Dynamical Symmetry Breaking and Mass Generation T

WDS'05 Proceedings of Contributed Papers, Part III, 436–441, 2005. ISBN 80-86732-59-2 © MATFYZPRESS

Dynamical Symmetry Breaking and Mass Generation T. Brauner and J. Hoˇsek Nuclear Physics Institute, AS CR, Reˇz,ˇ Czech Republic

Abstract. We give a brief review of the physics of spontaneous symmetry breaking. The ﬁeld-theoretic Goldstone theorem and its consequences are emphasized. In relativistic quantum theory, the emergence of particle masses is connected with spontaneous breaking of chiral symmetry. While in the Standard model of the electroweak interaction this is achieved by the Higgs mechanism, we show on a simple model that chiral symmetry may be broken dynamically by strong Yukawa interaction. The possibility of applying our results to electroweak interactions is discussed.

Introduction Understanding the origin of particle masses is one of the most challenging unsolved problems in current high-energy physics. All known elementary interactions besides gravity are described by the Standard model, which acquired its up-to-date form in late sixties through the work of Weinberg [1967] and many others. The Standard model has passed successfully all experimental probes, to which it has so far been exposed. Yet, and despite the ingenuity of its creators, it suffers from a couple of diseases that bother the minds of the world’s most prominent theorists. The presence of particle masses in the Standard model Lagrangian is prohibited by the SU(2)L × U(1)Y gauge symmetry of the electroweak interaction. To overcome this difficulty, the Higgs mechanism has been suggested which leads to spontaneous breakdown of the electroweak symmetry by condensation of a scalar Higgs field. Although this approach is formally well defined and absolutely consistent, it is rather unsatisfactory physically as it is introduced phenomenologically, that is more or less by hand. It has been a dream of many particle physicists ever since the invention of the Standard model to find a more physical, dynamical explanation for electroweak symmetry breaking, and thus the origin of particle masses. All such attempts inevitably lead to introduction of new particles and strong forces. Many of them, the first perhaps being the technicolor theory [Weinberg, 1979], introduce new strong gauge interactions, taking advantage of a rather good knowledge of quantum chromodynamics. In this contribution we present a modest attempt in a different direction. We wish to show that chiral symmetry might be broken dynamically with just the Yukawa interaction, which is already present in the Standard model Lagrangian, provided it is strong enough. The plan of the paper is following. We start with an introductory review of the physics of spontaneous symmetry breaking, emphasizing its general features and wide applications. We explain the connection to the problem of fermion masses in relativistic quantum field theory. Next we work out a simple model with Abelian chiral symmetry in order to demonstrate dynamical mass generation by means of the Yukawa interaction. Starting from general considerations of symmetry and its realization in quantum theory, we end up with the Schwinger–Dyson equations whose non-perturbative solution yields the desired masses. The extension to the Standard model, which involves a non-Abelian gauged chiral symmetry and several fermion species, is discussed in the conclusions.

Spontaneous symmetry breaking Symmetry plays a key role in analyzing most physical problems. It sometimes happens that the symmetry of the equations of motion is larger than the symmetry of their particular solution. In such a case we speak of spontaneous symmetry breaking (SSB). While in classical physics this seems rather trivial, in quantum field theory SSB has far-reaching consequences. Symmetry is introduced as an invariance of the Lagrangian or the action. By the Wigner theorem, it is implemented on the Hilbert space of states by unitary operators that commute with the Hamiltonian, leading to non-trivial degeneracy of the spectrum. On the other hand, the ground state is assumed to be non-degenerate, and it must therefore be invariant under the symmetry transformations. When SSB occurs, the ground state is by definition not invariant with respect to some of the symmetry operations. There are several examples of SSB both in non-relativistic and relativistic physics. The first class, which is probably more accessible to one’s intuition, includes for example the ferromagnet.

