Chiral Perturbation Theory with an Isosinglet Scalar
CChiral Perturbation Theory with an Isosinglet Scalar
Martin Hansen ∗ INFN Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, ItalyE-mail: [email protected]
Kasper Langæble
CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, DenmarkE-mail: [email protected]
Francesco Sannino
CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, DenmarkE-mail: [email protected]
We present an extension of chiral perturbation theory that explicitly includes an isos-inglet scalar in the Lagrangian. The dynamical effects from the scalar state is of phe-nomenological relevance in theories where the mass of the isosinglet scalar is compara-ble to the mass of the pseudo-Goldstone bosons. This near-degeneracy of states is forexample observed in certain near-conformal BSM models. From the Lagrangian we cal-culate the one-loop radiative corrections to the pion mass, the pion decay constant, andthe scalar mass. We then proceed and use the results to fit numerical lattice data for anSU(3) gauge theory with N f = XIII Quark Confinement and the Hadron Spectrum - Confinement 201831 July - 6 August 2018Maynooth University, Ireland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - ph ] J un hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen πσρ πσρ πσ K ρ
1. Introduction
In recent years there has been a renewed interest in studying the properties and thephenomenology of the isosinglet scalar, both in QCD and in near-conformal BSM mod-els, albeit for different reasons. In the former case, new advances in lattice field theoryhave made it possible to study the isosinglet channel in scattering processes, revealingmany interesting and non-trivial properties [1, 2]. In the latter case, the isosinglet scalaris interesting because of its relation to conformal symmetry. In particular, if the model ofinterest is sufficiently close to the conformal window, but still in the chirally broken phase,the isosinglet scalar can (at least partially) be identified with the Goldstone boson arisingfrom the spontaneous breaking of scale invariance. In this case, the scalar is commonlyknown as a dilaton, and because of its origin as a Goldstone boson, it might potentiallybe very light. This has indeed been observed in lattice simulations [3–9], where in somecases the isosinglet scalar is observed to be lighter than the pions. However, it shouldbe mentioned that, assuming these BSM models indeed are in the chirally broken phase,sufficiently close the chiral limit, the pions will always be the lightest states.At low energy, strongly coupled theories are described by chiral perturbation theory,under the assumption that there is a gap between the mass of the Goldstone bosons andthe heavier states, such that these heavy states can be integrated out. As shown on Fig. 1,in two-flavour QCD this is indeed the case, but in the previously mentioned case of near-conformal BSM models, there might not be any separation between the isosinglet scalarand the pions. Furthermore, when considering three-flavour QCD, the kaons are almostdegenerate with the isosinglet scalar, which again shows a lack of proper separation.1 hiral Perturbation Theory with an Isosinglet Scalar
Martin Hansen
Due to this lack of separation between the isosinglet scalar and the Goldstone bosons,in some cases chiral perturbation theory might not be a reliable description of the lowenergy physics. For this reason, there have been several attempts at constructing effectivefield theories that take both of these states into account [10–18]. While the different the-ories incorporate the isosinglet scalar in different ways, they are all extensions of eitherchiral perturbation theory or the linear sigma model. In these proceedings we will discussan extension of chiral perturbation theory described in [12].
