Nucl.Phys.B v

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Nuclear Physics B 777 (2007) 134

Two-loop renormalization in the Standard Model.Part I: Prolegomena

Stefano Actis a, Andrea Ferroglia b,c, Massimo Passera d,Giampiero Passarino e,f,

a Deutsches Electronen Synchrotron, DESY, Platanenallee 6, 15738 Zeuthen, Germanyb Fakultt fr Physik, Albert-Ludwigs Universitt, Freiburg, Germany

c Institut fr Theoretische Physik, Universitt Zrich, Zrich, Switzerlandd Dipartimento di Fisica, Universit di Padova and INFN, Sezione di Padova, Italy

e Dipartimento di Fisica Teorica, Universit di Torino, ItalyfINFN, Sezione di Torino, Italy

Received 13 December 2006; received in revised form 14 March 2007; accepted 5 April 2007

Available online 27 April 2007

Abstract

In this paper the building blocks for the two-loop renormalization of the Standard Model are introduced

with a comprehensive discussion of the special vertices induced in the Lagrangian by a particular diago-

nalization of the neutral sector and by two alternative treatments of the Higgs tadpoles. Dyson resummed

propagators for the gauge bosons are derived, and two-loop WardSlavnovTaylor identities are discussed.

In part II, the complete set of counterterms needed for the two-loop renormalization will be derived. In

part III, a renormalization scheme will be introduced, connecting the renormalized quantities to an input

parameter set of (pseudo-)experimental data, critically discussing renormalization of a gauge theory withunstable particles.

2007 Published by Elsevier B.V.

DOI of companion paper (part II): 10.1016/j.nuclphysb.2007.03.043.

DOI of companion paper (part III): 10.1016/j.nuclphysb.2007.04.027. Work supported by MIUR under contract 2001023713_006 and by the European Communitys Marie Curie Research

Training Network under contract MRTN-CT-2006-035505 Tools and Precision Calculations for Physics Discoveries at

Colliders.* Corresponding author at: Dipartimento di Fisica Teorica, Universit di Torino, Italy.

E-mail addresses: [email protected] (S. Actis), [email protected](A. Ferroglia),

[email protected] (M. Passera), [email protected] (G. Passarino).

0550-3213/$ see front matter 2007 Published by Elsevier B.V.

doi:10.1016/j.nuclphysb.2007.04.021
http://dx.doi.org/10.1016/j.nuclphysb.2007.03.043http://dx.doi.org/10.1016/j.nuclphysb.2007.04.027mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.nuclphysb.2007.04.021http://dx.doi.org/10.1016/j.nuclphysb.2007.04.021mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.nuclphysb.2007.04.027http://dx.doi.org/10.1016/j.nuclphysb.2007.03.043


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2 S. Actis et al. / Nuclear Physics B 777 (2007) 134

PACS: 11.10.-z; 11.15.Bt; 12.38.Bx; 02.90.+p

Keywords: Feynman diagrams; Multi-loop calculations; Self-energy diagrams; Vertex diagrams

1. Introduction

In a series of papers we developed a strategy for the algebraic-numerical evaluation of two-

loop, two-(three-)leg Feynman diagrams appearing in any renormalizable quantum field theory.

In [1] the general strategy has been designed and in [2] a complete list of results has been derived

for two-loop functions with two external legs, including their infrared divergent on-shell deriv-

atives. Results for one-loop multi-leg diagrams have been shown in [3] and additional material

can be found in [4]. Two-loop three-point functions for infrared convergent configurations have

been considered in [5], two-loop tensor integrals in [6], two-loop infrared divergent vertices in[7]. As a by-product of our general program we have developed a set of FORTRAN/95 routines

[8] for computing everything which is needed, from standard A0, . . . , D0 functions [9] to two-

loop, two-(three-)point functions. This new ensemble of programs, which includes the treatment

of complex poles [10], will succeed the corresponding library ofTOPAZ0 [11].

The next step in our project has been to introduce all those elements which are necessary

for a complete discussion of the two-loop renormalization of the Standard Model (SM). In this

paper we introduce basic aspects of renormalization which are needed before the introduction

of counterterms. In part III we will present a detailed analysis of the counterterms with special

emphasis to the cancellation of ultraviolet poles with non-local residues (the so-called problemof overlapping divergences), while in part III we will deal with finite renormalization deriving

renormalization equations, up to two loops, that relate the renormalized parameters of the model

to an input parameter set, which always includes the fine structure constant and the Fermi

coupling constant GF. Renormalization with unstable particles will also be addressed.

Having provided a derivation of the elements which are essential for constructing a renormal-

ization procedure, we will proceed in computing a first set of pseudo-observables, including the

running e.m. coupling constant and the complex poles characterizing unstable gauge bosons.

Several authors have already contributed in developing seminal results for the two-loop renor-

malization of the SM [12]. Here we want to present our own approach, from fundamentals to

applications. The whole set of results is completely independent from other sources; further-more, we wanted to collect in a single place all the formulas and algorithms that can be used for

many applications and are never there when you need them.

The code GraphShot [13] synthesizes the algebraic component of the project (for alternative

approaches see Ref. [14] and references therein) from generation of diagrams, reduction of tensor

structures, special kinematical configurations, analytical extraction of ultraviolet/infrared poles

[7] and of collinear logarithms and check of WardSlavnovTaylor identities (hereafter WST

identities) [15]. The corresponding output is then treated by a FORTRAN/95 code, LoopBack

[8], which is able to exploit the multi-scale structure of two-loop diagrams. Future applications

will include H

and H

gg, to give an example.

It is worth noticing that there are other solutions to the problems discussed in this paper;

noticeably, one can choose to work in the background-field formalism [16]; here we only stress

that our solution has been extended up to the two loops and has been implemented in a complete

and stand-alone set of procedures for two-loop renormalization.
http://dx.doi.org/10.1016/j.nuclphysb.2007.04.027http://dx.doi.org/10.1016/j.nuclphysb.2007.04.027http://dx.doi.org/10.1016/j.nuclphysb.2007.04.027http://dx.doi.org/10.1016/j.nuclphysb.2007.04.027


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S. Actis et al. / Nuclear Physics B 777 (2007) 134 3

The outline of the paper is as follows. In Section 2 we discuss the role of tadpoles in a

spontaneously broken gauge theory presenting two alternative schemes in Sections 2.2 and 2.3.

Diagonalization of the neutral sector in the SM is derived in Section 3. WST identities are

discussed in Section 4. Dyson resummation is analyzed in Section 5. Bases relevant for renormal-ization are introduced in Section 6. New sets of Feynman rules, required by our renormalization

procedure, are given in Appendices AC. In our metric, space-like p implies p2 = p2 + p24 > 0.Also, it is p4 = ip0 with p0 real for a physical four-momentum. Furthermore, upper and lowerindices are not distinguished.

2. Higgs tadpoles

Tadpoles in a spontaneously broken gauge theory have been discussed by many authors (see,

for instance, [17]). Here we outline those aspects which are peculiar to our approach.

2.1. The basics

Following notation and conventions of Ref. [18], the minimal Higgs sector of the SM is pro-

vided by the Lagrangian

(1)LS = (DK)(DK) 2KK (/2)

KK2

,

where the covariant derivative is given by

(2)DK =

i2

gB aa i

2gB0

K,

g/g = sin / cos , is the weak mixing angle, a are the standard Pauli matrices, Ba is atriplet of vector gauge bosons and B0 a singlet. For the theory to be stable we must require

> 0. We choose 2 < 0 in order to have spontaneous symmetry breaking (SSB). The scalar

field in the minimal realization of the SM is

(3)K = 12

+ i0

2 + i1

,

where and the HiggsKibble fields 0, 1 and 2 are real. For 2 < 0 we have SSB, K0 = 0.

In particular, we choose

+i0 to be the component of K to develop the non-zero vacuum

expectation value (VEV), and we set 00 = 0 and 0 = 0. We then introduce the (physical)Higgs fields as H = v. The parameter v is not a new parameter of the model; its value mustbe fixed by the requirement that H0 = 0 (i.e. K0 = (1/

2)(v, 0)), so that the vacuum does

not absorb/create Higgs particles. To see how this works at the lowest order, consider the part of

LS containing the Higgs field:

(4)(1/2)(H )2

2/2

(H + v)2 (/8)(H + v)4.These terms generate vertices that imply absorption of H in the vacuum, namely those linear

in H,

(5)2v (/2)v3H,

which correspond to the vertex H . This vertex gives a non-zero value to the diagrams

with one H line, and thus a non-zero VEV. We will set it to zero, i.e. v = (22/)1/2 (or v = 0,but then, no SSB).


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2.2. The parameterh

2.2.1. Definitions and Lagrangian

More complicated diagrams contribute to

H0

in higher orders of perturbation theory. The

parameter v must then be readjusted such that H0 = 0. First of all, let us introduce the newbare parameters M (the W boson mass), MH (the mass of the physical Higgs particle) and h(the tadpole constant) according to the following definitions:

(6)

M= gv/2,M2H = v2,h = 2 + 2 v2,

v = 2M/g, = (gMH/2M)2,2 = h 12 M2H.

This parameter h is the same as H of [18] and h of [21]. The new set of (bare) parameters

is therefore g, g, M, MH and h instead ofg, g, , , v. Remember that h (like v) is not an

independent parameter. In terms ofg, g, M, MH and h, LIS becomes (in Ref. [18] some terms

have been dropped)

LIS = 2KK (/2)

KK2

= h

2M2

g2+ 2M

gH + 1

2

H2 + 20 + 2+

+ M2HM22g2

12

M2HH2

(7) g M2H

4MH

H2 + 20 + 2+ g2 M2H

32M2

H2 + 20 + 2+

2,

with

=(1

i2)/

2. Note that (

2KK) is the only term ofLS containing h (actually,

the only term of the whole SM Lagrangian). Let us now set h such that the VEV of the Higgsfield H remains zero to each order of perturbation theory.

