Integrating out heavy particles with functional methods: a simplified framework
HJE
Integrating out heavy particles with functional methods: a simpli ed framework
Javier FuentesMart n 0 1 3
Jorge Portoles 0 1 3
Pedro RuizFemen a 0 1 2
0 Technische Universitat Munchen , D85748 Garching , Germany
1 Apt. Correus 22085, E46071 Valencia , Spain
2 Physik Department T31, JamesFranckStra e
3 Instituto de F sica Corpuscular , CSIC
4 Universitat de Valencia
We present a systematic procedure to obtain the oneloop lowenergy e ective Lagrangian resulting from integrating out the heavy We show that the matching coe cients are determined entirely by the hard region of the functional determinant involving the heavy elds. This represents an important simpli cation with respect the conventional matching approach, where the full and e ective theory contributions have to be computed separately and a cancellation of the infrared divergent parts has to take place. We illustrate the method with a descriptive toy model and with an extension of the Standard Model with a heavy real scalar triplet. A comparison with other schemes that have been put forward recently is also provided.
E ective eld theories; Beyond Standard Model

2
3
4
5
1 Introduction
The method
Examples
4.1
4.2
Conclusions
Comparison with previous approaches
Scalar toy model
Heavy real scalar triplet extension
A General expressions for dimensionsix operators
B The
uctuation operator of the SM
framework has pervaded the last fty years of research in particle physics.
Although the rationale and the procedure has been well developed long ago in the
literature (see for instance [1, 2]), the integration at nexttoleading order in the upper
theory, that is to say at one loop, is undergoing lately an intense debate [3{8] that, as we
put forward in this paper, still allows for simpler alternatives. There are two techniques to
obtain the Wilson coe cients of the EFT. The most employed one amounts to matching the
diagrammatic computation of given Green Functions with light particle external legs in the
full theory, where heavy states can appear in virtual lines, and in the EFT, at energies where
the EFT can describe the dynamics of the light particles as an expansion in inverse powers
of the heavy particle mass scale. Alternatively one can perform the functional integration of
{ 1 {
the heavier states without being concerned with speci c Green Functions, and later extract
the local contributions that are relevant for the description of the lowenergy dynamics of
the light elds. This last methodology was applied, for example, in refs. [9, 10], to obtain
the nondecoupling e ects of a heavy Higgs in the Standard Model (SM). The path integral
formulation has obvious advantages over the matching procedure as, for instance, one does
not need to handle Feynman diagrams nor symmetry factors, and one obtains directly the
whole set of EFT operators together with their matching conditions, i.e. no prior knowledge
about the speci cs of the EFT operator structure, symmetries, etc., is required.
One of the issues recently arisen involves the widely used technique to perform the
functional integration set up more than thirty years ago by the works of Aitchison and
Fraser [11{14], Chan [15, 16], Gaillard [17] and Cheyette [18]. As implemented by refs. [3,
4], this technique did not include all the oneloop contributions from the integration, in
particular those where heavy and light
eld quantum
uctuations appear in the same
loop. This fact was noticed in ref. [5], and
xed later on in refs. [7, 8], by the use of
variants of the functional approach which require additional ingredients in order to subtract
the parts of the heavylight loops which are already accounted for by the oneloop EFT
contribution. Here we would like to introduce a more direct method to obtain the oneloop
e ective theory that builds upon the works of refs. [9, 10], and that uses the technique
of \expansion by regions" [19{21] to read o the oneloop matching coe cients from the
full theory computation, thus bypassing the need of subtracting any infrared contribution.
In short, the determination of the oneloop EFT in the approach we propose reduces
to the calculation of the hard part of the determinant of e H , where e H arises from the
diagonalization of the quadratic term in the expansion of the full theory Lagrangian around
the classical eld con gurations, and the determinant is just the result of the Gaussian
integration over the heavy quantum
uctuations. In this way, the terms that mix light and
heavy spectra inside the loop get disentangled by means of a eld transformation in the path
integral that brings the quadratic
uctuation into diagonal form: the part involving only
the light quantum
elds remains untouched by the transformation and all heavy particle
e ects in the loops are shifted to the modi ed heavy quadratic form e H . This provides
a conceptually simple and straightforward technique to obtain all the oneloop local EFT
couplings from an underlying theory that can contain arbitrary interactions between the
heavy and the light degrees of freedom.
The contents of the paper are the following. The general outline of the method is
given in section 2, where we describe the transformation that diagonalizes the quadratic
uctuation which de nes e H , and then discuss how to extract the contributions from e H
that are relevant for determining the oneloop EFT. In section 3 we compare our procedure
with those proposed recently by [3, 7] and [4, 8]. The virtues of our method are better seen
through examples: rst we consider a simple scalar toy model in section 4, where we can
easily illustrate the advantages of our procedure with respect the conventional matching
approach; then we turn to an extension of the SM with a heavy real scalar triplet, that
has been used as an example in recent papers. We conclude with section 5. Additional
material concerning the general formulae for dimensionsix operators, and the expression
of the uctuation operator in the SM case is provided in the appendices.
{ 2 {
We outline in this section the functional method to determine the EFT Lagrangian
describing the dynamics of light particles at energies much smaller than mH , the typical mass of
a heavy particle, or set of particles, that reproduces the fulltheory results at the oneloop
level. The application of the method to speci c examples is postponed to section 4.
Let us consider a general theory whose eld content can be split into heavy ( H ) and
light ( L) degrees of freedom, that we collect generically in
= ( H ; L). For charged
degrees of freedom, the eld and its complex conjugate enter as separate components in H
and L. In order to obtain the oneloop e ective action, we split each eld component into
a background eld con guration, ^, which satisfy the classical equations of motion (EOM),
and a quantum
uctuation , i.e. we write
! ^ + . Diagrammatically, the background
part corresponds to tree lines in Feynman graphs while lines inside loops arise from the
elds; this means that terms higher than quadratic in the quantum
elds yield
vertices that can only appear in diagrams at higher loop orders. Therefore, at the oneloop
level one has to consider only the Lagrangian up to terms quadratic in :
L = L
tree(^) + L
(
2
) + O
The zeroth order term, Ltree, depends only on the classical eld con gurations and yields
the treelevel e ective action. At energies much lower than the mass of the heavy elds,
the background heavy elds ^H can be eliminated from the treelevel action by using their
EOM. The linear term in the expansion of L around the background elds is, up to a total
derivative, proportional to the EOM evaluated at
= ^, and thus vanishes. From the
quadratic piece
(2.1)
(2.2)
(2.3)
(2.4)
L
(
2
) =
2
O =
H XLyH !
