aa r X i v : . [ h e p - ph ] J u l October 1, 2018
Inevitable emergence of composite gauge bosons
Mahiko Suzuki
Department of Physics and Lawrence Berkeley National LaboratoryUniversity of California, Berkeley, California 94720 (Dated: October 1, 2018)
Abstract
A simple theorem is proved: When a gauge-invariant local field theory is written in terms ofmatter fields alone, a composite gauge boson or bosons must be formed dynamically. The theoremresults from the fact that the Noether current vanishes in such theories. The proof is carriedout by use of the charge-field algebra at equal time in the Heisenberg picture together with thewell-established analyticity of the form factor of the current. While there is no need of diagramcalculation for the proof, we demonstrate in the leading 1/N expansion of the existing models whatthe theorem means in diagrams and how the composite gauge boson emerges.
PACS numbers: 11.15.-q, 11.10.St . INTRODUCTION Some theories possess a local gauge symmetry, yet do not contain a gauge field explicitly.The CP N model [1] is one of the examples. It was shown in the leading 1/N expansion of the CP N − model that a U(1) gauge boson is indeed generated as a composite state of matterparticles.[2] The U(1) gauge symmetry of the CP N model was extended by Akhmedov[3]to the SU(2) symmetry. More recently, models were built with fermion matter alone.[4]Whether the symmetry is Abelian or non-Abelian, the models with fermion matter cannotbe reproduced by extension of the CP N model nor by means of the auxiliary field trick.[5, 6]Nonetheless, it was explicitly shown by the large N expansion of the diagram calculationthat these models indeed generate the composite gauge bosons as the massless bound statesof the matter particles.There is one peculiar feature common to the Lagrangian of composite gauge bosons. Thatis, the Noether current does not exist. This can be shown generally as a direct consequenceof local gauge invariance without referring to specific binding forces.[4] In fact, in the case ofthe non-Abelian gauge theory, if the Noether current existed, formation of composite gaugebosons would contradict with the theorem of Weinberg and Witten.[7]The diagrammatic study of the composite gauge bosons has been limited to the leadingorder of the 1 /N expansion which amounts to summing up an infinite series of loop diagramsof the matter particles[2, 4]. Because of the complexity of perturbative computation, wecannot keep such calculation under control beyond the leading order of 1/N. Nonetheless,it is natural to speculate that the composite gauge bosons are always formed irrespectivelyof specific details of the binding force when the total Lagrangian is gauge invariant withmatter particles alone.In this paper, we attempt to prove the formation of composite gauge bosons to all ordersof binding interactions without recourse to diagrams. The proof is based on the equal-time algebra of charges and fields in the Heisenberg picture, which incorporates all ordersof interactions. We show that a composite gauge boson must appear as a pole in the formfactor of the current carrying its quantum numbers. Although a diagrammatic verification isredundant for the proof, it is reassuring and also visually helpful to understand the proof interms of diagrams. After completing our proof, therefore, we demonstrate in the leading 1/Nexpansion of an existing model how the statement of our theorem is realized in diagrams.We organize the paper as follows: First the theorem is stated in Sec. II. After the nec-essary input of field theory is carefully reviewed in Sec. III, the theorem is proved in Sec.IV with the equal-time algebra of charges and fields for the non-Abelian gauge theories ofthe boson matter. In Sec. V, we demonstrate in diagrams how the statement of the the-orem is realized in the leading 1/N order of a concrete non-Abelian model. It is shown inSec. VI that the theorem holds just as well for the U(1) gauge theories. In order to applyour argument to the fermion matter, we discuss in Sec. VII on an issue in the canonicalquantization of the Dirac field, specifically, a problem related to quantization of constrainedsystems and a possibility of justifying the charge-field algebra without relying on the canon-ical quantization. We conclude with some perspectives in theory and phenomenology in Sec.VIII. 2 I. THEOREM
The theorem is stated as follows:
If a gauge-invariant Lagrangian field theory is written in terms of matter fields alone,there must be a composite gauge boson or bosons made of the matter particles.
The gist of the theorem is that formation of the composite gauge boson(s) is not apossibility but the necessity. The input crucial to prove this theorem is the absence ofthe Noether current in this class of theories. We study the form factor of the current inthe equal-time commutation relation of charges and fields by starting away from the gaugesymmetry limit. Then we approach the gauge symmetry by continuously varying a certainparameter and prove the theorem without referring to diagrams or details of binding forces.The theorem holds in the flat space-time of (3+1) dimensions for both the Abelian andnon-Abelian theories with boson or fermion matters. It is not dual to the Weinberg-Wittentheorem[7], which states that the non-Abelian massless gauge bosons cannot exist if thecorresponding Lorentz-covariant conserved currents exist. Their theorem is mute as towhether the non-Abelian gauge bosons must exist or not when such currents are absent.
III. NON-ABELIAN SYMMETRY WITH BOSON MATTER
All that we use for the proof is the basic quantum field theory and its simple applications.To emphasize specific subtleties relevant to our proof, however, we give a brief review onelementary subjects, some of which may have fallen into oblivion by now.