2.2.2. h fixing at the lowest order

At the lowest order, the only diagram contributing to H0 is(8)H

originated by the term in LIS linear in H, (2hM/g)H. Therefore, at the lowest order we willsimply set h = 0.

2.2.3. h fixing up to one loopDefine h = h0 + h1 g2 + h2 g4 + . The lowest-order h fixing of the previous section

amounts to h0 = 0. At the one-loop level, two types of diagrams contribute to the Higgs VEVup to O(g):

(9)T0: + T1:where the empty blob in the second term symbolically indicates all the one-loop diagrams con-

taining a scalar field (H, , 0), a gauge field (Z, W), a FaddeevPopov ghost field (X+, X,XZ), or a fermionic field. As an example, consider only the one-loop diagram containing the H

field: T(H )1 ; if this were the only T1 diagram, in order to have H0 = 0 it should cancel with T0,i.e.

(10)(2 )4i

(H )h

2M

g

g 3M

2H

4Mi 2A0(MH) = 0,


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where i 2A0(m) = 4n

dnq/(q2 + m2 i ). The solution of this equation is h0 = 0 and

(11)(H )h1

= 1(2 )4i

T

(H )1

2Mg

= 1

16 2

3M2H8M2

A0(MH)

.

Of course, (H )h1

is just the contribution to h1 arising from the one-loop tadpole diagram con-

taining the H field; the complete expression for h1 in the R gauge is

h1 = 1

16 2

3

2A0(M) + 3

4c2A0(M0) + M2 +

M20

2c2

(12)+ M2H

8M2

A0(Z M0) + 2A0(WM)

+ 3M2H8M2

A0(MH)

f

m2f

M2A0(mf)

,

where M0 = M/c and mf are the Z and fermion masses, and c = cos .2.2.4. h vertices in one-loop calculations

Beyond the lowest order, h is not zero and the Lagrangian LIS

contains the following vertices

involving a h factor (h vertices, from now on):

(13)H (2 )4i(2Mh/g),(14)H H (2 )4i(h),(15)0 0 (2 )

4i(h),(16)

+

(2 )4i(

h)

(as usual, the combinatorial factors for identical fields are included; see Appendix D of Ref. [21]).

Note that only scalar fields appear in the h vertices. These h vertices must be included in

one-loop calculations. Consider, for example, the Higgs self-energy at the one-loop level. The

diagrams contributing to this O(g2) quantity are

(17)H H + H H,where the empty blob in the second term represents all the one-loop contributions (two possible

topologies). The first diagram, containing a two-leg h vertex, should not be forgotten, and plays

an important role in the Ward identities (see later). One should also include diagrams containingtadpoles:

(18)H H + H H,but these diagrams add up to zero as a consequence of our choice for h.

2.2.5. h fixing up to two loops

Up to terms ofO(g3

), H0 gets contributions from the following diagrams:T0: (1) +

T1: (1/2) +


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T2: (1/6) + (1/4) + (1/4) +

T3

:(1/2)

+T4: (1/4) + (1/2) +

T5: (1/2) + (1) +

T6: (1/4) + (1/2) +

T7: (1/8) + (1/2) + (1/2).

The coefficients in parentheses indicate the combinatorial factors of each diagram when all fields

are identical. Owing to our previous choice for h0 and h1 , all the reducible diagrams add up to

zero: T4 = T5 = T6 = T7 = 0. The equation

(19)

3i=0

Ti = 0

provides then h2 :

(20)h2 =1

(2 )4i

T2 + T32Mg 3

.

2.2.6. h vertices in two-loop calculations

The two-leg h vertices in Eqs. (14)(16) should be included in all the appropriate diagrams

at the two-loop level, while all graphs (up to two loops) containing tadpoles will add up to zero

as a consequence of our choice for h0 , h1 and h2 . Note that two-leg h vertices will also

appear in the O(g4) self-energies of fields which do not belong to the Higgs sector; for example,

in diagrams like these:

which are representative of the only two irreducible O(g4) Z self-energy topologies containing

h vertices (excluding tadpoles, of course).

2.3. The t scheme

2.3.1. Definitions and Lagrangian

Tadpoles do not depend on any particular scale other than their internal mass, and cancel in any

renormalized self-energy. However, they play an essential role in proving the gauge invariance

of all the building blocks of the theory. In order to exploit this option, we will now consider a


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slightly different strategy to set the Higgs VEV to zero. Instead of using Eq. (6), the h scheme,

we will define the new bare parameters M (the W boson mass), MH (the mass of the physicalHiggs particle) and t (the tadpole constant) according to the following t scheme:

(21)

M(1 + t) = gv/2,(MH)

2 = (2M/g)2,0 = 2 + 2 (2M/g)2,

v = 2M(1 + t)/g, = (gMH/2M)2,2 = 12 (MH)2.

The new set of bare parameters is therefore g, g, M, MH and t instead of g, g and , , v

or g, g M, MH and h. Remember that t (like v and h) is not an independent parameter.Note that, contrary to h, the parameter t appears in the Higgs doublet K via = H + v, withv = 2M(1 + t)/g. As a consequence, all three terms of the Lagrangian LS in Eq. (1) dependon this parameter. In particular, the interaction part ofLS becomes

(22)LIS = 2KK (/2)(KK)2

= (1 + t)2

1 t(2 + t)M 2H M 2

2g2 t(t + 1)(t + 2)

M 2H M

gH

12

M 2HH2 1

4M 2H t(t + 2)

3H2 + 20 + 2+

(23) g(1 + t)M 2H4M

H

H2 + 20 + 2+ g2 M 2H

32M2

H2 + 20 + 2+2

,

while the term ofLS involving (DK)

(DK), yields a (lengthy) t-independent expression(see Refs. [18] and [21]), plus the following terms containing t:

t

igs MW+ +W A sc Z

gM2

H

2W+ W

+

ZZ

c2

(24)

M 2

2(t + 2)

2W+ W

+

ZZ

c2

+ M

cZ0 + MW+ + MW +

,

where, as usual, W = (B1 iB2)/

2, and

(25)

Z

A

=

c ss c

B3

B0

.

Where else, in the SM Lagrangian, does the parameter t appear? Wherever v doesas it

can be readily seen from Eq. (21). Let us now quickly discuss the other sectors of the SM:

YangMills, fermionic, FaddeevPopov (FP) and gauge-fixing. The pure YangMills Lagrangian

obviously contains no t terms.

The gauge-fixing part of the Lagrangian, Lgf, cancels in the R gauges the gauge-scalar mix-

ing terms Z0 and W contained in the scalar LagrangianLS. These terms are proportional

to gv/2, i.e., to M(1 + t) in the t scheme, and to M in the h scheme. The gauge-fixing La-grangian Lgf is a matter of choice: we adopt the usual definition

(26)Lgf = C+C 12C2Z

1

2C2A,


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with

(27)

CA = 1A

A, CZ = 1Z

Z + Z M

c0, C = 1

WW

+ WM

(note: no t terms), thus canceling the LS g-independent gauge-scalar mixing terms proportional

to M, but not those proportional to Mt (appearing at the end of Eq. (24)), which are ofO(g2).Alternatively, one could choose M(1 + t) instead of M in Eq. (27), thus canceling all LSgauge-scalar mixing terms, both proportional to M and Mt, but introducing then new two-legt vertices. In this latter case, as M= M(1+t), the gauge fixing Lagrangian would be identicalto the one of the h scheme. We will not follow this latter approach. Of course it is only a matter

of choice, but the explicit form ofLgf determines the FP ghost Lagrangian.

The parameter t shows up also in the FP ghost sector. The FP Lagrangian depends on the

gauge variations of the chosen gauge-fixing functions CA, CZ and C

. If, under gauge transfor-

mations, the functions Ci transform as

(28)Ci Ci + (Mij + gLij )j ,with i = (A,Z,), then the FP ghost Lagrangian is given by

(29)LFP = Xi (Mij + gLij )Xj .With the choice for Lgf given in Eq. (26) (and the relation gv/2 = M(1 + t)) it is easy to checkthat the FP ghost Lagrangian contains the t terms

(30)LFP = (M)2

t

WX+X+ + WXX + Z XZXZ /c2+ ,

where the dots indicate the usual t-independent terms. Had we chosen Lgf with M(1 + t)

instead ofM in Eq. (27), additional t terms would now arise in the FP Lagrangian.In the fermionic sector, the tadpole constant t appears in the mass terms:

(31)v

2(uu + dd) = (1 + t)(muuu + mddd)

(v = 2M(1 + t)/g), where and are the Yukawa couplings, and mu, md are the masses ofthe fermions. The rest of the fermion Lagrangian does not contain t, as it does not depend on v.

The Feynman rules for vertices involving a t factor (t vertices) are listed in Appendix B,

dropping the primes over M and MH. In the t scheme, contrary to the h one, we have (many)two- and three-leg t vertices containing also non-scalar fields. Note that three-leg t vertices

introduce a fourth irreducible topology for O(g4) self-energy diagrams containing t vertices,

namely:

.

2.3.2. t up to one loop

Define t =

t0 +

t1

g2

+

t2g4

+ . As we did for

h, we will now fix the parameter

tsuch that the VEV of the Higgs field H remains zero order by order in perturbation theory. At the

lowest order, the only diagram contributing to H0 is the same one depicted in Eq. (8), whichorigins from the term in LIS linear in H, t(t + 1)(t + 2)(M 2HM/g)H. Therefore, at thelowest order we can simply set t = 0, i.e. t0 = 0.


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Up to one loop, the diagrams T0 and T

1 contributing to the Higgs VEV are analogous to T0and T1 appearing in Eq. (9), so that t1 can be set in analogy with h1 :

(32)t1 =1

(2 )4i T1

2MgM 2H

.

