Low-energy constant L 10 in a two-representation lattice theory
Maarten Golterman, William I. Jay, Ethan T. Neil, Yigal Shamir, Benjamin Svetitsky
FFERMILAB-PUB-20-513-T
The low-energy constant L in a two-representation lattice theory Maarten Golterman, William I. Jay, EthanT. Neil, Yigal Shamir, ∗ and Benjamin Svetitsky Department of Physics and Astronomy,San Francisco State University, San Francisco, CA 94132, USA Theoretical Physics Department, Fermi NationalAccelerator Laboratory, Batavia, Illinois, 60510, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA Raymond and Beverly Sackler School of Physics and Astronomy,Tel Aviv University, 69978 Tel Aviv, Israel (Dated: October 6, 2020)
Abstract
We calculate the low-energy constant L in a two-representation SU(4) lattice gauge theory thatis close to a composite-Higgs model. From this we obtain the contribution of the new strong sectorto the S parameter. This leads to an upper bound on the vacuum misalignment parameter ξ whichis similar to current estimates of this bound. Our result agrees with large- N c scaling expectations,within large systematic uncertainties. ∗ [email protected] a r X i v : . [ h e p - l a t ] O c t . INTRODUCTION The composite-Higgs paradigm [1, 2] provides a solution to the problem of protecting theHiggs mass from large radiative corrections by supposing that the Higgs is a pseudo Nambu-Goldstone boson (pNGB) of some new strong interaction, dubbed hypercolor, operative atthe few-TeV scale. Often, one also supposes that the top quark is partially composite,meaning that it acquires its large mass by mixing with a top partner—a baryon of thenew strong force with the same Standard Model quantum numbers [3] (for reviews, seeRefs. [4–6]).A number of concrete realizations of the composite-Higgs scenario, based on asymptoti-cally free gauge theories, were proposed some time ago in Ref. [7] (see also Refs. [8–10]). Ina series of papers [11–15], we have studied the SU(4) gauge theory with two Dirac fermionsin the fundamental representation, together with two Dirac fermions—equivalently, 4 Ma-jorana fermions—in the sextet representation, which is a real representation. By itself, thisfermion content is not enough to accommodate a composite Higgs along with a top partner.Starting from here, however, we can reach two of the models proposed in Ref. [7]—denotedM6 and M11 in Ref. [10]—by increasing the number of fermion species in each representa-tion. In fact, the fermion content of our model is quite close to that of the M6 model, whichhas 3 fundamental Dirac fermions together with 5 Majorana sextet fermions. Values of low-energy constants (LECs) calculated in our model may thus be quite close to their values inthe M6 model. Our choice of two Dirac fermions in each representation allows us to usethe standard hybrid Monte Carlo (HMC) algorithm in our simulations, whereas simulatingthe actual M6 model would require the more costly rational HMC (RHMC) algorithm (see,for example, Ref. [17]). In this paper we focus on L , a next-to-leading order (NLO) LEC which, in chiralperturbation theory (ChPT) for fermions in a single representation, multiplies the operator[19, 20] O = − tr (cid:0) ( V µν − A µν )Σ( V µν + A µν )Σ † (cid:1) . (1.1)In the current model, as well as in the M6 model, Σ is the non-linear field for pNGBs madeout of the sextet fermions; by analogy with QCD, we will often refer to these pNBGs as“pions.” V µν and A µν are the field strengths of external gauge fields V µ and A µ which,in turn, couple to vector currents V µ and axial currents A µ . As in QCD, also for a realrepresentation the vector and axial currents are associated with unbroken and broken flavorgenerators, respectively. For more details, we refer to App. A. As we discuss in detail below, L can be extracted from (cid:104) V µ V ν − A µ A ν (cid:105) , the difference between the connected two-pointfunctions of the vector and axial currents. This paper is organized as follows. In Sec. II we give the necessary theoretical background.In Sec. III we describe the extraction of L from our lattice calculations. In Sec. IV we use L and the experimental value of the S parameter [22, 23] to obtain a bound on the scaleof the hypercolor theory, and we summarize. In App. A we briefly review the embeddingof the electroweak gauge fields of the Standard Model in the M6 composite-Higgs model[8, 9, 24, 25], and calculate the contribution of the hypercolor theory to the S parameter. In QCD, values of LECs typically change by a small amount when increasing the number of light flavorsin the simulation from 2 to 3 [16]. The M11 model has 4 fundamental Dirac fermions and 6 Majorana sextet fermions. We note that theSp(4) gauge theory, on which models M5 and M8 are based, is also currently under study [18]. ChPT for two fermion representations was developed in Ref. [21]. Another interesting LEC that can be extracted from (cid:104) V µ V ν − A µ A ν (cid:105) is C LR , which we have calculatedpreviously [15]. II. THEORETICAL BACKGROUND
In this section we summarize the theoretical background for our calculation. In Sec. II Awe give the basic definitions, and discuss partially-quenched ChPT at NLO. In Sec. II Bwe discuss corrections beyond NLO, and in Sec. II C we discuss lattice discretization effects.For relevant ChPT literature, see Refs. [19–21, 26–31]. For reviews, see Refs. [32, 33].
