Large N Scalars: From Glueballs to Dynamical Higgs Models
We construct effective Lagrangians, and corresponding counting schemes, valid to describe the dynamics of the lowest lying large N stable massive composite state emerging in strongly coupled theories. The large N counting rules can now be employed when computing quantum corrections via an effective Lagrangian description. The framework allows for systematic investigations of composite dynamics of non-Goldstone nature. Relevant examples are the lightest glueball states emerging in any Yang-Mills theory. We further apply the effective approach and associated counting scheme to composite models at the electroweak scale. To illustrate the formalism we consider the possibility that the Higgs emerges as: the lightest glueball of a new composite theory; the large N scalar meson in models of dynamical electroweak symmetry breaking; the large N pseudodilaton useful also for models of near-conformal dynamics. For each of these realisations we determine the leading N corrections to the electroweak precision parameters. The results nicely elucidate the underlying large N dynamics and can be used to confront first principle lattice results featuring composite scalars with a systematic effective approach. Preprint: CP-Origins-2015-035 DNRF90& DIAS-2015-35
Strong dynamics continues to pose a formidable challenge. Several analytical and numerical ingenious techniques have been invented, exploited and are routinely used to elucidate some of its physical properties. The large number of underlying colours limit is a time-honoured example ’tHooft:1973jz (); Witten:1979kh (); Witten:1980sp (). It has been extensively used in Quantum Chromodynamics (QCD) and string theory, and it constitutes the backbone of the gauge-gravity duality program. We will further highlight here the power of the large expansion by introducing a four-dimensional calculable framework permitting to investigate the dynamics of the lightest stable non-Goldstone large composite state.
t’ Hooft and Witten demonstrated that Yang-Mills theories at large number of colours admit an effective description in terms of an infinite number of non-interacting absolutely stable hadronic states of arbitrary spin ’tHooft:1973jz (); Witten:1979kh (); Witten:1980sp (). By capitalising on this central result we focus on the physics of the lightest massive scalar state that is known to play an important role in QCD Sannino:1995ik (); Harada:1995dc (); Harada:1996wr (); Harada:2003em (); Black:2000qq (); Caprini:2005zr (); GarciaMartin:2011jx (); Parganlija:2012fy (); Pelaez:2015zoa (); Ghalenovi:2015mha () and in various models of dynamical electroweak symmetry breaking as summarised in Sannino:2009za (). The framework can also be used to consistently determine quantum corrections to compare with first principle lattice simulations of composite dynamics featuring scalars Bursa:2011ru (); Fodor:2015vwa (); Kuti:2014epa (); Fodor:2012ty ().
The paper is organised as follows. In Section II we introduce the effective theory for the lightest massive glueball scalar state emerging within a pure Yang-Mills theory, and provide the associated counting scheme. Here we discuss the intriguing interplay between momentum and large expansions. The framework goes beyond the glueball example and lays the foundation of the subsequent analyses. We then extend the framework to several models of composite electroweak dynamics in Section III where the Higgs is identified with the lightest composite state. In Section IV we determine the large dependence, for each model, of the electroweak precision parameters Peskin:1990zt () stemming from different dynamical Higgs realisations. We summarise our results in Section V.
Ii Large N Effective Theory For the Lightest Glueball State
Consider the lightest scalar state stemming from an pure Yang-Mills theory, which is also expected to be the lowest lying state of the full theory. At infinite number of colours the effective Lagrangian for this state is simply the one of a free scalar field ’tHooft:1973jz (); Witten:1979kh (); Witten:1980sp ()
It is possible to go beyond the free-field limit by first defining with the intrinsic composite scale of the theory that permits to expand the effective Lagrangian both in and as follows
Here are dimensionless coefficients of order unity with . The expansion in takes care of the large suppression of higher point correlators, while the higher derivative terms take into account integrating out heavier states. Since the heavier states couple via suppressed interactions we must also have with .