Note that T1 and T1 have the same functional form, but depend on different mass parameters.

2.3.3. t up to two loops

The two-loop t fixing slightly differs from the h one. Up to terms ofO(g3), H0 gets

contributions from the following diagrams:

T0: (1) +

T1: (1/2) +T2: (1/6) + (1/4) + (1/4) +

T3: (1/2) + (1/2),

plus reducible diagrams (analogous to those appearing in T4T7 of Section 2.4) which add up

to zero because of our choice for t0 and t1 . Note the new diagrams in T

3 , with three-leg tvertices, not present in the h case (T3). The parameter t2 can be set in the usual manner,

requiring

(33)

3i=0

Ti = 0 t2 =1

(2 )4i

T2 + T3

2Mg3M2H

3

22t1 .

Note that T1,2 and T1,2 have the same functional form (but depend on different mass parameters)while T3 and T3 are different.

2.4. h andt: two comments

Consider the (doubly-contracted) WST identity relating the Z self-energy ,ZZ(p), the 0self-energy oo (p), and the Z0 transition ,Zo (p) (see Section 4):

(34)pp ,ZZ(p) + M20 oo (p) + 2ipM0,Zo (p) = 0.Both in the h and t schemes, each of the three terms in Eq. (34) contains contributions from

the tadpole diagrams, but they add up to zero, within each term. For example, at the one-loop

level, the first term in Eq. (34) contains the tadpole diagrams

(35)Z Z and Z Z

which cancel each other. In the h scheme at the one-loop level, only the second term of the

l.h.s. of Eq. (34) includes a diagram with a two-leg h vertex (Eq. (15)), while in higher orders,

two-leg h vertices appear in all three terms. In the t scheme, all three terms of Eq. (34) contain


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the two-leg t vertices already at the one-loop level. Similar comments are valid for the WST

identity involving the W self-energy.

Concerning renormalization, the constraints imposed on h and t in the previous sections

are the renormalization conditions to insure that 0|H|0 = 0, also in the presence of radiativecorrections. In particular, the renormalized h,t parameters are (R)h,t = h,t + h,t = 0. Theequivalent of Eqs. (6) and (21) for the renormalized parameters are just the same equations with

the tadpole constants set to zero. In the h scheme, the one-loop renormalization of the W and

Z masses involves the diagrams

(36)(a) (b) (c) .

(Diagrams (a) have two possible loop topologies.) Both (a) and (b) are gauge-dependent, but

their sum is gauge-independent on-shell. However, as we choose the h tadpole (c) to cancel (b),

the mass counterterm contains only (a) and is therefore gauge-dependent. On the contrary, in thet scheme, the one-loop renormalization of the W and Z masses involves the diagrams

(37)(a) (c) (b) (d) .

Once again, both (a) and (b) diagrams are gauge-dependent, their sum is gauge-independent

on-shell, and the t tadpole (d) is chosen to cancel (b). But, the mass counterterm is now gauge-

independent, as it contains both (a) and the two-leg t vertex diagram (c) (which is missing in

the h case).

3. Diagonalization of the neutral sector

3.1. New coupling constant in the h scheme

The Z transition in the SM does not vanish at zero squared momentum transfer. Although

this fact does not pose any serious problem, not even for the renormalization of the electric

charge, it is preferable to use an alternative strategy. We will follow the treatment of Ref. [22].

Consider the new SU(2) coupling constant g, the new mixing angle and the new W mass M in

the h scheme:

g = g(1 + ), g = (sin / cos )g,(38)v = 2M/g, = (gMH/2M)2, 2 = h 1

2M2H

(note: g sin / cos = g sin / cos ), where = 1g2 + 2g4 + is a new parameter yet to bespecified. This change of parameters entails new A and Z fields related to B

3 and B

0 by

(39)

Z

A

=

cos sin sin cos

B3

B0

.

The replacement g g(1 + ) introduces in the SM Lagrangian several terms containing thenew parameter . In our approach is fixed, order-by-order, by requiring that the Z transition

is zero at p2 = 0 in the = 1 gauge. Let us take a close look at these terms in each sector ofthe SM.


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The pure YangMills Lagrangian

(40)LYM = 14

Fa Fa

1

4F0 F

0 ,

with Fa = Ba Ba + g abc BbBc and F0 = B0 B0, contains the following new terms when we replace g by g(1 + ):

LYM = igc

Z

W+ W W+ W

ZW+ W W W+ + Z

W+ W

W W+

igs AW+ W W+ W A

W+ W

W W+

+ AW+ W W W+

+ g2 (2 + )

1

2W+ W

W

+ W

W+ W W+ W

+ c2

ZW+ Z W

ZZW+ W

+ s2 AW+ A W AAW+ W

(41)+ s c

AZ

W+ W + W+ W

2AZW+ W

,

where s = sin and c = cos . As these terms are ofO(g3) or O(g4), they do not contribute tothe calculation of self-energies at the one-loop level, but they do beyond it.

The Lagrangian LS, Eq. (1), contains several new terms when we employ the relationg = g(1 + ) and the h scheme of Eqs. (38). They can be arranged in the following threeclasses

(42)LS,h = L(nf=2)S,h + L(nf=3)S,h + L

(nf=4)S,h ,

according to the number of fields (nf) appearing in each interaction term (indicated by the su-

perscript in parentheses). The explicit expressions, up to terms ofO(g4), are

L(nf=2)S,h = M

1

2Ms2 AA

1

2M

2 + c2

ZZ

Ms

c

1 + c2

AZ + 0

s A + c Z

(43) M(2 + )W+ W + W + + W+

,

L(nf=3)S,h = g

MH

ZZ + s

cAZ + 2W+ W

+ 12

(s A + c Z)

H 0 0H + i+ i+

+ iW+ +W

s MA

s2 /c

MZ + 12

0

+ 12

W +

H + i 0+ 12

W+

H i0

(44) 12

H

+W + W+

,


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L(nf=4)S,h =

g2

2

1

2

H2 + 20

ZZ + s

cAZ + 2W+ W

+ +2s2 AA + 1 2c

2ZZ + (s /c 4s c )AZ 2W+ W + + s A s2 /cZ

(45) 0+W + W+ iH+W W+

.

The interaction part of the scalar Lagrangian, LIS = 2KK (/2)(KK)2, does not in-duce terms; these are only originated by the term involving the covariant derivatives,

(DK)(DK). On the other hand, as M/g = M/g, the h terms induced by LIS are givenby Eq. (7) expressed in terms of the ratio M/g of the barred parameters.

We choose the gauge-fixing Lagrangian Lgf of Eq. (26) with the following gauge functions:

(46)

CA = 1A

A, CZ = 1Z

Z + Z Mc

0, C = 1W

W + WM.

This R gauge -independent Lgf cancels the zeroth order (in g) gauge-scalar mixing terms in-

troduced byLS, but not those proportional to . Had one chosen gauge-fixing functions Eqs. (46)

with unbarred quantities, all the gauge-scalar mixing terms ofLS would be canceled, including

those proportional to , but additional new vertices would also be introduced.

New terms are also originated in the FaddeevPopov ghost sector. Studying the gaugetransformations (Eq. (28)) of the gauge-fixing functions CA, CZ and C defined in Eqs. (46), theadditional new terms of the FP Lagrangian (which is defined in Eq. (29)) in the

hscheme

are:

(47)LFP,h = L(nf=2)FP,h + L(nf=3)FP,h ,

where the two-field terms are

(48)L(nf=2)FP,h = M2

ZXZ

XZ + s

cXA

+ W(X+X+ + XX)

,

and the three-field terms are

L(nf=3)FP,h =

gi c W

+( XZ/Z )X ( X+/W)XZ

+ is W+

(XA/A)X (X+/W)XA

+ ic W

(X/W)XZ (XZ/Z )X+

+ is W

(X/W)XA (XA/A)X+

+ ic Z

(X+/W)X+ (X/W)X

+ is A

(X+/W)X+ (X/W)X

+ 12

WMi0(X+X+ XX) H (X+X+ + XX)+ 1

2cZMXZ[iX+ iX+ s H XA c H XZ]

(49)+ i2

WM

X(c XZ + s XA) X++(c XZ + s XA)

.


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(The bars over the FP ghost fields indicate conjugation. Obviously, the new FP fields XA and XZshould also be denoted with the bar indicating the field rediagonalization, just like the new fields

A and Z. However, this notation would be confusing and we will leave this point understood.)

Note that the FP ghostgauge boson vertices are simply the usual ones with g replaced by g.This is not the case, in general, for the FP ghostscalar terms.

Finally, the fermionic sector. The fermiongauge boson Lagrangian

Lf G = i2

2g

W+ u(1 + 5)d+ W d(1 + 5)u

(50)+ i2c

gZf

I3 2Qfs2 + I35

f + igsQfAf f(where I3 = 1/2 is the third component of the weak isospin of the fermion f, and Qf itscharge in units of

|e

|) becomes, under the replacement g

g(1

+ ) and the , A and Z

redefinitions,

LfG = i2

2g(1 + )W+ u(1 + 5)d+ W d(1 + 5)u

+ i2c

gZf

I3 2Qfs2 + I35

f + igs QfAf f

(51)+ i2

g (s A + c Z)I3f (1 + 5)f.The new neutral and charged current vertices are immediately recognizable. The CKM matrix

has been set to unity.

The fermion-scalar Lagrangian does not induce terms. Indeed, the Yukawa couplings and

in

(52)LfS = LKuR LKcdR + h.c.(where Kc = i2K is the conjugate Higgs doublet) are set by v/

2 = mu and v/

2 = md.

As v = 2M/g, it is = gmu/

2M and = gmd/

2M, and no appears in Eq. (52).

