integrations and to obtain the xed order terms in the similar fashion to eqs. (3.28)
and (3.50) in section 3.2.
D.2
A tailored basis for expanding
While the weighted Hermite polynomial basis presented in the main text is a systematic
expansion in an orthogonal basis, we can come up with a basis more closely tailored to
the behavior of the function f (t) =
[[1 tt]] , although it is not as systematic in that it is not
orthogonal and there is not a simple formula for the basis coe cients. This basis is:
Hermite basis with a weight for
The integral in eq. (3.12) can be done numerically but by series expanding (1 ix)= (ix)
directly. We then apply the same strategy that was used in section 3.2, but now use it
directly for f (t) =
Near x = 0, we have the Taylor expansion as
= ix(1
a0 x2)
2 Ex2(1
b0 x2) +
;
a~0 = 2 E2
~b0 =
2 E3 + 3
0:95723 :
HJEP04(218)9
fapp(t) =
X cn f (t; Nn; an; bn)
f (t; Nn; an; bn) = (t + 1)Nn ean(t+1)2+bn(t+1) = iNn (x
x0)Nn e an(x x0)2+ibn(x x0) ; (D.7)
ix( 0.2
Γ
n=2
n=4
n=6
Γ
ix) 1.0
1
([Γ0.5
m
I
n=2
n=4
Hermite polynomials up to 4th order.
where Nn are integers and an and bn are complex constants. In the 2nd equality, we set
t = c + ix and x0 = i(1 + c). This form has been deliberately chosen in anticipation of our
choice of c =
Because we are not aware of a systematic expansion in terms of this basis unlike the
weighted Hermite polynomial expansion, the values of an, bn, and Nn as well as cn in
eq. (D.6) should be determined by
tting to the exact function f (t). The integration
against the evolution kernel is given by
F0(a; b) =
FN (a; b) =
i
i
1
1
p
Z c+i1
A c i1
Z c+i1
A c i1
N
2
= !N (a; b) F0(a; b) ;
F0(a; b)
dt e A
(1+t)2 2L(1+t) f (t; 0; a; b) =
dt e A
(1+t)2 2L(1+t) (1 + t)N f (t; 0; a; b)
e A 4(b(1+2aLA)2)
1 + aA
The integral Ib is rewritten as
Ib =
q
T
2C1 X cn FNn (an; bn) ;
2
1
2
where the derivative on FN can be replaced by F0 multiplied by a coe cient dN;k.
Here, we show the result of using the basis in eq. (D.7) for Nn =0 and 2. The real part
of fapp(t =
1 + ix), which we call fR(t) is even, while the imaginary part fI (t) is an odd
function of x. We therefore write
fR(t) = g1(e g2x2
cos[g3x]) + g4x2e g5x2
= g1 f (t; 0; g2; 0)
g4f (t; 2; g5; 0)
(D.10)
fI (t) = h1 sin[h2x] + h3 sinh(h4x)
2i
h1 [f (t; 0; 0; h2)
f (t; 0; 0; h2)] +
2
h3 [f (t; 0; 0; ih4)
f (t; 0; 0; ih4)]
(D.11)
(D.8)
(D.9)
1 + ix. In practice we rst nd the t in the rst equality then, rewrite it in
terms of our basis.
The b space integral obtained by replacing f (t; Nn; an; bn) by FNn(an; bn).
Ib =
T
2C1 "g1 F0(g2; 0)
2
1
2
[F0(0; g3) + F0(0; g3)]
g4F2(g5; 0)
2
h1 [F0(0; h2)
F0(0; h2)] + i h3 [F0(0; ih4)
2
T
2C1
2 e AL2 g1e 1+g2A
4 p1 + g2A
2
g2A2L2
g5A2L2
g4e 1+g5A
(1 + g5A)3=2
A
2
A2L2
1 + g5A
F0(0; ih4)]
g1e g324A cosh[g3AL] + h1e h224A sinh[h2AL] + h3e 4 sin[h4AL]5
h24A
(D.12)
3
where we have de ned L = ln 2qT .
