Co-Supervisors: D r. H. W . Fearing, Dr. C. E. Picciotto. A b s t r a c t
P a r t I : The w° 7 7 form factor is studied in the context of chiral perturbation theory. The form factor is computed with all particles off-shell and up to one-loop level. The contribution of the complete 0 ( p e) parity-odd sector as constructed by Fearing and Scherer is computed for the first time. It is found that the vertex depends on only four independent low energy constants from this part of the chiral Lagrangian. Several processes are studied which involve this form factor and the low energy constants are constrained through comparison with experiment.
P a r t I I : A review is made of the construction and use of the soft photon approximation in particle physics. It is found that problems can arise in the practical application of the approximation which prevent one from making a model independent relation between a bremsstrahlung process and its non-radiative counterpart. The difficulty is traceable to the selection of expansion points which one is required to make during the construction of a soft photon amplitude. The body of literature which has employed the traditional Low form of the amplitude is found to be, for practical purposes, unaffected by this model- dependency. However, it is found th at certain recently proposed alternative forms of the soft photon approximation are subject to the problems and their usefulness in a model- independent analysis of, for instance, the off-shell behaviour of the pp elastic scattering amplitude is therefore called into question.
P a r t I I I : In several high energy hadron scattering experiments the production of low energy photons has been observed to be greatly in excess of the yield expected horn an application of the Low soft photon approximation. A study has been made of possible sources for these excess photons. The contribution of higher order terms in the soft photon ampL’tude is found to be negligible, as are interference effects between the bremsstrahlung amplitude and amplitudes describing the decay of hadronic resonances. Multiple photon emission within a simple multi-hadron production model is found to have a substantial effect on the soft photon spectrum, but cannot by itself explain the very large experimental yields. Examiners:
Dr. H. W. Fearing^ Co-Supervisor (Theory Group, TRIUMF)
Dr. C. E. Picciotto, Co-Supervisor (Physics Departmer t, University of Victoria)
Dr. IU K^Keel&c^ Departmental MemSer (Physics Department, University of Victoria)
Dr. C. R. MieWf^Outside Member (Mathematics Department, University of Victoria)
C ontents
A b s tr a c t... ii
Table of C o n te n ts ... iii
List of Figures ... vi
A cknow ledgem ents... ix
I T h e 7r° 7 7 F o r m F a c t o r 1 1 I n t r o d u c t io n 2 2 T h e C h ir a l L a g ra n g ia n 4 2.1 Parity even sector ... 4
2.2 Parity odd s e c t o r ... 10
2.2.1 The axial anomaly in chiral perturbation t h e o r y ... 10
2.2.2 The parity odd Lagrangian of 0 ( p 6) ... 13
3 R e n o r m a liz a tio n to O n e L o o p 19 3.1 Renormalization of £ 4 ... 20 3.2 Renormalization of £ $ ... 29 4 T h e jt°7 7 F o r m F a c to r 31 4.1 Tree l e v e l ... 31 4.2 One loop d ia g r a m s ... 36 4.3 R e n o rm a liz a tio n ... 44
4.4 Models of off-shell behaviour ... 47
5 P ro c e s s e s C o n ta in in g th e V e r te x 51 5.1 7T° -*• 7 7 ... 53
5.2 ir° —>7e+e ~ ... 54
C O N T E N T S iv 5.4 e+e~ —* ir°fi+fi~ ... 61 5.5 e~e~ —► i r ° e ~ e ~ ...63 5.6 e+e~ —* ir°e+e ~ ... 72 6 D is c u s sio n 80 I I V a lid ity o f S o ft P h o t o n A m p l i t u d e s 82 7 I n t r o d u c t i o n a n d B a c k g ro u n d 83 8 R e v ie w o f S o ft P h o to i. R e s u lts 8 6 8.1 Spin-0 derivations ... 86 8.1.1 Single expansion p o i n t ... 87
