Integrating out heavy particles with functional methods: a simplified framework

Journal of High Energy Physics, Sep 2016

We present a systematic procedure to obtain the one-loop low-energy effective Lagrangian resulting from integrating out the heavy fields of a given ultraviolet theory. We show that the matching coefficients are determined entirely by the hard region of the functional determinant involving the heavy fields. This represents an important simplification with respect the conventional matching approach, where the full and effective 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.

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Integrating out heavy particles with functional methods: a simplified framework

HJE Integrating out heavy particles with functional methods: a simpli ed framework Javier Fuentes-Mart n 0 1 3 Jorge Portoles 0 1 3 Pedro Ruiz-Femen a 0 1 2 0 Technische Universitat Munchen , D-85748 Garching , Germany 1 Apt. Correus 22085, E-46071 Valencia , Spain 2 Physik Department T31, James-Franck-Stra e 3 Instituto de F sica Corpuscular , CSIC 4 Universitat de Valencia We present a systematic procedure to obtain the one-loop low-energy 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 dimension-six 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 next-to-leading 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 low-energy dynamics of the light elds. This last methodology was applied, for example, in refs. [9, 10], to obtain the non-decoupling 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 one-loop 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 heavy-light loops which are already accounted for by the one-loop EFT contribution. Here we would like to introduce a more direct method to obtain the one-loop 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 one-loop matching coe cients from the full theory computation, thus bypassing the need of subtracting any infrared contribution. In short, the determination of the one-loop 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 one-loop 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 one-loop 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 dimension-six 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 full-theory results at the one-loop 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 one-loop 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 one-loop 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 tree-level e ective action. At energies much lower than the mass of the heavy elds, the background heavy elds ^H can be eliminated from the tree-level 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 one-loop 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 one-loop 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 one-loop 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 { one-loop 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 block-diagonal 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 heavy-particle block H , while leaving L untouched. This has the advantage that all heavy particle e ects in the one-loop 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 one-loop contributions with light particles running inside the loop, and heavy elds can appear only as tree-level lines through the dependence of L in ^H . We thus de ne the part of the one-loop 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 one-loop amplitudes with at least one heavy particle propagator in the loop. One-loop diagrams with n heavy propagators are reproduced from the n-th 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 low-energy scales in the theory, either p mL or any of the light-particle external momenta, pi mH . In dimensional regularization the two contributions can be computed separately by using the so-called \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 d-dimensional space of the loop momenta. In the hard region, all the low-energy 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 low-energy momenta and masses, with factors of 1=mH to adjust the dimensions. This part is therefore fully determined by the short-distance 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 one-loop 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 one-loop terms with light particles that arise from the Gaussian integral of L in eq. (2.8), yields the same one-loop amplitude as one would obtain using the Feynman rules of the e ective Lagrangian for the light elds obtained by tree-level 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 full-theory and EFT renormalized amplitudes at oneloop only the hard part of the full-theory amplitude remains, and one can read o the one-loop 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 one-loop 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 one-loop 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 loop-momentum 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 dimension-six 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 non-decoupling e ects (i.e. the O(m0H ) terms) introduced by a SM-like 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 , eld-strength 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 heavy-light 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 momentum-dependent and momentum-independent pieces, i.e. U = UH (x)+ULH (x; @x + ip) which, at the diagrammatic level, corresponds to a separation into pure heavy loops and heavy-light 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 dimension-six 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 dimension-4 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 heavy-light, 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 one-loop 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 dimension-six 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 one-loop dimension-six 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 dimension-six 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 gauge-invariant 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 one-loop 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 heavy-light loops from the rest. In the HLM method the determinant of the uctuation operator O which de nes the complete one-loop 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 block-diagonalizes 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 one-loop 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 one-loop terms from the EFT theory matched at tree-level. 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 one-loop 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 heavy-light 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, non-analytic 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 one-loop 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 dimension-6 e ective Lagrangian in the case that the heavy-light quadratic uctuation is derivative-independent has been written in terms of traces of manifestly gauge-invariant 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 heavy-light 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 one-loop 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 tree-level 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 2-suppressed operators in the EFT. Within this model this implies that we have to consider up to six-point 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 one-loop diagrams with only light elds, since they are present in both the full-theory 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 three-point Green functions involving heavy particles in the loop. The diagrams contributing to the matching of the four-point 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 non-analytic 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 one-loop: LEFT = LtErFeTe + 4! '^4 + 4!M 2 '^2@2'^2 + 6 6!M 2 '^ ; (4.10) which now includes the dimension-6 operator with four light elds, and the one-loop matching coe cient for the 4- and 6-light eld operators already present in LtErFeTe . The EFT We see that the soft components of the full-theory amplitude match the one-loop diagram in the e ective theory, and the matching coe cients of the '4 operators get thus determined by the hard part of the one-loop full-theory 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 one-loop e ective theory comes from the six-point Green i i hard = i + soft + O(M 4); contributions to the four-point 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 six-point e ective theory amplitude gives Again, we note that the soft terms of the full theory are reproduced by the one-loop 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 dimension-6 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 dimension-six 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 one-loop 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 heavy-light contributions arising from the quantum uctuations of the electroweak gauge bosons. The latter provide the matching contributions to the dimension-six 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 one-loop 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 one-loop 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 heavy-light 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 one-loop matching coe cients of the e ective theory. Within dimensional regularization these hard contributions are obtained by expanding out the low-energy 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 heavy-particle e ects does not require the subtraction of any one-loop 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 FPA2011-23778, FPA2014-53631-C2-1-P, PROMETEOII/2013/007, SEV-2014-0398]. J. F. also acknowledges VLC-CAMPUS for an \Atraccio del Talent" scholarship. A General expressions for dimension-six operators In this appendix we workout L1ElFooTp for the case of dimension-six 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 dimension-six 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 eld-strength 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 eld-strength 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 Gell-Mann 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 tree-level 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@A|bCC + 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 . 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Javier Fuentes-Martín, Jorge Portolés, Pedro Ruiz-Femenía. Integrating out heavy particles with functional methods: a simplified framework, Journal of High Energy Physics, 2016, 156, DOI: 10.1007/JHEP09(2016)156