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There, the Hamiltonian is invariant under rotations of electron spins. In the ground state, however, the spins of all electrons are aligned, breaking the full rotation SO(3) group down to the SO(2) subgroup of rotations about the axis of spontaneous magnetization, which constitutes the order parameter for the phase transition to the ferromagnetic state. The direction of the magnetization is arbitrary and is induced by some random external perturbation that slightly breaks the rotation symmetry. Other non-relativistic examples are the superconductor, where the broken symmetry is the electro- magnetic gauge invariance, or even any crystallic solid, where full translation symmetry is broken by the crystal lattice down to some discrete subgroup. Relativistic, especially particle physics offers a variety of examples of SSB, e.g. the gauge SU(2)L U(1)Y invariance of the electroweak interaction and its exten- × sions to larger symmetry groups in models of grand unification, or the color SU(3)c in superconducting phases of quantum chromodynamics. The experience gained from the above listed examples may be summarized in the observation that SSB is always characterized by the existence of an order parameter. This can acquire various values, which are formally connected by the broken symmetry transformations. There are therefore several degenerate ground states, each specified by the particular value of the order parameter. This is the intuitive picture of SSB. Formally, however, the broken symmetry is not implemented on the Hilbert space by unitary operators and thus is not manifested in the degeneracy of the spectrum. As a consequence, different ground states are not connected by a unitary transformation. They are unitarily inequivalent and the Hilbert spaces built above them are completely orthogonal — they represent different worlds with different values of the order parameter. Chiral symmetry We shall now in more detail comment on the connection of SSB and fermion mass generation. The 1 Lagrangian of a free relativistic spin- 2 (Dirac) fermion contains two separate pieces for the left- and right- handed chiral components of the Dirac field, 0 = ψ¯Ri∂ψ/ R + ψ¯Li∂ψ/ L. It is invariant under independent L phase transformations of the left and right fields, that is the Abelian chiral group U(1)R U(1)L. × Now the Dirac mass term, δ m = m(ψ¯RψL + ψ¯LψR), intertwines the left and right fields so that they can no more be rephased independently.L − The mass term is invariant only under the diagonal vector U(1)V . It follows that non-zero Dirac masses can arise in a chirally-invariant theory only after the chiral symmetry has been spontaneously broken. The mass m itself is the order parameter here. The ground state is chosen so that m is real and positive and corresponds directly to the physical mass of the particle. Goldstone theorem The Goldstone theorem [Goldstone et al., 1962] is one of a few exact and general results in quantum field theory, and is the starting point for any consideration about broken symmetry. It states that whenever a continuous symmetry is spontaneously broken, there is an excitation branch in the spectrum of the theory (the Nambu–Goldstone mode) for which E(k) 0 as k 0. For Lorentz-invariant theories, this statement considera→bly simplifies:→ the Nambu–Goldstone (NG) mode is a spinless particle with zero mass. It also holds that the number of NG bosons is equal to the number of broken symmetry generators. For non-relativistic systems the NG boson counting is more complicated [Nielsen and Chadha, 1976]. It turns out that the number of NG bosons may be smaller than the number of broken generators provided some of the symmetry generators develops non-zero density in the ground state. This is the case for the ferromagnet mentioned before — there are two broken generators, but a single NG mode, the magnon. Before concluding the review part of the contribution we note that there are also symmetries that are not exact, e.g. the chiral SU(3)R SU(3)L symmetry of quantum chromodynamics, which is slightly broken by u, d, and s quark masses.× When such a symmetry is spontaneously broken, one naturally expects a nearly massless particle (the pseudo-Nambu–Goldstone boson) to appear. However, no theorem analogous to Goldstone’s exists for approximate symmetries.

Abelian model of chiral symmetry breaking Now we shall introduce our model for dynamical breaking of chiral symmetry. We shall give a brief description and summarize the results, full details may be found in our recent paper [Brauner and Hoˇsek, 2005]. Trying to keep the idea as simple as possible, we consider the theory of a complex scalar ﬁeld φ with mass M and two massless Dirac fermions ψ1,2, coupled by the Yukawa interaction

Yukawa = y1 ψ¯1Lψ1Rφ + ψ¯1Rψ1Lφ† + y2 ψ¯2Rψ2Lφ + ψ¯2Lψ2Rφ† . (1) L

437 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION

A curious reader might wonder why we include two diﬀerent fermion species. The reason is that with just one fermion our model would suﬀer from an axial anomaly. This is no problem as long as the anomalous symmetry is global, as it actually is in our case. However, as we already mentioned above, our ultimate goal is to apply the same idea to electroweak symmetry breaking. For a gauge invariance the anomaly would be a disaster, and we therefore choose to remove it from the very beginning.