The value of the parameters gi; hi shifts as we shift the contour via the value of c,
so that the
nal result is independent of the contour chosen. In this paper we have
made the following choice for the contour and hence the corresponding parameters c =
1; g1 = 0:5532; g2 = 1:77; g3 = 2:465; g4 = 0:4582; g5 = 2:42; h1 = 0:0525; h2 = 4:09; h3 =
0:98; h4 = 0:793. The result of the t with these parameters is shown in
gure 14. It is to
be stressed that once the contour is xed, these parameters are also xed and hence can be
used for any observable in any kinematical regime. This is because the tting is only done
for the ratio of Gamma functions f (t) which, in no way involves the details of the speci c
observable or its kinematics. The only condition as we speci ed earlier that A be a small
number to ensure adequate suppression.
Let us check if our functions eq. (D.11) satis es the constraints in eq. (3.63)
fapp( 1) = g1(1
1) = 0
dt fapp( 1) = h1h2 + h3h4 = 0:992
This agrees with eq. (3.63) better than 1% that is acceptable at NNLL accuracy.
E
Mathematical proofs
E.1
A proof of the MellinBarnes identity for the Bessel function
Here we present a short proof of the key identity eq. (3.8) we use in the bspace integral
against the Bessel function:
dt
z
2t
:
we have
which are
This identity can be found, e.g. in [76], section 10.9.22, which is given as valid for J (z)
for
> 0. We brie y verify that it works also for
= 0.
For convenience let us change the integration variable in eq. (E.1) from t !
t. Then
J0(z) =
dt
1
2
z
2t
;
where now the contour lies to the right of all the poles of (t), i.e. c > 0. The contour can
be closed in the left half plane out at t !
1. The value of the integrand falls rapidly in
this limit, and the circular part of the contour contributes zero to the integral. Deforming
the contour, we pick up the residues at all the poles t =
n of the Gamma function
Then the integral in eq. (E.2) has the value
Res (t =
n) =
( 1)n
X1 ( 1)n
n=0
1
z 2n
;
(E.1)
(E.2)
(E.3)
(E.4)
which is precisely the series representation of the Bessel function J0, proving the identity.
E.2
Integral of complex Gaussian
We can evaluate the integral over a complex Gaussian using the contour integral:
Z
C
Z 1
1
Z
1
1
0 =
dz e z2 =
dx e x2 +
dx e (x iz0)2 = p
dx e (x iz0)2 ;
(E.5)
Z 1
1
where C is the contour shown in gure 15.
to the integral as e x
E.3
Proof of Gaussian integral of Hermite polynomials
Here we prove the result eq. (3.33) for the integrals Hn given in eq. (3.31). Starting from
the form of the result eq. (3.32), we have
H = H0
X1 tm
m=0
m! 1 + a0A
2 z0
t
1
2 z0
a0A
:
We need to identify the coe cient of each single power tn of t in order to read o the
coe cients Hn in eq. (3.30). Using the binomial theorem,
H = H0
1
X
Xm tm
m=0 k=0
m! k
2 z0
1 + a0A
t
1
2 z0
a0A
k
:
Using
k
m
and reindexing the k integral using n
m + k, we obtain
H = H0
1
X
2m
X
m=0 n=m
2 z0
n)! 1 + a0A
1
2 z0
a0A
n m
:
This is almost in the form eq. (3.30) where we can read o the coe cient of tn, but the
order of summation needs to be ipped. As illustrated by gure 16, the following sums
( t)n
1
1
X
2m
X
m=0 n=m
1
X
n
X
:
n=0 m=dn=2e
n)! 1 + a0A
n)! 1 + a0A
(E.6)
(E.8)
(E.10)
: (E.12)
are equivalent:
Thus,
H = H0
1
X
n
X
n=0 m=dn=2e
Hn = H0( 1)nn!
n
X
m=dn=2e
Now we can read o the coe cient of tn in the series in eq. (3.30), and obtain
2 z0
A 2
a0A
n m
2 z0
m
a0A
n m
1
2 z0
1
2 z0
For convenience, we reindex the sum over m by taking m ! n
m, and obtain
Hn = H0( 1)nn! X
bn=2c
m=0
1
2 z0
n m
1
2 z0
a0A
which after a rearrangement of factors gives the claimed result eq. (3.33).