A. Partially-quenched chiral perturbation theory at next-to-leading order
We begin with the two-point function of the vector-current, δ ab Π V V,µν ( q ) = (cid:90) d x e iqx (cid:104) V µa ( x ) V νb (0) (cid:105) , (2.1)and we define the axial-current correlator Π AA,µν similarly. We express their difference interms of two invariant functions,Π
LR,µν ( q ) = Π V V,µν ( q ) − Π AA,µν ( q ) (2.2)= ( q δ µν − q µ q ν )Π (1) ( q ) + q µ q ν Π (0) ( q ) . The transverse part, Π (1) ( q ), is an order parameter for chiral symmetry breaking. Havingin hand our lattice calculation of Π (1) [15], we compare in Sec. III the results with thepredictions of ChPT. We make similar use of the differenceΠ (1 − = Π (1) − Π (0) . (2.3)Our lattice calculation is based on different lattice formulations for the sea and valencefermions (see Sec. II C below), and we also allow for different sea and valence masses. Weare thus forced to consider partially-quenched (PQ) ChPT. Setting aside lattice correctionsfor now, we find that continuum PQ ChPT gives pole terms in leading order (LO),Π (1) = F vv q + ˆΠ (1) , (2.4)and Π (1 − = F vv q + M vv + ˆΠ (1) , (2.5)where ˆΠ (1) first arises in NLO. The poles arise from the creation and annihilation of a singlevalence pion; M vv and F vv are the valence pion mass and decay constant, respectively. The1 /q singularity in Π (1) is kinematical, and so its location is independent of M vv .In NLO, ˆΠ (1) arises from a pion loop, which introduces L as a counterterm. In our case,the loop is made of a mixed sea-valence pion. Explicitly,ˆΠ (1) ( q ) = G ( N )48 π (cid:20)
13 + log (cid:18) M vs µ (cid:19) − H ( s ) (cid:21) + 8 L . (2.6)3he ingredients of the NLO expression are the following. For N = 2 Dirac fermions in a realrepresentation, the group theoretical factor is G ( N ) = N + 1 = 3 [20]. In the continuum,the mass of the mixed pion is given to LO by M vs = (cid:0) M ss + M vv (cid:1) / , (2.7)where M ss is the mass of the sextet sea pion. Finally, the function H ( s ) is given by H ( s ) = 2 s + s log (cid:18) s − s + 1 (cid:19) , (2.8)where in turn s = (cid:112) M vs /q . (2.9)We use the same renormalization prescription for loop diagrams as in Refs. [20, 27]. Wechoose the renormalization scale to be µ = 1 /t , where t is the gradient-flow scale [34].Each of our lattice ensembles gives us values for Π (1) ( q ) and Π (1 − ( q ) in a range ofmomenta q and for a set of values of the valence fermion mass, giving two different approachesto ˆΠ (1) ( q ) via Eqs. (2.4) and (2.5); ˆΠ (1) ( q ) is supposed to satisfy Eq. (2.6), subject to NNLOand lattice corrections, described below. Likewise, each ensemble gives M vv and F vv , againas a function of the valence fermion mass, as well as an ensemble average of M ss . Then afit to Π (1) ( q ) or Π (1 − ( q ) gives L . B. Beyond next-to-leading order
The earliest determination of L in QCD was based on experimental input [19]. Thefirst lattice calculations, using ChPT at NLO, gave a similar value [35, 36]. The muchmore challenging calculation at next-to-NLO (NNLO) was performed by two groups [37, 38]several years later (see also Ref. [16]). The NNLO calculations, which combined latticeresults with experimental data, found a central value lower by some 30% than the earlyNLO calculations.In the continuum, an NNLO calculation of Π (1) and Π (1 − in the PQ theory will containnew loop diagrams, along with counterterms of the form1(4 πF ) (cid:0) b q q + b s m s + b v m v (cid:1) . (2.10)Here m s and m v are the masses of the sextet sea and valence fermions, and the parameters b q , b s and b v are linear combinations of the NNLO LECs. F is the sextet pion decay constantin the chiral limit. A full NNLO calculation is beyond the scope of this work; nonetheless,in view of the lesson from QCD calculations, we attempt below to estimate the systematicuncertainties of our calculation by exploring the effect of analytic terms similar in structureto the NNLO counterterms.In principle, Eq. (2.10) should contain an additional term proportional to the mass of thefundamental-representation sea fermions, m s, . We have found in previous work, however,that m s, has almost no effect on observables constructed from the sextet fermions [11], andhence we drop it. 4 . Lattice discretization Our lattice simulations employed Wilson fermions for the dynamical sea: two flavors inthe fundamental representation, along with two (Dirac) flavors in the sextet representation[11]. Because of the importance of chiral symmetry for the calculation of Π