;
XLH
L
we identify the uctuation operator O, with generic form
and which depends only on the classical elds ^.
The oneloop e ective action thus derives from the path integral
eiS = N
Z
D LD H exp i
Z
dx L(
2
) ;
which can be obtained by Gaussian integration. Our aim is to compute the oneloop heavy
particle e ects in the Green functions of the light
elds as an expansion in the heavy
mass scale mH . In terms of Feynman diagrams, the latter corresponds to computing all
oneloop diagrams involving heavy lines and expanding them in 1=mH . The latter can
be formally achieved by doing the functional integration over the elds
presence of mixing terms among heavy and light quantum
H . However, the
(
2
) (equivalently, of
elds in L
{ 3 {
oneloop diagrams with both heavy and light lines inside the loop), makes it necessary
to rst rewrite the
uctuation operator in eq. (2.3) in an equivalent blockdiagonal form.
A way of achieving this is by performing shifts (with unit Jacobian determinant) in the
elds, which can be done in di erent ways. We choose a
eld transformation
that shifts the information about the mixing terms XLH in the uctuation operator into
a rede nition of the heavyparticle block
H , while leaving
L untouched. This has the
advantage that all heavy particle e ects in the oneloop e ective action are thus obtained
through the computation of the determinant that results from the path integral over the
heavy elds. This shifting procedure was actually used in refs. [9, 10] for integrating out
the Higgs eld in the SU(
2
) gauge theory and in the SM. An alternative shift, which is
implicitly used in ref. [7], will be discussed in section 3.
The explicit form of the eld transformation that brings O into the desired
blockdiagonal form reads
and one immediately obtains
with
P =
P yOP =
I
L1XLH I
0
!
e H
0
0 !
L
;
;
e H =
XLyH L1XLH :
H
Z
The functional integration over the heavy elds H can now be carried out easily,
eiS =
det e H
c
N
D L exp i
Z
dx
1 y
2 L L L ;
with c = 1=2; 1 depending on the bosonic or fermionic nature of the heavy elds. For
simplicity, we assume that all degrees of freedom in the heavy sector are either bosons or
fermions. In the case of mixed statistics, one needs to further diagonalize e H to decouple
the bosonic and fermionic blocks. The remaining Gaussian integration in eq. (2.8)
reproduces the oneloop contributions with light particles running inside the loop, and heavy
elds can appear only as treelevel lines through the dependence of
L in ^H . We thus
de ne the part of the oneloop e ective action coming from loops involving heavy elds as
In order to compute the determinant of e H we use standard techniques developed in the
literature [15, 22]. First it is rewritten as
SH = i c ln det e H :
SH = i c Tr ln e H ;
{ 4 {
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
The derivatives in e H yields factors of ip upon acting on the exponentials.1 The symbol
tr denotes the trace over internal degrees of freedom only. Since e H contains the kinetic
term of the heavy elds, in the case of scalar elds it has the generic form
e H =
D^ 2
m2H
U ;
with D^ denoting the covariant derivative for the heavy elds with background gauge elds.
SH =
i
2
tr
Z
d Z
d x
ddp
(
2
)
d ln p
2
m2H
2ipD^
D^ 2
(2.13)
For fermions, the same formula, eq. (2.13), applies but with an overall minus sign and with
where Tr denotes the full trace of the operator, also in coordinate space. It is convenient
for our purposes to rewrite the functional trace using momentum eigenstates de ned in d
dimensions as
SH = i c tr
d h pj ln e H jpi
= i c tr
= i c tr
Z
Z
Z
ddp
(
2
)
d Z
d x
d Z
d x
ddp
(
2
)
ddp
(
2
)
d e ipx ln
d ln
(2.11)
(2.12)
(2.16)
i hD^= ; ei + i nD^= ; o
o
+ 2mH e +
( e
o) :
(2.14)
^
(
Here
e +
o is de ned by e H = iD=
mH
number of gamma matrices. Finally, we can Taylor expand the logarithm to get
, and
e ( o) contains an even (odd)
SH =
i Z
2
ddx X1 1 Z
n=1
n
ddp
(
2
)
d tr
2ipD^ + D^ 2 + U (x; @x + ip) !n )
p2
m2H
1
;
(2.15)
where we have dropped an irrelevant constant term, and the negative (positive) global sign
corresponds to the integration of boson (fermion) heavy elds.
The e ective action eq. (2.15) generates all oneloop amplitudes with at least one
heavy particle propagator in the loop. Oneloop diagrams with n heavy propagators are
reproduced from the nth term in the expansion of eq. (2.15). In addition the diagram can
contain light propagators, that arise upon expanding the term XLyH L1XLH in e H using
L
1 =
1
X (
1
)n
n=0
e L1XL
n
1
e L ;
1Note that e H can also depend in @x. Transpose derivatives are de ned from the adjoint operator,
which acts on the function at the left, and can be replaced by
@x, the di erence being a total derivative
as the whole diagonal of O.
which corresponds to the Neumann series expansion of
aration
L = e L + XL, with e L corresponding to the the
kinetic terms, i.e. e L1 is the light eld propagator. From the de nition of the uctuation
operator O, eq. (2.3), the terms in e L are part of the diagonal components of O. At the
L1, and we have made the
sep
uctuations coming from the
L1 using eq. (2.16) it is simpler to de ne e L directly
Loops with heavy particles receive contributions from the region of hard loop momenta
p
mH , and from the soft momentum region, where the latter is set by the lowenergy
scales in the theory, either p
mL or any of the lightparticle external momenta, pi
mH .