A. Gauge variation of Lagrangian
The reason to discuss the spinless boson matter first is mainly the notational and technicalsimplicity related to the spins. But there is one complication in the canonical quantizationof the Dirac field. Otherwise, no intrinsic difference exists between the boson matter andthe fermion matter.The Lagrangian is in the form of L tot = ∂ µ Φ † ∂ µ Φ − m Φ † Φ + L int . (1)A set of the scalar fields Φ / Φ † transform locally like an n/n -dimensional representation ofa Lie group; Φ → U Φ , Φ † → Φ † U † , (2)where U is given in terms of the n × n generator matrices T a as U = exp[ iT a α a ( x )] . (3)The matrices T a obey [ T a , T b ] = if abc T c with the structure constants f abc .We introduce N copies of the n -component complex scalar pairs Φ i / Φ † i ( i = 1 , , · · · N )since, after completing the proof, we make the large N-expansion in the diagram calculation3o demonstrate how the theorem works in the explicit model. However, we shall suppressthe copy index i hereafter unless we need to remind of it.The interaction Lagrangian L int is a functional of Φ, Φ † and their first derivatives in theknown models. We assume that L int does not contain time-derivatives of field higher thanthe first derivative. That is, L int should be just as singular as the free Lagrangian L inregard to the derivatives of field. Otherwise the gauge variation of L cannot be compensatedwith that of L int . Since the free Lagrangian L is not invariant under the local gauge transformation Eq.(2), the interaction Lagrangian L int must counterbalance the gauge variation δL of the freeLagrangian as δL int = − δL . (4)Since δL is known from the free Lagrangian in Eq. (1) as δL = ∂ µ Φ † ( U † ∂ µ U )Φ + Φ † ( ∂ µ U † U ) ∂ µ Φ + Φ † ( ∂ µ U † ∂ µ U )Φ , (5)the relation of Eq. (4) determines the gauge variation δL int uniquely even without knowing L int itself. We place an emphasis on this trivial but powerful constraint of gauge invariancesince it allows us to proceed in our proof without knowing an explicit form of L int . Wewould need the form of L int only when we carry out, as we shall do later, a diagrammaticdemonstration of the theorem in the interaction picture.Whereas we are interested in the gauge-invariant Lagrangian of Eq. (1), we insert aparameter λ in front of L int as L λtot = L + λL int , (6)and study how physics varies as λ approaches unity. The purpose of this seemingly redundantprocedure is the following: Since the composite gauge boson carries the same quantumnumbers J P C = 1 −− as the Noether current, we wish to study the gauge boson through theNoether current. However, if we stayed exactly in the gauge symmetry limit ( λ = 1), wewould not be able to do so since the Noether current vanishes there according to the generaltheorem. (cf Appendix A.) In order to study the pole of a composite gauge boson in theform factor, therefore, we must approach the gauge symmetry limit with L λtot of Eq. (6) bycontinuously varying the value of parameter λ to 1. By doing so, we can study where thebound-state pole of J P C = 1 −− is located off the gauge symmetry and how it moves to zeroturning into the massless gauge boson in the gauge limit. With L λtot as given in Eq. (6), weapproach the gauge limit along one special path in the functional space of Lagrangian. In fact, there is another reason for considering a large N. In our proof one-particle states will be treatedas the asymptotic states. If confinement occurs with the composite gauge bosons, the one-matter-particlestates are, strictly speaking, not the asymptotic states of the S-matrix. The simplest way to avoid thisinconvenience is to consider the case that there exist a sufficient number of matter multiplets to counterthe confinement. Higher derivatives would ruin causality in dynamics. Recall in classical physics that the solutions areacausal when the force contains a higher derivative. For instance, the radiation damping of a pointcharge. The same happens in classical field theory. In quantum theory we would not be able to quantizecanonically in the Heisenberg picture if L int is more singular. Obviously there are many different ways to approach the gauge limit. For instance, one may let λ → L tot = L + L int + (1 − λ ) L br where L br is some arbitrarily chosen gauge-breaking . Noether current The Noether current vanishes in the gauge-symmetric field theories for which the La-grangian consists only of matter fields. This is a simple inevitable consequence of gaugeinvariance, Abelian or non-Abelian. Since the Noether current due to the free Lagrangiancannot vanish by itself, this must happen such that the contribution from the interactionLagrangian cancels that from the free Lagrangian. The proof is very simple, as is given inAppendix A for the non-Abelian boson matter. Extension to other cases is trivial.In short, the gauge-symmetric Lagrangian L tot varies under the infinitesimal local phasetransformation by α a ( x ) of Eqs. (2) and (3) as δL tot = i ( ∂ µ J aµ ) α a + iJ aµ ∂ µ α a + O ( α ) , (7)after use of the equations of motion for Φ and Φ † in the first term. Since α a ( x ) is an arbitraryfunction of x , we can treat α a ( x ) and ∂ µ α a ( x ) as independent of each other. Consequentlythe first term of Eq. (7) leads to the definition of the Noether current and its conservation.The second term simply states that the Noether current must vanish.Both L and L int contribute to J aµ since both contain the first derivatives of Φ and Φ † in order to satisfy gauge invariance. When we modify L tot into L + λL int , it is no longergauge invariant off λ = 1 and therefore the Noether current J λaµ survives. It is simply given(cf Appendix A) by J λaµ = i (1 − λ ) (cid:16) Φ † T a ↔ ∂ µ Φ (cid:17) . (8)The factor (1 − λ ) in front indicates the fact that the Noether current vanishes in the gaugelimit. The Noether current thus takes the form identical with that of the free field theoryup to the factor (1 − λ ): J freeaµ = lim λ → (cid:16) − λ J λaµ (cid:17) . (9)However, we make a trivial but important remainder about Eq. (8). That is, J λaµ = (1 − λ ) J freeaµ . (10)The reason is that when we use Eq. (8) the fields in right-hand side are in the Heisenbergpicture, that is, the Φ / Φ † fields in J λaµ incorporate all the λ -dependence through the inter-action, while the Φ / Φ † fields in J freeaµ are independent of λ (= 0) by definition. It wouldbe clearer in this respect if we wrote the fields of the Heisenberg picture as Φ( x, λ ) andΦ † ( x, λ ). The implicit λ dependence of Φ and Φ † in the Heisenberg picture incorporates allinteractions and it is responsible for the formation of the bound states among others. C. Equal-time algebra of charges and fields