The Feynman rules for all these new vertices are listed in Appendix C, up to terms of

O(g4). Those corresponding to the pure YangMills Lagrangian (Eq. (41)) are not listed, as they

are identical to the usual YangMills ones, except for the replacement g g in the three-legvertices, and g2 g2 (2 + ) in the four-leg ones. In Appendix C, all bars over the symbols(indicating rediagonalization) have been dropped, except over g.

3.2. New coupling constant in the t scheme

The t scheme equations corresponding to Eqs. (38) are the following

g = g(1 + ), g = (sin / cos )g,(53)v = 2M(1 + t)/g, = (gMH/2M)2, 2 =

1

2(MH)

2.

(Note: g sin / cos = g sin / cos .) The analysis of the terms presented in the previous sec-tion for the h scheme can be repeated for the t scheme using Eqs. (53) instead of Eqs. (38).

The new fields A and Z are related to B3 and B

0 by Eq. (39). Thus, we obtain the following

results:


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The replacement g g(1 + ) in the pure YangMills sector introduces new verticescollected in LYM, which does not depend on the parameters of the h,t schemes. LYM has

already been given in Eq. (41).

The new terms introduced inL

S by Eqs. (53) can be arranged once again in the threeclasses

(54)LS,t = L(nf=2)S,t + L(nf=3)S,t + L

(nf=4)S,t ,

according to the number of fields appearing in the terms. The explicit expression for L(2)S,t is,

up to terms ofO(g4),

L(nf=2)S,t = M

1

2Ms2 AA

1

2M

2 + c2 + 4t

ZZ

M s

c

1 + c2

+ 2tAZ + 0(s A + c Z)(1 + t)(55) M(2 + + 4t)W+ W +

W

+ + W+

(1 + t)

with s = sin and c = cos , while, up to the same O(g4),(56)L

(nf=3,4)S,t = L

(nf=3,4)S,h (M M)

(L(nf=3)S,h

and L(nf=4)S,h

are given in Eqs. (44) and (45)). The subscripts t and h indicate the

t and h schemes. Note the presence of t factors in the new terms of Eq. (55). We will

comment on this in Section 3.3. Our recipe for gauge-fixing is the same as in the previous sections: we choose the R gaugeLgf to cancel the zeroth order (in g) gauge-scalar mixing terms introduced by LS, but not those

of higher orders (see discussions in Sections 2.3.1 and 3.1). Here, this prescription is realized by

Lgf (Eq. (26)) with

(57)

CA = 1A

A, CZ = 1Z

Z + Z M

c0, C = 1

WW

+ WM,

clearly -independent. The new terms of the FP ghost Lagrangian in the t scheme are:

(58)LFP,t = L(nf=2)FP,t + L(nf=3)FP,t ,where the two-field terms are

(59)L(nf=2)FP,t = (1 + t)M 2

Z XZ

XZ + s

cXA

+ W(X+X+ + XX)

,

and the three-field terms are the same as in the h scheme, with M replaced by M: L(nf=3)FP,t =

L(nf=3)FP,h (M M) (Eq. (49)). Like in the scalar sector, the and t factors are entangled. We conclude this analysis with the fermionic sector. As in the YangMills case, the fermion

gauge boson Lagrangian LfG

does not depend on the parameters of the h

or t

schemes. Its

expression in terms of the new coupling constant g contains new terms and is given in Eq. (51).

The neutral sector rediagonalization induces no terms in the fermion-scalar Lagrangian LfS(Eq. (52)), which contains, however, the t vertices discussed in Section 2.3 (Eq. (31)) (the ratio

M/g is now replaced by the identical ratio M/g).


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The Feynman rules for all vertices are listed in Appendix C, up to terms ofO(g4). All

primes and bars over A, Z, M, MH and have been dropped (but not over g). As we men-

tioned at the end of the previous section, the vertices of the pure YangMills sector need not

be listed.

3.3. The t mixing

A comment on the presence of t factors in the new vertices is now appropriate. Con-

sider the Lagrangian LS. As we already pointed out in Section 3.1, the interaction part LIS =

2KK (/2)(KK)2 does not induce terms, but gives rise to t terms: as M/g = M/g,these t terms are simply given by Eq. (23) expressed in terms of M/g instead of M/g. Onthe other hand, the derivative part ofLS, (DK)(DK), induces both and t vertices, plusmixed ones which we still call vertices (see the t factors in the two-leg terms ofL

(nf=

2)

S,t ).It works like this: first, we replace g g(1 + ) and g g(s /c ) in (DK)(DK),splitting the result in two classes of terms, both written in terms of g, with or without . Then

we substitute in both classes v 2M(1+t)/g: the class containing is, up to terms ofO(g4),LS,t (Eq. (54)), and includes also t factors, while the class free of has the same t vertices

as Eq. (24) with g, , M, A and Z replaced by g, , M, A and Z. The and t terms ofthe FaddeevPopov sector are intertwined just as in the case of the scalar Lagrangian.

3.4. Summary of the special vertices

The upshot of this first part of the paper lies in Appendices AC. There the readers find the full

set of SM (up to O(g4)) and h,t special vertices in the R gauges. All primes and bars over

A, Z, M, MH and have been dropped, but not over g, the SU(2) coupling constant of the

rediagonalized neutral sector. The readers can pick their preferred tadpole scheme, h or t, and

compute their Feynman diagrams including the h,t vertices ofAppendices A or B, respectively.

If they prefer to work with the rediagonalized neutral sector, they should simply replace g by g

in the h,t vertices, and add to them the ones ofAppendix C. There, vertices are listed for

the t scheme (note that and t terms are intertwinedsee Section 3.3); just set t = 0 to usethe h scheme instead.

Finally, Table 1 graphically summarizes which of the SM sectors provides each type of special

vertex. Note the overlap of and t terms in the scalar and FaddeevPopov sectors.

Table 1

Special vertices in the Standard Model

Sector h t

Scalar: (DK)(DK)

Scalar: 2KK + (/2)(KK)2 YangMills Gauge-fixingFaddeevPopov Fermiongauge boson FermionHiggs


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4. WST identities for two-loop gauge-boson self-energies

The purpose of this section is to discuss in detail the structure of the (doubly-contracted)

WardSlavnovTaylor identities (WSTI) for the two-loop gauge-boson self-energies in the SM,focusing in particular on the role played by the reducible diagrams. This analysis is performed in

the t HooftFeynman gauge.

4.1. Definitions and WST identities

Let ij be the sum of all diagrams (both one-particle reducible and irreducible) with two

external boson fields, i and j , to all orders in perturbation theory (as usual, the external Born

propagators are not to be included in the expression for ij )

(60)ij = n=1

g2n

(16 2)n(n)ij .

In the subscripts of the quantities (n)ij we will also explicitly indicate, when necessary, the

appropriate Lorentz indices with Greek letters. At each order in the perturbative expansion it is

convenient to make explicit the tensor structure of these functions by employing the following

definitions:

(61)(n),VV = D(n)V V + P(n)V Vpp , (n),V S= ipMSG(n)V S, (n)SS = R(n)SS ,

where the subscripts V and S indicate vector and scalar fields, MS is the mass of the HiggsKibble scalar S, and p is the incoming momentum of the vector boson (note:

(n),SV = (n),V S).

The quantities Dij , Pij , Gij , and Rij depend only on the squared four-momentum and are sym-

metric in i and j . Furthermore, D and R have the dimensions of a mass squared, while G and P

are dimensionless.

The WST identities require that, at each perturbative order, the gauge-boson self-energies

satisfy the equations

pp (n),AA = 0,

pp (n)

,AZ +ipM0

(n)

,A0 =0,

pp (n),ZZ + M20 (n)00 + 2ipM0

(n),Z0

= 0,(62)pp

(n),WW + M2(n) + 2ipM(n),W = 0,

which imply the following relations among the form factors D, P, G, and R

(63)D(n)AA + p2P(n)AA = 0,

(64)D(n)AZ + p2P(n)AZ + M20 G(n)A0 = 0,

(65)p2D(n)

ZZ +p4P

(n)

ZZ +M20 R

(n)

00 = 2M20 p

2G(n)

Z0

,

(66)p2D(n)W W + p4P(n)W W + M2R(n) = 2M2p2G(n)W .

The subscripts A, Z, W, and 0 clearly indicate the SM fields. We have verified these WST

Identities at the two-loop level (i.e. n = 2) with our code GraphShot [13].


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4.2. WST identities at two loops: the role of reducible diagrams

At any given order in the coupling constant expansion, the SM gauge-boson self-energies sat-

isfy the WSTI (62). For n 2, the quantities (n)

ij contain both one-particle irreducible (1PI)and reducible (1PR) contributions. At O(g4), the SM

(n)ij

functions contain the following irre-

ducible topologies: eight two-loop topologies, three one-loop topologies with a t1 vertex, four

one-loop topologies with a 1 vertex, and one tree-level diagram with a two-leg O(g4) t or

vertex (see figure at the end of Section 2.3.1). Reducible O(g4) graphs involve the product of

two O(g2) ones: two one-loop diagrams, one one-loop diagram and a tree-level diagram with

a O(g2) two-leg vertex insertion, or two tree-level diagrams, each with a O(g2) two-leg vertex

insertion. There are also O(g4) topologies containing tadpoles but, as we discussed in previous

sections, their contributions add up to zero as a consequence of our choice for t.

In the following we analyze the structure of the O(g4) WSTI for photon, Z, and W self-

energies, as well as for the photonZ mixing, emphasizing the role played by the reduciblediagrams.

4.2.1. The photon self-energy

The contribution of the 1PR diagrams to the photon self-energy at O(g4) is given, in the

t HooftFeynman gauge, by (with obvious notation)

(67)(2)R,AA =

1

(2 )4i

1

p2

(2)R,AA +

1

p2 + M20

(2)R,AA

,

where

(2)R,AA = (1),AA(1),AA, (2)R,AA = (1),AZ(1),ZA + (1),A0

(1),0A

.