In dimensional regularization the two contributions can be computed separately by using
the socalled \expansion by regions" [19{21]. In this method the contribution of each region
is obtained by expanding the integrand into a Taylor series with respect to the parameters
that are small there, and then integrating every region over the full ddimensional space
of the loop momenta. In the hard region, all the lowenergy scales are expanded out and
only mH remains in the propagators. The resulting integrand yields local contributions in
the form of a polynomial in the lowenergy momenta and masses, with factors of 1=mH
to adjust the dimensions. This part is therefore fully determined by the shortdistance
behaviour of the full theory and has to be included into the EFT Lagrangian in order
to match the amplitudes in the full and e ective theories. Indeed, the coe cients of the
polynomial terms from the hard contribution of a given (renormalized) amplitude provide
the oneloop matching coe cients of corresponding local terms in the e ective theory.
This can be understood easily since the soft part of the amplitude results upon expanding
the vertices and propagators according to p
mL
mH , with p the loop momentum.
This expansion, together with the oneloop terms with light particles that arise from the
Gaussian integral of
L in eq. (2.8), yields the same oneloop amplitude as one would
obtain using the Feynman rules of the e ective Lagrangian for the light
elds obtained
by treelevel matching, equivalently the Feynman rules from Ltree in eq. (2.1) where the
background heavy eld ^H has been eliminated in favour of ^L using the classical EOM.
Therefore, in the di erence of the fulltheory and EFT renormalized amplitudes at
oneloop only the hard part of the fulltheory amplitude remains, and one can read o the
oneloop matching coe cients directly from the computation of the latter. Let us nally
note that in the conventional matching approach, the same infrared regularization has to
be used in the full and EFT calculations, in order to guarantee that the infrared behaviour
of both theories is identical. This is of course ful lled in the approach suggested here,
since the oneloop EFT amplitude is de ned implicitly by the full theory result. Likewise,
the ultraviolet (UV) divergences of the EFT are determined by UV divergences in the soft
part, that are regulated in d dimensions in our approach. For the renormalization of the
amplitudes, we shall use the MS subtraction scheme.
Translated into the functional approach, the preceding discussion implies that the EFT
Lagrangian at oneloop is then determined as
Z
ddx L1ElFoTop = SHhard ;
{ 6 {
(2.17)
HJEP09(216)5
where SHhard, containing only the hard part of the loops, can be obtained from the
representation (2.15) by expanding the integrand in the hard loopmomentum limit, p
mH
mL; @x. In order to identify the relevant terms in this expansion, it is useful to introduce
the counting
p ; mH
;
(2.18)
and determine the order
k, k > 0, of each term in the integrand of eq. (2.15). For a
given order in
only a
nite number of terms in the expansion contributes because U is
at most O( ) and the denominator is O(
2
).2 For instance, to obtain the dimensionsix
e ective operators, i.e. those suppressed by 1=m2H , it is enough to truncate the expansion
up to terms of O
2 , which means computing U up to O
4 (recall that d4p
4).
Though it was phrased di erently, this prescription is e ectively equivalent to the one used
in refs. [9, 10] to obtain the nondecoupling e ects (i.e. the O(m0H ) terms) introduced by a
SMlike heavy Higgs.
Finally we recall that, although the covariance of the expansion in eq. (2.15) is not
manifest, the symmetry of the functional trace guarantees that the
nal result can be
rearranged such that all the covariant derivatives appear in commutators [16, 23]. As a
result, one can always rearrange the expansion of eq. (2.15) in a manifestly covariant way
in terms of traces containing powers of U , eldstrength tensors and covariant derivatives
acting on them. As noted in refs. [17, 22, 23], this rearrangement can be easily performed
when U does not depend on derivatives, as it is the case when only heavy particles enter
in the loop.3 However, for the case where U = U (x; @x + ip), as it happens in general in
theories with heavylight loops, the situation is more involved and the techniques developed
in refs. [17, 22, 23] cannot be directly applied. In this more general case it is convenient
to separate U into momentumdependent and momentumindependent pieces, i.e. U =
UH (x)+ULH (x; @x + ip) which, at the diagrammatic level, corresponds to a separation into
pure heavy loops and heavylight loops. This separation presents two major advantages:
ULH at most O
rst, the power counting for UH and ULH is generically di erent, with UH at most O ( ) and
0 , both for bosons and fermions, which allows for a di erent truncation
of the series in eq. (2.15) for the terms involving only pure heavy contributions and those
involving at least one power of ULH . Second, universal expansions of eq. (2.15) in a
manifestly covariant form for U = UH (x) have been derived in the literature up to O
i.e. for the case of dimensionsix operators [3, 22, 24, 25], that we reproduce in eq. (A.2).
2 ,
The evaluation of the remaining piece, corresponding to terms containing at least one power
of ULH can be done explicitly from eq. (2.17).
2The part of the operator U coming from
H arises from interaction terms with at least three elds.
If all three
elds are bosons, the dimension4 operator may contain a dimensionful parameter
or a
derivative, giving rise to a term in U of O( ). If two of the
dimension 4 and then
Contributions from XLyH
0, which yields a contribution in U of O( ) upon application of eq. (2.14).
L1XLH , in the following referred as heavylight, appear from the product of two
interaction terms and a light eld propagator and hence they generate terms in U of O( 0).
3With the exception of theories with massive vector
elds and derivative couplings among two heavy
elds are fermions the operator is already of
and one light elds.
{ 7 {
Let us end the section by summarizing the steps required to obtain the oneloop
matching coe cients in our method:
1. We collect all eld degrees of freedom in L, light and heavy, in a eld multiplet
= ( H ; L), where i and ( i) must be written as separate components for charged
elds. We split the elds into classical and quantum part, i.e
! ^ + , and identify
the
and
uctuation operator O from the second order variation of L with respect to
evaluated at the classical eld con guration, see eqs. (2.2) and (2.3),
Oij =
HJEP09(216)5
of the form:
Z
ddp
2. We then consider U (x; @x), given in eqs. (2.12) and (2.14), with e H de ned in eq. (2.7)
The computation of U requires the inversion of
L: a general expression for the
latter is provided in eq. (2.16). The operator U (x; @x + ip) has to be expanded up to
a given order in , with the counting given by p; mH
dimensionsix EFT operators, the expansion of U must be taken up to O
4 .