8.1.2 Many expansion points ... 95
8.2 The extension to higher spin ... 97
8.2.1 Non-radiative a m p litu d e ... 98
8.2.2 Soft photon a m p litu d e ... 99
9 T h e S oft P h o to n A m b ig u ity 104 9.1 A general f o r m a lis m ...104
9.2 Reduction to cases in the lite r a tu r e ...109
10 T h e P h a s e S p ace P r o b le m 114 11 P r o t o n - P r o t o n B r e m s s tr a h lu n g 122 11.1 Spin-0 d e r iv a tio n ... 123
11.2 Lim itation to elastic s c a t t e r i n g ... 126
11.3 Im pact of the phase space p r o b le m ...128
11.4 Sym m etrization p ro b le m ... 132
11.5 Extension to spin-^ s c a tte rin g ... 137
11.6 A pplication to pp b re m sstra h lu n g ...142
12 A S o lu tio n , fo r C o m m o n P ro c e s s e s 150 12.1 (s12,t„ ) ... 150
12.2 ,. t ) ... 155
C O N T EN T S v I I I S o ft P h o t o n E x c e s s in H a d r o n S c a t t e r i n g 15 9 14 I n tr o d u c tio n 160 15 R e v ie w 162 15.1 E x p e r im e n t...162 15.2 Q ualitative d is c u s s io n ... 165 15.3 T h e o ry ... 167 16 P o s s ib le S oft P h o to n S o u rc e s 169 16.1 Multiple photon e m i s s i o n ...169 16.1.1 Elastic channel—ir~p —... 172
16.1.2 M ulii-pion channels—A simple m o d e l ... 173
16.2 Decay of Hadronic Resonances ... 181
16.3 Higher order term s in the soft photon expansion ... 188
17 C o n c lu sio n s 191 B ib lio g ra p h y 193 A p p e n d ic e s A pp E la s tic P h a s e S h ift F o r m a lis m 204 A .l Helicity m atrix elements M ,, j M m>tTn in term s of phase s h ifts ...204
A.2 Invariant functions in terms of M , , / M m>tTn ... 208
B P h a s e S p a c e b y G a u s s - L e g e n d re Q u a d r a tu r e s 216 B .l T h e o ry ... 217 B.2 Im p lem en tatio n ... 218 B.3 U s a g e ... 219 C A n g u la r A v e ra g e o f P h o to n E m is sio n O p e r a to r s 224 D D e c o u p lin g a P r o c e s s a t a R e s o n a n c e L in e 229 E S o m e U se fu l I n te g r a ls 233
List o f Figures
4.1 The one-loop diagrams contributing at 0 ( p 6) to the 7T° 7 7 vertex... 36
4.2 Behaviour of our various calculational options with one photon taken off-shell in the space-like direction... 49
5.1 The regions of the vertex function A ( m l 0, k\) accessible to each of the considered processes... 52
5.2 Four-m om entum labelling for the process e+e_ —• tt° 7... 57
5.3 The cross section for e+e~ — 7r° 7... 59
5.4 An expansion of the low energy region of Fig. 5.3... . 60
5.5 Four-momentum labelling for the process e+e_ — ir°p+p ~ ...61
5.6 The cross section for e+e_ — x ° p +p ~ ... 64
5.7 The spectrum dcr/dM^ for the process e+e~ — ir°p+p ~ ... . 65
5.8 Identical to Fig. 5.7 but uses the VMD model of option (2) for the 7r°7‘7' form factor... 66
5.9 Identical to Fig. 5.7 but uses the one-loop result of option (3) for the ir°7*7' form factor... 6 7 5.10 Identical to Fig. 5.7 bu t uses the 0 ( p 6) fit to d ata of option (4) for the 7r°7*7* form facto r... 68
5.11 Four-m om entum labelling for the process e~e~ —> 7r°e"e_ ... 68
5.12 The differential quantity d2a/dcos d13 dcos 024 for the process e~e~ — 7r°e_ e_ as a function of the final state electrons’ scattering angles. The center of mass energy is 400 MeV... 72
5.13 Identical to Fig. 5.12 except th a t the VMD model of option (2) was used for the vertex factor... 73
5.14 Identical to Fig. 5.12 except th a t the one-loop result of option (3) was used for the vertex factor... 73
5.15 Identical to Fig. 5.12 except th a t the fitted 0 ( p a) result of option (4) was used for the vertex factor... 74
L IS T O F F IG U R E S
5.16 Four-momentum labelling for the process e+e_ ~ T°e+e~ ... 74 5.17 The differential quantity d2cr/dcos913 dcos 92i for the process e+e~ —
