Bled Workshops in Physics Vol. 14, No. 2
A
Proceedings to the 16th Workshop What Comes Beyond ... (p. 65) Bled, Slovenia, July 14-21, 2013
6 Coupling Electromagnetism to Global Charge
E. I. Guendelman*
Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
Abstract. It is shown that an alternative to the standard scalar QED is possible. In this new version there is only global gauge invariance as far as the charged scalar fields are concerned although local gauge invariance is kept for the electromagnetic field. The electromagnetic coupling has the form j ^(A^ + d^B) where B is an auxiliary field and the current j ^ is A^ independent so that no "sea gull terms" are introduced. In a model of this kind spontaneous breaking of symmetry does not lead to photon mass generation, instead the Goldstone boson becomes a massless source for the electromagnetic field. Infrared questions concerning the theory when spontaneous symmetry breaking takes place and generalizations to global vector QED are discussed. In this framework Q-Balls and other non topological solitons that owe their existence to a global U(1) symmetry can be coupled to electromagnetism and could represent multiply charged particles now in search in the LHC. Finally, we give an example where an "Emergent" Global Scalar QED can appear from an axion photon system in an external magnetic field.
Povzetek. Pokažem, da obstaja alternativa standardni skalarni teroriji kvantne elektrodinamike. V tej novi različici velja ža nabita skalarna polja samo globalna umeritvena invarianca, lokalno umeritveno invarianco pa zahtevamo ža eletromagnetno polje. Elektromagnetna sklopitev ima obliko	+ 3^B), kjer je B pomožno polje, tok je neodvisen od A^, žaradi česar so členi tipa "sea gull" v teoriji motenj enaki nič. Pri spontani žlomitvi simetrije ostane foton brež mase, Goldstoneov božon pa postane brežmasni ižvor elektromagnetnega polja. V prispevku obravnavam probleme, ki jih ima ob spontani žlomitvi simetrije ta teorija v infrardečem območju ter njeno posplositev do globalne vektorske teorije kvantne elektrodinamike. V tem okviru se lahko krogle "Q" in ostali netopolosi solitoni, ki dolgujejo svoj obstoj globalni simetriji U(1), sklopijo ž elektromagnetnim poljem in bi lahko predstavljali večkratno nabite delče, ki jih trenutno isčejo na LHC. Na konču podamo primer, kako lahko "porajajočo" globalno skalarno teorija kvantne elektrodinamike ižpeljemo iž sistema aksion-foton v žunanjem magnetnem polju.
6.1 Introduction
In this paper it will be shown that an alternative to the standard scalar QED is possible. In this new version there is only global gauge invariance as far as the charged scalar fields are concerned although local gauge invariance is kept for the electromagnetic field, we call this new model Global scalar QED. The electromagnetic coupling has the form	+ 3^B) where B is an auxiliary field
and the current is A^ independent so that no "sea gull terms" are introduced. In
* e-mail: guendel@bgu.ac.il
a model of this kind spontaneous breaking of symmetry does not lead to photon mass generation, instead the Goldstone boson becomes a massless source for the electromagnetic field, Infrared questions concerning the theory when spontaneous symmetry breaking takes place and generalizations to global vector QED are discussed.
In this framework Q-Balls [1] and other non topological solitons [2] that owe their existence to a global U(1) symmetry can be coupled to electromagnetism and could represent multiply charged particles now in search in the LHC [3].
Finally, we give an example where an "Emergent" Global Scalar QED can appear from an axion photon system in an external magnetic field.