3. The nal step consists on the evaluation of the traces of U (x; @x + ip) in eq. (2.15)
up to the desired order  O
2 for the computation of the oneloop dimensionsix
e ective Lagrangian {. For this computation it is convenient to make the separation
traces of UH (x), see eq. (A.2). The remaining contributions consist in terms involving
at least one power of ULH (x; @x + ip): a general formula for the case of dimensionsix
operators can be found in eq. (A.3). Their computation only requires trivial integrals
p 1 : : : p 2k
(
2
)d (p2)
p2
m2H
=
(
1
) + +k i
(4 ) 2
d
d2 + k
( )
d
2
d2 + k
k +
+
(2.20)
where g 1::: 2k is the totally symmetric tensor with 2k indices constructed from
g
tensors.
Terms containing open covariant derivatives, i.e. derivatives acting only at the
rightmost of the traces, should be kept throughout the computation and will either vanish
or combine in commutators, yielding gaugeinvariant terms with eld strength
tensors. A discussion about such terms can be found in appendix A.
3
Comparison with previous approaches
In ref. [7], a procedure to obtain the oneloop matching coe cients also using functional
integration has been proposed. We wish to highlight here the di erences of that method,
in the following referred as HLM, with respect to the one presented in this manuscript.
{ 8 {
The rst di erence is how ref. [7] disentangles contributions from heavylight loops
from the rest. In the HLM method the determinant of the
uctuation operator O which
de nes the complete oneloop action S is split using an identity (see their appendix B) that
is formally equivalent in our language to performing a eld transformation of the form
(3.1)
(3.2)
(3.3)
HJEP09(216)5
that blockdiagonalizes the uctuation operator as:
PHLM =
I
0
PHyLM OPHLM =
H1XLyH !
I
H
0
The functional determinant is then separated in the HLM framework into two terms: the
determinant of
H , that corresponds to the loops with only heavy particles, and the
determinant of e L, containing both the loops with only light propagators and those with
mixed heavy and light propagators. The former contributes directly to UH , and provides
part of the oneloop matching conditions (namely those denoted as \heavy" in ref. [7]),
upon using the universal formula valid for U not depending in derivatives, eq. (A.2), up
to a given order in the expansion in 1=mH . On the other hand, to obtain the matching
conditions that arise from e L (called \mixed" contributions in the HLM terminology), one
has to subtract those contributions already contained in the oneloop terms from the EFT
theory matched at treelevel. To perform that subtraction without computing both the
determinant of e L and that of the quadratic
uctuation of LtErFeeT, HLM argues that one
has to subtract to the heavy propagators that appear in the computation of det e L the
expansion of the heavy propagator to a given order in the limit mH ! 1. According to
HLM, the subtracted piece builds up the terms (\local counterparts") that match the loops
from LtErFeeT. These \local counterparts" have to be identi ed for each order in the EFT, and
then dropped prior to the evaluation of the functional traces. This prescription resembles
the one used in ref. [25] to obtain the oneloop e ective Lagrangian from integrating out a
heavy scalar singlet added to the SM.
While we do not doubt the validity of the HLM method, which the authors of
ref. [7] have shown through speci c examples, we believe the framework presented in this
manuscript brings some important simpli cations. Let us note rst that in the method of
ref. [7], contributions from heavylight loops are incorporated into det e L, which results
from the functional integration over the light elds. If the light sector contains both bosonic
and fermonic degrees of freedom that interact with the heavy sector (as it is the case in
most extensions of the SM), a further diagonalization of e L into bosonic and fermionic
blocks is required in order to perform the Gaussian integral over the light elds. That step
is avoided in our approach, where we shift all heavy particle e ects into e H and we only
{ 9 {
need to perform the path integral over the heavy elds. Secondly, our method provides
a closed formula (up to trivial integrations which depend on the structure of ULH ) valid
for any given model, from which the matching conditions of all EFT operators of a given
dimension are obtained. In this sense it is more systematic than the subtraction
prescription of the HLM method, which requires some prior identi cation of the subtraction terms
for the heavy particle propagators in the model of interest. Furthermore, in the HLM
procedure the light particle mass in the light eld propagators is not expanded out in the
computation of the functional traces, and intermediate results are therefore more involved.
In particular, nonanalytic terms in the light masses can appear in intermediate steps of
the calculation, and cancellations of such terms between di erent contributions have to
occur to get the infrared nite matching coe cients at one loop. Given the amount of
algebra involved in the computation of the functional traces, automation is a prerequisite
for integrating out heavy particles in any realistic model. In our method, such automation
is straightforward (and indeed has been used for the heavy real scalar triplet example given
in section 4). From the description of ref. [7], it seems to us that is harder to implement
the HLM method into an automated code that does not require some manual intervention.
An alternative framework to obtain the oneloop e ective Lagrangian through
functional integration, that shares many similarities with that of HLM, has been suggested in
ref. [8]. The authors of ref. [8] have also introduced a subtraction procedure that involves
the truncation of the heavy particle propagator. Their result for the dimension6 e ective
Lagrangian in the case that the heavylight quadratic uctuation is derivativeindependent
has been written in terms of traces of manifestly gaugeinvariant operators depending on
the quadratic
uctuation U (x), times coe cients where the EFT contributions have been
subtracted. Examples on the calculation of such subtracted coe cients, which depend on
the ultraviolet model, are provided in this reference. The approach is however limited, as
stated by the authors, by the fact that it cannot be applied to cases where the heavylight
interactions contain derivative terms. That is the case, for instance, in extensions of the
SM where the heavy
elds have interactions with the SM gauge bosons (see the
example we provide in subsection 4.2). Let us also note that the general formula provided in
the framework of ref. [8] is written in terms of the components of the original uctuation
operator where no diagonalization to separate heavy and light eld blocks has been
performed. This implies that its application to models with mixed statistics in the part of the
light sector that interacts with the heavy one, and even to models where the heavy and
light degrees of freedom have di erent statistics, must require additional steps that are not
discussed in ref. [8].
4
Examples
In this section we perform two practical applications of the framework that we have
developed above. The rst one is a scalar toy model simple enough to allow a comparison
of our method with the standard matching procedure. Through this example we can also
illustrate explicitly that matching coe cients arise from the hard region of the oneloop
amplitudes in the full theory. The second example corresponds to a more realistic case
where one integrates out a heavy real scalar triplet that has been added to the SM.
that, upon substituting in eq. (4.1), gives the treelevel e ective Lagrangian
LtErFeTe =
1
2
4!