LR,µν ( q ), weconstructed the current correlators using staggered valence fermions [15]. These are muchmore economical than other chiral fermion formulations—overlap and domain-wall—thathave been used for calculations of L in QCD [35, 36].We calculated the connected part of the vector and axial two-point functions as follows.At a formal level, we introduce two valence staggered fields in the sextet representation, andconsider the flavor non-singlet vector currents (see for example Ref. [39]). These are V µa ( x ) = η µ ( x )2 (cid:2) ¯ χ ( x ) U µ ( x ) T a χ ( x + ˆ µ ) + ¯ χ ( x + ˆ µ ) U † µ ( x ) T a χ ( x ) (cid:3) , (2.11)defined from one-component staggered fields χ, ¯ χ . Here U µ ( x ) is the SU(4) lattice gaugefield. The corresponding partially conserved axial currents include the sign factor (cid:15) ( x ) =( − x + x + x + x , and are given by A µa ( x ) = η µ ( x ) (cid:15) ( x )2 (cid:2) ¯ χ ( x ) U µ ( x ) T a χ ( x + ˆ µ ) − ¯ χ ( x + ˆ µ ) U † µ ( x ) T a χ ( x ) (cid:3) . (2.12)The other sign factors are, as usual, η ( x ) = 1 , η ( x ) = ( − x , η ( x ) = ( − x + x , η ( x ) = ( − x + x + x . (2.13)These currents correspond to the nearest-neighbor staggered action.We calculated the current–current correlation function with these staggered currents, andwe extracted lattice approximations to the invariant functions by the same method as inRefs. [15, 35, 40]. Introducing the chiral currents, J Lµa = V µa − A µa and J Rµa = V µa + A µa , wedefine the lattice correlator, δ ab Π lat µν ( q ) = 14 a (cid:88) x e iqx (cid:10) J Lµa ( x ) J Rνb (0) (cid:11) , (2.14)where a is the lattice spacing. The factor of corrects for the summation over the fourtastes contained in the staggered field. With Π lat µν ( q ) in hand, we extract the transverse andlongitudinal functions via Π (1) = (cid:80) µν P ⊥ µν Π lat µν q ) , (2.15)Π (0) = (cid:80) µν P (cid:107) µν Π lat µν (ˆ q ) , where the lattice projectors are P ⊥ µν = ˆ q δ µν − ˆ q µ ˆ q ν , (2.16) P (cid:107) µν = ˆ q µ ˆ q ν . Throughout this paper the traceless, hermitian flavor generators are normalized according to tr( T a T b ) = δ ab . Our sign convention here is opposite to that in our earlier paper [15]. q µ = (2 /a ) sin( aq µ / q = (cid:80) µ ˆ q µ . Following common practice, it is also conve-nient to replace q everywhere by ˆ q in the ChPT results of Sec. II A.Since we use different sea and valence lattice fermions, we need to generalize the partially-quenched results to mixed-action lattice ChPT. This entails the introduction of two newparameters. First, in place of Eq. (2.7), the mass of the mixed sea-valence pion becomes t M vs = t (cid:0) M ss + M vv (cid:1) / a ∆ mix , (2.17)where we have expressed all quantities in t units, and ˆ a = a/ √ t . Here ∆ mix is a new LOLEC of the mixed-action theory [28–30]. Following the reasoning of Ref. [31], ∆ mix must bepositive.In addition, at NNLO there is one more analytic term. The full set of analytic NNLOterms we use is t (cid:0) b q q + b ss M ss + b vv M vv (cid:1) + b a ˆ a . (2.18)This involves two technical changes compared to Eq. (2.10). First, instead of using the decayconstant F for the reference scale, it is more convenient for us to use the gradient flow scale t . Also, we replace the term linear in m s ( m v ) by a term linear in M ss ( M vv ), noting thatthey are interchangeable at LO in ChPT. The new element in Eq. (2.18) is the last term: adiscretization term proportional to a . In App. B we explain why the discretization term is ∼ a , and not ∼ a . III. FITS TO NUMERICAL RESULTS
We begin with a brief description of the ensembles. In this work we use 12 ensembles withvolume 16 ×
32, the same set of ensembles we used for our study of the baryon spectrum[12]. In addition, we use 3 ensembles with volume 24 ×
48, numbered 40, 42 and 43 inRef. [11]. For each ensemble, we calculated the connected two-point function of the (partially)conserved vector and axial staggered currents of the sextet representation for 7 valencemasses: am v = 0 . , aM vv , and its decay constant aF vv , again as afunction of am v . The sextet (Wilson) sea pion mass, aM ss , and the gradient flow scale t /a , which are also used in our analysis, were previously obtained in Ref. [11]. Fixing t as the scale of the theory gives us the lattice spacing a for each ensemble: For the presentensemble set, t /a is in the range 0.9–2.7, while √ t M ss is in the range 0.2–0.58. On eachensemble, correlations of all valence observables as well as M ss were calculated using single-elimination jackknife. The only exceptions are correlations of t with other observables,which we ignore because the fluctuations in t are very small.While all 7 valence masses are used in our analysis, we restrict our fits to Π (1) ( q )and Π (1 − ( q ) to the smallest momentum (which is timelike on our asymmetric lattices).On the 16 ×
32 lattices, this momentum is aq (cid:39) a ˆ q (cid:39) . am v = 0 . A fourth ensemble with this volume turned out to be an outlier, and is not included in our analysis. All the calculations described to this point—staggered valence spectra and current correlators—werecarried out for the analysis of C LR presented in Ref. [15], which can be consulted for further details. As a first approach, we perform correlated fits of Π (1) to Eq. (2.4), and of Π (1 − toEq. (2.5), using the NLO expression for ˆΠ (1) , Eq. (2.6). At this stage, we do not include anyof the analytic NNLO terms of Eq. (2.18) in the fits. We find that these fits have a good p -value if and only if they include data from only the smallest valence mass, am v = 0 . L are statistically consistent whether we fit Π (1) or Π (1 − , andregardless of whether we include the parameter ∆ mix in the formula (2.17) or set it to zero.While statistically satisfactory, the inability to perform an NLO fit with any larger