We use the equal-time algebra of the charges and fields in the Heisenberg picture for ourproof of the theorem. With the “canonical momentum” defined by Π ≡ ∂L/∂ ( ∂ Φ), thefield Φ obeys the equal-time commutation relation,[Φ r ( x , t ) , Π s ( y , t )] = iδ rs δ ( x − y ) . (11) interaction. Instead we have chosen here the specific form L λtot for which the Noether current off λ = 1takes the simple form determined by the free Lagrangian L alone. r, s ) refer to components of the n -dimensional representation. Eq. (11) holdsseparately for each of N copies. Φ † and Π † obey the same form of commutation relation, andall other equal-time commutators among Φ , Φ † , Π and Π † vanish. In terms of these canonicalvariables, the charge component of the Noether current is expressed as J λa = i (Φ † T a Π † − Π T a Φ)= i (1 − λ )(Φ † T a ↔ ∂ Φ) , (12)where the summation over the N copies is understood. Notice that the factor (1 − λ ) appearswhen J a is written in Φ, Φ † and their time-derivatives. But Eq. (12) does not mean thatΠ and Π † are proportional to 1 − λ . (cf Appendix B) The Noether charge is defined by Q λa = Z d x J λa ( x , t ) . (13)It is independent of time since the Noether current is conserved. By use of the canonicalcommutation relations, one can show that the charges form the Lie algebra,[ Q λa , Q λb ] = if abc Q λc . (14)The commutation relations of Q λa with the fields Φ / Φ † form the charge-field algebra,[ Q λa , Φ r ( x )] = − ( T a ) rs Φ s ( x ) , (15)and the hermitian conjugates. It should be emphasized that both Eqs. (14) and (15) arethe direct consequences of the canonical commutation relations Eq. (11) and therefore validirrespectively of L int . The peculiarity of the matter gauge theories to be emphasized hereis that the Noether charge operator Q λa vanishes in the gauge symmetry limit according toEq. (12).Now here comes the key point. One might notice that something does not look quiteright about Eqs. (14) and (15) at least superficially. Let us take the matrix elements of theboth sides of Eq. (15), for instance. When the charge Q λa is expressed with the Noethercurrent as written in the second line of Eq. (12), it looks as if its matrix element were alwaysproportional to (1 − λ ). If so, when it is substituted in Eq. (15), the left-hand side would beinfinitesimally small like (1 − λ ) near λ = 1. On the other hand the matrix element of theright-hand does not vanish at λ = 1. The same superficial inconsistency appears as (1 − λ ) vs (1 − λ ) from Eq. (14) too. How should we answer to this question ?There is no computational error here. The fact that charge operator Q λ is proportionalto (1 − λ ) is a manifestation of the absence of the Noether current in the gauge invarianttheories that consist only of matter fields. Then, how can the charge-field commutationrelation Eq. (15) hold valid near λ = 1 ?We shall find that this is the place where the formation of the composite gauge bosonsenters and solves the puzzle. By examining the form factor of the Noether current in thefollowing section, we shall find that a composite vector bound-state is formed in the channelof J λaµ , and therefore that the matrix element of i (Φ † T a ↔ ∂ µ Φ) at zero momentum transferturns out to be proportional to 1 / (1 − λ ) and compensates the factor (1 − λ ) in front of theoperator (Φ † T a ↔ ∂ µ Φ). 6 . Dispersion relation for form factor of Noether current
To study consistency of the powers of (1 − λ ), we need to examine the matrix elementsfor the both sides of Eq. (15) between the vacuum h | and the one-particle state | p i , inparticular, the one-particle matrix element of J λaµ near the zero momentum-transfer limit.We define the Lorentz-scalar form factor F ( t, λ ) by separating (1 − λ ) from J λaµ as11 − λ h p ′ , s | J λaµ (0) | p , r i = h p ′ , s | i (Φ † T a ↔ ∂ µ Φ) | p , r i = s E p ′ E p ( p ′ + p ) µ ( T a ) sr F ( t, λ ) , (16)where the variable t is the invariant momentum transfer t = ( p ′ − p ) . Even after the factor(1 − λ ) is removed from the Noether current, the form factor F ( t, λ ) still depends on λ . This λ dependence comes from the multiple interaction of L λint of Eq. (6), which is implicit in theHeisenberg operator i (Φ † T a ↔ ∂ µ Φ), as we have already pointed out.Analyticity of the function F ( t, λ ) is well known. F ( t, λ ) is analytic in the variable t withthe branch points on the positive real axis of the complex t -plane. The lowest branch point t is located at the invariant mass squared of the lowest two-particle threshold. If there isa bound state of J P C = 1 −− with mass m bound , the function F ( t, λ ) has a simple pole at m bound below t and, barring a tachyon, above t = 0 for λ = 1. (See the left-side figure inFig. 1.) Im t Im t Re t Re tF(t, λ) λ) FIG. 1: Analyticity of F ( t, λ ) and 1 /F ( t, λ ) in the complex t plane. The cross in the left-side figureindicates the pole due to a bound state of J P C = 1 −− for F ( t, λ ). The crosses in the right-sidefigure are due to possible poles of 1 /F ( t, λ ), that is, zeros of F ( t, λ ). The inverse of the form factor 1 /F ( t, λ ) possesses the cuts at the same locations as F ( t, λ ),but a bound-state pole of F ( t, λ ) becomes a zero of 1 /F ( t, λ ) and therefore does not generatea singularity. The dispersion relation for 1 /F ( t, λ ) therefore takes the form of F ( t, λ ) = 1 π Z ∞ t Im(1 /F ( t ′ , λ )) t ′ − t − iǫ dt ′ + X i c i ( λ ) t i ( λ ) − t + c ( λ ) , (17) If F ( t, λ ) has a zero, it turns into a pole of 1 /F ( t, λ ), which would have to be taken into account inwriting the dispersion relation for 1 /F ( t, λ ). Such zeros can appear in general on the real axis of t and/orpairwise symmetrically above and below the real axis because of the relation F ( t, λ ) ∗ = F ( t ∗ , λ ), where t i ( λ )’s ( i = 1 , , · · · ) are the locations of zeros of F ( t, λ ) and c i ( λ )’s are constantsindependent of t with c ( λ ) = 1 /F ( ∞ , λ ). We are interested in the formation of a compositevector boson with small mass ( → λ → /F ( t, λ ) on the positivereal axis in the neighborhood of t = 0. Given Eq. (17), we can expand 1 /F ( t, λ ) in theTaylor series in t in the neighborhood of t = 0 off λ = 1 as1 F ( t, λ ) = a ( λ ) + a ( λ ) t + O ( t ) , ( λ = 1) , (18)where a ( λ ) and a ( λ ) are some real finite constants that may depend on λ . Having expressedthe behavior of 1 /F ( t, λ ) in the form of Eq. (18), we are ready to prove the theorem. IV. PROOF OF THEOREM