'^4 +
2
To proceed at one loop we use the background eld method as explained in section 2:
! ^ +
and ' ! '^ + '. We have
= ( ; ') and we consider the same counting as in
eq. (2.18): p ; M
. The uctuation operator in eq. (2.3) is given by
Let us consider a model with two real scalar elds, ' with mass m and
with mass M ,
whose interactions are described by the Lagrangian
L('; ) =
4!
'
4
3!
'
m we wish to determine the e ective eld theory resulting from integrating
eld: LEFT('^). We perform the calculation up to and including 1=M 2suppressed
operators in the EFT. Within this model this implies that we have to consider up to
sixpoint Green functions. This same model has also been considered in ref. [7].
At tree level we solve for the equation of motion of the
eld and we obtain
(4.2)
(4.3)
(4.4)
5) :
(4.5)
'^
2
(4.6)
(4.7)
H =
L =
XLH =
'^2 ;
M 2 ;
m2
2
'^
2
'^ ^ ;
1
p2
2
4
'^
2
+O(
m2
1 +
1
p2
5) :
that only depends on the classical eld con gurations. In order to construct e H (x; @x + ip)
in eq. (2.7) we need to determine
L1(x; @x + ip) up to, and including, terms of order
4
:
m2
p2
+
1
p4
2
'^
2
1 +
m2
p2
+
1
p4
2
'^
2
4
p p
p6
Inserting this operator in eq. (2.15), we notice that at the order we are considering only
the n = 1 term contributes, with
LEFT
(
2
)d
p2
:
The momentum integration can be readily performed: in the MS regularization scheme
with
= M we nally obtain
LEFT
'^
4
1
'^
6 :
(4.8)
Let us recover now this result through the usual matching procedure between the
full theory L('; ) in eq. (4.1) and the e ective theory without the heavy scalar eld .
Our goal is to further clarify the discussion given in section 2 on the hard origin of the
matching coe cients of the e ective theory by considering this purely academic case. In
order to make contact with the result obtained in eq. (4.8) using the functional approach,
we perform the matching o shell and we use the MS regularization scheme with
= M .
We do not consider in the matching procedure oneloop diagrams with only light elds,
since they are present in both the fulltheory and the e ective theory amplitudes and,
accordingly, cancel out in the matching.
For the model under discussion there is no contribution to the two and threepoint
Green functions involving heavy particles in the loop. The diagrams contributing to the
matching of the fourpoint Green function are given by
hard
=
i
soft
soft
(4.9)
where we have explicitly separated the contributions from the hard and soft
loopmomentum regions. Note that a nonanalytic term in m can only arise from the soft
region, since in the hard region the light mass and the external momenta are expanded
out from the propagators. For the corresponding EFT computation we need the e ective
Lagrangian matched at oneloop:
LEFT = LtErFeTe +
4!
'^4 +
4!M 2 '^2@2'^2 +
6
6!M 2 '^ ;
(4.10)
which now includes the dimension6 operator with four light elds, and the oneloop
matching coe cient for the 4 and 6light eld operators already present in LtErFeTe . The EFT
We see that the soft components of the fulltheory amplitude match the oneloop diagram in
the e ective theory, and the matching coe cients of the '4 operators get thus determined
by the hard part of the oneloop fulltheory amplitude:
=
3
;
=
3
in agreement with the result for the '4 terms in eq. (4.8).
function. The full theory provides two diagrams for the matching:
The next contribution to the oneloop e ective theory comes from the sixpoint Green
i
i
hard
=
i
+
soft
+ O(M 4);
contributions to the fourpoint Green function read
=
i
= i
i
3M 2 (s + t + u) :
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
where once more we have explicitly separated the hard and soft contributions from each
diagram. The sixpoint e ective theory amplitude gives
Again, we note that the soft terms of the full theory are reproduced by the oneloop diagram
in the e ective theory. The local contribution is determined by the hard part of the full
theory amplitude and thus reads
that matches the result found in eq. (4.8) for the '^6 term.
i
=
+ O(M 4) ;
= i
soft
2
=
45
As a second example, we consider an extension of the SM with an extra scalar sector
comprised by a triplet of heavy scalars with zero hypercharge,
a; a = 1; 2; 3, which interacts
with the light Higgs doublet [26]. A triplet of scalars are ubiquitous in many extensions
of the SM since the seminal article by Gelmini and Roncadelli [27]. However, we are not
interested here in the phenomenology of the model but in how to implement our procedure
in order to integrate out, at one loop, the extra scalar sector of the theory, assumed it is
much heavier than the rest of the spectrum. Partial results for the dimension6 operators
involving the light Higgs doublet that are generated from this model have been provided
in the functional approaches of refs. [7, 8].
The Lagrangian of the model is given by
L = LSM +
1
2
D
aD
a
M 2 a a
1
2
4
( a a)2 +
y a
a
y
Here
as D
is the SM Higgs doublet and the covariant derivative acting on the triplet is de ned
a
ac + g"abcW b
c
. Within the background eld method we split
the elds into their classical (with hat) and quantum components:
and W a ! W^ a + W a. Given as an expansion in inverse powers of its mass, the classical
a ! ^ a + a
,
! ^+
eld of the scalar triplet reads
^ a =
^y ^ i ^y a ^ + O
M 6
:
(4.17)
respectively, as H =
from eqs. (2.2) and (2.3),
Following the procedure described in the section 2 we divide the elds into heavy and light,
a and L = f ; ; W ag. The uctuation matrix is readily obtained
H =
XLyH =
ab ;
Xa
XWc
0
B
B
L = BB X
y
Xa
Xy

XWc

XWda  ;
XWd y1
XWd CCC ;
cd A
C
W
with
W
ab =
= (
ab =
W
ab
SM
)SM +
D^ a2b + ab
h
XWab = g abc D^ ^ c
Xa
=
a ^
2 ^ ^ a;
+ g2 g
acm bdm ^ c ^ d;
a ^ a
^ c ^ c
+ g acd ^ cD^ db;
2
^y ^ i
2
^ a ^ b ;
(4.19)
e
ab =
ab
h Xa
Xb
+ c:c:i
where c:c: is short for complex conjugation and we have used the following de nitions:
XW
ca 
X
1 =
=
XW
ab = XWab
y
cd
W
1 Xb
1
XW
db + O
+ Xa
 X
5 ;
1 +
1 Xy
1 
X
XWa
1 ;
1 Xb
1 
X
1 ;
+ c:c: :
and the rest of uctuations in
L involving only the light
elds are contained in the
quadratic piece of the SM Lagrangian, which we provide in eqs. (B.5) and (B.7). The
quadratic term containing all uctuations related to the heavy triplet is given by our
formula (2.7),
e
=
The expansion in inverse powers of the heavy mass of the triplet requires a counting
analogous to the one in eq. (2.18), i.e. p
and M
. For the counting of the dimensionful
parameter
we choose
and then, from eq. (4.17) we have ^ a
1
. As we are
interested in dimensionsix e ective operators we can neglect contributions O
smaller. This is because in eq. (2.15) the propagator in the heavy particle provides an
5 and
extra power
2. Hence we only need the numerator up to O(
4).
