= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
Jaw,wa functions IN PASSARINO-VELTMAN reduction
© 2010 D. Yu. Bardin1)*, L. V. Kalinovskaya1)**, V. K. Kolesnikov1)***, W. von Schlippe2)****
Received March 23, 2010
We continue to study a special class of Passarino—Veltman functions J arising at the reduction of infrared divergent box diagrams. We describe a procedure of separation of two types of singularities, infrared and mass singularities, which are absorbed in simple C0 functions. The infrared divergences of C0's can be regularized then by any method: photon mass, dimensionally or by the width of an unstable particle. Functions D0 are represented as certain linear combinations of the standard C0 Passarino—Veltman functions and infrared finite functions J. Then mass singularities are extracted from J to other combinations of C0. The rests are free of both types of singularities and are expressed as explicit and compact linear combination of logarithms and dilogarithm functions. The extensive comparison of numerical results with those obtained with the aid of the LoopTools package is presented.
1. INTRODUCTION
In the standard Passarino—Veltman reduction [1] of 4-point box functions with an internal photon line connecting two external lines on the mass shell there appears an infrared and mass singular D0 function (see, for example, [2]). A typical example of these diagrams arising in the calculation of one-loop electroweak (EW) corrections to ff — ZZ(ZA) processes (f is a fermion) was considered in [3], where a universal approach to the calculation of such diagrams was proposed. The main idea is to introduce an auxiliary, infrared finite function J and derive a relation between infrared divergent D0, J and the infrared divergent C0 function.
In this paper we describe how this approach works for t - bff, fib - tfl, and ff - tb (fi is a massless fermion) Charged Current (CC) processes. For these processes one meets eight such box functions, four direct and four cross ones. Cross boxes are trivially derived from direct ones by a permutation of arguments. Boxes for t decays are related to those of t decays, see [4]. So, it is sufficient to consider only one pair of boxes shown in figure.
For the processes under consideration the basic definition of the function J reads:
in2J (Q2,T2; mb,mt,md,mu,Mw) = (1)
!)JINR, Dubna, Russia.
2)PNPI, RAS, Gatchina.
E-mail: bardin@nu.jinr.ru
E-mail: kalinov@nu.jinr.ru
E-mail: kolesnik@nu.jinr.ru
E-mail: wvsch@sw4989.spb.edu
_ 4-n f jn —2q ■P2
J dodid2dz
where
do = q2, di = (q + pi)2 + m2, (2)
d2 = (q + Pi + P2)2 + M2, d3 = (q — p^)2 + m2d.
Using the standard Passarino—Veltman reduction it is possible to derive relations between infrared and mass singular functions D0(-m2, —mf, —m2a, -md, Q2,T2;0,mb ,Mw ,md) and Co(—m2, —m2,T2; md, 0,mb) and an infrared finite but mass-singular auxiliary function JAW(Q2, T2; mb, mt, md, ma, MW) and another C0(—ml, —m2d,Q2; MW ,md, 0) with mass singularity. These basic relations, exact in masses, are:
Jaw(Q2,T2; mb, mt, md, ma, Mw) = (3)
= (MW + Q2)Do (—ml —m2, —mU, —md,
Q2,T2;0,mb ,Mw ,md) +
+ Co(—m2a, —m2d, Q2; Mw, md, 0) —
— Co(—md, —m2,T2; md, 0,mb), Jwa(Q2,T2; mt, mb, ma, md, Mw) =
= (MW + Q2)Do (—mt, —m2, —ml, —md, Q2,T2; Mw, mt, 0,ma) + + Co(—ml, —md, Q2; 0, mu, Mw) —
— Co(—m2, —ml, T2; mt, 0,mu).
Let us emphasize that we have changed the ordering of mass arguments of JWA as compared to D0. For JdfA they are ordered into two pairs of heavy (b, t)
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and light (d, u) quarks so that the first mass in each pair corresponds to the fermion coupled to the photon, leading to the appearance of a contribution logarithmically singular in this mass.