We take the matrix element of Eq. (15) between the vacuum h | and the one-matter-particle state | p , s i , and insert a complete set of states P | n ih n | between Q λa and Φ( x ).Since Q λa is a generator of a Lie group, only the one-particle state that belongs to the samerepresentation as | p , s i survives in the sum. Use Eq. (16) to express h p , s | Q λa | p , r i in termsof the form factor. We also use the relations, h | Φ r ( x ) | p , s i = s E p p Z δ rs e − ipx , h | Q λa = 0 , (19)where Z is the wave-function renormalization of the matter particle (0 < Z < √ Z , we are simply left with(1 − λ ) F (0 , λ ) = 1 , (20)or F (0 , λ ) = 11 − λ . (21)This is what the charge-field algebra imposes on the form factor F ( t, λ ) at t = 0. Since thecharge-field algebra is just as fundamental as quantum field theory itself, the form factor F ( t, λ ) must obey Eq. (21) no matter what the interaction of matter particles may be.How can the form factor of i (Φ † T a ↔ ∂ µ Φ) satisfy Eq. (21) ? There must be some dynamicalreason for it. The only possibility allowed by analyticity is that a bound state is present inthis channel with the mass square proportional to (1 − λ ) so that F ( t, λ ) ∼ / ( m bound − t )near t = 0. No other possibility exists according to the behavior of the form factor allowedby analyticity. the asterisk indicates a complex conjugate. But a zero does not appear for F ( t, λ ) at t = 0. The reasonfor F (0 , λ ) = 0 is that (1 − λ ) F (0 , λ ) is equal to the nonvanishing charge of the global symmetry for λ = 1,which must be nonzero. /F ( t, λ ) near t = 0,we obtain a ( λ ) = 1 − λ, (22)therefore, 1 F ( t, λ ) = (1 − λ ) + a ( λ ) t + O ( t ) . (23)The coefficient a ( λ ) cannot be determined by the group theory alone. Eq. (23) means that F ( t, λ ) has a dynamical pole at t = − − λa ( λ ) . (24)We call this pole dynamical since it is not an artifact due to a definition or a kinematicalchoice of amplitude. The value of a ( λ ) that determines the location of the pole dependsnot only on λ but also on details of the binding force. Therefore this pole in t possesses allthe properties of a physical bound state. It ought to be a composite vector-meson.Analyticity of the form factor follows from local field theory. With the help of analyticity,the charge-field algebra thus requires that a bound state be formed in the channel of J P =1 −− with the mass squared proportional to (1 − λ ). When this happens, the multiplicativefactor 1 − λ of the charge operator Q λa coming from the Noether current is canceled by thedynamical factor 1 / (1 − λ ) due to the bound-state pole ∼ / ( m bound − t ) in F ( t, λ ), where m bound ∝ (1 − λ ). There is no other possibility. The puzzle is thus solved and the proof hasbeen completed.It should be pointed out that the crucial relation Eq. (21) for our proof can also be ob-tained in the form of [(1 − λ ) F (0 , λ )] = (1 − λ ) F (0 , λ ) by taking the one-particle expectationvalue for the both sides of the charge algebra Eq. (14).We add a few remarks before closing this short Section.The preceding argument gives us one interesting byproduct: Although the local Noethercurrent vanishes in the gauge limit, the conserved Noether charge can still be defined for thematter particles through the limiting value lim λ → (1 − λ ) F (0 , λ ). The value of this chargeis equal to what we would naively assign as the global charge to the matter particle. It isreassuring that we still have the global Noether charge as the conserved quantum numberin the gauge symmetry limit even though the Noether current operator itself disappears.Existence of the non-Abelian Noether charges as the limiting values has no conflict withthe Weinberg-Witten theorem. To rule out the non-Abelian gauge-boson formation by theWeinberg-Witten theorem, we must have a Lorentz-covariant conserved current density thatis capable of transferring spatial momentum. [7] In the gauge theories that consist onlyof matter fields, such a local current density does not exist in the gauge symmetry limit.Therefore the global charge as defined above does not interfere with the Weinberg-Wittentheorem.Once a set of massless vector-bound states are formed in a gauge invariant theory, thesebosons ought to be the gauge bosons of the underlying Lie group. The argument leadingto this conclusion is, in short, that there is no other way known in field theory to accom-modate such massless vector bosons in conformity with the gauge symmetry built in thetotal Lagrangian. When the couplings of higher dimension are included, perturbative renor-malizability does not hold in the space-time dimension of four. Nonetheless, when they arewritten in terms of effective gauge fields, all interactions up to the dimension four are exactlythe same as in the standard renormalizable gauge theory. The couplings of higher dimension9or the matter fields can be combined and cast into gauge-invariant combinations with theeffective vector gauge fields. The explicit demonstration was given through diagram com-putation of the higher dimensional couplings up to the dimension six in the 1/N expansionof the known Abelian and non-Abelian models. [4] V. DIAGRAMMATIC STUDY