Symmetry and Noether currents There are altogether three independent phase transformations of the scalar and fermion fields that leave the action invariant. The theory thus possesses a global U(1)V 1 U(1)V 2 U(1)A symmetry. The two vector U(1)’s stand for independent transformations of the Dirac× spinors.× The associated Noether currents, µ ¯ µ µ ¯ µ jV 1 = ψ1γ ψ1, jV 2 = ψ2γ ψ2, correspond to separate conservation of the numbers of fermions of the first and second type, respectively. The axial U(1) is the symmetry that interests us most. It prohibits a straightforward introduction of Dirac mass terms into the Lagrangian and will therefore be the subject of our attempts at dynamical breaking. The Yukawa interaction term Eq. (1) enforces that the scalar field transforms simultaneously with the fermions according to

+iθγ5 iθγ5 2iθ ψ1 e ψ1, ψ2 e− ψ2, φ e− φ. → → → Consequently, both the fermions and the scalar contribute to the axial Noether current,

µ µ µ µ µ j = ψ¯1γ γ5ψ1 ψ¯2γ γ5ψ2 + 2i (∂ φ)†φ φ†∂ φ . (2) A − − Higgs mechanism In the Standard model symmetry is broken by means of the Higgs mechanism. We now show how it works (or, rather, would work) in case of our model for later comparison with our dynamical method of breaking the symmetry. The key idea is to change to sign of the scalar ﬁeld mass squared so that the scalar potential reads

2 λ 2 V (φ) = M φ†φ + (φ†φ) . − 2 The scalar ﬁeld then develops a non-zero vacuum expectation value at tree level, 0 φ 0 = v/√2, where v = 2M 2/λ. In order for the perturbation theory to be meaningful, φ musth be| shifted| i to the new − 1 minimum,p φ = √2 (v + φ1 + iφ2). φ2 is now the NG boson, φ1 is the massive Higgs, and the Yukawa interaction Eq. (1) yields the fermion mass terms,

vy1 vy2 Yukawa ψ¯1ψ1 + ψ¯2ψ2 + other terms. L → √2 √2

Note also that when the axial current Eq. (2) is rewritten in terms of the new ﬁelds φ1,2, it contains a term linear in the NG ﬁeld φ2,

µ µ µ µ µ µ j = ψ¯1γ γ5ψ1 ψ¯2γ γ5ψ2 + 2v∂ φ2 + 2 (φ1∂ φ2 φ2∂ φ1) . A − − This is another characteristic feature of SSB — the NG mode is created from the vacuum by the broken symmetry current.

Ward identities Within the Higgs mechanism all calculations are quite simple as one just has to do perturbation theory above the shifted ground state. On the other hand, any attempt at dynamical symmetry breaking necessarily involves non-perturbative techniques and life becomes more complicated. This is the price one has to pay for the more physical picture of breaking the symmetry by microscopic dynamics. It is therefore desirable to extract as much information as possible from the symmetry of the problem. The basic technical tool in quantum ﬁeld theory is the formalism of Green’s functions. The symmetry properties are implemented in the language of Green’s functions by the set of Ward identities for the correlation functions of the Noether currents. They represent the conservation of the currents inserted in a time-ordered product of operators.

438 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION

= = ψ1L ψ1R ψ1L ψ1R ψ2L ψ2R ψ2L ψ2R

ψ1R ψ1L ψ2R ψ2L

ψ1L ψ1R ψ2R ψ2L

= + φ φ φ φ φ φ

ψ1R ψ1L ψ2L ψ2R Figure 1. One-loop Schwinger–Dyson equations for the symmetry-breaking parts of the fermion and scalar propagators. The dashed circles denote the one-particle-irreducible self-energies, while the ﬁlled circles stand for the full self-energies.

We shall be concerned with three-point (vertex) functions of the axial current and a pair of ei- ther the fermion or the scalar ﬁelds. There are altogether three such vertex functions, deﬁned by µ µ ¯ µ µ ¯ the formulas GAψ1 (x, y, z) = 0 T jA(x)ψ1(y)ψ1(z) 0 , GAψ2 (x, y, z) = 0 T jA(x)ψ2(y)ψ2(z) 0 , and µ µ h | { 1 }| i h | { }| i GAφ(x, y, z) = 0 T jA(x)Φ(y)Φ†(z) 0 . The correspondingh | { Ward identities}| i may be derived in an elementary way by direct evaluation of the divergence of the Green’s functions, making use of the canonical commutation relations,

µ 1 1 qµΓAψ1 (p + q, p) = S1− (p + q)γ5 + γ5S1− (p), µ 1 1 qµΓ (p + q, p) = S− (p + q)γ5 γ5S− (p), (3) Aψ2 − 2 − 2 µ 1 1 qµΓ (p + q, p) = 2D− (p + q)Ξ + 2ΞD− (p). Aφ − The symbols S and D denote the full propagators of the fermions and the Nambu scalar doublet Φ, respectively, while the Γ’s represent proper vertex functions, that is the G’s with the full propagators on the external legs cut oﬀ. It is crucial to observe that the Ward identities hold whether the symmetry is spontaneously broken or not. Indeed, for the Higgs mechanism sketched above their validity can be explicitly checked as both their right- and left-hand sides are calculable perturbatively.

