#### Heavy-quark form factors in the large \(\beta _0\) limit

Eur. Phys. J. C
Heavy-quark form factors in the large β0 limit
Andrey G. Grozin 0 1 2 3
0 THEP, Institut für Physik, Universität Mainz , Mainz , Germany
1 Institut für Theoretische Teilchenphysik, Karlsruher Institut für Technologie , Karlsruhe , Germany
2 Budker Institute of Nuclear Physics , Novosibirsk , Russia
3 Novosibirsk State University , Novosibirsk , Russia
Heavy-quark form factors are calculated at β0αs ∼ 1 to all orders in αs at the first order in 1/β0. Using the inversion relation generalized to vertex functions, we reduce the massive on-shell Feynman integral to the HQET one. This HQET vertex integral can be expressed via a 2 F1 function; the nth term of its ε expansion is explicitly known. We confirm existing results for nlL−1αsL terms in the form factors (up to L = 3), and we present results for higher L.
1 Introduction
Quark form factors are building blocks for various production
cross sections and decay widths in QCD. Massive quark form
factors are known up to two loops [
1
]; recently they have been
calculated at three loops in the large Nc limit [
2
].
We shall consider heavy-quark form factors in the large β0
limit, where β0αs ∼ 1, and 1/β0 is an expansion parameter
(see the reviews [
3–5
]). A bare form factor can be written as
F = 1 +
∞ L−1
L=1 n=0
aLn β0n
g02
(4π )d/2
L
Keeping terms with the highest degree of β0 in each order of
perturbation theory, we get
1
F = 1 + β0 f
β0g02
(4π )d/2
+ O
The leading coefficients aL,L−1 can easily be obtained from
n L−1 terms (Fig. 1). We shall consider only the first 1/β0
f
order.1
1 In some cases it is possible to obtain results for 1/β02 corrections;
see, e.g. [
6–8
].
(1)
(2)
k1,2
10]:
J (μ) =
2 Heavy-quark bilinear currents
We consider the QCD currents
J0 = Q¯ 0 Q0 = Z (αs(n f )(μ)) J (μ),
= γ [μ1 · · · γ μn],
where Q0 is a bare heavy-quark field. The antisymmetrized
product of n γ matrices has the property
γ μ γμ = η(d − 2n) , η = (−1)n.
All results for form factors of this current will explicitly
depend on n and η.
In situations when the initial heavy-quark momentum p1
and the final one p2 can be written as p1,2 = mv1,2 + k1,2 (m
2
is the on-shell mass, v1,2 = 1) with small residual momenta
m, these currents can be expanded in HQET ones [9,
2
i=0
Gi (μ, μ )O˜ i (μ ) + O
where the leading HQET currents are
J˜i0 = h¯v20 i hv10 = Z˜ (αs(nl )(μ)) J˜i (μ),
i =
, /v1
+ /v2, /v1 /v2,
and the O˜ i are local and bilocal dimension-4 HQET
operators with appropriate quantum numbers. Here hv1,20 are
two (unrelated) bare fields describing HQET quarks with
the velocities v1,2 having small (variable) residual momenta;
the HQET Lagrangian explicitly contains v1,2. These
reference velocities can be changed by arbitrary small vectors
of order ki /m (reparametrization invariance). The HQET
(3)
(4)
(5)
(6)
mv
k1
k2
a
k3
k4
mv
ω
k1
k2
b
k3
k4
ω
Hi (μ, μ ) = Hˆi
current renormalization constant Z˜ does not depend on the
Dirac structure and is a function of the Minkowski angle ϑ :
v1 · v2 = cosh ϑ = w.
For our purpose it is convenient to choose v1,2 = p1,2/m,
i. e., both residual momenta k1,2 = 0. Then the matrix
elements of O˜ i vanish: non-zero expressions for these matrix
elements (having dimensionality of energy) cannot be
constructed, because we have no non-zero dimensionful
parameters. The coefficients Hi in (5) can be obtained by matching
the on-shell matrix elements (k1,2 = 0) in QCD and HQET:
2
i=0
Q( p2 = mv2)| J0|Q( p1 = mv1) =
Fi u¯2 i u1,
Q(k2 = 0)| J˜i0|Q(k1 = 0) = F˜i u¯2 i u1, F˜i = 1,
(7)
where u1,2 are the Dirac spinors of the initial quark and
the final one (all loop corrections to F˜i vanish because they
contain no scale). Therefore the bare matching coefficients
(in the relation similar to (5) but for the bare currents) are
H 0
i = Fi /F˜i = Fi . The renormalized matching coefficients
are
Hi (μ, μ ) = H 0 Z˜ (αs(nl )(μ )) Fi Z˜
i Z (αs(n f )(μ)) = F˜i Z
UV divergences cancel in the ratio Fi /Z as well as in the ratio
F˜i /Z˜ . Both Fi and F˜i contain IR divergences which cancel in
the ratio Fi /F˜i because HQET is constructed to reproduce the
IR behaviour of QCD (F˜i have no loop corrections because
their UV and IR divergences cancel each other).