6.2 Conventional scalar QED and its sea gulls
In conventional scalar QED, we "minimally couple" a globally invariant action (under global phase transformations). To be concrete, for a complex scalar field ^ with mass, m whose Lagrangian density can be represented in relativistic invariant form in the absence of interactions to electromagnetism as
L = AVv	- m2c2^>	(6.1)
' 9 9x^ 9xv	^ ^	v '
Then, in the standard scalar QED model we introduce the electromagnetic interaction with scalar charged particles by introducing the minimal coupling in
the Lagrangian for charged particles (see Eq. 6.1). As we recall, minimal coupling requires that we let the momentum p^ be replaced by p^ —» p^ — eA^ where p^ = —i h-dv and where A^ is the electromagnetic 4-vector whose Lagrangian is given by
lem = —	(6.2)
with F^v = 9^Av — 9vA^. We can now write the total Lagrangian after using the minimal coupling substitution into Eq. 6.1
Lt = g^
9	9
(A— — ieA^:][( A— + ieAv M

m2c>> — ^F^v (6.3)
This leads to the equation of motion for the scalar field ^
(i — e^)2^ = (^V — eA)2^ + m2c> ot	i
(6.4)
This equation and the lagrangian density from which it is derived are invariant under local gauge transformations:
A —> A' = A + Vx;	' = $ — with exp [ieX(6.5)
c ot	h
Furthermore the electromagnetic field satisfies the Maxwell's equations where the electric charge density p and the current density j (x) are given by (now set
c = h = 1).
p(x) = i(X*— - X —) -	and j(x) = -i(X*VX - XVX-) - 2eAX*X
ot ot
(6.6)
There is an example, the BCS theory of superconductivity [4], where the effective theory in terms of the composite Cooper pairs retains the local gauge invariance which involves the local phase transformations of the composite scalar, however we may ask if this is a general rule, may be not.
When thinking of the electromagnetic interactions of pions, the quadratic dependence of the interactions on the potentials characterises the sea gull behaviour of standard scalar QED. As pointed out by Feynman [5], it is somewhat puzzling that spinor electrodynamics does not lead to any of such sea gulls . Considering that the microscopic description of charged pions is really the spinor electrodynamics of quarks, shouldn't we search for an effective scalar electrodynamics devoid of sea gulls?, is this possible?. In the next section we will see that this can be achieved in global scalar QED. The Global Scalar QED could address other questions as well. like the electromagnetic coupling of Q-balls and can "emerge" as an effective description of a system of axions and photons in an external field.
6.3 Global Scalar QED
There are many possible motivations for departing from the scheme implied by the minimal coupling, which leads to scalar QED. For example, if the complex scalar field is to describe a pion, since the macroscopic hadron is a very non local construction in terms of the fundamental quark fields and gluon fields as has been revealed from both the theoretical point of view [6] and from the experimental point of view [7] and in fact we may have several alternative candidates for the pion wave function (and any such proposal could give rise to a different effective theory), we do not necessarily have to keep a local gauge invariance in terms of the composite scalar fields (that would describe the hadrons), although global phase invariance must be respected. Also local gauge transformations for the photon should be mantained. Other possible use of deviating from the the minimal coupling scheme, as we will see, could be to couple Q-Ball type solitons to electromagnetism. Finally, we will give an example where an "Emergent" Global Scalar QED can appear from an axion photon system in an external magnetic field.
We work therefore with the following lagrangian density
L = dX-dX -	-	+ MA" + 9"B)	(6.7)
where
j" = ie(X*d" - xdX-)	(6.8)
and where we have also allowed an arbitrary potential U(X*X) to allow for the possibility of spontaneous breaking of symmetry. The model is separately
invariant under local gauge transformations
A^ + 3^A; B^B - A	(6.9)
and the independent global phase transformations
>exp(ix)^	(6.10)
The use of a gauge invariant combination (A^ + 3^B) can be utilized for the construction of mass terms[8] or both mass terms and couplings to a current defined from the gradient of a scalar in the form (A^ + 3^B)3^A [9]. Since the subject of this paper is electromagnetic couplings of photons and there is absolutely no evidence for a photon mass, we will disregard such type of mass terms and concentrate on the implications of the (A^ + 3^B j couplings.
