On electric and magnetic charges in gauge theories

I l[l[g i ='~ :| "~-"fk'5 [~,1 :!

PROCEEDINGS SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 133 (2004) 277-280

ELSEVIER

www.elsevierphysics.com

On Electric and Magnetic Charges in Gauge Theories E. Bagan, ~ i%. Horan b, A. Kovner b, M. Lavelle b*, Z. Maznmder

b, D. McMullan

b and S. Tanimura c

~Grnp de F~sica Tebrica, Departament de Fisica and IFAE, Edifici Cn, Universitat Autbnoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain bSchool of Mathematics and Statistics, University of Plymouth, Plymouth, PL4 8AA, United Kingdom CDept. of Engineering Kyoto University Kyoto 606-8501 Japan

Physics and Mechanics

In this talk we will summarise a systematic approach to constructing physical, i.e., gauge-invariant, fields describing charged particles. We will show that this can be done in a well defined way in abelian theories and perturbatively in non-abelian theories. Various pertnrbative tests in three and four dimensions will be reported. The existence of a non-perturbative obstruction to the observability of quarks will be demonstrated. Finally the extension to magnetic charges is discussed and their non-existence in pure SU(2) is shown.

1. P h y s i c a l C h a r g e d V a r i a b l e s Any description of a physical field should be gauge invariant. Thus the fermions in interacting QED cannot be directly interpreted as electrons. However, it was pointed out many years ago by Dirac [1] that this can be corrected by associating a photonic cloud as follows

exp{-i~/ d4zf,(x-z)A'(z)} ~(x).

(1)

This is gauge invariant if O ~ f ~ ( w ) (~(4)(%d). There are of course a wide range of possible functions f~ that fulfill this, for example the stringlike ansatz e x p { - i e d z i A i ( x o , z~)}¢(x). However, this has a hard to interpret path (F) dependence. This problematic property can be factored out [2] and one extracts a gauge invariant and path-independent result =

fr

(

OiAi )

D(X) = e x p / - i e - - ~ -

~ ~b(x).

*Talk presented by M. Lavelle.

(2)

This dressed charge has the attractive property that e x i -- Yi

which indicates that this describes a static abelian charge. Knowledge of the LiSnard-Wiechert potentials can be similarly used co produce descriptions of moving charges [3,@ To take this beyond QED a more systematic approach is required. The approach presented here is based on the heavy effective theory. For simplicity consider a s t a t i c dressed charge, ~, which must obey 00~ = 0. Combined with the effective theory's equation of motion, ( O o + g A o ) ¢ = 0 and writing the dressed charge as ~ = h-l~b we obtain the requirement Ooh - 1 = g h - l A o ,

(4)

which we call the (static) dressing equation. This and the requirement that h-l~p is gauge invariant, i.e., if we demand that under a gauge transformation h -1 ~ h -1 U, this allows us to systematically construct the dressing.

0920-5632/$ see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.04.178

278

E. Bagan et aL/Nuclear Physics B (Proc, SuppL) 133 (2004) 27~280

In QED it is possible to solve these two demands (gauge invariance and kinematical) and we then find that for an asymptotic charge with an arbitrary velocity the correct form of the dressing is h -1 = e-~eZ~e-iex

(5)

where = - Jr d s ( r / + . ) . O" F~ ,

-

g'A o

(6) (7)

and g~ = (r/+ v)~(U - v). O - O" with P the past (future) trajectory of an incoming (outgoing) partide. Here we have written the four velocity u , as ~(r/+ v), with r/the unit temporal vector and v = (0,v). We see from this equation that the dressing factorises into two parts: a gauge invariant structure (K), which is essential if we are to fulfill the dressing equation, and a minimal part (X) which is needed to ensure that the dressed matter field is locally gauge invariant. This approach can be extended to non-abelian theories order by order in perturbation theory. An algorithm for the construction of the minimal dressing is presented in the appendix of [5]. In the static limit the minimally dressed expression has been investigated by Haller and collaborators [6]. We are as yet unaware of a closed expression. After this review of the construction of physical fields, we now want to consider how these dressed descriptions can be used in QED and QCD calculations.