2. Approach
In this section we introduce a version of chiral perturbation theory augmented withan isosinglet scalar field. To keep the discussion short, we will only describe the mainpoints, while the details can be found in the original paper [12].Before we can write down the Lagrangian, we have to adopt an appropriate count-ing scheme that includes both the pion and the scalar mass. To this end, we choose thesimplest possible extension, where the scalar counts in the same way as the pions O ( m π ) = O ( m σ ) = O ( p ) . (2.1)There are two reasons for choosing this particular counting scheme. First of all, whenchoosing a counting scheme where the scalar counts differently from the pions, in theperturbative expansion, a Feynman diagram including both scalars and pions, does nolonger contribute to a single order in the chiral expansion, but in fact to different neigh-bouring orders. While this is only a mathematical problem, it does make the calculationsmore complicated. The second reason is phenomenologically motivated, because in latticesimulations of near-conformal BSM models, it is observed that the scalar mass in fact doesseem to scale similarly to the pion mass. This is of course only true in some intermediaterange of quark masses, because in the chiral limit the pions are massless, while the scalaris not, i.e. m π = Am q , m σ = m + Bm q . (2.2)Here m is the scalar mass in the chiral limit and m q is the quark mass. This means that,sufficiently close to the chiral limit, one should indeed integrate out the scalar and usenormal chiral perturbation theory.Having established the counting scheme, we can write down the Lagrangian for oureffective theory. Let G be the global flavour symmetry and let H denote the stability groupafter spontaneous chiral symmetry breaking. The Goldstone boson manifold G / H is thenparametrized by u = exp (cid:32) i √ f π φ a X a (cid:33) , (2.3)where f π is the tree-level pion decay constant and X a are the broken generators. Fromthis definition we can build the two primary invariants (i.e. objects invariant under the2 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen stability group) used to construct the Lagrangian u µ = i ( u † ( ∂ µ − ir µ ) u − u ( ∂ µ − il µ ) u † ) , (2.4) χ ± = u † χ u † ± u χ † u . (2.5)The first invariant u µ is used to construct the kinetic term, with l µ and r µ being the externalcurrents. The second invariant χ ± is used to construct the mass term, with χ the diagonalmass matrix. In fact, at leading order the chiral Lagrangian is simply given by L = f π (cid:104) u µ u µ + ˜ χ + (cid:105) , (2.6)where ˜ χ + = χ + − ( χ + χ † ) is the mass term without the constant piece (this subtractionis needed later on) and (cid:104)·(cid:105) denotes the trace in flavour space. At higher order, the chiralLagrangian contains many more terms, each associated with an unknown low-energyconstant (LEC). For example, at next-to-leading order (NLO) we have terms like L = L (cid:104) u µ u ν u µ u ν (cid:105) + L (cid:104) u µ u µ (cid:105)(cid:104) u ν u ν (cid:105) + L (cid:104) u µ u ν (cid:105)(cid:104) u µ u ν (cid:105) + · · · . (2.7)The exact expression for the Lagrangian is not important for the current discussion andwe refer to [12, 19] for details. While the chiral Lagrangian only depends on the globalflavour symmetry, the LECs encode information about the underlying strong dynamics.For this reason, they can be divided into contributions from various sources, such as heav-ier resonances L i = ˆ L i + ∑ R L Ri . (2.8)Here L Ri is the contribution from a resonance R and ˆ L i is a remainder not directly relatedto any resonance. For example, under the assumption of vector meson dominance, thecontributions L Ri can be written in terms of the decay constants and masses of these heavyvector resonances [20, 21]. In the same way, the isosinglet scalar might contribute to theLECs, but when the scalar is light, the contribution is dynamical and not just a constant.With the previous discussion in mind, we will now introduce the isosinglet scalar σ inthe chiral Lagrangian as a non-trivial background field [10, 22]. In practice this is done byexpanding each coefficient in the Lagrangian in powers of σ / f π . Because we are interestedin calculating the radiative corrections to the two-point functions at next-to-leading order,the expansion is only needed for the leading-order Lagrangian and we can stop the seriesexpansion at second order. L = f π (cid:34) + S (cid:18) σ f π (cid:19) + S (cid:18) σ f π (cid:19) + · · · (cid:35) (cid:104) u µ u µ (cid:105) + f π (cid:34) + S (cid:18) σ f π (cid:19) + S (cid:18) σ f π (cid:19) + · · · (cid:35) (cid:104) ˜ χ + (cid:105) (2.9)It is now evident that the subtraction in the mass term is needed to avoid terms that onlyinclude the scalar field. The associated Lagrangian for the scalar field can be written as L σ = ∂ µ σ∂ µ σ − m σ σ (cid:34) + S (cid:18) σ f π (cid:19) + S (cid:18) σ f π (cid:19) (cid:35) . (2.10)3 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen
In principle we should also perform a series expansion in front of the kinetic term, butsince we will only consider on-shell quantities, these terms are related to the potential viathe equations of motion. Because the scalar field parametrise the fluctuations around thevacuum, it must have vanishing expectation value, and this leads to certain constraintson the two parameters S and S controlling the potential S ≥ − (cid:112) S , S ≥ S i between the scalar and the pions, as shown later on. Moreover, whenperforming the calculations at NLO, the results are largely independent of the pattern ofchiral symmetry breaking, and for this reason, the results can easily be applied to anymodel of interest.