The proof of our theorem is complete in the preceding section. Nothing needs to beadded mathematically. Since the proof does not refer to any specific group property ofthe matter fields or their interactions, the theorem should hold for all non-Abelian gaugetheories of boson matter. Nonetheless, it is reassuring to see that the bound-state pole isindeed generated in the form factor and that the pole migrates with the value of parameter λ in the way as we have asserted. It will help us to envision the theorem in terms of diagramssince the diagrams often give us better or more intuitive understanding of physics.For diagrammatic demonstration, we choose the SU(2) doublet model and make the largeN expansion. Except for keeping the leading 1/N terms, the diagrammatic calculation belowmakes no approximation. To work in the large N expansion, we introduce the N doublets ofmatter. The interaction Lagrangian of the SU(2) gauge symmetry is given by [3, 4] L λint = λ ( P i Φ † i τ a ↔ ∂ µ Φ i )( P j Φ † j τ a ↔ ∂ µ Φ j )4 P k Φ † k Φ k , (25)where the summations over i, j and k run from 1 to N . When the free Lagrangian of Φ andΦ † is added to this L λint , the total Lagrangian L + L λint is SU(2) gauge invariant at λ = 1.When the value of λ is in a right range, this interaction generates an SU(2) triplet of boundstates in the channel of J P C = 1 −− . In the gauge symmetry limit, the force is just right tomake the bound states exactly massless in the leading 1/N order. When we perform the diagram calculation, we express the denominator of Eq. (25) insum of its vacuum expectation value and normal-ordered product and expand it around thevacuum expectation value in the power series of the normal-ordered terms,[4] L λint = λ ( P i Φ † i τ a ↔ ∂ µ Φ i )( P j Φ † j τ a ↔ ∂ µ Φ j )4 P k h | Φ † k Φ k | i × X l =0 ( − l (cid:16) P k : Φ † k Φ k : P k h | Φ † k Φ k | i (cid:17) l , (26)where : Φ † Φ : denotes the normal-ordered product of Φ † Φ. To obtain the form factor F ( t, λ )of i (Φ † τ a ↔ ∂ µ Φ) defined in Eq. (16), we follow the leading 1/N computation of the two-bodyscattering amplitude performed in Ref. [4]. It amounts to iteration of the bubble diagrams,as shown in Fig. 2.After the group-theory coefficients have been factored out, the form factor F ( t, λ ) isobtained as the solution of the simple algebraic equation F ( t, λ ) = 1 + K ( t ) F ( t, λ ) , (27) We should remark here that the form of L int appears to be unique up to addition of terms that are gaugeinvariant by themselves e.g., globally invariant nonderivative interactions. It is easy to show that suchnonderivative interactions do not affect the composite gauge-boson mass nor coupling in the leading 1/Norder.[4] p’ p’ pq qQ Q FIG. 2: The form factor F ( t, λ ) in the leading 1 /N order ( t = q ). Each bubble in the left-sidefigure gives the function K ( t ) in Eq. (27) and its iteration generates a vector bound-state in theright-side figure. where K ( t ) comes from the single bubble in the left side figure of Fig. 2. Since we areinterested in F ( t, λ ) near t = 0, we need K ( t ) also near t = 0 in Eq. (27). We carry out theloop integral of the bubble with the dimensional regularization to preserve gauge invariance.The result is K ( t ) = λ (cid:16) − D/ t m (cid:17) + O ( t ) , (28)where m is the matter-particle mass and D is the space-time dimension. With this functionK(t), the inverse form factor is given by1 F ( t, λ ) = (1 − λ ) − λ (1 − D/ t m + O ( t ) . (29)This form of 1 /F ( t, λ ) clearly shows that a vector-boson pole exists in F ( t, λ ) and that thepole goes to zero as λ →
1. By comparing Eq. (29) with the coefficients defined in Eq. (18)in the preceding section, we find a ( λ ) = 1 − λ,a ( λ ) = − λ (1 − D/ / m . (30)The coefficient a ( λ ) = 1 − λ agrees with what we have obtained in Eq. (22) in the precedingsection. This is no surprise since it is a requirement of the Noether charge being the generatorof the global symmetry group off λ = 1. The coefficient a ( λ ) determines the location of thebound-state pole m bound as a function of λ and the matter-particle mass m . As we expect,the location of the pole reaches zero as we approach the gauge symmetry limit, λ → m bound = 6(1 − λ ) λ (1 − D/ m . (31)This exercise in the SU(2) model illustrates how our theorem works. While the Noethercurrent operator disappears like (1 − λ ) as we approach the gauge limit, the location of thebound-state pole converges to zero so as to cancel this (1 − λ ) factor with 1 /m bound ∝ / (1 − λ )at t = 0.The diagrammatic exercise presented here indicates that up to a proportionality constantthe Noether current acts like a composite vector-boson field V µ whose mass turns to zero inthe gauge limit. This may remind some theorists of the field-current identity of Kroll, Lee,and Zumino[10] that identified the gauge current of hadrons with the (massive) gauge field.They attempted to equate the electromagnetic current J EMµ to the ρ ◦ - ω or ρ ◦ - ω - φ field upto a scale factor; J EMµ = f V ρ − ωµ . But there is a fundamental difference. Being massive, the11 ◦ /ω mesons are not gauge bosons of the flavor SU(2) × U(1). The photon being compositewas not their option. Our passing remark here is only that if one lets m ρ , m ω → Q λa connects a one-particle state only to anotherone-particle state that belongs to the same multiplet. This would not be the case if themomentum transfer q is nonvanishing across the current. The spatial Fourier components Q λa ( q , t ) of the charge density J λa ( x , t ) do not form a finite algebra:[ Q λa ( q , t ) , Φ( q ′ , t )] = − T a Φ( q + q ′ , t ) . (32)When we insert a complete set of states P | n ih n | between Q λa ( q , t ) and Φ( q ′ , t ), all mul-tiparticle states also contribute as long as their quantum numbers are right. In this case,the one-particle matrix element h p ′ | Q λa ( q , t ) | p i ∼ / ( m bound + | q | ) vanishes like (1 − λ )as λ → q = 0. Then, comparing the matrix elements on both sides of Eq. (32),it may look as if our power dependence argument of (1 − λ ) would fail like (1 − λ ) vs / (1 − λ ) in the left-hand side. In this case,however, multiparticle states in P | n ih n | contribute as well without a constraint of energyconservation. In particular, the composite vector-boson enters the continuum and its polar-ization sum generates the mass singularity ∼ ( − g µν + k µ k ν /m bound ) through its longitudinalpolarization. [11] This mass singularity would be canceled out if the vector-boson mass isgenerated by spontaneous symmetry breaking [12, 13] and if the matrix elements are a setof physically observable scattering amplitudes. Since our matrix elements satisfy neitherconditions, it ought to happen that the mass singularity proportional to 1 / (1 − λ ) of thelight vector composite survives and restores consistency in the (1 − λ ) powers. We do notattempt computation of the mass singularities here. VI. U(1) GAUGE THEORIES