valencemasses should be taken as a warning sign. As we already mentioned in Sec. II B, in QCDthe NNLO corrections are crucial. It is thus important to gauge their significance here. Thebest we can do is to explore the effect of including various combinations of the analyticNNLO terms in the fit. We are led to repeat these fits, still only with the smallest valencemass, while trying out all possible combinations of the NNLO analytic terms (2.18). Thisturns out to have a dramatic effect on the result: both the mean value and the error of L vary substantially depending on the subset of the analytic NNLO terms that we include inthe fit. We conclude that these fits to NLO ChPT are not under control.More details of these fits may be found in App. C. This appendix also reports otherexploratory studies that we have carried out.In order to constrain the theory better, we turn to correlated fits that include data fromall seven valence masses. All fits of Π (1) to Eq. (2.4) give a p -value that is practically zero,and will not be considered any further. By contrast, fits of Π (1 − to Eq. (2.5) turn outto give a good p -value as long as the NNLO parameter b vv is present in the fit. Given themuch larger valence masses included in the new fits, the need for an NNLO ingredient is notsurprising. As for the difference between Π (1) and Π (1 − , we do not conclude that ChPTaccounts for Π (1 − better than Π (1) . Rather, this difference stems primarily from the betterbehaved correlation matrix of the Π (1 − data.The results of the Π (1 − fits are summarized in Table 1 and plotted in Fig. 1. All 16 fitsinclude L and b vv , and together they cover all combinations of the remaining parameters,∆ mix , b q , b ss and b a . With 15 ensembles and 7 valence masses, we have altogether 105 datapoints. The number of parameters is between 2 and 6, so that there are between 103 and 99degrees of freedom. In spite of the fairly strong correlations still present in the Π (1 − data,the p -value is always good.If we look at the NNLO parameters, we see that the results for all of them are nicelyconsistent across all fits. Indeed we find that b vv is particularly stable. The NLO mixed-action parameter, ∆ mix , is always consistent with zero. The presence of ∆ mix in the fit hasvirtually no effect on the mean value of L , and a very small effect on its error. We thusbase our final result on the fits that do not include ∆ mix .Turning to L itself, we see that, like the fits at the smallest valence mass (see App. C 1),different combinations of the NNLO parameters again give rise to results that vary signifi-cantly. This means that our main source of uncertainty is systematic. In order to estimatethis uncertainty, we momentarily disregard the statistical errors and consider the spreadof mean values of L reported in Table 1. The lowest mean value comes from fit 7, andthe highest from fit 9. The two results have similar statistical errors. We take the finalmean value to be the average of fits 7 and 9, and the systematic uncertainty to be half their The calculation of C LR in Ref. [15] involved integration of Π (1) ( q ) over all lattice momenta. it p -value − L ∆ mix b vv b q b ss − b a − . − . − . − . (1 − to Eq. (2.5) using all 7 valence masses. The NNLO parameter b vv is included in all fits. The 16 fits cover all combinations of ∆ mix and the three remaining NNLOparameters. difference. Adding in the statistical error of the two fits, our final result is L = − . stat (35) syst . (3.1)We note that fits 11 and 15 have a big statistical error that largely overlaps with thesystematic error of our final result. These fits include all, or all but one, of the NNLOparameters, and so their statistical error probably reflects a growing redundancy among thefit parameters. As it happens, the central value stated for L coincides with the results offits 1 and 2, where only b vv is added to the NLO parameters, and the error band in Eq. (3.1)covers all the points plotted in Fig. 1. We believe that Eq. (3.1), in which the dominanterror is systematic, accounts well for the behavior of L reported in Table 1.The dominant finite-volume effects in our calculation originate in the NLO loop of themixed valence-sea pion. Since in practice ∆ mix vanishes, M vs can be approximated byEq. (2.7). We find that in all cases M vs L > . M vs L > M vs L >
4. The p -value of these fits is better than 0.75, and mostly above 0.9. In all fits, L changes by much less than 1 σ . Finally, finite-volume effects in the sea sector were shown tobe well under control in Ref. [11].In QCD, it is customary to quote L at the ρ meson mass [16]. We can change therenormalization scale µ in Eq. (2.6) from 1 / √ t to the sextet vector meson mass M V . InRef. [11] we found M V √ t ≈ . L by about − . . . . . . L . . . . . mix . . . . . b vv b q . . . . . b ss . . . . . b a FIG. 1. Sixteen fits of Π (1 − to data from all 7 valence masses. All fits include L and b vv asparameters but have different combinations of ∆ mix and the other NNLO parameters b q , b ss and b a . Fits without ∆ mix are shown in purple, and with ∆ mix in orange. The index i = 1 , . . . ,
16 onthe ordinates corresponds to the rows of Table 1.