For practical reasons we choose to work in the Landau gauge for the quantum
uctuations, i.e. the renormalizable gauge with W = 0 in eqs. (B.8) and (B.11). The computation
is much simpler in this gauge because the inverse of the propagators are transverse.
Rearranging the expression in eq. (4.20), we can write
U =
D^ 2
eq. (4.21) up to O(
To proceed we now come back to eq. (2.15) (with negative sign), with mH = M and
only the classical eld con guration for the gauge bosons is involved. Then by computing
. Remember that the hat on the covariant derivatives indicates that
4) one can obtain the oneloop e ective theory that derives from the
model speci ed in eq. (4.16) upon integrating out the triplet of heavy scalars.
We do not intend here to provide the complete result of the generated
dimensionsix operators. As a simple example and for illustrative purposes, we provide details on
the computation of the heavylight contributions arising from the quantum
uctuations
of the electroweak gauge bosons. The latter provide the matching contributions to the
dimensionsix operators with Higgs elds and no eld strength tensors proportional to g2,
which were not obtained with the functional approach in ref. [8] due to the presence of
\open" covariant derivatives. The computation of such contributions was also absent in
the approach of ref. [7]. The relevant term in U (x; @x + ip) for this calculation is
XW
ca 
W
cd
1
XW
(4.23)
(4.20)
(4.21)
(4.22)
The rst operator in eq. (4.23) simply reads
ig abc ^ cp +
p2 ig abc ^y c ^ p
where, in the last line, we used the EOM for the heavy triplet, eq. (4.17), and we de ned
the hermitian derivative terms
yD
h(D
)y i
;
+
i
2
+ O
= g abc D^ ^ c
+ g abc D^ ^ c
+ g acd ^ cD^ db
D^ ^ y a b ^
^y a bD^ ^ +
^y a d ^ D^ db + c:c: + O
2
i
2
g
p2
i ab ^yD^ ^
$
1
p2
g
p2
i
2
g
p
2
M 2 acd ^ d D^ cb + ip
with the covariant derivative acting on the Higgs eld as speci ed in eq. (B.2). The
contributions from the heavy triplet to the
uctuation
W , see eq. (4.19), do not a ect
the computation of
in eq. (B.11) (with W = 0) for the latter. As a result we obtain
XWca 
cd
W
= g2
1
g
p2 +
+ O
p p
p4
5 ;
h
ab D^ ^ c
D^ ^ c
D^ ^ a
D^ ^ b
2
ab p4
$
^yD^ ^
$
^yD^ ^
and we dropped the terms proportional to (p2
M 2) since they yield a null contribution
in the momentum integration, as explained below.
Only the rst term of the series in eq. (2.15) contributes in this case:
1loop
LEFT
W
=
i Z
2
ddp
(
2
)d
h
XWca 
p2
1
(4.27)
From eq. (4.27) it is clear that terms proportional to (p2
M 2) yield scaleless terms that are
set to zero in dimensional regularization, which justi es having dropped them in eq. (4.26).
After evaluating the integral in the MS regularization scheme, using the heavy triplet EOMs
and rearranging the result through partial integration we nally get for
= M
1loop
LEFT
2 ^y ^ +
4
5 h ^y ^
^yD^ 2 ^ + h:c:i
4
5 ^yD^ ^
2
:
(4.28)
In order to compare this result with previous calculations done in the literature, we focus
on the heavy triplet contributions to Q D =
yD
. From the result in eq. (4.28) we
2
nd for its oneloop matching coe cient
C D( = M )
O(g2)
=
1
16 2 M 4 4 g ;
which agrees with the result given in ref. [5] for the term proportional to g2. The remaining
contributions to C D(
= M ) have also been calculated with our method. However their
computation is lengthy and does not provide any new insight on the method. The nal
result reads
HJEP09(216)5
C D( = M ) =
2
M 4
2 +
1
4
5 g2 + 16
3
20
:
(4.30)
In eq. (4.30) we have also included the term arising from the rede nition of
that absorbs
the oneloop contribution to the kinetic term,
!
1
3 2=64 2M 2
. This result is in
agreement with the one provided in ref. [5] once we account for the di erent convention in
the de nition of : our
equals 2 in that reference.
5
Conclusions
The search for new physics in the next run at LHC stays as a powerful motivation for a
systematic scrutiny of the possible extensions of the SM. The present status that engages
both collider and precision physics has, on the theoretical side, a robust tool in the
construction, treatment and phenomenology of e ective eld theories that are the remains of
ultraviolet completions of the SM upon integration of heavy spectra.
Though, traditionally, there are two essential procedures to construct those e ective
eld theories, namely functional methods and matching schemes, the latter have become
the most frequently used. Recently there has been a rediscovery of the functional methods,
initiated by the work of Henning et al. [3]. The latter work started a discussion regarding
the treatment of the terms that mix heavy and light quantum
uctuations, that was nally
clari ed but which, in our opinion, was already settled in the past literature on the
subject. In this article we have addressed this issue and we have provided a framework that
further clari es the treatment of the heavylight contributions and simpli es the technical
modus operandi.
The procedure amounts to a particular diagonalization of the quadratic form in the
path integral of the full theory that leaves untouched the part that entails the light elds.
In this way we can integrate, at one loop, contributions with only heavy
elds inside the
loop and contributions with mixed components of heavy and light
elds, with a single
computation and following the conventional method employed to carry out the rst ones
only. We have also showed that in the resulting determinant containing the heavy particle
e ects only the hard components are needed to derive the oneloop matching coe cients
of the e ective theory.