We can repeat our argument made for the non-Abelian theories and show that the theoremworks for the U(1) gauge theories as well. Since the U(1) Noether current also vanishes in thegauge limit, we approach the U(1) gauge symmetry limit by multiplying the same parameter λ on L int as we have done. To avoid arbitrariness in the overall U(1) charge scale, we definethe Noether current as J λµ = − i ∂L λ ∂ µ Φ Φ + i Φ † ∂L λ ∂ µ Φ † , = i (1 − λ )(Φ † ↔ ∂ µ Φ) , We end up with a sum rule which involves a continuum of states all the way up to infinite energies. Someexamples using the charge density algebra are found in the Reference [8]. See also Reference [9]. λ = Z J λ ( x , t ) d x . (33)Just as in the non-Abelian case, the factor (1 − λ ) does not appear in J λ when we expressit by use of Π / Π † ; J λ = i (Φ † Π † − ΠΦ) . (34)Consequently the charge-field commutation relation does not have an explicit dependenceon (1 − λ ); [ Q λ , Φ( x , t )] = − Φ( x , t ) , (35)in spite that Q λ = i (1 − λ ) R (Φ † ↔ ∂ Φ) d x .We take the matrix element between the vacuum h | and the one-particle state | p i forthe both sides of Eq. (35). When we insert a complete set of states P | n ih n | between the Q λ and Φ( x , t ), we are immediately led to h p | Q λ | p i = 1 . (36)The reasoning goes from here exactly as in the non-Abelian case: When h p ′ | Q λ | p i is writtenas (1 − λ ) F ( t, λ ) with the form factor F ( t, λ ) of the Heisenberg operator i (Φ † ↔ ∂ µ Φ), Eq.(36) requires that the function F ( t, λ ) must behave like F ( t, λ ) → − λ + O ( t ) (37)near λ = 1 in the neighborhood of t = 0. This is realized only if F ( t, λ ) has a bound-statepole, µ / ( m bound − t ), on the real axis in the complex t -plane and if m bound reaches zero at λ → m bound = µ (1 − λ ). VII. FERMION MATTER
The Noether theorem is based on the invariance of Lagrangian under the phase rotation offields. Therefore, whether fields are canonically independent or not, the conserved Noethercurrent consists of all the fields that enter Lagrangian, J aµ = − i ∂L∂ ( ∂ µ Ψ) T a Ψ + i Ψ † T a ∂L∂ ( ∂ µ Ψ † ) . (38)If we want to treat Ψ and Ψ † on the equal footing, we may choose the free Lagrangian inthe form L = i ↔ ∂ Ψ − m ΨΨ , (39)by adding a total divergence term. With L λint added to this L , it may look trivial to repeatour proof for the boson matter to prove the theorem for the fermion matter. But it is notthe case.If we formally defined the conjugate momentum by Π = ∂L/∂ ( ∂ Ψ) with L + L λint andsimilarly for Π † , the Noether charge density would take the form of J λa = i (Ψ † T a Π † − Π T a Ψ) , (40)13here T a = τ a for the SU(2) doublet and T a → † , Π † ) as all independentof each other, it looks that we would obtain the charge-field algebra at equal time,[ Q λa , Ψ] = − T a Ψ (41)and its hermitian conjugate just as in the case of bosons. Then, with Eq. (41), our prooffor the boson models would apply to the fermion models with no modification. However,we encounter one problem: This naive derivation of Eq. (41) is incorrect although the finalresult is most likely correct. There is a subtlety special to the canonical formalism of theDirac field.[14–18].The problem arises from the fact that the Lagrangian of the Dirac field is linear in thetime derivative and therefore that only two of those four variables above can be treated ascanonically independent. For instance, if one chooses Ψ and Π as independent variables,Ψ † and Π † are functions of Ψ and Π. This turns the equal-time anticommutator { Ψ , Ψ † } + nontrivial and dependent on the interaction, in general.In the matter gauge theories, the interaction L int contains the derivatives of field in orderto counterbalance the gauge variation of the free Lagrangian L . In a such case, unlikethe Dirac field interacting with a nonderivative interaction, we do not have an option ofsetting Π † = 0 by choosing L asymmetric in Φ and Φ † . Consequently the equal-timeanticommutator between Ψ and Ψ † may become dependent on L int in general. Althoughthe prescription to determine the anticommutators has been known when this happens, onehas to go through cumbersome steps. The canonical quantization is thus not best suited forour purpose in the case of the Dirac field since we would have to check each model one byone to make sure that the algebra Eq. (41) is indeed valid for a given interaction.In some cases we can circumvent this procedure. For instance, in the known model of theU(1) symmetry [4], we can remove the time-derivative of Ψ † entirely and realize Π † = 0 byan appropriate rewriting of the Lagrangian. Then the independent canonical variables areonly Ψ and Π, and they obey the simple equal-time anticommutator { Ψ , Π } + = iδ ( x − y ).It is interesting to note that in this case Ψ turns out to be twice as large as what we wouldobtain formally by ignoring the interdependency of the variables. Since the Noether chargeis given by a single term J λ = − i ΠΨ in the case of Π † = 0, the correct charge-field algebra[ Q λ , Ψ] = − Ψ immediately follows in the same form as that for the bosons. We shall describein Appendix C how it works for the U(1) model.In the case of the boson matter the charge-field algebra is an immediate consequence ofthe canonical quantization. In contrast, its derivation through the canonical quantizationrequires some knowledge of the interaction in advance in the case of the Dirac field. Ourgoal is to prove the theorem as generally as possible without referring to specific propertiesof the interaction or without knowing the interaction at all. For this purpose, it is desirableto derive the charge-field algebra Eq. (41) in a way that does not rely on the canonicalquantization.In fact, a line of argument can be made to advocate validity of the charge-field algebrairrespectively of the interaction. It goes as follows: The charge-field algebra Eq. (41) isobtained as the O ( α ) terms of the global symmetry rotation of the fields by angle α , e − iQα Ψ( x ) e iQα = e iα Ψ( x ) (42)for the field of a unit U(1) charge. For non-Abelian symmetries, Q and α should be modifiedappropriately by attaching relevant group-component indices. Then going from Eq. (42)14ackward, ask what kind of operator the Q can be. The operator Q must be a space-time independent Lorentz-scalar since the