To leading order in the double expansion we have:
This shows that the trilinear coupling of the scalar is naturally suppressed in this limit. Expanding a little further we have:
We have ordered the terms in such a way that the order counts as , however, we can imagine several different situations. For example we can work in the very low momentum region. In this limit we can order and therefore to the next leading order we drop the derivative terms and obtain
The effective theory features small self couplings, even though it stems from a highly nonperturbative underlying gauge theory. The glueball mass receives corrections at the fundamental theory level. By computing the one loop corrections to the two-point function one can check that the effective theory correctly reproduces the expected large corrections. Of course, the effective Lagrangian is not renormalizable in the usual sense but because the and ordering it is possible to organise and subtract the divergences order by order in this double expansion. Futhermore the coefficients of this effective theory can be determined, for a given underlying Yang-Mills gauge theory, via lattice simulations Lucini:2013qja ().
Iii Large N Scalars for Dynamical Higgs Models
We now extend the framework presented above in order to introduce consistent effective descriptions of dynamical Higgs models . We are not concerned with fitting the latest experimental data but focus instead on elucidating the associated large dynamics and effective theory structure.
iii.1 The Dynamical Higgs as the lightest Large N glueball
We start by considering the logical possibility that the dynamical Higgs state is the lightest glueball state of a new fundamental composite theory. Besides pure Yang-Mills one can also consider theories with matter displaying large distance conformality and then add an explicit source of conformal breaking, such as fermion masses. This has been show to occur via lattice simulations Bursa:2011ru () and via controllable perturbative examples Grinstein:2011dq (); Antipin:2011aa ().
Within this scenario one can envision the newly discovered particle at the Large Hadron Collider (LHC) to be the lightest glueball state of a new Yang-Mills theory with a new -independent string tension proportional to . The scale is not automatically related, here, with the electroweak symmetry breaking scale GeV or . Therefore dynamical spontaneous breaking of the electroweak symmetry is triggered by either another strongly coupled sector or, if within the same theory, via a distinct dynamically induced chiral symmetry scale
where is the SM Lagrangian without Higgs and Yukawa sectors, the ellipses denote Yukawa interactions for SM fermions other than the top-bottom doublet , and includes higher-dimensional operators, which are suppressed by powers of . In this Lagrangian is the usual exponential map of the Goldstone bosons produced by the breaking of the electroweak symmetry, , with covariant derivative , are the Pauli matrices, with . The kinetic term and potential of the SM Higgs have been replaced by the effective theory for the lightest glueball state. The tree-level SM is recovered for and . Here we will keep these couplings of order unity. Also we have not speculated on the hidden sector providing the link between the new glueball theory and the SM sector, but required it to respect the large counting for the insertion of an extra Glueball-Higgs. We have also ordered the higher derivatives on such that they are subleading when compared to the operators retained here.
If we consider the infinite number of colours limit of the new Yang-Mills theory first we arrive at a perturbative self-interacting glueball state coupled to the SM also via perturbative couplings. We have, therefore, at our disposal an organisation structure that allows to go beyond the tree-level Hansen (). We shall investigate the dependence on the number of new colors in the section dedicated to the electroweak parameters.
iii.2 The Large N physics of the Dynamical Higgs
In time-honoured models of dynamical electroweak symmetry breaking Weinberg:1975gm (); Susskind:1978ms () the Higgs can be identified with a fermion-antifermion meson
Let’s assume for definitiveness that we have an underlying theory featuring a doublet of techniquarks transforming according to the representation of the composite theory and therefore we can set . For a generic it is sufficient to replace with . For example for the fundamental representation and . Differently from the previous Glueball-Higgs example here also the pion sector is affected by the large scaling since the self-interactions among the composite pions are also controlled by . Choosing for definitivness the underlying fermions to belong to the fundamental representation we have
We have therefore:
with a given quark, the weak coupling and the Yukawa coupling. Substituting in terms of and the weak coupling via
in the effective Lagrangian we arrive at
Nicely the cost of introducing an extra power of the composite field is monitored by the corresponding power in .
iii.3 The large dynamical pseudo-dilaton
It is possible to imagine that the Higgs state of the SM is associated to the spontaneous breaking of a conformal symmetry. There are several possible realisations according to which the breaking of the conformal dynamics can be either associated to a nonperturbative sector Sannino:1999qe (); Goldberger:2008zz () or a perturbative one Grinstein:2011dq (); Antipin:2011aa ().