It is interesting to consider separately the reducible diagrams that involve an intermediate photon

propagator ((2)R,AA) and those including an intermediate Z or 0 propagator (

(2)R,AA). By

employing the definitions given in the previous subsection and Eq. (63) with n = 1, one verifiesthat 2R,AA obeys the photon WSTI by itself,

(68)pp (2)R,AA = p2

D

(1)AA + p2P(1)AA

2 = 0.

This is not the case for (2)R,AA, although most of its contributions cancel when contracted by

pp as a consequence of Eq. (64) (n = 1),

(69)pp (2)R,AA = p2M20

p2 + M20

G

(1)A0

2.

The only diagrams contributing to the A0 mixing up to O(g2) are those with a W or FP

ghosts loop, and the tree-level diagram with a insertion. Their contribution, in the t Hooft

Feynman gauge, is

(70)G(1)A0

=(2 )4is c 2B0p

2, M , M +16 21.

A direct calculation (e.g. with GraphShot) shows that this residual contribution of the reducible

diagrams to the O(g4) photon WSTI, Eq. (69), is exactly canceled by the contribution of the

O(g4) irreducible diagrams, which include two-loop diagrams as well as one-loop graphs with a

two-leg vertex insertion.


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4.2.2. The photonZ mixing

We now consider the second of Eqs. (62) for n = 2. Reducible diagrams contribute to bothAZ and A0 transitions. Following the example of Eq. (67), we divide these contributions in

two classes: the diagrams that include an intermediate photon propagator and those mediated bya Z or a 0, namely, for the photonZ transition in the t HooftFeynman gauge,

(2)R,AZ =

1

(2 )4i

1

p2

(2)R,AZ +

1

p2 + M20

(2)R,AZ

,

(2)R,AZ = (1),AA(1),AZ,

(71)(2)R,AZ = (1),AZ(1),ZZ + (1),A0

(1),0Z

,

and, for the photon0 transition in the same gauge,

(2)R,A0 = 1(2 )4i

1p2

(2)R,A0 + 1p2 + M20(2)R,A0

,

(2)R,A0

= (1),AA(1),A0 ,(72)

(2)R,A0

= (1),AZ(1),Z0 + (1),A0

(1)00

.

The reducible diagrams with an intermediate photon propagator satisfy the WSTI by themselves.

Indeed,

(73)pp (2)R,AZ + iM0p(2)R,A0 = 0,

as it can be easily checked using Eq. (63) with n = 1. On the contrary, the remaining reduciblediagrams must be added to the irreducible O(g4) contributions in order to satisfy the WSTI for

the photonZ mixing:

(74)pp

(2)R,AZ

(2 )4i(p2 + M20 )+ (2)I,AZ

+ iM0p

(2)R,A0

(2 )4i(p2 + M20 )+ (2)I,A0

= 0.

4.2.3. The Z self-energy

Also in the case of the WSTI for the O(g4) Z self-energy it is convenient to separate the

reducible contributions mediated by a photon propagator from the rest of the reducible diagrams.

In the t HooftFeynman gauge it is

(2)R,ZZ =

1

(2 )4i

1

p2

(2)R,ZZ +

1

p2 + M20

(2)R,ZZ

,

(2)R,ZZ = (1),ZA(1),AZ,

(75)(2)R,ZZ = (1),ZZ (1),ZZ + (1),Z0

(1),0Z

,

(2)R

,Z0 =1

(2 )4i 1

p2

(2)R

,Z0 +1

p2 + M20

(2)R

,Z0,

(2)R,Z0

= (1),ZA(1),A0 ,(76)

(2)R,Z0

= (1),ZZ(1),Z0 + (1),Z0

(1)00

,


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(2)R00

= 1(2 )4i

1

p2

(2)R00

+ 1p2 + M20

(2)R00

,

(2)R00

=

(1),0A

(1),A0

,

(77)(2)R00

= (1),0Z (1),Z0

+ (1)00 (1)00

,

and, once again, the reducible diagrams mediated by a photon propagator satisfy the WSTI by

themselves, i.e.

(78)pp (2)R,ZZ + M20 (2)R00 + 2ipM0

(2)R,Z0

= 0,as it can be easily checked using the one-loop WSTI for the photon Z mixing (Eq. (64) with

n = 1).

4.2.4. The W self-energyAll the O(g4) 1PR contributions to the WSTI for the W self-energy are mediated, in the

t HooftFeynman gauge, by a charged particle of mass M. A separate analysis of their contri-

bution does not lead, in this case, to particularly significant simplifications of the structure of

the WSTI. However, some cancellations among the reducible terms occur, allowing to obtain a

relation that will be useful in the discussion of the Dyson resummation of the W propagator. The

1PR quantities that contribute to the O(g4) WSTI for the W self-energy have the following form:

(2)R,WW =

1

(2 )4i(p2 + M2) D(1)W W2 + pp2D(1)W WP(1)W W + p2P(1)W W2 + M2G(1)W 2,

(2)R,W =

ipM(2 )4i(p2 + M2) G

(1)W

D

(1)W W + p2P(1)W W + R(1)

,

(79)(2)R =

1

(2 )4i(p2 + M2)

p2M2

G(1)W

2 + R(1) 2.Contracting the free indices with the corresponding external momenta, summing the three con-

tributions and employing Eq. (66) with n = 1, we obtain(2 )4ipp (2)R

,WW +M2

(2)R

+2ipM

(2)R

,W

(80)= p2M2G(1)W 2 R(1) D(1)W W + p2P(1)W W.5. Dyson resummed propagators and their WST identities

We will now present the Dyson resummed propagators for the electroweak gauge bosons.

We will then employ the results of Section 4 to show explicitly, up to terms ofO(g4), that the

resummed propagators satisfy the WST identities.

Following definition Eq. (60) for ij , the function Iij represents the sum of all 1PI diagrams

with two external boson fields, i and j , to all orders in perturbation theory (as usual, the external

Born propagators are not to be included in the expression for Iij

). As we did in Eq. (61), we

write explicitly its Lorentz structure,

(81)I,VV = DIV V + PIV Vpp , I,V S= ipMSGIV S, ISS = RISS,


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where V and S indicate SM vector and scalar fields, and p is the incoming momentum of

the vector boson (note: I,SV = I,V S). We also introduce the transverse and longitudinalprojectors

t = ppp2

, l = ppp2

,

tt = t , ll = l , tl = 0,(82)I,VV = DIV Vt + LIV Vl, LIV V = DIV V + p2PIV V.

The full propagator for a field i which mixes with a field j via the function Iij is given by the

perturbative series

ii

=ii

+ii

n=0

n+1

l=1 kl

Ikl

1klkl kl

(83)= ii + ii Iii ii + ii

k1=i,jIik1 k1k1

Ik1i

ii + ,

where k0 = kn+1 = i , while for l = n + 1, kl can be i or j . ii is the Born propagator of thefield i . We rewrite Eq. (83) as

(84)ii = ii

1 ()ii1

,

and refer to ii as the resummed propagator. The quantity ()ii is the sum of all the possible

products of Born propagators and self-energies, starting with a 1PI self-energy Iii , or transition

Iij , and ending with a propagator ii , such that each element of the sum cannot be obtained as a

product of other elements in the sum. A diagrammatic representation of()ii is the following,

()ii = + + + where the Born propagator of the field i (j ) is represented by a dotted (solid) line, the white blob

is the i 1PI self-energy, and the dots at the end indicate a sum running over an infinite number of

1PI j self-energies (black blobs) inserted between two 1PI ij transitions (gray blobs).

It is also useful to define, as an auxiliary quantity, the partially resummed propagator for the

field i , ii , in which we resum only the proper 1PI self-energy insertions I

ii , namely,

(85)ii = ii

1 Iii ii1

.

If the particle i were not mixing with j through loops or two-leg vertex insertions, ii would

coincide with the resummed propagator ii . ii can be graphically depicted as

ii = + + + .Partially resummed propagators allow for a compact expression for ()ii ,

(86)()ii = Iii ii +

Iij jj

Ij i ii ,

so that the resummed propagator of the field i can be cast in the form

(87)ii = ii

1 Iii + Iij jj Ij iii 1.


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We can also define a resummed propagator for the ij transition. In this case there is no corre-

sponding Born propagator, and the resummed one is given by the sum of all possible products

of 1PI i and j self-energies, transitions, and Born propagators starting with ii and ending with

jj . This sum can be simply expressed in the following compact form,

(88)ij = ii Iij jj .

5.1. The charged sector

We now apply Eqs. (85), (87), (88) to W and charged Goldstone boson fields. The partially

resummed propagator of the charged Goldstone scalar follows immediately from Eq. (85). The

Born W and propagators in the t HooftFeynman gauge are

(89)

W W =

p2 + M2 , =1

p2 + M2 ,where, for simplicity of notation, we have dropped the coefficients (2 )4i . In the same gauge,

the partially resummed and W propagators are

(90) =

1 I 1 = p2 + M2 RI1,

(91)W W =

1

p2 + M2 DIW W

+ p

p PIW W

p2 + M2 DIW W p2PIW W

.

Eq. (91) assumes a more compact form when expressed in terms of the transverse and longitudi-

nal projectors t and l ,

(92)W W =

t

p2 + M2 DIW W+ l

p2 + M2 LIW W.

The resummed W and propagators can be then derived from Eq. (87),

(93) =

p2 + M2 RI p2M2(GIW )

2

p2 + M2 LIW W

1,

(94)

W W =t

p2 + M2 DIW W + l

p2

+ M2

LIW W

p2M2(GIW )2

p2 + M2 RI1

.

The resummed propagator for the W transition is provided by Eq. (88),

(95)W =

ipMGIWp2 + M2 RI

p2 + M2 LIW W

p2M2(GIW )2

p2 + M2 RI

1.