The dependence of Hi (μ, μ ) on μ and μ is determined
by the RG equations. Their solution can be written as
×
αs(n f )(μ)
αs(n f )(μ0)
αs(nl )(μ )
αs(nl )(μ0)
γn0/(2β0(n f ))
−γ˜0/(2β0(nl ))
Kγ(nn f )(αs(n f )(μ))
K −(nγ˜l)(αs(nl )(μ )),
where for any anomalous dimension γ (αs ) = γ0αs /(4π ) +
γ1(αs /(4π ))2 + · · · we define
Kγ (αs ) = exp
Matrix elements of the currents with n = 0, 1 can be
written via smaller numbers of form factors:
Q(mv2)| J |Q(mv1) = F Su¯2u1,
F S = F0 + 2F1 + (2w − 1)F2,
where Fi with n = 0, η = 1 are used, and
Q(mv2)| J μ|Q(mv1) = (F1V + F2V )u¯2γ μu1
(v1 + v2)μ
−F2V u¯2u1 2 ,
F V
1 = F0 + 2F1 − (2w − 3)F2,
F V
2 = −4(F1 + F2),
where Fi with n = 1, η = −1 are used.
3 Inversion relations
On-shell massive self-energy integrals with one massive line
and any number of massless ones in some cases can be
expressed via similar off-shell HQET integrals. Suppose all
massless lines can be drawn at one side of the massive
one and the resulting graph is planar (e.g., the diagram in
Fig. 2a). Lines of such a diagram subdivide the plane into
a number of polygonal cells (plus the exterior); with each
cell we can associate a loop momentum (flowing
counterclockwise). Then outer massless edges of the diagram
correspond to the denominators −ki2 − i 0; inner massless edges to
−(ki − k j )2 − i 0; and massive edges to m2 − (ki + mv)2 − i 0
(Table 1). The corresponding HQET diagram (Fig. 2b) has
HQET denominators −2ki · v − 2ω − i 0 instead of massive
ones. First we perform a Wick rotation of all loop momenta
ki0 → i ki0 (in the v rest frame). Then, in Euclidean
momentum space, we invert each loop momentum [
11
]:
(10)
(11)
(12)
(13)
(14)
(9)
ki → kk2i .
i
see Table 1. As a result, a massive on-shell diagram (Fig. 2a)
becomes m− ni (the sum runs over all massive line
segments, ni are their indices, i. e. the powers of the
denominators) times the off-shell HQET diagram (Fig. 2b) with
ω = −(2m)−1 (15). The indices of all inner massless edges,
as well as of all massive edges (which become HQET ones),
remain intact (see Table 1). From the same table it is clear
that the index of an outer massless edge becomes d − ni ,
where the sum runs over all edges of the cell to which this
outer edge belongs (they can be all massless, or one of them
can be massive). If there is a cell ki bounded only by inner
massless edges, and maybe one massive one, then the
denominator (ki2)d− n j will appear (Fig. 3). This denominator does
not correspond to any line, and hence the resulting integral is
not a Feynman integral at all; in this case, the discussed
relation becomes rather useless (though formally correct). The
inversion relations [
11
] were used, e.g., in [
12–14
]).
The inversion relations can be generalized to similar vertex
integrals; the masses of the initial particle and the final one
may differ. At one loop (Fig. 4), the integrals
I (n1, n2, n; ϑ; ω1, ω2) =
M(n1, n2, n; ϑ; m1, m2) =
dd k
iπd/2
1
× [−k2 − 2m1v1 · k − i0]n1 [−k2 − 2m2v2 · k − i0]n2 (−k2 − i0)n
dd k
iπd/2
(16)
Euclidean
ki2
(ki − k j )2
The integrals I (17) have been investigated in [
15
]. Here
we need only the integrals M (16) with m1 = m2; they reduce
to the integrals I (17) with ω1 = ω2, which are especially
simple [
15
]:
I (n1, n2, n; ϑ ; ω, ω) = (−2ω)d−n1−n2−2n I (n1 + n2, n)
× 3 F2 nn1 +12,nn2,2,n1d2+−n22+n1 1 − c2osh ϑ ,
where
I (n1, n) =
(−d + n1 + 2n) (d/2 − n)
(n1) (n)
(17)
(19)
(20)
k + m2v2 m2v2 − q
is the one-loop HQET self-energy integral. We only need
integer n1,2; in this case all I reduce by IBP to 2 master
integrals [
15
]: I (1, 0, n) (trivial) and I (1, 1, n) (given by (19)).