ir°e+e_ as a function of th e electron and positron scattering angles. The center of mass energy is 400 MeV... 77 5.18 Identical to Fig. 5.17 except th a t the vertex factor is defined by the VMD
model of option (2)... 78 5.19 Identical to Fig. 5.17 except th a t the vertex factor is defined by the one-
loop calculation, option (3)... 78 5.20 Identical to Fig. 5.17 except th at the vertex factor is defined by the fitted
0 ( p 6) result of option (4)... 79
8.1 Feynman diagrams representative of terms in the soft photon approxim ation. 88 8.2 Labelling of four-m om enta for processes involving b oth spin-j and spin-0
particles... 98 10.1 This chapter’s definitions for m om enta and Lorentz invariants of the ra
diative process... 116 10.2 Example showing the phase space problem for a soft photon am plitude. . 117 10.3 Diagrams (c) and (d) of Fig. 10.2...118 10.4 The phase space problem in proton-proton brem sstrahlung... 119 10.5 Region of phase space covered in the pjrj experiment of Ref. [1]...120 11.1 Trajectories of radiative variable pairs through non-radiative phase space. 129 11.2 Identical to Fig. 11.1 but with 03 = 35°, 9t = 35° and beam kinetic energy
of 157 MeV... 130 lx .3 Identical to Fig. 11.1 but with 93 = 12.4°, 04 = 12° and beam kiretic
energy of 280 MeV...131 11.4 Comparison of different pp phase shift sets, used to com pute brem sstrahlung. 144 11.5 The same as Fig. 11.4 but w ith beam kinetic energy 200 MeV and outgc .ng
proton angles at 16.4° on either side of the beam direction... 145 11.6 The same as Fig. 11.4 but with beam kinetic energy 280 MeV and outgoing
proton angles a t 10° on either side of the beam direction... 146 11.7 Trajectories of radiative variable pairs projected into the (kCM, c o s( 9 Cm ) )
plane... 147 11.8 Symmetry properties of the Low-(s, t) and the T u T t s am plitudes...149 16.1 M ultiple photon emission as a percentage of single photon emission. . . .1 7 2 16.2 Rapidity distributions in a simple model of m ulti-pion production... 174
LIST OF FIG U R E S viii
16.3 Transverse m om entum distributions in a simple m odel of m ulti-pion pro duction...175 16.4 D istribution of values for the function a A/ i r in our simple model, w ith
initial state particles assumed neu tral... 177 16.5 The function a A / ir , w ith initial state particles taken to be oppositely
charged... 179 16.6 F it to ./o-prong, inelastic ir~p cross section... 184 16.7 Upper bound on photon spectrum due to the A(1232) resonance in the
process ir~p —> i r ' j r f ... 186 16.8 As in Fig. 16.7 but for an interm ediate 7V'(1440) resonance... 186 16.9 As in Fig. 16.7 but for an interm ediate p(770) resonance... 187 B .l A program to calculate the muon decay rate and th e resulting electron
energy spectrum , w ritten using the C++ library P H A S E + + ... 220 B.2 A listing of the decay topologies supported by the package P H A S E+ + . . 221 B.3 A section of the output of the muon decay rate program shown in Fig. B .l .223 C .l The electro-magnetic factor A as a function of ir~p centre-of-mass scat
tering an g b for the process ir~p —► x~p~f, for several incident pion lab m om enta...226 C.2 The electro-magnetic factor A as a function of pp contre-of-mass scattering
angle for the process pp —*■ p p j , for several incident proton lab m om enta. . 227 C.3 The electro-magnetic factor A as a function of np centre-of-mass scat
tering angle for the process np —> njr/, for several incident proton lab m om enta...228 D .l Four-m om entum and am plitude labels for a general scattering process
A cknow ledgem ents
Thanks go to my supervisors, Dr. Harold W. Fearing (7earing@triumf.ca) and Dr. Charles Picciotto (pic@uvvm.uvic.ca). who have given their tim e and insights gener ously during this degree.