The large counting together with the request to satisfy the conformal relations can be both ensured by imposing
with , in general, a new scale. The Lagrangian in (6) then becomes:
This framework can be immediately extended to any dilatonic-like interpretation of the state , such as the one coming from a near conformal-like technicolor dynamics where () is identified with . Because we would like to investigate the explicit dependence we hold fix the independent scale . Clearly this limit corresponds to the Glueball-Higgs case with extra constraints for the couplings.
We are now ready to investigate the first consequences of the large counting.
Iv S and T parameters
As a relevant application of the formalism introduced above we study two important correlators, i.e. the and parameters Peskin:1991sw () for the three different types of dynamical Higgs models discussed above.
Defining by the difference between in the full theory , and the value of in the SM, i.e. we arrive at Foadi:2012ga ()
where is the coefficient of the linear term in multiplying the operator which in the SM is equal to two, and
where we have absorbed the finite part of the counterterm in the actual value of . is the reference value of the Higgs mass, is the Weinberg angle, and is given in equation (22) of Foadi:2012ga ().
We can now determine the large behaviour of these relevant parameters coming from the various large dynamical Higgs models introduced earlier.
iv.1 S and T for the Large Glueball and Dilaton Higgs Models
In the glueball case we have
and the SM limit is recovered when . The precision parameters are then
We have chosen GeV and GeV and is a counterterm that depends on the specific underlying theory. For the Glueball-Higgs theory we absorbe the unknown counterterm in the definition of and therefore set it to zero in the numerical evaluation. We expect that higher glueball states will not modify the counting.
The first observation is that if the glueball-Higgs scale is larger than the electroweak scale there is a positive contribution to the parameter and an associated negative one for the parameter. Vice-versa we observe a reduction (increase) of the () parameter if is smaller than . This is an intriguing general result given the fact that the scale of compositeness is can be kept above the electroweak scale.
Increasing while keeping fixed and one arrives at the following -independent results
The corrections appear at .
In Figure 1 we plot and as function of the number of colors for different values of . Because the corrections are in the large limit is approached quickly. We have also assumed that is its natural order of magnitude and, in any event, can be partially reabsorbed in . We compare the result with the experimental value of precision data in Fig. 2 for GeV (magenta) and . In red we have GeV and . Finally we plot the GeV (green) case for . The composite scale is always higher than the electroweak scale of GeV. The left most point on each curve corresponds to the smallest . The experiments prefer smaller values of with in the range two to four. Larger values of require to be away from the large N limit and therefore we cannot conclude on the viability of the GeV case. Increasing further it is clearly not preferred by precision observables. If, therefore, a Glueball-Higgs model does describe the Higgs we expect soon new states to be discovered with masses in the range GeV.
We stress that by requiring to be in agreement with precision measurements the couplings of the Higgs to the standard model gauge bosons are also close to the experimental values. This occurs because the product is constrained to be near the electroweak scale.
For the Dilaton-Higgs example we have
that corresponds to the results above but now with exactly equal to one.
iv.2 S and T for the large Dynamical Higgs
It is interesting to explore what happens for the large dynamical Higgs. The main difference with respect to the previous case is that the electroweak scale and the dynamical Higgs scale are now identified. Among the possible underlying models that can lead to this kind of effective dynamics there are time-honoured examples such as minimal models of (near-conformal) technicolor Appelquist:1999dq (); Sannino:2004qp (); Dietrich:2005jn ().
Differently from the Glueball-Higgs case we have at our disposal only the dependence of the effective coupling which goes as for the fundamental representation (chosen here) or as for two-index representations Sannino:2007yp (). We show in Fig. 3 the comparison with the precision electroweak constraints for the dynamical-Higgs for GeV and . The left most point on each curve corresponds to the smallest , and further assumed (left panel), (central panel), (right panel). It is clear from the results that it is possible to abide the electroweak precision constraints for larger number of colours provided that is larger than in the SM.