Explicitly, the rst several Hn given by eq. (3.33) are
H0 = e 1+Aa0A (L i =2)2
p
1
1+a0A
2z0
1+a0A H0
H1 =
H2 =
H3 =
H4 =
H5 =
H6 =
H1
(1+a0A)2 4 2z02 +6(A( 2 a0) 1)(1+a0A)
H0
(1+a0A)4 16 4z04 +48 2z02(A( 2 a0) 1)(1+a0A)+12(A( 2 a0) 1)2(1+a0A)2
H1
(1+a0A)4 16 4z04 +80 2z02(A( 2 a0) 1)(1+a0A)+60(A( 2 a0) 1)2(1+a0A)2
H0
(1+a0A)6 64 6z04 +480 4z04(A( 2 a0) 1)(1+a0A)
+720(A( 2 a0) 1)2(1+a0A)2 2z02 +120(A( 2 a0) 1)3(1+a0A)3
E.4
Recursion relation for Hn derivative
Here we prove the recursion relation eq. (3.49) for derivatives of the integrals Hn of Hermite
polynomials in eq. (3.28). We can prove the relation either from this integral eq. (3.28)
directly, or from the nal result eq. (3.33) of the integration. We present both
computations here.
(E.13)
(E.14)
Beginning from the integral de nition of Hn in eq. (3.28), we obtain
1
e A(L i =2)2Z 1
dx Hn( x)e a0x2 1
A (x+z0)2
1
2A L
i
(E.15)
where we used z0 = A( 2 + iL). The rst term in brackets then cancels with the z0 term,
1
A
e A(L i =2)2Z 1
1
dx ( 2ix)Hn( x)e a0x2 A1 (x+z0)2 :
(E.16)
2x can be expressed as a derivative on the Gaussian:
+H0( ; a0) (1+a0A)n
( 1)nn! bn=2c
X
m=0
i2z0
o ( 1)nn! bn=2c [A( 2 a0) 1](1+a0A) m(2 z0)n 2m
X
m!(n 2m)!
m! (n 2m)! n[A( 2 a0) 1](1+a0A)om
1 1
@Ln(2 z0)n 2mo
Using the all orders result for Hn in eq. (3.33). We then compute
+H0( ; a0) (1+a0A)n
m!(n 1 2m)!
( 1)n(n 1)! b(n 1)=2c i2An [A( 2 a0) 1](1+a0A) m(2 z0)n 1 2m
and we have simply
so
which proves eq. (3.49).
E.4.2
Second proof
where in the last line we integrated the d=dx term by parts, using that the boundary terms
1. Now we can use the known recursion relation for derivatives of
Hn0(x) = 2nHn 1(x) ;
2iz0
1 + a0A Hn
2in A
1 + a0A Hn 1 ;
e A(L i =2)2Z 1
2iz0
1 + a0A Hn
dx Hn( x)
1
e A(L i =2)2Z 1
d
dx
1
2z0
A
e a0x2 A1 (x+z0)2
dx Hn0( x)e a0x2 A1 (x+z0)2 ;
d
dx
e a0x2 A1 (x+z0)2 =
2z0 e a0x2 A1 (x+z0)2 ;
m=0
X
m=0
(E.17)
(E.18)
(E.19)
(E.20)
(E.21)
where we have used
2iz0
1 + a0A H0( ; a0)
(E.22)
For the second term in @LHn( ; a0), we can make the following observations
For even values of n, the term m = n=2 does not contribute to the derivative.
For odd values of n, bn=2c is the same as b(n
1)=2c.
We can then immediately write down
@LHn( ; a0) =
2iz0
1 + a0A Hn( ; a0)
2iAn
1 + a0A Hn 1( ; a0)
(E.23)
which proves eq. (3.49).
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