3. Results
Having defined the Lagrangian, we are now able to calculate the two-point functionsneeded to define the renormalized pion mass, pion decay constant, and scalar mass, atnext-to-leading order. For both the pion mass and the pion decay constant there are fourdiagrams in total; two diagrams with scalars in the loop, one contact term and one di-agram only with pions. We define the renormalized pion mass as the pole mass in thepropagator, and the result readsˆ m π = m π + m π f π ( a + a L π + a J πσπ ) + m σ f π ( a L σ + a J πσπ )+ m π m σ f π ( a + a L π + a L σ + a J πσπ ) , (3.1)while the result for the pion decay constant readsˆ f π = f π + m π f π ( b + b L π + b J πσπ ) + m σ f π ( b + b L σ + b J πσπ )+ H πσπ f π ( b m π + b m σ + b m π m σ ) . (3.2)Here a i and b i are specific combinations of the various low-energy constants. In the caseof normal chiral perturbation theory, the only non-zero constants are a and b whichmeans that including the scalar vastly increases the complexity of the results. In the equa-tions we use the following auxiliary functions as shorthand notation for the chiral logs4 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen and the unitarity corrections. We refer to the appendix of [12] for the definition of thebarred functions. L x = π log (cid:18) m x µ (cid:19) J xyz = π (cid:34) J ( m x , m y , m z ) + (cid:35) H xyz = π (cid:34) H ( m x , m y , m z ) (cid:35) (3.3)We also calculated the renormalized scalar mass, which readsˆ m σ = m σ + m σ f π ( c L σ + c J ππσ + c J σσσ ) + m π f π ( c L π + c J ππσ )+ m π m σ f π ( c L π + c J ππσ ) . (3.4)The calculation of the scalar self-energy includes a diagram with an intermediate pionloop. Because of this diagram, when the scalar is sufficiently heavy, the pions are able togo on-shell, corresponding to the σ → ππ decay channel being kinematically allowed. Inthe analytical expression for the renormalized scalar mass, this results in a branch cut inthe function J ππσ as shown in Fig. 2. The branch cut starts at m σ = m π and above thisthreshold the decay width of the scalar can be extracted from the imaginary part Γ = n π π m σ f π (cid:18) S (cid:18) m σ − m π (cid:19) + S m π (cid:19) (cid:115) − m π m σ . (3.5)Here n π is the number of pions for the given pattern of chiral symmetry breaking and S and S are the two low-energy constants parametrizing the decay width. Because S parametrize the interaction between the scalar and the pion mass term, in the chiral limit,this coefficient is irrelevant, and the decay width only depends on S . We performed several consistency checks of the previous results to ensure their va-lidity. For the pion we checked that the renormalized mass ˆ m π vanishes in the limit chirallimit where m π →
0. This is a non-trivial check, because the m σ term only vanishes due toan exact cancellation in the chiral limit.For the pion decay constant we checked that we obtain a finite and non-zero value inthe chiral limit, which indeed is the caseˆ f π = f π + m σ f π (cid:18) b + ( b + b ) L σ − b π (cid:19) . (3.6)Because of the scalar corrections, we observe that ˆ f π and f π no longer coincide in the chirallimit. In fact, the entire right-hand side corresponds to what is denoted f π in normal chiralperturbation theory. 5 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen
RealImaginary
Figure 2: Behaviour of J ππσ as a function of the ratio m σ / m π . A branch cutis developed when m σ = m π after which the imaginary part describes thedecay width of the isosinglet scalar in the σ → ππ channel.Finally we checked that all the results are independent of the renormalization scale.This means that changing the renormalization scale corresponds to a shift in all the LECs.Again, this is a non-trivial property, because it depends on the specific combinations ofthe functions defined in Eq. (3.3).