2. Applications and Implications The IR problem has long been argued to preclude a particle interpretation for the electron [7]. However, the dressed fields can be shown to yield such an interpretation [8,9]. In explicit perturbative calculations it has been demonstrated that the on-shell dressed Green's functions are IR finite. Unsurprisingly this only holds if each leg is taken to be on the point of the mass shell appropriate to its dressing, i.e., at p = rn-),(r/+ v) ~. The detailed form of the cancellations of the IR divergences reflect the factorised nature of the

dressing: the minimal part removes the soft IR divergences while the additional part is needed to cancel the phase IR divergences which occur in pair production. It is also noteworthy that despite the necessarily non-local and non-covariant form of the dressing the dressed propagator has been shown to be multiplicatively UV renormalisable. Three dimensional QED has recently attracted much interest as an effective theory (see, e.g., [10]) in condensed matter systems. The extraction of physical predictions from these theories has been hampered by the gauge dependence of the Lagrangian fermion. This has sparked a search for a gauge invariant description of the effective particles. Motivated by this we have studied the on-shell propagator of our dressed matter in 2 + 1 dimensions (where by power counting the IR divergences are much worse). The one loop logarithmic IR singularities in the mass shift are found to have the form:

×

v.k

ff:

'

(9)

where V~ = (r/+ v ) , ( r / - v)- k - k, comes from the minimal dressing and W = [(r/+ v) • k k , (r/+ (r/+ v) comes from the massless dressing. These divergences are clearly gauge invariant as one would expect. They also vanish at the appropriate point on shell since, at this point, p~ = m y ( r / + v ) ' , the linear combination -

p" p.k

V" V.k

W" V.k '

(10)

vanishes exactly. A similar cancelation takes place for both the linear and the logarithmic IR divergences in the wave function renormalisation constant. This is strong evidence for the relevance of this dressed field to such systems. It is interesting to note that both components of the dressing are required for this cancelation in 2 + 1 dimensions. It can also be shown that any description of coloured quarks and gluons must be locally gauge invariant. This is because in QCD the colour charge operator, f d a x ( J 3 - fab'EbA~), is only

279

E. Bagan et al./Miclear Physics B (Proc. SuppL) 133 (2004) 277-280 gauge invariant when it acts upon locally gauge invariant states [11]. Only then may we use the non-abelian version of Gauss' law and rewrite

~/0a

--

#abcFbAc LoiEa

J

--i--i

----+

(11)

g

From the simpler gauge transformation of E~ (and using Gauss' theorem and demanding that gauge transformations must be in the centre of SU(3) at spatial infinity) one can show that it is possible to define coloured objects in this way. Just as we saw from (3) that the dressed fields in QED have the correct electromagnetic fields, so in QCD can we demonstrate that our dressed quarks have the correct static potential. The procedure is as follows: we solve the dressing equation and gauge invariance properties order by order in perturbation theory. We then take the expectation value of the Hamiltonian between appropriate (statically dressed) gauge invariant states and the potential, V, is the separation dependent part of the energy. Using the minimal dressing one finds at leading order in g v(,.) -

g2C F

02)

This is of course the QED result with a colour factor. (It is amusing to note that the stringy ansatz of connecting the matter fields by an exponential factor leads already to a linear confining potential with a divergent coefficient in QED! This shows that it corresponds to an infinitely excited state.) The additional dressing makes no contribution at this order. At the next order in g, the significance of the factorisation of the dressing makes itself more visible. It has long been known that the factor 11 in the one loop beta function in QCD can be decomposed as 12 - 1 where the factor of 12 is the result of anti-screening while the - 1 comes from screening by physical glue. Using just the minimal dressing one recovers at this order Vg4(r) =

g4 (4~) 2

N C F C A 12 27rr V log/#r; ,r ,

(13)

which is exactly the full anti-screening result! These calculations have been repeated in threedimensional QCD and it was again found that

the minimal dressing produced the effects of antiscreening. We thus have the following intuitive picture: the minimal dressing is required for gauge invariance and produces the anti-screening effects, while the extra gauge invariant part of the dressing, which we originally introduced to satisfy the dressing equation, produces the screening of charges by physical glue. This attractive picture might appear to indicate that quarks could be observed, however, it can be shown that there is a non-perturbative obstruction to this [5]. If we try to construct a colour charged object, we need a locally gauge invariant description and, as we have seen, this imposes a requirement on the allowed gauge transformations at spatial infinity. This restriction can be shown to imply that it is impossible to nonperturbatively fulfill the minimal requirement of the dressing transforming as h -1 -+ h-lU, since the dressing could be used to construct a unique gauge fixing and the Oribov ambiguity tells us that this cannot be done with our boundary conditions. We conclude that colour charges cannot be observed in unbroken non-abelian gauge theories. 3. M a g n e t i c C h a r g e s ? For several years it has been claimed that confinement may be linked to a dual Meissner effect. However, studies of magnetic charges in non-abelian theories are often reported in specific gauges and the physical content of what is happening is often quite unclear. It is tempting to try to remedy this by constructing a gauge invariant magnetic charge via