6.4 A Double Charge Theory
As we will see the scalar QED model has two charge conservation laws associated with it. We see that Maxwell's equations are satisfied with j ^ being the source, that is
= j^ (6.11)
of course this implies 3V3^FV|X = 3 j = 0. The same conclusion can be obtained from the equation of motion obtained from the variation with respect to B.
The Noether current obtained from the independent global phase transformations ^ —» exp(ix)^, x being a constant, is
3^ 3^ 3x^ - ^ 3X1
Therefore
V = ie(^*^ -^^) + 2e(A^ +	(6.12)
jB = V - j^ = 2e(A^ +	(6.13)
is also conserved, that is 3^((A^ +	= 0
6.5 No Klein Paradox
An interesting difference between standard scalar QED and global scalar QED appears in the case of strong fields Consider the global scalar QED equations with an external electromagnetic field potential step-function: e(A0 + 3oB) = V (x);eAt + e3tB = 0.
v, ( 0 for x < 0 V (x) = \ Vo for x>0
The Global QED equation in the presence of this potential is (h = 1, c = 1)
-+ V2^ - m2^ = 0	(6.14)
3t2
for x < 0.
- 0 + 2iVo ^ + V2^ - m2^ = 0	(6.15)
for x > 0. To solve the equation with this potential, we try solutions of the form:
= ^ = e-iEt [eipx + Re-ipx] for x < 0 = ^ = Te-iEteip 'x for x > 0	(6.16)
where represents a wave like solution for the Klein-Gordon field for x < 0 and represent the field for wave like solution for x > 0. R is the amplitude of that part of wave that is reflected wave while T is that part that is transmitted. We substitute and Eq. 6.16 into Eqs. 6.14 and 6.15 respectively. We thus find
E2 - p2 - m2 = 0 -> E = +Vp2 + m2 for x < 0	(6.17)
since for incident wave for x < 0 we chose the positive sign in the square root as our boundary condition. and
E2 - 2EV0 - p '2 - m2 = 0->p' = E(E - 2V0) - m2 for x>0 (6.18)
We see here that from a certain positive value of V0, V0crit = (E2 - m2)/2E and higher,p' becomes imaginary and therefore there is no transmited wave for large values of V0, totally opposite to the behaviour of standard scalar QED, where for large enough barrier a transmitted wave is restored once again, leading to the "Klein paradox", the transmitted wave is interpreted there as pair creation process, no such process appears in global scalar QED.
6.6 Behaviour under Spontaneous breaking of symmetry, new couplings of Goldstone Bosons to Electromagnetism and associated infrared problems
The absence of quadratic terms in the vector potential implies that no mass generation for the photon takes place. Furthermore the Goldstone boson that results from this s.s.b. /writing ^ = pexp(i6), where p is real and positive, we obtain that the phase of the ^ field, is not eaten, it remains in the theory, in fact it couples derivatively to (A^ + 3^B), like the A field studied in [9] and it produces a gradient type charge. In fact under s.s.b. regarding p as a constant, j^ = 2ep23^6 the coupling (A^ + 3^B j implies the coupling of (A^ + 3^B) to a gradient current, as discussed in [9].
It should be pointed out that this type of gradient current j ^ = 2ep 2 3 ^ 6 for p = constant generates an infrared problem, since the 6 field now represents a massless field, which instead of being eaten becomes a source of electromagnetism. The normal way of solving for the electromagnetic field, using the Green's function method does not work straightforwardly, since the source now in Fourier space has support only in the light-cone and the Green's function has a pole like behaviour at
the light-cone as well, so we encounter an undefined product of distributions. This is very similar to the solution of a forced harmonic oscillator when the external force has exactly the same frequency to that of the oscillator, that is the resonant case.