symmetry at λ = 1 is global but unbroken. Theoperator Q is dimensionless and has a negative charge parity since it generates a phaseof the opposite sign for Ψ † as Ψ † e − iα . The only possible candidate for Q is a charge ofsome conserved vector current J µ . Up to an overall proportionality constant, therefore, thiscurrent ought to be the Noether current that arises from the phase rotation of the fields.It is the only candidate that we have at hand. The Noether current has the right scale ofproportionality constant since its scale is fixed by Eq. (42) that corresponds to the rotationper a unit angle of α . This argument is a little wordy, but it is almost equally as good asthe derivation based on the canonical quantization. It works for the boson matter too.Once Eq. (41) has been accepted in one way or another, we can repeat what we havedone for the boson matter. Define the electric and magnetic form factors in the standardway as 11 − λ h p ′ | J λaµ (0) | p i = h p ′ | Ψ T a γ µ Ψ | p i = s m E p ′ E p u p ′ T a (cid:16) γ µ F ( t, λ ) + iσ µν q ν m F ( t, λ ) (cid:17) u p , (43)where we have suppressed the indices for spins, copies and multiplet components of thefermion. Compare the one-particle matrix elements for the both sides of the charge algebraEq. (41) near λ = 1. The consistency in the power of (1 − λ ) on the both sides requires thatthe electric form factor F ( t, λ ) must obey F (0 , λ ) = 11 − λ . (44)It means existence of a pole of the composite gauge boson in F ( t, λ ) at t = m bound ∝ (1 − λ ).The magnetic form factor F ( t, λ ) does not enter the ( q µ = 0) limit because of the kinematicalfactor iσ µν q ν . Refer to Reference [4] more for the Pauli term F ( t, λ ), the dimension-fiveinteraction, in the leading 1/N order.Our proof ought to hold for any SU(2) multiplet other than the doublet and for any grouphigher than SU(2) as well, if such a model is built .The diagrammatic demonstration is a little less simple for the fermion matter since twochannels S and D couple to form the vector bound state.[4] But it is no more than asmall technical complication. VIII. SUMMARY AND DISCUSSION
We can realize gauge invariance without introducing a fundamental vector gauge-fieldof any kind. In order to connect between the matter fields at separate space-time pointsin such theories, the interaction Lagrangian must be carefully concocted by including thederivatives of matter fields. In this paper we have proved that such matter interactionsinevitably generate composite gauge bosons.The proof is based on the three properties:(1) Most importantly, the Noether current vanishes in the gauge symmetry limit of suchtheories. 152) The equal-time charge-field algebra holds in the Heisenberg picture.(3) The form factor of current obeys the well-established analyticity.In our proof we have started with a globally invariant but not locally invariant theory( λ = 1) and then have approached the gauge symmetry by continuously varying the value ofparameter λ . When we follow this path to the gauge symmetry, consistency of the charge-field algebra requires that a bound state must be present in the channel of J P C = 1 −− and turn massless in the gauge symmetry limit. The proof has been given step by step indetail for the non-Abelian gauge theories of the boson matter. The proof has been triviallyextended to the Abelian theories. The theorem holds for the fermion matter as well. But wehave cautioned about the issue that we encounter if we rely on the canonical quantizationof the Dirac field. Our proof is valid to all orders of interactions since the theorem has beenproved in the Heisenberg picture.This theorem gives us another way to understand why the composite state of J CP =1 −− cannot be massless if the Noether current exists: Because, if a massless bound statewere formed in the presence of the nonvanishing Noether current, it would lead to theinconsistency O (1 / (1 − λ )) = O (1) as λ → × U(1). Butwhat shall we do about the composite gluons ? Is the so-far unsuccessful attempt to builda matter gauge-theory beyond the SU(2) doublet only for a technical reason or for a morefundamental reason ? In the past we saw a few cases in which physics cannot be extendedbeyond SU(2). One is the G-parity ( G = C exp[ iT π ]) of low-energy hadron physics. Weknow why it cannot. Another is the instanton solution of the non-Abelian gauge theory[20]. This is because of the winding number arising from mapping of the SU(2) solutiononto the sphere S of the four-dimensional space-time. Recall that the QCD instanton isno more than the SU(2) instantons embedded into the SU(3) parameter space. In our caseunlike the instanton, there seems to be nothing topological in our case. In the no-Abelianmodels so far invented, the special property of τ a for the SU(2)-doublet plays a crucial role.If an extension is possible beyond the SU(2)-doublet, it appears that we shall need a verydifferent approach to model building.Once we have proved formation of composite gauge bosons, it is not necessary everytime to go back to the original matter Lagrangian as far as the gauge boson interactions ofdimension four are concerned. An obvious question is how to handle the effective interactionsof dimension higher than four. This is the place where we expect to see difference between16he elementary gauge bosons and the composite ones phenomenologically. It is too early tospeculate on it. Appendix A: Noether current