The great advantage of basic relations (3) is the following. The complex object D0, containing an infrared divergence, is excluded in favor of an explicitly computed J function and simple objects, C0 functions, whose infrared divergences can be regularized by any method: by a photon mass, by dimensional regularization or by the width of an unstable particle. Examples of the latter C0 functions, regularized by the width, may be found in [4].
We use the following definitions:
Q2 = (pi + P2 )2, T2 = (P2 + P3 )2, (4)
U2 = (P2 + P4 )2.
Section 2 is devoted to the calculation of one of these functions — JdAW = J. From here we omit indices of J since the list of arguments uniquely defines its type.
In Section 3 we present a similar calculation of the J function for the process ud — tb. The basic relations for direct AW and WA functions in this case are:
Jaw(Q2, T2; md, mu, mb, mt) = (5) = (Q2 + MW)Do(-md, -m2u, -m2, -m\, Q2 ,T2; 0, md ,Mw m) + + Co(-m2, -mb, Q2; Mw, mb, 0) -
- CO (-mb, -md, T2; mb, 0, md),
Jwa(Q2, T2; mu, md, mt, mb) = = (Q2 + MW)Do(-md, -mU, -m2, -m2b, Q2 ,T2; Mw ,mu, 0,mt) +
Jwa (right) functions.
+ Co(-m2, -ml, Q2; 0, mt, Mw) -
- Co (-mU, -mt, T2; mu, 0, mt).
Again, we limit ourselves to the presentation of function JAW.
In Section 4 we briefly discuss the J functions for the t-channel process bu — td.
For all processes we take the limit of vanishing light quark masses. The mass of the quark which is not coupled to the photon may be set equal to zero, while that for the quark coupled with the photon develops a mass singular logarithm. We keep logarithmic terms and neglect quark masses everywhere else. This approximation results in different expressions for J functions for the three channels under consideration, and this is why the results of the derivation must be presented for three channels separately.
Every Section ends by a numerical comparison with results obtained with the aid of the SANC [5] packages and the LoopTools package [6] for zero width and IR regularization by infinitesimal photon mass.
Section 5 contains a short description of the work with the FORTRAN packages, which realize the calculation of "doubly subtracted" J functions (see Sections 2—4 for their definition).
In Section 6 we make our conclusions.
2. CALCULATION OF THE J FUNCTION FOR t bud DECAY
2.1. Representation in the Form of a Triple Integral
We proceed with calculation of integral J, Eq. (1). Introducing Feynman parameters x, y, z, as shown in figure, one can pass to a (3 + n)-multiple integral
over x, y, z and over the internal momentum q. In n-dimensional space we have:
cTq-
= in
q2 — 2qp + m2)a
(6)
In our case we have m2 = Lz, p = zkxy, a = n = 4, where the variable L and the vector kxy are
L = (Q2 + MW)xy + (p2 + mf) (1 - x)y + (7)
+ (P4 + md) (1 - y), kxy = P4(1 - y) - Piy(1 - x) - (pi + P2)xy. Therefore Eq. (6) becomes
d4q
= in
r(4)
(q2 — 2zqkxy + Lz) (.Lz — z2
kxy) z(kxy)ß
J = J dx J ydyNxy J 0 0 0
(L — zkXy)2
and
Ny = T 2(1 — y) + ym2 + Q2 + m2,
kly = —m2x2 y2 + Ny xy — T2, L = P 2xy — it, P2 = Q2 + m3, T2 = T 2y(1 — y) + m2y + m4(1 — y).
2.2. Integration with Respect to z
The integration with respect to z is straightforward:
dz
( L — zkx2 ) 2
(16)
( kx2 ) 2
ln(L — k2 ) — ln L +
k2 xy
L kly
(8)
In terms of Feynman variables the denominator takes the form
D = d0(1 - z)+dizy(1 - x)+ (9)
+ d2zxy + d3z(1 - y).