IV. CONCLUSION
Phenomenologically, L appears in the dimension-6 lagrangian L that controls the lead-ing deviations of Higgs decay rates from their Standard Model value [4, 6, 41]. Since onlyone linear combination of the parameters in L is determined by L , however, we do notpursue this calculation. On the other hand, we can use our result for L to obtain thecontribution of the hypercolor sector to the S parameter, which we denote by S HC . Thecalculation is relegated to App. A. The result is S HC = ξS NLO , S
NLO = 0 . . (4.1)The error of S NLO is dominated by the systematic error of L . In contrast with technicolormodels, in composite-Higgs models S HC is suppressed [4, 6] by the vacuum misalignmentparameter ξ = 2 v /F , where v = 246 GeV is the vacuum expectation value of the Higgsfield in the Standard Model, and F is the decay constant of pNGBs made of the sextetfermions in the chiral limit. In arriving at Eq. (4.1) we took F = 1 . ξ ≤ . M = M h , with M h = 125 GeV the physical Higgs mass, thus obtaining an over-estimate of S NLO for thegiven F .The current experimental estimate is S = − . σ upper bound of0.09 [23]. Together with Eq. (4.1) this gives an independent 1 σ bound ξ ≤ . . . . (4.2) See, for example, Refs. [35, 36, 42]. The factor of 2 in the definition of ξ stems from our normalization convention for F . F [TeV] S H C FIG. 2. Plot of S HC , the contribution of the hypercolor theory to the S parameter, as a functionof the sextet decay constant F , assuming all pNGBs have the same mass: M = M h = 125 GeVfor the blue band, and M = 10 M h for the orange band. S HC depends on F mainly through thevacuum misalignment parameter ξ , and hence the curves are approximately linear, with slope of −
2. The horizontal line is the 1 σ upper bound on the S parameter, while the vertical line gives thelower bound on F compatible with the bound ξ ≤ . S parameter of the hypercolor theory does not lead to a stronger boundon ξ . Our new bound is compatible with the bound ξ ≤ . S parameter of the hypercolor theory does not lead to a more stringent constraint on thescale of new physics.In Fig. 2 we plot S HC as a function of the sextet decay constant F in physical units,for the simplified case of degenerate pNGBs. The blue band is obtained assuming that thecommon pNGB mass is M = M h , while for the orange band M = 10 M h = 1 .
25 TeV. Webelieve that, together, these bands provide an idea on S HC for the realistic case of non-generate pNGB masses. Coming from the right, the bands cross the line F = 1 . S parameter, which illustrates the point thatour new bound on the S parameter (4.2) is essentially the same as the existing experimentalbound ξ ≤ . L in a prototype composite-Higgs model, using staggered valence fermions to define the sextet-representation currentcorrelators. We used the full NLO ChPT expressions for (cid:104) V µ V ν − A µ A ν (cid:105) , along with the an-alytic NNLO terms. The error in our final result (3.1) is dominated by systematic uncertain-ties. We believe that these uncertainties can be significantly reduced only by a full-fledgedNNLO calculation, a demanding task both theoretically and numerically. At a modest cost,the present calculation provides an indication of the size that L could have in similarcomposite-Higgs models.For the fundamental representation, large- N c considerations suggest that, like F π , L will scale with N c . For other representations, the expectation is that F π and L scalewith the dimension of the representation [11]. In N f = 3 QCD, the current best value is10 = − . × − [16, 37, 38]. Thus, our result (3.1) is reasonably consistent with theanticipated scaling.In Ref. [14] we showed that the same prototype composite-Higgs model is unable to inducea realistic top mass via its coupling to the top partner. With two Dirac fermions in boththe fundamental and sextet representations, our model is close to the M6 model of Ferrettiand Karateev [7, 10]. This suggests that a realistic top mass might not be attainable in M6either. The prospects are brighter for M11, which has more fermions in both the fundamentaland sextet representations. The bigger fermion content places M11 closer to the conformalwindow. This, in turn, may significantly enhance the coupling between the top quark andits partner. Acknowledgements
Our calculations of staggered fermion propagators and currents were carried out withcode derived from version 7.8 of the publicly available code of the MILC collaboration[43]. Computations for this work were carried out with resources provided by the USQCDCollaboration, which is funded by the Office of Science of the U.S. Department of Energy.This material is based upon work supported by the U.S. Department of Energy, Office ofScience, Office of High Energy Physics, under Awards No. DE-SC0010005 (Colorado) andDE-SC0013682 (SFSU), and by the Israel Science Foundation under grant No. 491/17 (TelAviv). Fermilab is operated by the Fermi Research Alliance, LLC under contract No. DE-AC02-07CH11359 with the U.S. Department of Energy.