The computation elucidate another important point, i.e. that at large the leading corrections, expected to be proportional to , do not come from the dynamical-Higgs sector but rather from the tree-level exchange of spin one resonances Foadi:2012ga (). A simple way to understand this point is to observe that the dynamical-Higgs corrections appear, at the effective Lagrangian level, from loops of and technipions, i.e. the longitudinal components of the SM gauge bosons. At the more fundamental level these corrections are subleading in and therefore respect the counting provided by our effective approach. Furthermore these dynamical-Higgs corrections were not taken into account in Peskin:1991sw (), have been amended in Foadi:2012ga () and here we provide the intrinsic dependence.
V Conclusions and Top Corrections
We introduced effective field theories and associated counting schemes to consistently describe the lightest massive large stable composite scalar state emerging in any theory of composite dynamics. The framework allows for systematic investigations of composite dynamics featuring non-Goldstone (and Goldstone) scalars. As time-honoured examples we discussed the lightest glueball state stemming from Yang-Mills theories. We further applied our effective approach to models of (near-conformal) dynamical electroweak symmetry breaking. In particular we considered the following three possibilities: the Higgs is the lightest glueball of a new composite theory; it is a large scalar meson in models of dynamical Higgs such as technicolor, and finally we considered it to be a large pseudodilaton in the form of a conformal compensator. For each of these models, we provided the leading corrections to the precision parameters.
We observe that it is straightforward to show that in this framework the top corrections to the Glueball and dynamical Higgs mass can be reliably estimated in the large limit by rescaling in equation (4) of Foadi:2012bb (); Cacciapaglia:2014uja () by the opportune power of , for each model, and simultaneously replacing the cutoff scale by either or .
The results provide useful insight stemming from the large dynamics of these models and can be viewed as the stepping stone for consistent determination of quantum corrections at the effective Lagrangian level containing massive scalar states. The effective approach is directly applicable also to models of composite Goldstone Higgs dynamics Kaplan:1983fs (); Kaplan:1983sm () when including the first massive scalar state Cacciapaglia:2014uja (); Arbey:2015exa (); Cacciapaglia:2015eqa (), as well as to investigate interesting flavour properties Ghosh:2015gpa (); Altmannshofer:2015esa (). Finally holographic studies of the spectrum and large N properties of strongly coupled theories Arean:2013tja (); Alho:2013hsa (); Alho:2015zua () can benefit from a model independent large N computation that can be performed with the effective theories constructed here.
Acknowledgements.CP-Origins is partially supported by the Danish National Research Foundation grant DNRF:90.
- We remind the reader that in theories with an intact centre group the confining and the chiral scale are well separated Mocsy:2003qw ().
- This does not always have to be the case, meaning that the lightest state can be, in principle, made by multi-fermion states.
- G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72, 461 (1974).
- E. Witten, “Baryons in the 1/n Expansion,” Nucl. Phys. B 160, 57 (1979).
- E. Witten, “Large N Chiral Dynamics,” Annals Phys. 128, 363 (1980).
- B. Lucini and M. Panero, “Introductory lectures to large- QCD phenomenology and lattice results,” Prog. Part. Nucl. Phys. 75, 1 (2014) [arXiv:1309.3638 [hep-th]].
- F. Sannino and J. Schechter, “Exploring pi pi scattering in the 1/N(c) picture,” Phys. Rev. D 52, 96 (1995) [hep-ph/9501417].
- M. Harada, F. Sannino and J. Schechter, “Simple description of pi pi scattering to 1-GeV,” Phys. Rev. D 54, 1991 (1996) [hep-ph/9511335].
- M. Harada, F. Sannino and J. Schechter, “Comment on ‘Confirmation of the sigma meson’,” Phys. Rev. Lett. 78, 1603 (1997) [hep-ph/9609428].
- M. Harada, F. Sannino and J. Schechter, “Large N(c) and chiral dynamics,” Phys. Rev. D 69, 034005 (2004) [hep-ph/0309206].