From expression (9) we derive
D = q2 — 2zqkxy + Lz — ie. (10)
Since —ie is an infinitesimal addition, it is possible to replace it by —iez and redefine L and D:
L = L — ie, D = q2 — 2zqkxy + Lz. (11)
The triple integral over Feynman parameters may be expressed by the same Eqs. (13), (16), (17) as given in [3]:
1 1 1
z
dz—-t^tt, (12)
2.3. Integration with Respect tox
In Eqs. (12), (16) we carry out, first of all, a change of variables
xy = x', ydx = dx', (17)
hence the ingredients become
Nxy = —2m2x + Ny, (18)
kXy = —m2x2 + Ny x — T2, L = P2x — ie.
y
The key identity is
N -—k2-y ~ dx yl
(19)
it allows to fulfill the integration by parts in full analogy with Section 2.3 of [3]:
y y y
I(y) = y dUV = UV — J dVU,
(20)
where we have introduced an infinitesimal parameter 5, because both parts in Eq. (20) diverge separately. Let
(21)
where we have neglected the light quark mass mu which does not lead to a mass singularity, and changed the notation of the other masses as follows:
mb — m1, mt — m2, MW — m3, md — m4. (13)
The ingredients entering Eq. (12) are
Nxy = -2kxy P2 = -2m2xy + Ny, (14)
L* = L — kly
be a quadratic expression in x:
L* = ALx2 + Blx + Cl = = Al(x — xli )(x — xl2 ),
(22)
with coefficients
Al = m2,
(23)
Bl = (T2 — m2)y — T2 — m2 + m2, Cl = T 2y(1 — y) + m\y + m|(1 — y) — it,
and discriminant
Dl = BL — 4AlCl.
(24)
(15)
Next, we introduce the following notation: L*\y = = L*(x = y,y), L*\o = L*(x = 0,y), and kxy\y = = k,x.y (x = y, y). The two binomials are
L*\y = m^y + ml(1 - y) - ie, (25)
2
z
0
1
0
d
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kXy\y = Q2y — mi(1 — y). For Eq. (20) one has
du= dk2xy
BARDIN et al.
The quadratic trinomial CL (see (23)), Cl = aT y2 + br y + ct , aT = —T2, bT = T2 + m\ — m\, ct = m\ — ie,
(kxy)2
(26)
DT = bT — 4aT ct ,
dV p 2kXy
kxy dL
dx LL* (L*)2 dx and the integral I(y) becomes:
has the roots:
WZ!, =
br ±
2T 2
(33)
I(y) = —
1
kxy \y
[ln(L*\y) — ln(L\y)] —
y
-¿[ln(L*|0)-ln(L|i)]+ J
P2dx LL* '
After some more calculations we arrive at a one-dimensional integral over y, where the infinitesimal with parameter 5 cancels out.
2.4. Integration overy We return to the one-dimensional integral:
(27) The following integration with respect to y deviate from the presentation given in [3].
2.4.1. Splitting into three parts. Let us redistribute terms in Eqs. (29), (30) into three parts:
I (y)= Io (y)+Ii(y) + h(y), (34)
+
Io(y) = 1 f 1
kxy\y
+
(35)
i
J = j dyl(y), 0
y/drkv-uti v — vt2
(28)
\n[ 1
In (p2y~le
yio
2
m4 — ie
with integrand
1
Ii(yi) =
HV) = -T<rrlHL*\y) ~ HP'y)} + (29)
1 1 /Dryi
N(yi) +
k2
kxy \y
+
1
1
1
y — yT! y — yr2
Ip(y),
and
Ip(y) = HCl) + ln(L*\y) — 2 ln(P2y) + (30)
+
BL VdE
-In
2 CL + y(BL-VDE)Y CL J '
+ (M(yb y/Di) - M{y±, -y/Dl)
h(y2) = -Ii(y2).
For Ii(y{) and I2(y2) we have changed the variables,
yi = y -
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