Appendix A: S parameter For a real representation, the non-linear field is symmetric, Σ = Σ T , and takes values inSU( N M ), where N M is the number of Majorana fermions. The symmetry breaking patternis SU( N M ) → SO( N M ) [44], and, assuming that the vacuum (cid:104) Σ (cid:105) is aligned with the identitymatrix, the generators of SO( N M ) are antisymmetric. L couples to the NLO operator [20] O real10 = tr (cid:0) B µν Σ B Tµν Σ ∗ (cid:1) , (A1)where the external gauge field B µν promotes the full SU( N M ) flavor symmetry group to alocal symmetry. For the calculation of L , as well as the S parameter, we only need thelinearized part of B µν . Writing B µν = V µν − A µν , with V µν ( A µν ) for the unbroken (broken)generators, we arrive at Eq. (1.1), which has the same form as in the familiar QCD case. In terms of (cid:104) V µ V ν − A µ A ν (cid:105) , the S parameter may be defined for any fermion representationas [20, 22] S = − π lim q → ∂∂q q Π (1) = − π lim q → ˆΠ (1) . (A2)At NLO this gives S NLO = − G ( N )24 π (cid:18) (cid:18) M µ (cid:19)(cid:19) − πL , (A3) For more details, see Refs. [20, 21, 45] and references therein. q → H ( s ) = − /
3. Starting from the renormalizationscale µ = 1 / √ t used in Sec. III, it will be convenient to reexpress log( t M ) = log( M /F )+log( t F ), where F is the decay constant of the sextet fermions in the chiral limit, using √ t F = 0 . L in the sextet sector of thishypercolor theory is close to what we find in our lattice model, Eq. (3.1). When applyingEq. (A3) to the M6 model, we will use G ( N ) = G (5 /
2) = 7 / L × SU(2) R symmetry of the Standard Model is identified with an SO(4) sub-group of the unbroken SO(5), with the SU(2) L gauge fields W iµ , i = 1 , ,
3, coupled to thegenerators T iL , and the U(1) Y gauge field B µ coupled to T R . The Higgs doublet is identifiedwith 4 pNGBs of the coset SU(5)/SO(5). After electroweak symmetry breaking, the vacuumof the sextet sector becomes (cid:104) Σ (cid:105) = Ω ( ζ ), where the argument of Ω is ζ = √ h/F , with h the expectation value of the pNGB field associated with the physical Higgs particle. Theexplicit form of Ω, as well as of the SU(2) L,R generators T iL,R , may be found in Appendix Bof Ref. [8].Experimentally, the S parameter is defined as the contribution of new physics beyond theStandard Model to Eq. (A2), where, instead of (cid:104) V µ V ν − A µ A ν (cid:105) , the transverse function Π (1) is defined from the correlator (cid:10) J W µ J Bν (cid:11) [4, 6, 22, 23]. The contribution of the hypercolortheory to the S parameter, denoted S HC , is given at NLO by [compare Eq. (4.1)] S HC = ξS NLO , (A4)where S NLO is calculated in the hypercolor theory using Eq. (A3); the vacuum misalignmentparameter is ξ ≡ v F = sin ( √ h/F ) . (A5)In arriving at Eq. (A4) we used tr (cid:0) T iL Ω T jR (Ω ) ∗ (cid:1) = ξδ ij . (A6)In Eq. (A3) we have made the simplifying assumption that the 14 pNGBs of the cosetSU(5)/SO(5) are all degenerate in mass. In reality, apart from any explicit mass terms in thehypercolor theory, the coupling to Standard Model fields will generate an effective potential[4–6, 8–10, 24, 25], whose minimization must generate an expectation value for the Higgsfield. Then the pNGBs split into several distinct multiplets of the SU(2) V diagonal subgroupof SU(2) L × SU(2) R . The Higgs doublet contains the physical Higgs field and the 3 NGBsof SU(2) L × SU(2) R → SU(2) V symmetry breaking. The other 10 pNGBs split into twosinglets, a triplet, and a quintet of SU(2) V . Furthermore, the coupling to the U(1) Y gaugefield breaks explicitly SU(2) R , and thus also SU(2) V , generating additional mass splittingsdepending on the electric charges, which in turn range from 0 to ±
2. The electricallycharged pNGBs must have large masses to evade detection. Thus, calculating S NLO for therealistic non-degenerate case is tedious, and the resulting expression will depend on several The 3 exact NGBs turn into the longitudinal components of the W ± and Z bosons. S NLO in the conclusion section for the degeneratemass case using Eq. (A3). To get an idea of the variation of the S parameter as a functionof the pNGB masses, we calculate it for two distinct choices of the common pNGB mass M . Appendix B: NNLO discretization effects