We will now show explicitly, up to terms ofO(g4), that the resummed propagators defined above

satisfy the following WST identity:

(96)pp

W W + ipM

W ip MW + M

2

= 1,which, in turn, is satisfied if

(97)p2M2

GIW 2 + M2RI + p2LIW W RI LIW W + 2p2M2GIW = 0.


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This equation can be verified explicitly, up to terms ofO(g4), using the WSTI for the W self-

energy: at O(g2) Eq. (97) becomes simply

(98)M2R(1)

+p2L

(1)W W

+2p2M2G

(1)W

=0,

which coincides with Eq. (66) for n = 1. To prove Eq. (97) at O(g4) we can combine the last ofEqs. (62) with n = 2 and Eq. (80) to get1

(99)p2M2

G(1)W

2 + M2R(2)I + p2L(2)IW W R(1) L(1)W W + 2p2M2G(2)IW = 0.5.2. The neutral sector

The SM neutral sector involves the mixing of three boson fields, A, Z and 0. As the

definitions for the resummed propagators presented at the beginning of Section 5 refer to the

mixing of only two boson fields, we will now discuss their generalization to the three-field case.Consider three boson fields i, j and k mixing up through radiative corrections. For each of

them we can define a partially resummed propagator ll (l = i,j, or k) according to Eq. (85).For each pair of the three fields, say (j,k), we can also define the following intermediate propa-

gators

(100)jj (j,k) = jj

1 Ijj + Ij k kk Ikjjj 1,(101)j k (j,k) = jj (j,k)Ij k kk ,

where the parentheses on the l.h.s. indicate the chosen pair of fields. [kk (j,k) and kj (j,k)

can be simply derived from the above definitions by exchanging j k.] The reader will imme-diately note that the r.h.s. of the above Eqs. (100), (101) are almost identical to those of Eqs. (87),(88), with the appropriate renaming of the fields. Eqs. (100), (101), introduced in the context of

three-field mixing, define however only intermediate propagators (denoted by the tilde), while

Eqs. (87), (88), presented in the analysis of the two-field mixing case, define the complete re-

summed propagators (denoted by the bar). Indeed, the definition of full resummed propagator in

the three-field mixing scenario requires one further step: the resummed propagator for a field i

mixing with the fields j and k via the functions Iij

, Iik and Ij k

can be cast in the following

form

(102)ii = ii1 Iii +l,m

Iil lm(j,k)

Imi

ii1

,

where l and m can be j or k, while the resummed propagator for the transition between the fields

i and k is

(103)ik = ii

l=j,kIil lk (j,k).

Armed with Eqs. (100)(103), we can now present the A, Z and AZ propagators. First of

all, the Born A, Z and 0 propagators in the t HooftFeynman gauge are

(104)AA =

p2, ZZ =

p2 + M20, 00 = 1

p2 + M20,

1 For simplicity of notation, in this section we dropped the coefficients (2 )4i.


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where, for simplicity of notation, we have dropped once again the coefficients (2 )4i. The

partially resummedpropagators (three) can be immediately computed via Eq. (85) and the inter-

mediate ones (twelve) via Eqs. (100) and (101). Finally, after some algebra, Eqs. (102) and (103)

provide us with the fully resummed propagators: V V = tT

V V + lL

V V, with V = A, Zand

(105)TAA =

p2 DIAA (DIAZ)

2

p2 + M20 DIZZ

1,

(106)TZZ =

p2 + M20 DIZZ (DIAZ)

2

p2 DIAA

1,

(107)TAZ = DIAZ

p2 DIAA

p2 + M20 DIZZ DIAZ21.

These specific results are not new since they have already been documented in earlier studies,

noticeably in Refs. [19,20]. The expressions of the longitudinal components of these propagatorsare more lengthy and we will only present them up to terms ofO(g4):

(108)LAA =

p2 +Og61,(109)LZZ =

p2 + M20 LIZZ

(LIAZ)2

p2

p2M20 (GIZ0

)2

p2 + M20+Og6

1,

(110)LAZ =LIAZ

p2(p2 + M20 LIZZ )+ M

20

(p2 + M20 )2GIA0 G

IZ0

+Og6.Eq. (108) achieves its compact form due to the use of the WSTI (63) and (64) with n = 1, 2. AlsoEq. (110) has been simplified using L

(1)AA = 0 (i.e. Eq. (63) with n = 1). We point out that if we

use the one-loop WSTI for the photon self-energy, Eq. (63), the transverse part of the resummed

AZ propagator becomes, up to terms ofO(g4),

(111)TAZ = DIAZ

p2

1 + PIAA

p2 + M20 DIZZ1 +Og6,

thus showing a pole at p2 = 0 ifDIAZ(p2 = 0) were not vanishing because of the rediagonaliza-tion of the neutral sector.

In order to show explicitly, up to terms ofO(g4), that the above resummed propagators satisfy

their WSTI, we also present the resummed propagators involving the neutral HiggsKibble scalar0:

(112)A0

= ip M0p2

GIZ0

LIAZ

(p2 + M20 )2+

GIA0

p2 + M20 RI00

+Og6,

(113)Z0

= ipM0

p2 + M20 LIZZ

GIA0

LIAZ

p2(p2 + M20 )+

GIZ0

p2 + M20 RI00

+Og6,

(114)00 =

p2 + M20 RI00 M20 GIA0

2 p2M20

p2

+M2

0G

IZ0

21

+Og6

.With these results, and with the WSTI (63)(65), (Eqs. (74) and (78)), we can finally prove, up

to O(g4), the following WSTI for the resummed A, Z and AZ propagators,

(115)pp AA = 1,


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(116)pp AZ + ipM0A0 = 0,

(117)pp ZZ + M20 00 + 2ipM0Z0 = 1.

6. The LQ basis

For the purpose of the renormalization, it is more convenient to extract from the quantities

defined in the previous sections the factors involving the weak mixing angle . To achieve this

goal, we employ the LQ basis [18], which relates the photon and Z fields to a new pair of fields,

L and Q:

(118)

Z

A

=

c 0

s 1/s

L

vs Q

.

Consider the fermion currents j

A and j

Z coupling to the photon and to the Z. As the Lagrangianmust be left unchanged under this transformation, namely jZ Z + j A A = j L L + j QQ, the

currents transform as

(119)

j

Z

jA

=

1/c s2 /c0 s

j

L

jQ

.

If we rewrite the SM Lagrangian in terms of the fields L and Q, and perform the same transfor-

mation (Eq. (118)) on the FP ghosts fields (from (XA,XZ) to (XL,XQ)), then all the interaction

terms of the SM Lagrangian are independent of . Note that this is true only if the relation

M/c = M0 is employed, wherever necessary, to remove the remaining dependence on . In thisway the dependence on the weak mixing angle is moved to the kinetic terms of the L and Qfields which, clearly, are not mass eigenstates.

The relevant fact for our discussion is that the couplings of Z, photon, XZ and XA are related

to those of the fields L and Q, XL and XQ by identities like the ones described, in a diagrammatic

way, in the following figure:

= 1c

s2

c

= sc

s3

c.

As the couplings of the fields L, Q, XL and XQ do not depend on , all the dependence on this

parameter is factored out in the coefficients in the r.h.s. of these identities.

Since appears only in the couplings of the fields A, Z, XA and XZ (once again, the relation

M/c = M0 must also be employed, wherever necessary), it is possible to single out this parame-ter in the two-loop self-energies of the vector bosons. Consider, for example, the transverse part

of the photon two-loop self-energy D(2)AA

(which includes the contribution of both irreducible and

reducible diagrams). All diagrams contributing to D(2)AA can be classified in two classes: those

including (i) one internal A, Z, XA or XZ field, and (ii) those not containing any of these fields.

The complete dependence on can be factored out by expressing the external photon couplings

and the internal A, Z XA or XZ couplings of the diagrams of class (i) in terms of the couplings


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of the fields L, Q, XL and XQ, namely

(120)D(2)AA = s2

1

c2

fAA1 + fAA2 + s2 fAA3

,

where the functions fAAi (i = 1, 2, 3) are -independent. Similarly, we can factor out the dependence of the transverse part of the two-loop photonZ mixing and Z self-energy,

(121)D(2)AZ =

s

c

1

c2fAZ1 + fAZ2 + s2 fAZ3 + s4 fAZ4

,

(122)D(2)ZZ =

1

c2

1

c2fZZ1 + fZZ2 + s2 fZZ3 + s4 fZZ4 + s6 fZZ5

,

where, once again, the functions f

AZ

i and f

ZZ

i (i = 1, . . . , 5) do not depend on . Analogousrelations hold for the longitudinal components of the two-loop self-energies.We note that D

(2)AZ and D

(2)ZZ also contain a third class of diagrams containing more than one

internal Z (or XZ ) field (up to three, in D(2)ZZ

). However, the diagrams of this class involve the

trilinear vertex ZH Z (or XZH XZ), which does not induce any new dependence.

However, from the point of view of renormalization it is more convenient to distinguish be-

tween the dependence originating from external legs and the one introduced by internal legs.

We define, to all orders,

DAA = s2

QQ;extp2

= s2

n=1

g2

16 2

n

(n)

QQ;extp2

,

DAZ = sc

AZ;ext = sc

n=1

g2

16 2

n

(n)AZ;ext,

(123)DZZ = 1c2

ZZ;ext = 1c2

n=1

g2

16 2

n

(n)ZZ;ext,

(n)AZ

;ext

=

(n)3Q

;ext

s2

(n)QQ

;extp

2,

(124)(n)ZZ;ext = (n)33;ext 2s2 (n)3Q;ext + s4 (n)QQ;extp2.