Schwinger–Dyson equations Spontaneous symmetry breaking is a non-perturbative phenomenon and thus cannot be revealed at any finite order of perturbation theory. One of the most useful non-perturbative methods is the technique of the Schwinger–Dyson equations. These form an infinite set of coupled integral equations for the Green’s functions of the theory. In practice, the system is always closed at a certain order, rendering a finite set of equations. Our aim is to generate masses for the fermions by means of SSB and we shall be primarily interested in the two-point Green’s functions. In order for our calculations to be simple and transparent, we shall ignore radiative corrections to the interaction vertices and consider just the truncated Schwinger–Dyson equations for the propagators, as depicted in Fig. 1.

Model calculations To demonstrate that the model of Eq. (1) does indeed lead to chiral symmetry breaking, we make further simplifying assumptions that allow us to proceed analytically as far as possible. We neglect all symmetry-preserving radiative corrections to the propagators (that is wave function renormalization).

1 We will show that spontaneous breaking of U(1)A induces mixing of φ and φ†. In such a case, it is convenient to φ introduce the Nambu doublet notation, Φ = , allowing us to treat φ and φ† on the same footing. The upper and φ† 1 0 lower components are distinguished by the diagonal matrix Ξ = , which is analogous to γ5 in the fermion 0 −1 sector.

439 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION

The propagators of ψ1,2 and Φ then take the form

2 2 1 1 p M Πp S1−,2 (p) = /p Σ1,2p,D− (p) = − 2− 2 . − Π∗ p M − p −

The symmetry-breaking self-energies Σ1,2 and Π are found as a self-consistent solution to the Schwinger– Dyson equations in Fig. 1. In letters this is

4 2 d k Σ1,k Πk p Σ1 = iy − , ,p 1 4 2 2 2 2 2 2 Z (2π) k Σ1,k [(k p) M ] Πk p − − 4 − − − | | 2 d k Σ2,k Πk∗ p Σ2 = iy − , ,p 2 4 2 2 2 2 2 2 (4) Z (2π) k Σ2,k [(k p) M ] Πk p − − 4 − − − | | 2 d k Σj,k Σj,k p Πp = 2iyj 4 2 2 2 − 2 . − =1 2 Z (2π) k Σj,k (k p) Σj,k p jX, − − − − We temporarily put oﬀ the actual solution of Eqs. (4) and simply assume that a non-trivial solution does exist. Note that the desired fermion masses are determined by the poles of the full propagators, that is they are found from the pole condition,

2 2 2 m1,2 = Σ1,2(p ) at p = m1,2.

Next we plug the fermion and scalar self-energies into the Ward identities (3) and calculate the properties of the NG boson. At low momentum transfer, or near the NG boson mass shell, the vertex functions are dominated by the pole coming from the NG boson coupling to the broken current [Jackiw and Johnson, 1973]. The Ward identities are then written exclusively in terms of the now known fermion and scalar propagators, and the eﬀective vertices of the NG boson with a pair of fermions or scalars. ¯ These vertices, denoted as Pψ1,2ψ1,2 (p + q, p) and PΦΦ(p + q, p), are the yield of the Ward identities. They are given by the formulas 1 P ¯ (p + q, p) = [Σ1,p+q + Σ1,p] γ5, ψ1ψ1 −N 1 P ¯ (p + q, p) = [Σ2 + + Σ2 ] γ5, ψ2ψ2 N ,p q ,p 2 0 Πp+q + Πp PΦΦ(p + q, p) = . −N Π∗+ Π∗ 0 − p q − p