In this appendix we explain why the NNLO discretization term in Eq. (2.18) is O ( a ),and not O ( a ). In itself, an a discretization term is consistent with the usual power count-ing of staggered ChPT. Since, however, our mixed-action calculation also includes Wilson(sea) fermions, the question arises whether there should be an O ( a ) discretization term inEq. (2.18).Wilson ChPT comes with two alternative power counting schemes for the discretizationeffects, known as GSM, where m ∼ a , and LCE, where m ∼ a (see for example Ref. [33]).Here we follow the GSM scheme, as we did in our spectroscopy study [11]. The LO potentialfor pions made out of Wilson fermions is then [46] L m = − F Bm (cid:48) † ) . (B1)The shifted mass m (cid:48) is defined by Bm (cid:48) = Bm + W a , (B2)where B is the usual continuum LEC, while W is a new LEC peculiar to Wilson ChPT.A central feature is the relation between the shifted mass m (cid:48) and the axial Ward identitymass m AWI . The latter is defined by imposing the following identity in the Wilson theory, ∂ (cid:104) A a ( t ) P a (0) (cid:105) = 2 m AWI (cid:104) P a ( t ) P a (0) (cid:105) , (B3)where the correlation functions are evaluated at zero spatial momentum. A aµ and P a arethe renormalized (and, possibly, improved) axial current and pseudoscalar density of theWilson theory. In Ref. [47] the following relation was proved between the shifted mass m (cid:48) and m AWI , m AWI = m (cid:48) + O ( m ) + O ( ma ) + O ( a ) . (B4)Notice the absence of an O ( a ) term on the right-hand side; the O ( a ) term from Eq. (B2)has been absorbed into m (cid:48) . Physically, Eq. (B3) implies that the mass of the Wilson pionsatisfies M π ∼ m AWI , whereas Wilson ChPT at LO implies the relation M π ∼ m (cid:48) . Thus,Eq. (B4) expresses the consistency of Wilson ChPT with the underlying theory.If we tune the shifted mass to zero, it follows that m = O ( a ), and so m AWI = O ( a ) , m (cid:48) → . (B5)In words, m AWI vanishes simultaneously with the shifted mass, up to a residual O ( a ) part.The leftover O ( a ) term is important, as it leaves room for the Aoki phase [46, 48]. Inparticular, within the so-called 1st-order scenario [46], m AWI and M π do not vanish at m (cid:48) = 0. Rather, they attain O ( a ) minimum values there, and their m (cid:48) derivatives arediscontinuous.In our mixed-action case, only two terms in the chiral lagrangian are relevant to thisdiscussion, namely,( ˜ Bm + ˜ W a ) tr (cid:16) P s (Σ + Σ † ) (cid:17) tr (cid:16) ( V µν − A µν ) P v Σ( V µν + A µν ) P v Σ † (cid:17) , (B6)13here ˜ B and ˜ W are new NNLO LECs. The chiral field Σ now accounts for the sea, valence,and ghost quarks. The corresponding projectors— P s , P v and P gh , respectively—satisfy P s + P v + P gh = 1. With these projectors in place, the first trace has the same form as theWilson LO potential (B1), whereas the second trace reduces to the L operator (1.1) actingon the valence entries of the Σ field.The key observation is that [by Eq. (B5)] when we tune m (cid:48) → O ( a ). In order that no O ( a ) violations willsurvive in this limit, we must have ˜ Bm + ˜ W a = ˜ Bm (cid:48) (B7)in Eq. (B6). To see this, we may consider the Ward–Takahashi identity ∂ (cid:10) A a ,s ( t ) P as (0) J Lv,µb J Rv,µb (cid:11) = 2 m AWI (cid:10) P as ( t ) P as (0) J Lv,µb J Rv,µb (cid:11) , (B8)where the subscripts s, v refer to sea and valence operators, respectively [cf. Eq. (2.14)]. Inthe mixed-action theory, this Ward–Takahashi identity corresponds to an axial transforma-tion in the Wilson sea sector only. Since the valence operators in Eq. (B8) are inert underthis transformation, consistency with Eq. (B3) requires that the coefficient on the right-handside must be 2 m AWI . Furthermore, in order that the identity be reproduced in mixed-actionChPT, Eq. (B7) must be true. The prefactor in Eq. (B6) is therefore proportional to theshifted mass, and a separate O ( a ) term cannot be present. Appendix C: Other methods
In this appendix we briefly describe several alternative fits that we have carried out forthe determination of L . As will be clear, the results are inferior in quality to the preferredanalyses presented in the body of the paper.
1. Fits with the lightest valence mass
We mentioned in Sec. III that we attempted fits using only data at the lightest valencemass, am v = 0 .
01. We performed correlated fits of Π (1) to Eq. (2.4); alternatively, we fitΠ (1 − to Eq. (2.5). The left panel of Fig. 3 shows the values of L obtained from fits ofΠ (1) with all combinations of the NNLO parameters [Eq. (2.18)], both with and withoutthe parameter ∆ mix [Eq. (2.17)]. The four columns of + / − signs indicate which NNLOparameters are present/absent in each fit. In each case we plot a fit that includes ∆ mix inblue, and a fit where we set ∆ mix = 0 in red. The results of fitting Π (1 − are presentedsimilarly in the right panel of Fig. 3, using the same color scheme. All fits are of goodquality, with p -values in the range 0.25–0.8.A comparison of the two panels of Fig. 3 shows that there is generally good agreementbetween the values of L obtained from each fit to Π (1) , and from the corresponding fitto Π (1 − . The pure NLO fits (the topmost fit in each panel) have tiny statistical errors—roughly the size of the symbol. Nonetheless, the spread of L values obtained with varioussubsets of the NNLO parameters shows that relying on the NLO result would be misleading.Indeed both the mean value and the error of L vary substantially depending on whichadditional parameters are present in the fit. As we explained in Sec. II B and Sec. III, any14 .05 0.04 0.03 0.02 0.01 0.00 b q b ss b a b vv b q b ss b a b vv FIG. 3. Results for L , using only the smallest valence mass am v = 0 .