We show that the Noether current is identically zero in gauge theories which consist onlyof matter fields.[4] Since this is the basis of our theorem, we reiterate it in the simplestway. We choose the non-Abelian gauge theory of boson matter as an example. Extensionto fermion matter involves only minor modifications due to spins and anticommutativity.Gauge invariance of the action of the total Lagrangian L tot requires to the first order in α a ( x ) ∂ µ (cid:16) ∂L∂ ( ∂ µ Φ) T a Φ − Φ † T a ∂L∂ ( ∂ µ Φ † ) (cid:17) α a + (cid:16) ∂L∂ ( ∂ µ Φ) T a Φ − Φ † T a ∂L∂ ( ∂ µ Φ † ) (cid:17) ∂ µ α a + 0( α ) = 0 , (A1)where the equation of motion has been used in the first term as usual. Since α a are arbitraryfunctions of x µ , the terms proportional to α a and ∂ µ α a must vanish separately in Eq. (A1).The terms proportional to α a allow us to define the Noether current J µa and lead us to itsconservation: J aµ ≡ − i ∂L∂ ( ∂ µ Φ) T a Φ + i Φ † T a ∂L∂ ( ∂ µ Φ † ) , (A2) ∂ µ J aµ = 0 . (A3)Then the requirement that the terms proportional to ∂ µ α a be zero in Eq. (A1) is nothingother than the vanishing of the Noether current: J aµ = 0 . (A4)When L int is multiplied with λ and turned into L λint , L int → λL int ≡ L λint , (A5)it breaks gauge invariance of the total Lagrangian L λtot ≡ L + λL int so that the Noethercurrent J aµ no longer vanishes for λ = 1. However, we do not need an explicit form of L int to obtain the Noether current for λ = 1 since the variation of L int is determined by that ofthe free Lagrangian L alone through gauge invariance of L + L int . To obtain the Noethercurrent in this case, split the Lagrangian as L λtot = (1 − λ ) L + λ ( L + L int ) . (A6)The second term does not contribute to the Noether current since it is gauge invariant. TheNoether current arises only from the first term and takes the form of (1 − λ ) times theNoether current due to L ; J λaµ = i (1 − λ ) (cid:16) Φ † T a ↔ ∂ µ Φ (cid:17) . (A7)17 ppendix B: Effect of interaction in equal-time algebras The equal-time algebras of the charge Q λa are free of an explicit dependence on the factor(1 − λ ). It is because this factor does not appear in Q λa when it is written in terms of Π andΠ † instead of ∂ Φ and ∂ Φ † . The purposes of Appendix B is to show how the charge densityacquires the factor (1 − λ ) when we switch from Π and Π † to ∂ Φ and ∂ Φ † , but that Π norΠ † vanishes individually as λ → q i , p j ] = iδ ij , and make the correspondence q i ( t ) → Φ( x , t ) and p i ( t ) → Π( x , t ) = ∂L tot /∂ ( ∂ Φ( x , t )). According to the standard quantization rule, a pair of the canonical“coordinate” and “momentum” obeys the equal-time commutation relation,[Φ( x , t ) , Π( y , t )] = iδ ( x − y ) , (B1)and so forth. The unit matrices are to be understood in the right-hand side of Eq. (B1)with respect to the components of the group indices, the copies and so forth.According to Eq. (A2), the charge density can be expressed as J λa = i (Φ † T a Π † − Π T a Φ) . (B2)A factor of (1 − λ ) does not appear in the right-hand side of Eq. (B2). Consequently, thecelebrated equal-time algebra of the charge densities results [8] as[ J λa ( x , t ) , J λb ( y , t )] = if abc J λc ( x , t ) δ ( x − y ) (B3)without (1 − λ ). Similarly[ J λa ( x , t ) , Φ( y , t )] = − T a Φ( y , t ) δ ( x − y ) . (B4)When the Noether charge is written with ∂ Φ and ∂ Φ † instead of Π and Π † , the factorof (1 − λ ) appears. But this does not mean that Π and Π † are proportional to (1 − λ ). It isinteresting to see in the known model how the factor (1 − λ ) appears in the charge densityupon switching from Π and Π † to ∂ Φ and ∂ Φ † .Take the SU(2) doublet model [4] as an example. The interaction is given by L int = λ (Φ † τ a ↔ ∂ µ Φ)(Φ † τ a ↔ ∂ µ Φ)4(Φ † Φ) . (B5)The momenta conjugate to Φ and Φ † are given byΠ = ∂L λtot ∂ ( ∂ Φ)= ∂ Φ † + λ (Φ † τ a ↔ ∂ Φ)2(Φ † Φ) Φ † τ a , (B6)and its hermitian conjugate, respectively. Notice that neither Π nor Π † vanishes as λ → † τ a Π † − Π τ a Φ and using [ τ a , τ b ] = 2 δ ab , we obtain i (cid:16) Φ † τ a † − Π τ a (cid:17) = (1 − λ ) (cid:16) Φ † τ a ↔ ∂ Φ (cid:17) . (B7)Dependence on the interaction enters the Noether current through Π and Π † . However, inthe combination of (Φ † τ a Π † − Π τ a Φ), the contribution of the interaction turns out to besimply λ times (Φ † τ a ↔ ∂ Φ) with a minus sign.18 ppendix C: Canonical quantization of Dirac field
The complication in the canonical quantization of the Dirac field is due to the fact thatthe Lagrangian is linear in the time derivative and therefore the hermitian conjugate fieldΨ † is no longer canonically independent of (Ψ, Π) after Ψ and Π are chosen as the canonicalvariables. This is an example of the so-called constrained dynamical systems.[14–19].Let us first recall the free Dirac field. When we choose the Lagrangian in the asymmetricform, L = i Ψ ∂ Ψ − m ΨΨ , (C1)we obtain Π = ∂L /∂ ( ∂ Ψ) = i Ψ † and impose { Ψ , Π } + = iδ ( x − y ) at equal time. Thecanonical quantization is complete with this condition since Π † = ∂L/∂ ( ∂ Ψ † ) = 0.We may add a total divergence term to L and antisymmetrize it with respect to ∂ µ Ψand ∂ µ Ψ † as L = i ↔ ∂ Ψ − m ΨΨ . (C2)In this case we cannot proceed with the naive rule of quantization by treating both Ψ andΨ † as independent coordinates.Let us consider the interacting Dirac fields. We can sometimes circumvent the difficultyby modifying L int without changing physics. Consider the U(1) matter model [4] as anexample. The interaction is given by L λint = − iλ γ µ Ψ)(Ψ ↔ ∂ µ Ψ)(ΨΨ) , (C3)We add a total derivative term∆ L λint = − iλ ∂ µ (cid:16) (Ψ γ µ Ψ) log(ΨΨ) (cid:17) , (C4)to the original interaction Eq. (C3) and turn it into L λint + ∆ L λint = − iλ (Ψ γ µ Ψ)(Ψ ∂ µ Ψ)(ΨΨ) . (C5)Here we have used ∂ µ (Ψ γ µ Ψ) = 0. The purpose of adding ∆ L λint is to remove the term ∂ Ψ † from the interaction. Now the total Lagrangian reads L λtot = i Ψ ∂ Ψ − m ΨΨ − iλ (Ψ γ µ Ψ)(Ψ ∂ µ Ψ)(ΨΨ) . (C6)Since Π † = ∂L/∂ ( ∂ Ψ † ) = 0 for this Lagrangian, we can now choose Ψ and Π as canoni-cally independent variables and treat Ψ † as a trivial dependent variable, i.e., the constraintvariable. The variable Π defined by Π = ∂L/∂ ( ∂ Ψ) with the Lagrangian of Eq. (C6) turnsout to be twice as large as what we would obtain for Π by pretending (Ψ, Π, Ψ † , Π † ) as allindependent in the original Lagrangian. Since the simple canonical quantization relation { Ψ( x , t ) , Π( y , t ) } + = iδ ( x − y ) (C7)19olds, we are led to the desired result, Eq. (41) for [ Q, Ψ]. Its hermitian conjugate correctlygives what we want for [ Q, Ψ † ].Alternatively we can choose Ψ and Ψ † , instead of Ψ and Π, as the canonical variables forthe original L λtot . To do so, we must take account of the interdependency of the variablesby making sure that Hamilton’s equation of motion should hold correctly. The generalprescriptions of this procedure have been discussed in length, but the case of the Lagrangianlinear in the time-derivative can be presented in a compact mathematical form, which isfound, for instance, in the lecture note, “Constrained Quantization Without Tears” byJackiw [19]. Acknowledgments
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