Almira Blazek (almira@ unixg.ubc.ca) deserves my gratitude for her tolerance of my moodiness, my failure to help sufficiently around the house, and my general neglect of her during the latter stages of this degree. I hope I can be as supportive during the final stages of her own Ph.D.
Thanks and praise go to
Richard Stallman (rms@ gnu.ai.mit.edu) for GNU-EMACS
Donald E. K nuth for T^X
Leslie Lam port (lamport@ gatekeeper.dec.com) for TAT^X
Bill Dimm (billd@ lnssunl.tn.cornell.edu) for FeynDiagram.
W ithout these program s the writing and editing phase of this degree would have been considerably more difficult, and I may have been forced 13 resort to the use of such non-optim al devices as typewriters, drafting tools, or perhaps even the TECO editor.
Part I
2
C hapter 1
Introd u ction
Q uantum chromo-dynamics (QCD) is the sector of the standard model of particle physics which is believed to describe the strong interaction. QCD is w ritten in term s of quark degrees of freedom and. due to a property called confinement, these quarks are not d'rectly observable in experiments. We instead observe hadrons such as pious and kaons which are thought to be strongly bound collections of quarks and their antiparticles and of the force carrying gauge bosons of QCD, gluons. Confinement arises because the effective coupling of quarks to gluons is very large for interactions w ith a small characteristic m om entum transfer squared and tends to zero as this m om entum transfer is increased. At small m om entum , where the coupling is large, a p erturbative treatm ent of QCD will fail to converge.
One may tackle this problem by using non-perturbative techniques like lattice field theory, or using models such as the Namfcu-Jona-Lasinio model [2] or QCD sum rules [3]. Still another approach is to distill the symmetries obeyed by QCD into another field theory which has as its dynamic fields the physical hadrons observed in the laboratory. One m ay then recover the powerful framework of perturbatio n theory by expanding this field theory simultaneously about the chiral lim it, at which the pseudo Scalar meson octet containing the pions and kaons becomes massless, and about the low energy lim it in a m om entum expansion. So long as the m om enta involved sue small enough this perturbative expansion will be convergent. Something is lost in the transition from QCD to this chiral pertu rb ation theory, since there is more dynam ical structure to QCD th an is contained in its symmetries. This loss of inform ation is param eterized in the chiral Lagrangian by low energy constants which act as weights for each of the Lagrangian’s terms. While a t the lower orders in the momentum expansion there are only a few such low energy constants, as one goes to higher orders their num bers increase rapidly—in principle, to all orders in the m om entum expansion, there are an infinite num ber of
Chapter 1 Introduction 3
them . Their values are not predicted within the context of chiral p ertu rb atio n theory (x P T ) but m ust be taken from a model or by fitting x P T calculations to experim ent.
The chiral Lagrangian which we shall consider here contains only the mesons of the lowest mass pseudosraiar octet as dynamic fields. In Chp. 2 we w rite down the chiral Lagrangian to 0 ( p 4) in the parity-even sector and C?(p6) in the parity-odd. W ith only parity conserving external fields present, such as the electrom agnetic field, the parity even and parity odd term s describe respectively the interactions of even and odd numbers of mesons at a vertex.