01, for all combinations ofthe four NNLO parameters and ∆ mix (see App. C 1). Left panel: Fits of Π (1) . Right panel: fits ofΠ (1 − . The columns with + / − signs indicate which NNLO parameters are present/absent in eachfit. Fit results with ∆ mix free are shown in blue, and fits with ∆ mix = 0 in red. The topmost pairof points in each plot represents the NLO fit. combination of NNLO parameters is as good as any other, when it comes to estimating thesystematic effect of the missing non-analytic NNLO terms. With their error bars, the resultsdisplayed in Fig. 3 would allow L to be basically anywhere in the range [ − . , . L with amuch smaller systematic error, at the modest price of always having to include the NNLOparameter b vv in the fit.
2. Using ChPT for F vv The pole parts in Eqs. (2.4) and (2.5) are proportional to the valence decay constantsquared, F vv . Instead of taking F vv from data, we may alternatively use the NLO expression, √ t F vv, = ˚ F (cid:2) (1 − ) + t (cid:0) L vs M ss, + L vs M ss, + L vv M vv, (cid:1)(cid:3) + L latt6 ˆ a . (C1)The notation here is similar to Ref. [11] (see also Ref. [21]). The subscripts 4 and 6 refer tothe fundamental and sextet representations, respectively. ˚ F is the sextet decay constant inthe chiral limit in t units, while L vs , L vs , L vv and L latt6 are linear combinations of variousNLO LECs. The NLO logarithm is∆ = t M vs, π ˚ F log( t M vs, ) . (C2)This logarithm is the same as in Ref. [11], except that the pion in the loop is now a mixedsea-valence pion. Note the absence of a discretization term ∼ a , which can be proved usingarguments similar to those in App. B.Fits using the above expression for F vv = F vv, are largely consistent with our final resultfor L , Eq. (3.1). However, the uncertainty in the value of L turns out to be much largerhere, and therefore we do not include fits that make use of Eq. (C1) in our main analysis.15 .04 0.06 0.08 0.10 q t ( ) L R ( m v ) m v a ( ) L R E21E22E24E26E31E33E34E35E36E37E38E39E40E42E43
FIG. 4. Left: Π (1) in the chiral-valence limit am v → q t for all 15 ensembles.The horizontal grey band shows the contribution 8 L to Π (1) ( q ), where for simplicity we combinedthe statistical and systematic errors of our final result in quadrature [see Eq. (3.1)]. Right: Thelinear fits of Π (1) in each ensemble that give the limits shown on the left. Ensemble numbers areas in Ref. [11]. The sextet decay constant in the chiral limit, ˚ F , which is one of the fit parameters inEq. (C1), was already determined in Ref. [11]. The result we find here for ˚ F is consistentwith the value reported in Ref. [11], albeit with a larger error.One could similarly carry out fits using the NLO expressions for both F vv and M vv inthe pole term of Π (1 − . In view of the limited success of the fits using Eq. (C 2) we donot pursue such fits. We comment that in the case of M vv one expects larger finite-volumeeffects, originating from “hairpin” diagrams with a valence pion in the loop. In the fitsreported in Sec. III we take both F vv and M vv from data, and hence this issue does notarise.
3. Extrapolation to the m v → We showed above the result of fitting only the smallest valence mass for each ensemble.This was motivated by a desire to distill highly correlated data down to a single data pointfor each ensemble. An alternative, similarly motivated, is to extrapolate Π (1) to the chiral-valence limit, leaving us with Π (1) ( am v →
0) for each ensemble. We have performed theextrapolations via uncorrelated linear fits. The linearity of the extrapolation, and the useof independent fit parameters for each ensemble, both mean that this is not ChPT. Still,the linear extrapolations turn out to have some interesting features all by themselves.The results are shown in the left panel of Fig. 4, plotted against ˆ q t for each of the 15ensembles. Note that while the dimensionless q µ a is always the smallest time-like momentum,the gradient-flow scale t /a varies considerably between ensembles. The jaggedness ofthe plot is because Π (1) ( am v →
0) depends not only on ˆ q t , but also on the sea sextetfermion mass, as well as (weakly) on the sea fundamental fermion mass. To give a visualimpression, the contribution 8 L to Π (1) ( q ) is shown as a horizontal grey band, using the16nal result (3.1), with statistical and systematic error added in quadrature [see Eqs. (2.4)and (2.6)].The actual linear extrapolations are shown in the right panel of Fig. 4. The first thingto notice is that, visually, the linear fits describe the data well. A distinct feature of theselinear fits is that almost the same slope is found for all the 16 ×
32 ensembles; the threelarger 24 ×
28 ensembles (E40, E42 and E43) again exhibit a similar slope, which in turnis bigger than that of the 16 ×
32 ensembles.The dependence of the slopes on the lattice size appears to arise primarily from thekinematical pole, F vv / ˆ q (recall we are using only the smallest timelike momentum). Wehave checked that ( aF vv ) is also roughly linear in am v , with again, almost the same slopefor all the 16 ×
32 ensembles, and with a different slope for all the 24 ×
28 ensembles. Theslope for the 16 ×
32 ensembles is in fact larger than the slope for the 24 ×
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