- D. Black, A. H. Fariborz, S. Moussa, S. Nasri and J. Schechter, “Unitarized pseudoscalar meson scattering amplitudes in three flavor linear sigma models,” Phys. Rev. D 64, 014031 (2001) [hep-ph/0012278].
- I. Caprini, G. Colangelo and H. Leutwyler, “Mass and width of the lowest resonance in QCD,” Phys. Rev. Lett. 96, 132001 (2006) [hep-ph/0512364].
- R. Garcia-Martin, R. Kaminski, J. R. Pelaez and J. Ruiz de Elvira, “Precise determination of the f0(600) and f0(980) pole parameters from a dispersive data analysis,” Phys. Rev. Lett. 107, 072001 (2011) [arXiv:1107.1635 [hep-ph]].
- D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, “Meson vacuum phenomenology in a three-flavor linear sigma model with (axial-)vector mesons,” Phys. Rev. D 87, no. 1, 014011 (2013) [arXiv:1208.0585 [hep-ph]].
- J. R. Pelez, T. Cohen, F. J. Llanes-Estrada and J. Ruiz de Elvira, “Different Kinds of Light Mesons at Large ,” Acta Phys. Polon. Supp. 8, no. 2, 465 (2015).
- Z. Ghalenovi, F. Giacosa and D. H. Rischke, “Masses of Heavy and Light Scalar Tetraquarks in a Non-Relativistic Quark Model,” arXiv:1507.03345 [hep-ph].
- F. Sannino, “Conformal Dynamics for TeV Physics and Cosmology,” Acta Phys. Polon. B 40, 3533 (2009) [arXiv:0911.0931 [hep-ph]].
- F. Bursa et al., “Improved Lattice Spectroscopy of Minimal Walking Technicolor,” Phys. Rev. D 84, 034506 (2011) [arXiv:1104.4301 [hep-lat]].
- Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, “Toward the minimal realization of a light composite Higgs,” PoS LATTICE 2014, 244 (2015) [arXiv:1502.00028 [hep-lat]].
- J. Kuti, “The Higgs particle and the lattice,” PoS LATTICE 2013, 004 (2014).
- Z. Fodor, K. Holland, J. Kuti, D. Nogradi, C. Schroeder and C. H. Wong, “Can the nearly conformal sextet gauge model hide the Higgs impostor?,” Phys. Lett. B 718, 657 (2012) [arXiv:1209.0391 [hep-lat]].
- M. E. Peskin and T. Takeuchi, “A New constraint on a strongly interacting Higgs sector,” Phys. Rev. Lett. 65, 964 (1990).
- B. Grinstein and P. Uttayarat, “A Very Light Dilaton,” JHEP 1107, 038 (2011) [arXiv:1105.2370 [hep-ph]].
- O. Antipin, M. Mojaza and F. Sannino, “Light Dilaton at Fixed Points and Ultra Light Scale Super Yang Mills,” Phys. Lett. B 712, 119 (2012) [arXiv:1107.2932 [hep-ph]].
- A. Mocsy, F. Sannino and K. Tuominen, “Confinement versus chiral symmetry,” Phys. Rev. Lett. 92, 182302 (2004) [hep-ph/0308135].
- M. Hansen, K. Langaeble, C. Pica and F. Sannino, in preparation.
- S. Weinberg, “Implications of Dynamical Symmetry Breaking,” Phys. Rev. D 13, 974 (1976).
- L. Susskind, “Dynamics of Spontaneous Symmetry Breaking in the Weinberg-Salam Theory,” Phys. Rev. D 20, 2619 (1979).
- E. Corrigan and P. Ramond, “A Note on the Quark Content of Large Color Groups,” Phys. Lett. B 87, 73 (1979).
- F. Sannino and J. Schechter, “Alternative large N(c) schemes and chiral dynamics,” Phys. Rev. D 76, 014014 (2007) [arXiv:0704.0602 [hep-ph]].
- E. B. Kiritsis and J. Papavassiliou, “An Alternative Large Limit for QCD and Its Implications for Low-energy Nuclear Phenomena,” Phys. Rev. D 42, 4238 (1990).