Inversion relations can be generalized to diagrams with
more external legs. For example, the one-loop massive box
diagram with two on-shell legs and the corresponding
offshell HQET one (Fig. 5)
M(n1, n2, n3, n4; ϑ; m1, m2; q2, q · v1, q · v2) =
1
× (−k2 − 2m1v1 · k)n1 (−k2 − 2m2v2 · k)n2 (−(k + q)2)n3 (−k2)n4 ,
(21)
I (n1, n2, n3, n4; ϑ; ω1, ω2; q2, q · v1, q · v2) =
1
× (−2k · v1 − 2ω1)n1 (−2k · v2 − 2ω2)n2 (−(k + q)2)n3 (−k2)n4
dd k
iπd/2
dd k
iπd/2
We need only terms with the highest degree of n f ; therefore,
there is no need to distinguish between n f and nl = n f − 1,
or any n f + const. The gluon propagator can be written as
Dμν (k) = k2(1 −
1
(k2))
gμν −
kμkν
k2
,
where the gluon self-energy is
(k2) = β0 (4πg)02d/2 e−γ ε D(ε) (−k2)−ε,
ε
m2v2
Fig. 5 Box diagrams
D(ε) = eγ ε (1 − ε) (1 + ε) 2(1 − ε) 5
(1 − 2ε)(1 − 23 ε) (1 − 2ε) = 1 + 3 ε + · · ·
(the same function appears also in the one-loop self-energy
integral with arbitrary masses m1,2 and arbitrary p2, where
At this leading large β0 order, the coupling constant
renormalization is simple:
β0 (4πg)02d/2 e−γ ε = b Zα(b)μ2ε,
αs (μ) 1
b = β0 4π , Zα = 1 + b/ε .
The bare QCD matrix elements can be written in the
form [
6,16
]
1 ∞ fi (ε, Lε)
Fi = δi0 + β0 L=1 L
(−m2)L + O
. (27)
It is convenient to write the functions fi (ε, u) in the form
usual for on-shell massive QCD problems (see [
5
])
eγ ε
fi (ε, u) = CF D(ε)
(1 − 2u) (1 + u)
(3 − u − ε)
Ni (ε, u).
(28)
We calculate the vertex function (Fig. 1) and multiply it by
Z oQs with the 1/β0 accuracy (see [
5
]). Reducing on-shell
massive QCD integrals to off-shell HQET ones by the inversion
relation (18) and then to the master integrals by IBP [
15
], we
obtain
N0(ε, u) =
−ηu n − 2 + ε − 2(w + 1)u(n − 2)2
w − 1
−u ηu + 4(w + 1)ε (n − 2)
+2(2 − u) w + (w + 1)u
−(6w + 2u + ηu2)ε
+2 w − (w + 1)u ε2 F
+ηu n − 2 + ε + 2(n − 2)2 + 4ε(n − 2)
w − 1
−6(1 − u2) + 2(1 − u)(5 + 2u)ε
−2(1 − 2u)ε2,
n − 2 + ε
N1(ε, u) = u ηw
w − 1
− ηu(n − 2) − 2 + u + ε
−ηuε F − ηu n − 2 + ε ,
w − 1
n − 2 + ε
N2(ε, u) = ηu
w − 1
×[1 − (1 + (w − 1)u)F ],
(26)
(29)
(30)
(22)
(23)
(24)
(25)
where
F = 2 F1
1, 1 + u 1 − w
3/2 2
both indices are equal to 1 [
17
]). At ϑ = 0 this result agrees
with the result of [
18
] at m1 = m2; see also [
5
].2
Re-expressing the bare form factors (27) via the
renormalized coupling we obtain
1 ∞ fi (ε, Lε)
Fi = δi0 + β0 L=1 L
D(ε)
is regular at the origin; expanding (b/(ε+b))L in b, we obtain
a quadruple sum. In the coefficient of ε−1 all fnm except
fn0 cancel; differentiating this coefficient in log b (and using
the fact that F (30) at u = 0 is ϑ/ sinh ϑ ) we obtain the
anomalous dimension corresponding to Z /Z˜ [
6,16
]:
These anomalous dimensions at the 1/β0 order are [
19,20
]
γn = 4CF
b (1 + 23 b) (2 + 2b)
β0 (1 + b)2(2 + b) 3(1 + b) (1 − b)
× (n − 1)(3 − n + 2b) + O
γ˜ = 4CF
b (1 + 23 b) (2 + 2b)
β0 (1 + b) 3(1 + b) (1 − b)
(37)
(38)
(39)
(40)
In the coefficient of ε0 all fnm except fn0 and f0m cancel.