M=expli/d3zfi(z)E~(z)}

,

(14)

where E is the field strength and fi is the desired monopole potential. However, in the nonabelian theory the chromo-electric field strength, E'~, is not gauge invariant. This can, though, be dressed. An appropriate choice of dressing (which means that the dressed object does not generate an electric field) involves solely the chromoelectric field. This can be constructed and the

280

E. Bagan et aL/Nuctear Physics B (Proc. SuppL) i33 (2004) 277-280

details can be found in [12]. In pure SU(2) though one can quite directly show that no gauge invariant magnetic charge exists [13]. There are no charge conjugation odd observables in the theory. Another way of seeing this is to realise that the sign of the charge conjugation operator can be flipped by a gauge transformation. We conclude that there are no magnetic monopoles in unbroken SU(2). It may well be that lattice simulations are seeing the effects of vortices. 4. C o n c l u s i o n s The work reported here shows that it is possible to construct (perturbatively in QCD) gauge invariant descriptions of charged particles with a well defined velocity in both 3 + 1 and 2 + 1 dimensions. The dressings around the matter fields have a structure that factorises into two parts, one of which is itself gauge invariant. The dressed fields have been show to have good IR properties and to be multiplicatively renormalisable in the UV domain. We suggest that they may be useful in condensed matter theory. Only the use of such fields allows us to talk about coIour in a meaningful way and it was shown that the structure of the dressing is directly reflected in the anti-screening and screening interactions between charges. Furthermore it was demonstrated that there is a topological obstruction to the construction of a non-perturbative description of a gauge invariant quark or gluon. This non-observability, we believe, underlies colour confinement. There are still many open questions raised by this work. For example, it would be very important to extend the construction of the dressing to massless charges - the simplest arena being massless QED. This limit is rather subtle (for more details see [14]) and more work is needed. A description of massless charges would offer a solution to the problems of collinear divergences which are a major difficulty in QCD. Finally it is very imporrant to see how, and at what scale, the Gribov ambiguity prevents the construction of gauge invariant quarks in QCD. If it can be constructed, a closed form solution would be extremely useful here.

A c k n o w l e d g m e n t s : ZM was supported by EPSRC studentship grant number 00309451. REFERENCES

1. P . A . M . Dirac, Can. J. Phys. 33, 650 (1955). 2. M. Lavelle and D. McMullan, Phys. Lett. B471, 65 (1999), hep-ph/9910398. 3. E. Bagan, M. Lavelle, and D. McMulIan, Phys. Lett. BAT0, 128 (1996), hep-

th/9512118. 4. E. Bagan, M. Lavelle, and D. McMullan, Phys. Rev. D56, 3732 (1997), hepth/9602083. 5. M. Lavelle and D. McMullan, Phys. Rept. 279, 1 (1997), hep-ph/9509344. 6. L. Chen, M. Belloni, and K. Hailer, Phys. Rev. D55, 2347 (1997), hep-ph/9609507. 7. P.P. Kulish and L. D. Faddeev, Theor. Math. Phys. 4, 745 (1970). 8. E. Bagan, M. Lavelle, and D. McMullan, Phys. Lett. B477, 396 (2000), hepth/0003087. 9. E. Bagan, M. Lavelle, and D. McMullan, Annals Phys. 282, 471 (2000), hep-ph/9909257. i0. D. V. Khveshchenko, Nucl. Phys. B642, 515 (2002). 11. M. Lavelle and D. McMullan, Phys. Lett. B371, 83 (1996), hep-ph/9509343. 12. R. Horan, M. Lavelle, D. McMullan, and A. Khvedelidze, (2002), hep-th/0204258. 13. A. Kovner, M. Lavelle, and D. MeMullan, J H E P 12, 045 (2002), hep-lat/0211005. 14. R. Horan, M. Lavelle, and D. McMullan, Pramana J. Phys. 51, 317 (1998), hepth/9810089, Erratum-ibid, 51 (1998) 235.