To resolve this problem, we note first that considering FV| as an antisymmetric tensor field (without at first considering whether this field derives from a four vector potential), then a solution of the equation 9VFV| = j^ is 1
FV|| —
rl
dÀÀ2 (xv j|(Ax)- X|jv(Ax))	(6.19)
0
For a generic current the above FV| does not derive from a potential, however if the current is the gradient of a scalar field, the above FV| derives from a potential and provides a solution of the problem, where the Green's function method fails. Notice that the similarity with the the resonant case of the forced harmonic oscillator is very close, there the solution is of the form of an oscillating function times time and in the above solution we see the similar xv dependence appearing.
The resulting gauge potentials displays also a linear dependence on xv, which is interesting, since the central issue in the confinement problem for example is how to obtain potentials with linear dependence on the coordinates, although it is not clear how the very specific solution studied here is relevant to the confinement problem.
Axions are an example of Goldstone bosons with non trivial electromagnetic interactions
6.7 Global Vector QED
In this case we consider a complex vector field W^ and consider the action
1 |IV a|3p p*	1
g 1 Gi«Gvp - 4 with G|v — 9^Wv - dvW1 and where
L — -4g|VgapG|«GVp - ^F^ + j^A1 + 9^B) + M2W|W^ (6.20)
ji = ie(W*aGK| - W^G^)	(6.21)
This model displays global phase invariance for the complex vector field W^ and local gauge invariance for the photon and B fields (6.7), as was the case of global scalar QED. Once again, no sea gull terms are present here.
6.8 Q Balls and other global U(1) solitons as electromagnetically charged Particles
An interesting situation could present itself when considering solitons as in the case of Q-Balls [1] or other non topological solitons [2], that depend on the existence of a U(1 ) symmetry.
11 want to thank R. Tabensky for pointing this to me
These solitons have been found using actions like that used in Global scalar QED for the case e=0. The idea is minimizing the energy under the constraint that the charge of the system is given. This leads us to time dependent configurations with time dependence of the form
^(r,t) = p(r)exp(iwt)	(6.22)
We see that if there was a local gauge transformation that involve a local phase transformation of the complex scalar field then the phase of ^ is a totally unphysical quantity and the above eq. 6.22 becomes totally meaningless. That is not the case in global QED, for which 6.22 is meaningful.
Furthermore, the standard Q-Balls hold in the limit e —» 0 and also the small e case can be treated in perturbation theory, The introduction of a non zero e tends to destabilize the soliton as a consequence of the Coulomb repulsion that appears from the Q-ball having an electric charge. This effect is small for the case of small e, so we know there must be a range of parameters for which electrically coupled Q-Ball solitons exist.
6.9 "Emergent" scalar QED from a system of photons and axions in an external magnetic field
In this section we will consider how an "Emergent" scalar QED from a system of photons and axions in an external magnetic field. Such analysis was considered in [10] and in [11], where a "scalar QED analogy" was recognized. As we will discuss here, although the system of photons and axions in an external magnetic field does indeed have features that resemble scalar QED, the more close correspondance is with Global Scalar QED.
The action principle describing the relevant light pseudoscalar coupling to the photon is
S =
d4x
VvF,v +	- !mV - g
4
2
2
8

(6.23)
We now specialize to the case where we consider an electromagnetic field with propagation along the y and z directions and where a strong magnetic field pointing in the x-direction is present. This field may have an arbitrary space dependence in y and z, but it is assumed to be time independent.