The coefficients fn0 form Kγn−γ˜ (αs (μ)), see (9); we have [
6
]
where
where the Borel images of the perturbative series for Hˆi are
1
Si (u) = u
The integral (37) is not well defined because of poles at the
integration contour. The leading renormalon ambiguities are
given by the residues at u = 1/2 [
21
] (see also [
5
]). It is easy
to calculate these residues because F (30) at u = 1/2 is just
2/(w + 1):
As demonstrated in [
21
], matrix elements of the QCD
currents between ground-state mesons (pseudoscalar or vector)
are unambiguous: the IR renormalon ambiguities of the
leading matching coefficients Hi are compensated by the UV
renormalon ambiguities in the matrix elements of the 1/m
suppressed HQET operators O˜ i in (5) (see also [
5
]).
The hypergeometric function F (30) has been expanded in
u to all orders [
17
], the coefficients are expressed via Nielsen
polylogarithms Snm (x ). The result [
17
] is written for the
case of an Euclidean angle3; its analytical continuation to
Minkowski angles is
1
F = sinh ϑ (2 cosh(ϑ/2))2u
n
∞
sinh(ϑ u)
u
un
(−2)n−m Sm,n−m+1(−eϑ )
+ O
Our results satisfy this requirement ( f1,2(−b, 0) = 0 because
the QCD current J does not mix with currents with other
Dirac structures).
2 There are a few typos in Sect. 8.8 of [
5
]. The unnumbered formula
below (8.93) should read
R0 = cosh(Lu), R1 =
sinh[(1 − 2u)L/2]
sinh(L/2)
In the second formula in (8.95), the coefficient of R0 should contain an
extra factor 3. In both formulae in (8.96), their right-hand sides should
be 1 + αs correction.
−e−ϑu
+eϑu
n=1
∞
un
n=1
m=1
m=1
n
(−2)n−m Sm,n−m+1(−e−ϑ ) .
(41)
It is possible to re-express this expansion in terms of Nielsen
polylogarithms of just one argument, see [
23
], but then the
symmetry ϑ → −ϑ will not be explicit.
3 M. Yu. Kalmykov has informed me that there is a typo: the power of
cos ϑ in (2.7) should be 1 + 2ε. This typo has been corrected in [
22
].
Acknowledgements I am grateful to M. Steinhauser for useful
comments and hospitality in Karlsruhe, where the major part of this work
was done; to J. M. Henn for useful discussions and hospitality in Mainz;
and to M. Yu. Kalmykov for bringing Ref. [
17
] to my attention and
discussions related to it.
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Appendix A: Anticommuting γ5 and ’t Hooft–Veltman γ5
For flavour-nonsinglet currents one may use the
anticommuting γ5 without encountering contradictions; they are related
to the currents with the ’t Hooft–Veltman γ5 by a finite
renormalization [
24–26
]:
(q¯ γ5AC nτ q)μ = Z2−n(α(n f )(μ))(q¯ γ5HV nτ q)μ,
(A.1)
where τ is a flavour matrix with Tr τ = 0. The currents with
γ5AC n have anomalous dimensions γn, because they can be
obtained from the case of massless quarks; γ5HV n is just
4−n with reshuffled components. Equating the derivatives
in d log μ we obtain
Z2−n(αs ) = Kγ(nn −f)γ4−n (αs ),
where the anomalous dimensions γn and γ4−n differ starting
from two loops. In particular, Z0(αs ) = 1. In HQET currents
with γ5AC and with γ5HV have the same anomalous dimension
γ˜ , and the finite renormalization factor similar to (A.2) is 1.
In the large β0 limit (see (35))
8n
Zn(αs ) = exp − β0
(1 + 23 b) (2 + 2b)
(1 + b)2(2 + b) 3(1 + b) (1 − b)
0
b
×
+O
db
1
β02
.
At the leading 1/β0 order we may use these formulae for
flavour singlet currents, too. The matrix γ5AC n has the same
property (4) but with η = −(−1)n. From our results (27)–
(29) we see that, indeed,
Hˆγ5AC n = Hˆ n η→−η = Hˆγ5HV n = Hˆ 4−n .