4. Origins of the scalar
As already mentioned, our approach for including the scalar is completely generic.However, if we assume a specific physical origin for the scalar, we can make predictionsfor the couplings S i between the scalar and the pions. As an example, here we will con-sider the case where the scalar emerges as a pseudo-dilaton [23, 24]. In this scenario, thescalar is introduced as the conformal compensator, and the Lagrangian reads L = f π (cid:20) (cid:104) u µ u µ (cid:105) exp (cid:18) σ f π (cid:19) + (cid:104) χ + (cid:105) exp (cid:18) y σ f π (cid:19)(cid:21) . (4.1)Although we use f π as the compensating scale for the pseudo-dilaton in the exponential,de facto, depending on the microscopic realization it can differ, but our results still apply.Expanding the exponential to second order we find that our couplings are given by S = S = S = y , S = y y = − γ ∗ with γ ∗ being the anomalous dimension of the fermion mass in the under-lying gauge theory. It is now evident that γ ∗ is the only new parameter in the expressionfor the pion mass and the pion decay constant, when the scalar field is a pseudo-dilaton.6 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen m q L × T ˆ m σ Table L × T ˆ m π ˆ f π Table0.012 42 ×
56 0.151(27) XVII 42 ×
56 0.16362(43) 0.04542(27) XXI0.015 36 ×
48 0.162(59) XVII 42 ×
56 0.18614(44) 0.05054(15) XXI0.020 36 ×
48 0.190(36) XVII 36 ×
48 0.22052(33) 0.05848(15) XXII0.030 30 ×
40 0.282(39) XVII 36 ×
48 0.28084(39) 0.07137(20) XXII0.040 30 ×
40 0.365(51) XVII 30 ×
40 0.33501(21) 0.08264(10) XXIII0.060 24 ×
32 0.46(13) XVII 30 ×
40 0.43035(44) 0.10118(28) XXIIITable 1: Numerical data used for the fitting procedure. The table column referto the table in [4] from where the data was taken. For simplicity we averagedthe upper and lower errors for the scalar mass.With this example it is evident that the value of the couplings S i can be used to makepredictions about the origin of the scalar field. The original paper [12] contains a few moreexamples of different physical origins.
5. Fitting lattice data
As an example, we will now use the results from section 3 to fit a set of lattice datafrom the LatKMI collaboration. The simulated model is an SU(3) gauge theory with N f = a M and a F are known numbers, but at non-zero lattice spacing thesewill receive unknown corrections. For this reason, it is usually very difficult to fit latticedata when using the continuum values, and this is still true even with the additionalparameters introduced by the scalar extension. This is why we also include these as freeparameters. { B , f π , a M , b M , a F , b F , m σ , S , S , S , S , S , S } (5.1)The result of the fit is shown in Fig. 3. During the fitting procedure we found two differentminima with the same value of χ /dof = hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen
Figure 3: Fit to the numerical data in Table 1. Two different minima werefound with the same value of χ . For the two minima there is no visual differ-ence for the pion quantities, but there is a small difference for the scalar mass,because this quantity has significantly larger errorbars.scalar is slightly different. This is easily understood, because the uncertainty on the scalarmass is significantly larger, such that the value of χ is completely determined by the pionquantities. Close to the chiral limit, the fit for the scalar mass is very different for the twominima, however, this is not important because the scalar is unstable in this region. Asa consequence, the fit cannot be used to extrapolate the scalar mass, and we are satisfiedthat the fit for the scalar mass is consistent in the region of quark masses where we havedata.For one of the minima, we see the branch cut in the scalar mass close to the chirallimit, while in the other case, the coefficients in front of the J ππσ function are so small thatwe do not see the branch cut. The fact that the coefficients are quite different for the twominima proves that the uncertainty on the scalar mass is too large to properly constrainthe S i parameters. This is an important observation because, as previously discussed,the values of the fitted parameters can be used to distinguish between different physicalorigins of the scalar, but unfortunately this is not possible with the currently availabledata.We finally remark that, for both minima, the constraint on the scalar potential in Eq.(2.11) is satisfied because both coefficients are positive. Furthermore, in the chiral limit thedifference between the renormalized and the bare pion decay constant is relatively small,8 hiral Perturbation Theory with an Isosinglet Scalar Martin Hansen namely: ˆ f π − f π ˆ f π ∼
5% (5.2)This is expected when the scalar only acts as a small perturbation of chiral perturbationtheory.
6. Conclusion
We presented a simple extension of chiral perturbation theory that accounts for thedynamical effects of a light isosinglet scalar state. After discussing the chosen countingscheme, we introduce the Lagrangian and calculate the radiative one-loop corrections tothe pion mass, the pion decay constant, and the scalar mass. Our approach is very generic,it makes no assumptions about the physical origin of the scalar and the results are validfor different patterns of chiral symmetry breaking. For this reason, the framework canbe used for a large class of interesting models. After presenting the results, we arguethat different physical origins of the scalar correspond to imposing constraints on someof the low-energy constants, and as such, in principle one can make predictions about thenature of the scalar by fitting these constants to data. For this reason, we use the results tofit numerical data from a lattice simulation, and while this is possible, the uncertainty onthe scalar mass is too large to properly constrain the interesting coefficients.
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