Furthermore, our procedure is such that

(125)(n)3Q;ext = (n)3Q;extp2,

with (n)3Q;ext regular at p

2 = 0. At O(g2) the external quantities are -independent while, atO(g4) the relation with the coefficients of Eqs. (120)(122) is

(2)QQ;extp

2

=1

c2f

AA1 + f

AA2 + f

AA3 s

2 ,

(2)3Q;ext =

1

c2

fAA1 + fAZ1

fAA1 + fAZ2 + s2 fAA2 + fAZ3 + s4 fAA3 + fAZ4 ,


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(2)33;ext =

1

c2

fAA1 + 2fAZ1 + fZZ1

fAA1 2fAZ1 + fZZ2 + s2 fAA1 + 2fAZ2 + fZZ3

(126)+ s4 fAA

2 + 2fAZ3 + fZZ4 + s6 f

AA3 + 2fAZ4 + fZZ5 ,

and s , c in Eq. (126) should be evaluated at O(g0), in any renormalization scheme, for two-

loop accuracy.

Consider the process f f hh; taking into account Dyson resummed propagators and ne-glecting, for the moment, vertices and boxes we write

M(f f hh)= (2 )4i

e2QfQh

TAA +eg

2cQf

(vh + ah5)TZA

(127)

+eg

2c Qh

(vf + af5)

TZA +

g2

4c2

(vf + af5)

(vh + ah5)TZZ

where f and h are fermions with quantum numbers QI, I3i , i = f, h; furthermore we haveintroduced

(128)vf = I3f 2Qfs2 , af = I3f,with e2 = g2s2 . Always neglecting terms proportional to fermion masses it is useful to introducean effective weak-mixing angle as follows:

(129)s2eff= s2

1 AZ;ext1 s2 AA;ext

, Vf = I3f 2Qfs2eff.

The amplitude of Eq. (127) can be cast into the following form:

M(f f hh) = (2 )4i

11 s2 AA;ext

e2QfQh

p2

(130)+ g2

4c2(Vf + af5) (Vh + ah5)TZZ

.

The functions AA;ext, AZ;ext and ZZ;ext start at O(g2) in perturbation theory. Eq. (130)shows the nice effect of absorbingto all ordersnon-diagonal transitions into a redefinition of

s2 and forms the basis for introducing renormalization equations in the neutral sector, e.g. the one

associated with the fine-structure constant . Questions related to gauge-parameter independence

of Dyson resummation, e.g. in Eq. (129), are not addressed here, but we will present a detailed

discussion in part III, where their relevance will be investigated.

The results in Eqs. (127)(130) have already been documented in earlier studies, noticeably

in Refs. [19] and [20].

7. Conclusions

In this paper we prepared the ground to perform a comprehensive renormalization procedure

of the Standard Model at the two-loop level; with minor changes our results can be extended to

an arbitrary gauge theory with spontaneously broken symmetry.
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The same set of problems that we encountered in this paper may receive different answers; for

instance, one could decide to work in the background-field method and treat differently the prob-

lem of diagonalization of the neutral sector in the SM. Our solution has been extended beyond

one-loop and it is an integral part of a renormalization procedure which goes from fundamen-tals to applications. The whole set of new Feynman rules of our appendices has been coded in

GraphShot and has proven its value in several applications, including the proof of the WST

identities.

In this paper we outlined peculiar aspects of tadpoles in a spontaneously broken gauge

theory and extended beyond one-loop a strategy to diagonalize the neutral sector of the SM,

order-by-order in perturbation theory. The obtained results have been used as the starting point

in the construction of the renormalized Lagrangian of the SM and in the computation of

(pseudo-)observables up to two loops.

Acknowledgements

We gratefully acknowledge several important discussions with Dima Bardin, Ansgar Denner,

Stefan Dittmaier and Sandro Uccirati. The work of A.F. was supported in part by the Swiss

National Science Foundation (SNF) under contract 200020-109162.

Appendix A. Feynman rules for h vertices

In this appendix we present the new set of diagrammatic rules induced by our approach. The

Feynman rules for the h vertices are extremely simple and can be immediately derived from

Eq. (7):

H 2Mh/gH H h0 0 h+ h,

where h = h1 g2 + h2 g4 + and M is the bare W mass. If working with the rediagonalizedneutral sector, simply replace g by g. Multiply each vertex by a factor (2 )4i . As usual, we have

included the combinatorial factors for identical fields (see Appendix D of Ref. [21]).

Appendix B. Feynman rules for t vertices

In this appendix we present the t vertices. They can be read off the Lagrangian terms of

Eqs. (23), (24), (30) and (31), including the combinatorial factors for identical fields. Also, t =t1 g

2 + t2 g4 + . Simply replace g by g if working with the rediagonalized neutral sector. Thetwo-leg t vertices are:

H H 3M2H/2t(t + 2)0 0

M2H/2t(t +

2)

+

M2H/2

t(t + 2)Z Z M20 t(t + 2)W+ W

M2t(t + 2)


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Z 0 ipM0t

W+ ipMt

W + ipMt

f f mftX+ X+ WM2tX X WM2tXZ XZ ZM20 t

where M0 = M/ cos and t(t + 2) = 2t1 g2 + (2t1 + 2t2 )g4 + O(g6). Each vertex must bemultiplied by the usual factor (2 )4i.

The three-leg t vertices are:

gt

3M2H/2M

gt

M2H/2M

gt

M2H/2M

+ igts M

igts M

igts2 M0

+ igts2 M0

gtM

gt

M/c2


29/401


where s = sin , c = cos and, once again, each vertex must be multiplied by the factor (2 )4i.The t tadpole H is:

(2 )4

i

M2HM

1

g t(t + 1)(t + 2)= (2 )4iM2HM2t1 g 32t1 + 2t2

g3 + Og5.

Appendix C. Feynman rules for vertices

In this appendix we present the vertices. The new vertices introduced by the replacement

g g(1 + ) in the SM Lagrangian are listed here up to terms ofO(g4) in the R gauges. Allprimes and bars over A, Z, M, MH and have been dropped, except over g. Also, =1

g2

+2

g4

+ . As usual, each vertex must be multiplied by the factor (2 )4i. The following

two-leg vertices are in the t scheme. For the h scheme, just set t = 0.A A

g4s2 M

221

Z Z 2

g2M21 + g4M2

2 + 21t1 + c2 21 /2

A Z (s /c )

g2M21 + g4M2

2 + 21t1 + c2 21

W+ W 2

g2M21 + g4M2

2 + 21t1 + 21 /2

A 0 ips M

g21 + g4(2 + 1t1 )

Z 0 ipc M

g

2

1 + g4

(2 + 1t1 )W+ ipM

g21 + g4(2 + 1t1 )

W + ipM

g21 + g4(2 + 1t1 )

X+ X+ WM2

g21 + g4(2 + 1t1 )

X X WM2

g21 + g4(2 + 1t1 )

XZ XZ ZM2

g21 + g4(2 + 1t1 )

XZ XA

Z(s /c )M

2

g21

+ g4(2

+1t1 )

The three-leg vertices are (all momenta are flowing inwards):

g31[2M]

g31(s /c )M

g

3

1[2M]

g31(is /2)(q k)


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g31(s /2)(q k)

g31(ic /2)(q k)

g31(c /2)(q k)

g31[is M]

g3

1[is M]

g31is 2 M/c

g31

is 2 M/c

g31(q

k)/2

g31(q k)/2

g31i(q k)/2

g31i(q

k)/2.

The trilinear vertices with FP ghosts are:

g31s p /W

g31(s p /W)

g3

1c p /W

g31(c p /W)


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g31(c p /W)

g31(s p /W)

g31(c p /Z)

g31(s p /A)

g31c p /Z

g31c p /W

g31s p /A

g31s p /W

g31(iMW/2)

g31(iMW/2)

g31(MW/2)

g31(MW/2)

g31(iMZ /2c )

g31(iMZ /2c )

g3

1(s MZ /2c )

g31(MZ/2)


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g31(ic MW/2)

g31(is MW/2)

g31(ic MW/2)

g31(is MW/2).

The three-leg vertices introduced by the pure YangMills Lagrangian are not listed here as

they can be immediately derived from the usual YangMills vertices (see, e.g., Appendix D ofRef. [21]) by simply replacing g g.

The trilinear vertices with fermions are:

g31(is I3/2)(1 + 5)

g31(ic I3/2)(1 + 5)

g31(i/2

2)(1 + 5)

g31(i/2

2)(1 + 5).

The four-leg vertices are:

g41

g41(s /2c )

g41

g41

g41(s /2c )

g41


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g412s2

g41

1 2c2

g41(s /2c 2s c )

g41

g41(s /2)

g41(s /2)

g41(is /2)

g41(is /2)

g41s2 /2c

g41s2 /2c

g41

is 2 /2c

g41is2 /2c .

The four-leg vertices introduced by the pure YangMills Lagrangian are not listed here as

they can be immediately derived from the usual YangMills vertices (see, e.g., Appendix D of

Ref. [21]) by simply replacing g2 g2 (2 + ).

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Nuclear Physics B 777 (2007) 3599

Two-loop renormalization in the Standard Model.Part II: Renormalization procedures

and computational techniques

Stefano Actis

a

, Giampiero Passarino

b,c,

a Deutsches Electronen Synchrotron, DESY, Platanenallee 6, 15738 Zeuthen, Germany

b Dipartimento di Fisica Teorica, Universit di Torino, Italyc INFN, Sezione di Torino, Italy

Received 13 December 2006; accepted 14 March 2007

Available online 19 April 2007

Abstract

In part I general aspects of the renormalization of a spontaneously broken gauge theory have been in-

troduced. Here, in part II, two-loop renormalization is introduced and discussed within the context of the

minimal Standard Model. Therefore, this paper deals with the transition between bare parameters and fields

to renormalized ones. The full list of one- and two-loop counterterms is shown and it is proven that, by a

suitable extension of the formalism already introduced at the one-loop level, two-point functions suffice in

renormalizing the model. The problem of overlapping ultraviolet divergencies is analyzed and it is shown

that all counterterms are local and of polynomial nature. The original program of t Hooft and Veltman

is at work. Finite parts are written in a way that allows for a fast and reliable numerical integration with

all collinear logarithms extracted analytically. Finite renormalization, the transition between renormalized

parameters and physical (pseudo-)observables, will be discussed in part III where numerical results, e.g.for the complex poles of the unstable gauge bosons, will be shown. An attempt will be made to define the

running of the electromagnetic coupling constant at the two-loop level.