The renormalized low energy constants of 0 ( p 4) paiity-even and 0 ( p 9) parity-oc will enter during our calculation of the 7r° 7 7 ^ d r?87 7 vertex functions. In Chp. 3 we use the background field m ethod to construct an effective action containing the one-loop structure of the lowest order p art of the chiral Lagrangian. The divergent term s of this one-loop structure are then isolated using the results of the heat kernel m ethod and are shown to renormalize the 0 ( p 4) parity-even sector of the Lagrangian. We then briefly describe the m ethod of extending this work to the 0 ( p 6) parity-odd sector, which would form an interesting future project.
In Chp. 4 we com pute the vertex factors ir°7 7 and Tfe7 7 w ith sill the interacting particle;- off mass shell. Taking the virtual meson off shell introduces th° controversial issue of whether term s from the Lagrangian proportioned to the classical equation of m otion should be included in the cedculation. It was iecently noted [4, 5] for the 7 ^ 7 form factor th a t though such term s appear in the calculated form fa c t''’- they vanish when one uses this form factor to compute a measurable process such as Com pton scattering. We will consider the contribution of such term s though they are Irrelevant for our later calculations of physical processes, all of which have the neutral meson on-shell.
In Chp. 5 we com pute several decays and cross sections which involve the 7r° 7 7 vertex in order to constrain the values of the low energy constants upon which it depends. Most of these processes have either been measured or will be accessible when the high luminosity e+e“ collider D A $N E [6] comes on-line.
4
C hapter 2
T he Chiral Lagrangian
In this chapter we will define our notation for the chiral Lagrangian and, where appropri ate, relate our notation to th a t of other authors. We shall consider only the pseudoscalar meson octet as dynamic fields of the Lagrangian, and we use the conventional m om entum power counting scheme of Weinberg [7] to define the ordering of term s in the perturbative expansion.
We write the chiral Lagrangian in the form
C = L 2 + £4 + £4 -f- Cq + £g + . ..
where the subscripts denote the order in powers of m om entum of the contained term s. The terms w ith superscript A are parity odd or anomalous while the remaining term s are parity even. P arity odd or even terms, in the absence of parity-violating external fields, define th e interactions of odd or even numbers of mesons.
2.1
P arity even sector
The lowest order term of the parity even sector is
C2 = ^ - ( D llUD»U' + XV' + x 1u )
where
• ( ...) = T r ( ...), a trace over flavour degrees of freedom.
• U = exp (i \ a<t>a/ F ), where a sum m ation on index a is implied.
• X = 2B m , where B is a constant and m = d iag(m „,m <J,m ,) is a m a trix of quark masses.
2.1 P arity even sector 5
• To this order in x P T we have F = Fr = (92.4 d 0.3) MeV, the charged pion decay constant. Care m ust be taken with this definition since the charged pseudoscalar meson decay constants are often defined to contain an extra factor of \ /2—see, for example, Ref. [8, pp. 1443]. This would imply for the charged pion a value f , = y/2F'K = (130.7 ± 0.5) MeV.
• D jtU = dyJJ — ir^U + W l ^ , where and are right and left handed external fields.
• A„ are th e Gell-Mann matrices of flavour SU(3); we use the representation of Ref. [8, pp. 1288]. = f 7 5 ^ + * " )> r t * * U K * + K ~ )’ U K * - K r )' j . ( K ° + K °), - K c ), 1) , }. Hence, / 7T° + ^ Tf s y/2n+ V2K+ \ 4> = \ a(f>a = I y/2ir~ -7T° + \ / 2 K ° V V 2 K - y/2K° J
where 7r+, K ° . . . are the pseudoscalar meson fields. In particular, t/8 is the flavour-SU(3) octet p art of th e physical tj and r/1 mesons. They are related by
77 = t]6 c o s Op - Tfa sin Op 77' = T}& sin Op -|- r)i cos Op
where 771 is a flavour-SU(3) singlet, and 0P is a mixing angle. Combined fits to experiment give this mixing angle as Op ~ —14° [9] where SU(3) sym m etry breaking effects are included and Op ~ -2 0 ° [10] assuming SU(3) symmetry. We shall not consider the rfr singlet contribution and compute only the octet component of the meson-7 -7 vertices. This clearly restricts our ability to compare with experiment to any great degree of accuracy in th e case of the 17. We shall, however, be mainly concerned w ith th e 7r°7 7 vertex.