For the small perturbations, we consider only small quadratic terms in the action for the axion and the electromagnetic fields, considering a static magnetic field pointing in the x direction having an arbitrary y and z dependence and specializing to y and z dependent electromagnetic field perturbations and axion fields. This means that the interaction between the background field , the axion and photon fields reduces to
Si =

d4x [|3^EJ,
(6.24)
where (3 = gB(y,z). Choosing the temporal gauge for the photon excitations and considering only the x-polarization for the electromagnetic waves (since only this polarization couples to the axion) we get the following 2+1 effective dimensional action (A being the x-polarization of the photon, so that Ex = — dtA)
S2 =
dydzdt
1 1 1->-> -9^A9^A +	— ^mV + p^9tA
(6.25)
Since we consider only A = A(t,y,z), ^ = ^(t,y,z), we have avoided the integration over x. For the same reason p runs over t, y and z only . This leads to the equations
9„9^ + m2^ = pdtA
(6.26)
and
9^A = — (9t^.	(6.27)
As is well known, when choosing the temporal gauge the action principle cannot reproduce the Gauss constraint (here with a charge density obtained from the axion photon coupling) and has to be imposed as a complementary condition. However this constraint is automatically satisfied here just because of the type of dynamical reduction employed and does not need to be considered anymore.
Without assuming any particular y and z-dependence for p, but still insisting that it will be static, we see that in the case m = 0, we discover a continuous axion photon duality symmetry (these results were discussed previously in the 1+1 dimensional case, where only z dependence was considered in [10] and generalized for the case of two spatial dimensions in [11]), since
1.	The kinetic terms of the photon and axion allow for a rotational O (2) symmetry in the axion-photon field space.
2.	The interaction term, after dropping a total time derivative, can also be expressed in an O(2) symmetric way as follows:
S-=2
dydzdtp [$9tA — A9t^].	(6.28)
It is easy to see that after introducing an appropriate complex field this
coupling is exactly of the global scalar QED form. The U(1) axion photon symmetry is (in the infinitesimal limit)
SA =	= — eA,	(6.29)
where e is a small number. Using Noether's theorem, this leads to the conserved current j with components given by
jN = A9t^ — ^9tA — 3p (A2 +	(6.30)
and
jN = A9t^ - ^9tA.	(6.31)
Here i = y, z coordinates. In order to have the exact correspondence with Global scalar QED, we must define the complex field ^ as
^ = + iA),	(6.32)
we see that in terms of this complex field, the Noether charge density takes the form
jN = i(^3t^ -	(6.33)
which, as in Global scalar QED does not coincide with the current that enters in the interaction lagrangian, which is
jo =	-^3t(6.34)
We observe that the correspondance with standard scalar QED is approximate, only to first order in since (6.28) which represents the interaction of the magnetic field couples with the "axion photon density" 6.34, that does not contain | dependence.
This interaction has exactly the same form as that of the global scalar QED with an external "electric " field. In fact the magnetic field (or more precisely |/2) appears to play the role of external electric potential of Global scalar QED e(A0 + 90B) = V(x) that couples to the axion photon density,6.34 which plays the role of an electric charge density, exactly as in Global Scalar QED.
From the point of view of the axion-photon conversion experiments, the symmetry (6.29) and its finite form, which is just a rotation in the axion-photon space, implies a corresponding symmetry of the axion-photon conversion amplitudes, for the limit w >> m.
In terms of the complex field, the Noether current takes the form
jN =	-	(6.35)
6.10 Discussion and Conclusions
Discussing the new global QED makes sense from both the purely theoretical point of view, since it provides a new type of viewing interactions of charged scalar particles with electromagnetism, as well as from a phenomenological point of view, since standard scalar QED contains the sea gull contributions for which apparently do not represent any known physical process in the electrodynamics of charged pions for example, so it makes sense to build a theory without such sea gulls.
In this framework Q-Balls [1] and other non topological solitons [2] that owe their existence to a global U(1) symmetry can be coupled to electromagnetism and could represent multiply charged particles now in search in the LHC.
Finally we have shown an example of an "Emergent" global scalar QED from a system of photons and axions in an external magnetic field.
Acknowledgements
I am very grateful to Holger Nielsen, Anatoli Romaniouk, Norma Susana Mankoc Borštnik and Jon Chkareuli for very useful conversations, as well as to all the participants of the conference "What is beyond the Standard Model", 16th Conference , Bled, July 14-21, 2013 for additional feedback on my presentation at this conference on the subject of this paper.
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