Matrix elements of the currents with γ5AC and n = 0, 1
can be written via smaller numbers of form factors:
Q(mv2)| J |Q(mv1) = F P u¯2γ5ACu1,
F P = F0 − 2F1 − (2w + 1)F2,
(A.2)
(A.3)
(A.4)
(A.5)
5 Appendix B: Expansion of the hypergeometric
function F
We can also find several terms of this expansion using
the Mathematica package HypExp [
27,28
] (which uses
HPL [
29,30
]). This results in
1
F = sinh ϑ
u2
ϑ − H−+(τ )u − (H−+−(τ ) − 2H−+(τ )l) 2
3
−(H−+−−(τ ) − 2H−+−(τ )l + 2H−+(τ )l2) u
3
H−+−−−(τ ) − 2H−+−−(τ )l + 2H−+−(τ )l2
−
where Fi with n = 0, η = −1 are used, and
,
where Fi with n = 1, η = 1 are used.
The divergence of the axial current is
i ∂μ(Q¯ 0γ5ACγ μ Q0) = 2m0 Q¯ 0γ5AC Q0,
where the bare mass m0 = Z mosm. Taking the matrix element
of this equation we obtain
F A
1 +
w − 1 F A
2 = Z mos F P .
2
The on-shell mass renormalization constant Z mos at the first
1/β0 order is given by the formula similar to (27), (28) with
Nm (ε, u) = −2(3 − 2ε)(1 − u); see, e.g., [
5
]. And indeed,
from (29), (A.5)–(A.6) we obtain
N A
1 +
w −2 1 N2A = N P + Nm .
(A.6)
(A.7)
(A.8)
(A.9)
(B.10)
where
τ = tanh ϑ2 , l = 21 H−(τ ) = log cosh ϑ2
H+(τ ) = ϑ,
,
and H···(τ ) are harmonic polylogarithms (see [
29–31
]). Only
one new polylogarithm appears at each order.
In order to compare the expansion coefficients in (41) and
in (B.10), we need to transform them to harmonic
polylogarithms of the same argument, which we choose as x = e−ϑ .
In (41), we first rewrite Snm (−x −1) via Snm (−x ) using the
formula from [
23
]; then we rewrite Snm (−x ) via H···(−x )
and then via H···(x ); we rewrite log cosh(ϑ/2) (B.11) via
H···(x ); and finally we re-express products of harmonic
polylogarithms via their linear combinations. In (B.10) we rewrite
harmonic polylogarithms with ± indices [
30
] via normal ones
with indices 0, ±1; substitute τ = (1 − x )/(1 + x ) and
re-express via H···(x ); and finally convert products of
harmonic polylogarithms to sums. All these steps are done in
Mathematica using HPL [
29,30
]. We have checked that
all the coefficients presented in (B.10) agree with (41).
(B.11)
Appendix C: Vector form factors
The vector form factors F1V,2 (13) can be written in the
form (27), (28); from (29), (13) we obtain
N1V (ε, u) = 2[2w + u − 3u2 − 3wε + 2wuε − (w − 3)u2ε
+wε2 − (w + 1)uε2]F
−2 2 + u − 3u2 − 3ε + 2uε
+2u2ε + ε2 − 2uε2 ,
N2V (ε, u) = 4u(1 + u − 2uε)F.
All loop corrections to F1V vanish at ϑ = 0, and hence N V
1 =
0 at w = 1.
The form factor F1V = H1V /Z˜ , where Z˜ at the 1/β0
order is determined by the anomalous dimension (36), and
H1V contains only non-negative powers of ε. We choose
μ = μ = μ0 = m. H1V at ε = 0 is given by the
coefficients fn0 (which produce K−γ˜ (10)) and f0n (which
produce Hˆ1V (37)); εn terms (n > 0) require all fnm . Writing the
expansion (B.10) as F = f0 − f1u − f2u2/2 − f3u3/3 − · · ·
we obtain up to four loops
(C.12)
(C.13)
−2w f1 + (3w + 1) f0 − 4
H V
1 = 1 + CF β0
b
−
w f2 + (3w + 1) f1 −
2
− 3 w f3 +
3w + 1
2
f2 +
2 π 2 π 2
3 ζ3w − 4 w − 12 −16w
2 π 2 π 2
3 ζ3w − 4 w − 12 − 16w
f1
π 4w +
92
nlL−1αsL terms with L = 1, 2, 3 in F1V,2 from [
2
].
Using HPL [
29, 30
] we have successfully reproduced all
π 4 + 269ζ3 +
206
3
+ · · · .
(C.15)
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