- F. Sannino and J. Schechter, “Chiral phase transition for SU(N) gauge theories via an effective Lagrangian approach,” Phys. Rev. D 60, 056004 (1999) [hep-ph/9903359].
- W. D. Goldberger, B. Grinstein and W. Skiba, “Distinguishing the Higgs boson from the dilaton at the Large Hadron Collider,” Phys. Rev. Lett. 100, 111802 (2008) [arXiv:0708.1463 [hep-ph]].
- A. Belyaev, M. S. Brown, R. Foadi and M. T. Frandsen, “The Technicolor Higgs in the Light of LHC Data,” Phys. Rev. D 90, 035012 (2014) [arXiv:1309.2097 [hep-ph]].
- M. E. Peskin and T. Takeuchi, “Estimation of oblique electroweak corrections,” Phys. Rev. D 46, 381 (1992).
- R. Foadi and F. Sannino, “S and T parameters from a light nonstandard Higgs particle,” Phys. Rev. D 87, no. 1, 015008 (2013) [arXiv:1207.1541 [hep-ph]].
- T. Appelquist, P. S. Rodrigues da Silva and F. Sannino, “Enhanced global symmetries and the chiral phase transition,” Phys. Rev. D 60, 116007 (1999) [hep-ph/9906555].
- F. Sannino and K. Tuominen, “Orientifold theory dynamics and symmetry breaking,” Phys. Rev. D 71, 051901 (2005) [hep-ph/0405209].
- D. D. Dietrich, F. Sannino and K. Tuominen, “Light composite Higgs from higher representations versus electroweak precision measurements: Predictions for CERN LHC,” Phys. Rev. D 72, 055001 (2005) [hep-ph/0505059].
- F. Sannino and J. Schechter, “Alternative large N(c) schemes and chiral dynamics,” Phys. Rev. D 76, 014014 (2007) [arXiv:0704.0602 [hep-ph]].
- R. Foadi, M. T. Frandsen and F. Sannino, “125 GeV Higgs boson from a not so light technicolor scalar,” Phys. Rev. D 87, no. 9, 095001 (2013) [arXiv:1211.1083 [hep-ph]].
- G. Cacciapaglia and F. Sannino, “Fundamental Composite (Goldstone) Higgs Dynamics,” JHEP 1404, 111 (2014) [arXiv:1402.0233 [hep-ph]].
- D. B. Kaplan and H. Georgi, Phys. Lett. B 136, 183 (1984).
- D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. B 136, 187 (1984).
- A. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre and F. Sannino, “Fundamental Composite Electroweak Dynamics: Status at the LHC,” arXiv:1502.04718 [hep-ph].
- G. Cacciapaglia, H. Cai, A. Deandrea, T. Flacke, S. J. Lee and A. Parolini, “Composite scalars at the LHC: the Higgs, the Sextet and the Octet,” arXiv:1507.02283 [hep-ph].
- D. Ghosh, R. S. Gupta and G. Perez, “Is the Higgs Mechanism of Fermion Mass Generation a Fact? A Yukawa-less First-Two-Generation Model,” arXiv:1508.01501 [hep-ph].
- W. Altmannshofer, S. Gori, A. L. Kagan, L. Silvestrini and J. Zupan, “Uncovering Mass Generation Through Higgs Flavor Violation,” arXiv:1507.07927 [hep-ph].
- D. Areán, I. Iatrakis, M. Järvinen and E. Kiritsis, JHEP 1311, 068 (2013) [arXiv:1309.2286 [hep-ph]].
- T. Alho, M. Järvinen, K. Kajantie, E. Kiritsis, C. Rosen and K. Tuominen, “A holographic model for QCD in the Veneziano limit at finite temperature and density,” JHEP 1404, 124 (2014) [JHEP 1502, 033 (2015)] [arXiv:1312.5199 [hep-ph]].
- T. Alho, M. Jarvinen, K. Kajantie, E. Kiritsis and K. Tuominen, “Quantum and stringy corrections to the equation of state of holographic QCD matter and the nature of the chiral transition,” Phys. Rev. D 91, no. 5, 055017 (2015) [arXiv:1501.06379 [hep-ph]].