Within dimensional regularization these hard contributions are
obtained by expanding out the lowenergy scales with respect the hard loop momentum
which has to be considered of the same order as the mass of the heavy particle. In this way,
our determination of the EFT local terms that reproduce the heavyparticle e ects does
not require the subtraction of any oneloop contributions from the EFT, as opposed to the
conventional (diagrammatic) matching approach or to the recently proposed methods that
use functional techniques. We have included two examples in section 4: a scalar toy model,
that nicely illustrates the simplicity of our approach as compared to the diagrammatic
approach, and a heavy real scalar triplet extension of the SM, which shows that our method
can be applied also to more realistic cases.
Acknowledgments
We thank Antonio Pich and Arcadi Santamaria for comments on the manuscript. P. R.
thanks the Instituto de F sica Corpuscular (IFIC) in Valencia for hospitality during the
completion of this work. This research has been supported in part by the Spanish
Government, by Generalitat Valenciana and by ERDF funds from the EU Commission [grants
FPA201123778, FPA201453631C21P, PROMETEOII/2013/007, SEV20140398]. J. F.
also acknowledges VLCCAMPUS for an \Atraccio del Talent" scholarship.
A
General expressions for dimensionsix operators
In this appendix we workout L1ElFooTp for the case of dimensionsix operators. Following the
guidelines in section 2, we make the separation U (x; @x + ip) = UH (x) + ULH (x; @x + ip)
and expand eq. (2.15) up to O(
2
). The Lagrangian L1ElFooTp then consists of two pieces:
L1ElFooTp = LEFT H
1loop
1loop
+ LEFT LH
:
(A.1)
The rst term comes from contributions involving UH (x) only and, since UH (x) is
momentum independent, it can be obtained from the universal formula provided in the
literature [3, 22, 24, 25] (see also [4] for the case when several scales are involved) which, for
completeness, we reproduce here:
HJEP09(216)5
tr fUH g
ln
2
m2H tr nF^ F
^
tr n(D^ UH )2o
1
90
1
40
1
12
1
60
1loop
=
cs
+
+
+
1
2
1
m2H
1
m4H
1 + ln
m2H
2
ln
m2H tr U H2
tr U H3
+
1
24
+
1
6
1
60
tr U H4
1
120
2
m2H
+
+
1
12
1
12
1
12
tr nUH F^
^
F
o
tr n(D^ F^ )
2o
tr nF^
F^ F^
o
tr nUH (D^ UH )2o
+
tr nF^ (D^ UH )(D^ UH )
o
tr n(D^ 2UH )2o
+
tr nU H2 F^ F
^
o
+
tr n(UH F^ )
2o
1
60
where cs = 1=2; 1=2 depending, respectively, on the bosonic or fermionic nature of the
heavy elds. Here F
[D ; D ] and the momentum integrals are regulated in d
dimensions, with the divergences subtracted in the MS scheme. The second term in eq. (A.1) is
built from pieces containing at least one power of ULH . Given that UH is at most O( )
and ULH at most O( 0) in our power counting, the series in eq. (2.15) has to be expanded
up to n = 5 for the contributions to dimensionsix operators
1loop
LEFT LH
=
ics
Z
ddp
(
2
)
d
1
m2H trs fU g +
1
We have introduced a subtracted trace which is de ned as
trs ff (U; D )
g
tr ff (U; D )
f (UH ; D )
f g ;
where f is an arbitrary function of U and covariant derivatives, and
f generically denotes
all the terms with covariant derivatives at the rightmost of the trace (i.e. open covariant
derivative terms) contained in the original trace. The terms involving only UH that are
subtracted from the trace were already included in eq. (A.2) while all open derivative terms
F
from the di erent traces are collected in LEFT. The latter combine into gauge invariant
pieces with
eldstrength tensors, although the manner in which this occurs is not easily
seen and involves the contribution from di erent orders in the expansion.
+
+
1
1
o
+ 2ip trs nU D^ U 2o
+ trs nU 2D^ 2U o
+ trs nU D^ 2U 2o
4 p p trs nU D^ D^ U
o
+2ip trs nU D^ 2D^ U
o
+ 2ip trs nU D^ D^ 2U o
+trs nU (D^ 2)2 U
o
+ 2ip trs nU 3D^ U
o
+ 2ip trs nU 2D^ U 2o
+2ip trs nU D^ U 3o
4p p trs nU 2D^ D^ U
o
4p p trs nU D^ U D^ U
o
4p p trs nU D^ D^ U 2o
8i p p p trs nU D^ D^ D^ U
o
+
+
tr U H6
+ O
(A.2)
(A.3)
(A.4)
F
With the purpose of illustration, we compute LEFT that results from the integration
of the real scalar triplet extension of the SM presented in subsection 4.2. In this case,
F
gauge invariance of the nal result guarantees that the leading order contribution to LEFT
should contain at least four covariant derivatives, as terms with two covariant derivatives
cannot be contracted to yield a gauge invariant term. As it is clear from eq. (2.15), traces
with j derivatives and a number k of U operators have a power suppression of O
(we recall that ddp
4). The expansion of the operator ULH can yield in addition `
4 j 2k
covariant derivatives, and each of these receives a further suppression of
1 because they
are accompanied with a light eld propagator, see eq. (A.6). Since ULH is at most O( 0)
we then nd that, in general, terms with k insertions of UHL and a total number of j + `
derivatives have a power counting of at most O( 4 j ` 2k). As a result, the only gauge