The normalization factor, N = Jψ1 (0) + Jψ2 (0) + Jφ(0), is determined by the loop integrals, p 4 µ µ 2 d k (k q) Σ1,k Σ1,k + Σ1,k q − iq Jψ1 (q ) = 8 4 −2 2 2 2 , − Z (2π) k Σ1,k (k q) Σ1,k q − − − − 4 µ µ 2 d k (k q) Σ2,k Σ2,k + Σ2,k q iq Jψ2 (q ) = 8 4 −2 2 2 2− , − Z (2π) k Σ2,k (k q) Σ2,k q − − − −

4 µ 2 2 Πk∗ q Πk + Πk q µ 2 d k (2k q) (k M ) ℜ − − iq Jφ(q ) = 8 − − h i . 4 2 2 2 2 2 2 2 2 − Z (2π) (k M ) Πk (k q) M Πk q − − | | − − − | − | It is notable that the NG boson eﬀective vertices are proportional to the symmetry-breaking fermion and scalar self-energies. This is in accord with the general feature of broken symmetries that when symmetry is restored, the NG boson decouples and vanishes from the spectrum.

Numerical results It remains to show that the Schwinger–Dyson equations (4) really do have a non-trivial solution. This was done numerically. For simplicity we considered just the special case y1 = y2. The Lagrangian is then invariant under the discrete symmetry ψ1 ψ2, φ φ†. As far as the induced anomalous scalar self-energy is real, we may assume that this discrete↔ symmetry↔ is not spontaneously broken and the self-energies Σ1 and Σ2 are therefore equal. We are thus left with two coupled equations for Π and just one Σ.

440 BRAUNER AND HOSEK:ˇ DYNAMICAL SYMMETRY BREAKING AND MASS GENERATION

0.5

0.4

0.3 Π(p2)/M 2 0.2 Σ(p2)/M 0.1

0 0.0001 0.001 0.01 0.1 1 10 100 p2/M 2

Figure 2. The results of our numerical computation of the self-energies Σ and Π. The self-energies and momentum squared are all made dimensionless by dividing by an appropriate power of the scalar bare mass M.

The reduced set of equations may now be solved iteratively. We switched to the Euclidean space- time by a formal Wick rotation and integrated over two of the three spherical angles analytically. The resulting double integrals were calculated numerically in the process of iterating the Schwinger–Dyson equations. With an initial ansatz for the Σ we calculated the Π and took these two functions as the ﬁrst iteration to the self-energies. Our results are summarized in Fig. 2. We found that non-zero solution exists provided the Yukawa coupling is strong enough, the critical value being roughly ycrit 35. Numerical calculation also revealed that when a symmetry-breaking solution existed, the trivial≈ solution was unstable, which justiﬁes that the non-trivial solution really represents the ground state.

Conclusions We have demonstrated that chiral symmetry can be spontaneously broken by a strong Yukawa interaction. To this end, we have solved numerically the one-loop Schwinger–Dyson equations for the fermion and scalar self-energies. A natural question arises whether the same idea might be applied to electroweak symmetry breaking in the Standard model. In our previous work [Brauner and Hoˇsek, 2004] we showed that it is possible, at least in principle. One just has to introduce two scalar SU(2) doublets instead of the usual one Higgs. However, a lot of work must be done before an ultimate answer can be given. To meet experimental bounds, the scalars must be considerably heavier than the fermions. Also the problem of the inter-family hierarchy of the fermion masses and the CKM mixing come into question. All these issues will be the subject of our future work.

Acknowledgments. The authors are indebted to Petr Beneˇsfor doing the numerical calculation of the self-energies. The present work was supported in part by the Institutional Research Plan AV0Z10480505, and by the GACR doctoral project No. 202/05/H003.

References Brauner, T. and Hoˇsek, J., A model of ﬂavors, hep-ph/0407339, 2004. Brauner, T. and Hoˇsek, J., Dynamical fermion mass generation by a strong Yukawa interaction, hep-ph/0505231, 2005. Goldstone, J., Salam, A., and Weinberg, S., Broken symmetries, Phys. Rev., 127 , 965–970, 1962. Jackiw, R. and Johnson, K., Dynamical model of spontaneously broken gauge symmetries, Phys. Rev., D8 , 2386–2398, 1973. Nielsen, H. B. and Chadha, S., On how to count Goldstone bosons, Nucl. Phys., B105 , 445, 1976. Weinberg, S., A model of leptons, Phys. Rev. Lett., 19 , 1264–1266, 1967. Weinberg, S., Implications of dynamical symmetry breaking: An addendum, Phys. Rev., D19 , 1277–1280, 1979.

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