2007 Published by Elsevier B.V.

DOI of companion paper (part I): 10.1016/j.nuclphysb.2007.04.021.

DOI of companion paper (part III): 10.1016/j.nuclphysb.2007.04.027. Work supported by MIUR under contract 2001023713_006 and by the European Communitys Marie-Curie Research

Training Network under contract MRTN-CT-2006-035505 Tools and Precision Calculations for Physics Discoveries at

Colliders.* Corresponding author at: Dipartimento di Fisica Teorica, Universit di Torino, Italy.

E-mail addresses: [email protected] (S. Actis), [email protected] (G. Passarino).

0550-3213/$ see front matter 2007 Published by Elsevier B.V.

doi:10.1016/j.nuclphysb.2007.03.043
http://dx.doi.org/10.1016/j.nuclphysb.2007.03.021http://dx.doi.org/10.1016/j.nuclphysb.2007.03.043http://dx.doi.org/10.1016/j.nuclphysb.2007.03.027mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.nuclphysb.2007.03.043http://dx.doi.org/10.1016/j.nuclphysb.2007.03.043mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.nuclphysb.2007.03.027http://dx.doi.org/10.1016/j.nuclphysb.2007.03.043http://dx.doi.org/10.1016/j.nuclphysb.2007.03.021


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36 S. Actis, G. Passarino / Nuclear Physics B 777 (2007) 3599

PACS: 11.10.-z; 11.15.Bt; 12.38.Bx; 02.90.+p; 02.60.Jh; 02.70.Wz

Keywords: Renormalization; Feynman diagrams; Multi-loop calculations

1. Introduction

After the end of the Lep era it became evident that including estimates of higher-order ra-

diative corrections into one-loop calculations for physical (pseudo-)observables could not satisfy

anymore the need for precision required by the forthcoming generation of experiments. Since

LHC will be an arena for discovery physics, high precision will not be mandatory during its first

phase. However, according to some predestinate design, hadron machines are alternating with

electronpositron ones and, hopefully, ILC will come into operation: here the highest available

theoretical precision will play a fundamental role.As a matter of fact, it is not clear what kind of scenario will arise after the first few months of

running at LHC. Any evidence of new physics will lead to a striking search for new theoretical

models and their Born predictions; the quake could be so strong to remove any interest in quan-

tum effects of the Standard Model. On the contrary, we could be back to the familiar landscape:

effects of new physics hidden inside loops.

Since we have no firm opinion, we decided to follow the old rule si vis pacem para bellum,

building the environment that allows for a complete two-loop analysis of a spontaneously broken

gauge field theory. This construction requires several elements, and it is difficult to characterize

our strategy with a single acronym; although our work implies a lot of analytical aspects, the

final step (computing arbitrary two-loop diagrams) can only be done with an algebraic-numericalapproach.

If one thinks for a while, everything is in the old papers of t Hooft and Veltman [1], but

translating few formal properties into a working scheme is far from trivial. Most of the times it

is not a question of How do I do it?, rather it is a question of bookkeeping: Can I do it without

exhausting the memory of my computer?, or, Is there any practical way of presenting my results

besides making my codes public?.

We devoted a first paper [2] (hereafter part I) to deal with general aspects of a spontaneously

broken gauge theory. First of all, we showed how to treat tadpoles; although everybody knows

how to do it, general results are rarely presented in a way that everyone can use them. In addition,

we analyzed how to perform an order-by-order diagonalization of the neutral sector of a theoryof fundamental interactions; once again, one needs a comprehensive collection of results which

allows for practical applications.

Alternative solutions to solving these problems exist, noticeably in the background-field for-

malism [3] (compare also with Ref. [4]); our claim here is restricted to the construction of a set

of procedures which do not rely on other sources and cover broadly all the aspects, from genera-

tion of diagrams and renormalization to evaluation (mostly numerical) of physical processes. In

particular, our results show that the structure of the counterterms at the two-loop level, as well

as of the whole set of renormalized Green functions and the WardSlavnovTaylor identities [5],

has (in the conventional approach) a degree of simplicity comparable to the one obtained at one

loop in the background-field approach.

Another perennial question is: what about renormalization, with or without counterterms? In

a way, it is a fake question, since the two approaches are fully equivalent as far as S-matrix

elements are concerned. In this paper we focus on the transition from bare parameters to renor-


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malized ones, and in a third paper (hereafter part III) we will then discuss the ultimate step in

any renormalization procedure: the transition from renormalized parameters to a set of physical

(pseudo-)observables. Perhaps, one should try to make a clear vocabulary of renormalization in

quantum field theory. A renormalization procedure is designed to bring you from a Lagrangianto theoretical predictions; it includes regularization (nowadays dimensional regularization [1] is

easy to understand), a renormalization scheme and the choice of an input parameter set. The

scheme, being a transitory step, is almost irrelevant; it can be on-mass-shell or MS or based on

complex poles, but unless you do something illegal (resummations that are not allowed or similar

things) it really does not matter. Admittedly, one can define MS quantities as convenient tools,

but it is the last step that matters, at least as long as we have a convenient subtraction point (which

we miss in QCD).

Renormalized quantities should always be expressed in terms of a set of physical quantities.

One may indulge to the introduction of an MS running electromagnetic coupling constant (im-

porting a concept from QCD to QED, which sounds strange anyway) but, at the end of the day,only cross sections matter.

In this paper, we have done one thing: all the two-loop Greens functions of the theory are

made finite by introducing non-logarithmic counterterms and respecting unitarity. In addition,

one can easily check that renormalized WardSlavnovTaylor identities are satisfied. Actually,

we have done more, since all ultraviolet-finite parts have been classified and an algorithm has

been designed for their evaluation at any scale. What is innovative in our approach, as well as in

other modern approaches [6], is the idea that everything can be generated (is generated) by a set

of automatized procedures which deals satisfactorily with the somewhat greater complexity of a

two-loop calculation.

Furthermore, classification of ultraviolet divergencies is dynamically linked to a well-definedcomputational scheme. In other words, in our approach, the ultraviolet-finite parts are back in

their privileged position where they can play the role of creating the predictive power of the

theory. There are, of course, preliminary steps (not always the easy ones), but it is only the full

control on the multi-scale level that pays off.

The outline of this paper is as follows. After introducing our notation and conventions in

Section 2, we outline the strategy of our calculation in Section 3, where we classify all the

ultraviolet-divergent parts of the needed loop integrals. Next, in Section 4, we choose the gauge-

fixing Lagrangian and we define our renormalization scheme. In Section 5 we review one-loop

renormalization, and in Section 6 we show explicit results for tadpole renormalization and

neutral-sector diagonalization at two loops. Finally, we analyze the ultraviolet structure of two-loop self-energies in Section 7 and we show explicit results for the two-loop counterterms in

Section 8. Section 9 contains the summary, and several technical details connected with the rel-

evant kinematical limits, which represent the backbone of our computational techniques, are

discussed in Appendices AE.

2. Notation and conventions

Regularization. We employ dimensional regularization [1], denoting the number of the

spacetime dimensions by n = 4. In addition, we use a short-hand notation for regularization-dependent factors,

(1)UV = + ln + ln M2

2, UV(x) = UV ln M

2

x,


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38 S. Actis, G. Passarino / Nuclear Physics B 777 (2007) 3599

where = 0.5772156 . . . is the Euler constant, is the t Hooft unit of mass, M stands forthe bare (renormalized) W-boson mass (we do not distinguish unless strictly needed) and x is

a positive-definite kinematical variable. In our conventions the logarithm has a cut along the

negative real axis and it is understood that for all masses: M

2

M2

i , with 0+.Masses. We introduce a compact notation for ratios of squared masses,

(2)xH =M2HM2

, xl,i =m2l,i

M2, xu,i =

m2u,i

M2, xd,i =

m2d,i

M2.

Here MH is the Higgs-boson bare (renormalized) mass and ml,i , mu,i and md,i are the bare

(renormalized) masses of the charged lepton and the up and down quarks of the ith fermion

doublet, with i = 1, . . . , 3. We consider the minimal representation for the Standard Model scalarsector, defining M0 = M/c for the Z-boson bare (renormalized) mass. Here c (s ) is the bareor renormalized cosine (sine) of the weak-mixing angle . For notational clarity we frequently

employ the notation

(3){m}12...N = m1, m2, . . . , mN.

3. Outline of the calculation

Our calculation builds upon an automatic strategy for generating Feynman diagrams and eval-

uating the necessary one- and two-loop integrals. In addition, it does not rely on any black-box

tool: diagrams are generated through a set of FORM [7] routines implemented in the GraphShot

package [8] and loop integrals are computed through the FORTRAN/95 LoopBack [9] code.

The present work uses a set of results derived in a series of previous papers. The general strat-egy for handling multi-loop multi-leg Feynman diagrams was designed in Ref. [10] and the whole

body of results necessary for evaluating two-loop two-point integrals can be found in Ref. [11].

The calculation of two-loop three-point scalar integrals is considerably more involved: infrared-

convergent configurations are discussed in Ref. [12] and infrared- and collinear-divergent ones

are analyzed in Ref. [13]. Finally, our method for dealing with two-loop tensor integrals can be

found in Ref. [14] and results for one-loop multi-leg integrals are shown in Ref. [15].

It is worth noting that

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