invariant object involving ULH and four derivatives that one can construct at O(
2
)
includes only one power of ULH (i.e. j + ` = 4 and k = 1). Moreover, since ULH has to
be evaluated at leading order, the only relevant piece from ULH for the computation of
F
LEFT reads
ULFH = XL(1H) y
L1 ^=0 XL(1H) :
Here XL(1H) is de ned as the part of XLH that is O ( ), and we remind that ^ stands for the
classical eld con gurations. Using the expressions in eqs. (4.18) and (4.19) we have
2 4 "
X
m=0
^y a
2ipD^ + D^ 2 !m
p2
p2
b ^
+ ^( a)
2ipD^ + D^ 2 !m
( b) ^
#
;
where the covariant derivatives have to be expanded by applying the identities
D
a
=
a (D
) + c Dca ;
D ( a)
= ( a) (D
) + ( c)
Dca ;
with D denoting the Higgs eld covariant derivative, see eq. (B.2), and with Dca as de ned
F
in section 2. For the computation of LEFT up to O
eq. (2.15) with up to four open covariant derivatives and just one power of ULFH . These are
2 we need to isolate the terms in
ddp
n + 1
tr
8
:
2ipD^ + D^ 2 !n k
p2
2ipD^ + D^ 2 !k9
=
p2
;
;
(A.5)
(A.6)
(A.7)
(A.8)
and using the cyclic property of the trace we get4
ddp
2ipD^ + D^ 2 !n)
p2
Finally, keeping only terms with up to four covariant derivatives, performing the momentum
integration (see eq. (2.20)) and evaluating the SU (
2
) trace we arrive at the nal result
F
LEFT =
1
2
2
3
^y ^
^
W
a W^ a + g ^y iD^ a ^
D
^ W
^
a
$
gg0
2
^y a ^ W^ a B^
with the eldstrength tensors de ned in eq. (B.3) and
$
y iDa
= i
y aD
i (D
)y a
:
B
The
uctuation operator of the SM
In this appendix we provide the uctuation operator for the SM Lagrangian. The SM
Lagrangian in compact notation is given by
LSM =
+
1
4
G
iD=
G
1
4
1
4
W a W
a
B
B
+ (D
)y D
m2
y
e yu PuPR +
yd PdPR + h:c:
+ LGF + Lghost :
Here,
= q; `, Pu (Pd) project into the up (down) sector, yu;d is a Yukawa matrix for up
(down) elds, LGF and Lghost are the gauge xing and ghost Lagrangians, respectively, and
the covariant derivatives are de ned as
=
D
D
igcG= T Pq
igW aT a
1
2
igW= aT aPL
ig0B
:
ig0B= Y
;
reads Y
given by
In eq. (B.2), T a =
a=2 and T
=
=2 with
a and
the Pauli and the GellMann
matrices, respectively, Pq denotes a projector into the quark sector, and the hypercharge
= Y L PL + Y uR PuPR + Y dR PdPR. Accordingly, the eld strength tensors are
+ gf
G G ;
4The use of the cyclic property when derivative terms are involved is only justi ed for the functional
functional determinant, which is a gauge invariant object, the trace over internal degrees of freedom `tr'
can be recast into the full trace through the use of the identity (we recall that S = R ddx L)
Trff (x^)g =
Z ddx trfhxjf (x^)jxig =
Z ddx trff (x)g d(0) ;
and then reverted to a trace over internal degrees of freedom after the application of the cyclic property.
Following the same procedure as in section 2, we separate the elds into background,
^, and quantum
eld con gurations, , and expand the SM Lagrangian to second order in
the quantum
uctuation:
LSM = LtSrMee(^) + L(SM2) + O
3 ;
where LtSrMee is the treelevel SM e ective Lagrangian, and L(SM2) is computed using eq. (2.2):
L(SM2) =
1
2
y  Aa 
0
B
B
B
B X
 BBB XAa
B
BB X
B
X
XAa
X
X
with Aa = G
W a B
 denoting the gauge elds and
A
ab = BB
;
XA a = BBXWa CC
;
XAb y
XAb 
ab
A
X Ab
X Ab 
X
X
X Aa
0
0
HJEP09(216)5
(B.4)
(B.5)
1
C
A
X
0
1

X Aa  CCCCCCCCCC BBBBBBBB@AbCC + Lghost;
X CC
C (
2
)
C
0 ACC
1
X B
X Aa = BBX
Wa CCA ; (B.6)
g abcW^
c
;
where, generically, X = Xy 0. The pieces in the quadratic uctuation are de ned as
0
W
ab
a
BW
1
0
aC
BW CA
B
g
D^ 2 +
1
^y ^
^ ^y;
2
1 g0 2 ^y ^
^y a ^ ;
G D^ D^
G
g
2
1 2 ^y ^
+
1
+
B
gc
1
B
^
G
;
W D^ D^
W
^ yu Pu + ^ yd Pd + h:c: ;
0
W
G
0
0
=
=
ab = ab g
D^ 2 +
B = g
BWa = gg0g
= iD^=
=
^ ^y ;
XWa = ig ^y aD^
X
XB
= ig0 ^yD^
X G = 2 gc
X Wa = g a
Pq
^ ;
PL ^ ;
D^ ^ y a ;
D^ ^ y ;
X
X
=
=
P uPLyuy ^t i 2
i 2 yuPuPR ^
yd PdPR ^ ;
P dPLydy ^t :
The superscript t in the fermion elds denotes transposition in isospin space. Additionally,
we have
xed the gauge of the quantum
elds using the background
eld gauge, which
ensures that the theory remains invariant under gauge transformations of the background
elds. This choice corresponds to the following gauge xing Lagrangian:
LGF =
1
D^ G
2
1
D^ W a 2
1
Finally we also provide the expansion for the inverse operators
^y ^
^^y ;
+ O ( ) ;
D^ D^ D^ D^
D^ o
+ O
3 ;
(B.7)
(B.8)
(B.9)
(B.10)
7 ;
(B.11)
! D
^
while
from where, and de ning
it is straightforward to get
1 +
W
and analogously for
can be obtained from
G
derivative term.
= D^ 2 +
^y ^ +
^^y ;
m2
p2 +
m4 !
p4
m2 !
1 + 2 p2
p
+ 2i p4
4
p p
p6
1 + 2 p2
1 + 3 p2
m2 !
m2 !
^
D
D^ D^
1
p2
1
+ p4
+ 2i p6
p n ^
D
+
D^ o +
p p p D^ D^ D^ + 16
8i
4
g
p8
p8
p p nD^ D^
p2 + (1
B)
g
p2 + (1
+
p p
p4
W )
1
p6
2
p p p p
p10
D^ D^ + D^
+ O
p p
p4
3 ;
+ O
X = f
; B; W g, when p
. We have:
m2
2ipD^
W
g p
2
2
1
B
1
B
W
W p p
p p + O ( ) :
The inverse operator [
(x; @x + ip) 1 by making the substitution D^
(x; @x + ip)] 1 share the same expression, up to a total
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