We also define field strength tensors for the left and right handed external fields. F ? = d ^ r d vl^
2.1 P arity even sector 6
While the external fields could potentially include contributions from, for instance, the W ± vector bosons of the weak interaction we shall require for our calculation only the external vector field of the photon. This gives
r lt = vli + all = - e A ^ Q
Vp ^ cApQ
where v a ^ are vector and axial components of the external fields, Q = d ia g (|, — — | ) is a m atrix of light quark charges, and e = y/4wa is the positron charge. The covariant derivative and the field strength tensors then have rath er simple forms.
DpU = dfJJ — ieAp [U,Q]
D ^ = d„Ux -* eA „ [U \ Q ]
F f = F g = —eQ {d*Av - d vA*) = - e Q F ^
W ith this notation defined we are in a position to write down the chiral counting rules which determ ine the order, in powers of m om entum , of the term s in the chiral Lagrangian. Following Weinberg [7] and Gasser and Leutwyler [11] we assign
V ~ OijP)
D^U, tp, — O(p)
K *
-B - 0 ( p )
m u, m d, m, 0( p) .
As they stand these rules would lead to two concurrent expansions: an expansion in powers of m om entum about the low energy lim it, and an expansion in quark masses about the chiral lim it at which m u = m d = m , - 0 and flavour SU(3) is am exact symmetry. We follow the conventional course of combining these expansions by imposing th a t the constant B and the mass m atrix m only appear together in the Lagrangian in the com bination \ = 2 5 m .
A recently proposed alternative framework is th a t of Generalized * P T (see [12] for a recent review) where m is allowed to appear independently of 5 in the Lagrangian. This leads to b oth even and odd powers of m om entum in the Lagrangian, gives a modified prediction for th e Gell-M ann-Okubo mass relation, and allows the value of the quark condensate in the chiral lim it, a quantity proportional to 5 , to become much smaller th an the value given by conventional * P T . The unmodified mass relation for the pseudoscalar mesons, which is replicated by conventional ^ P T ,
2.1 P arity ev en sector 7
appears, however, to be in good agreement w ith m easured masses. Also in apparent contradiction of Generalised x P T are the recent lattice calculations of Refs. [13, 14] which indicate th a t the quark condensate (0|gg|0) in the chiral lim it is close in value to the prediction of the conventional expansion of the chiral Lagrangian [15].
We shall now rewrite this lowest order p art of the chiral Lagrangian, £2, in term s of th e physical pseudoscalar meson masses. This will, to 0 ( p 2), give us th e relationship between the constants B m ^ , B m d, B m , and these physical masses, and will also allow us to reproduce the Gell-M ann-Okubo mass formula for the pseudoscalar meson octet. We set the external fields to zero, implying — d^, and expand U in term s of <f> = Xa<f>a.
£ 2 = + +
+2B m + ~ <f > 7 + . . . ) + 2 B m
= ~ ^ ( m ^ 2) + (constants) + 0((f>4)
Here, as in several later calculations, we have used the algebraic m anipulator M a th e -
M A T I C A [16] to m ultiply and take tracer of explicit representations of the m atrices <f>, m
and, when it is required, Q. The result is
£ 2 = ( d ,ti r + d f l Tr~ - (2R m ) i r +7r “ ) + i ( d Mi r ° d Mt t 0 - (22? m )7r ° 2)
+ { d ^ K + d ^ K ' - B ( m u + m , ) K +K ~ ) + - B ( m d + m , ) K ° K °)
+ \ ( d liT)fid liT)6 - | B (f n + 2 m , ) r / 82) + 7$ B { m d - m u )7r°7/8 + . . . ( 2 . 2 )
where m = \ { m u + m d). These are the kinetic and mass term s for three complex scalar fields and two real scalar fields. The term implies th a t the fields we have de^'m ated as 7T° and q8 are shifted from the physical fields. Ignoring this term for a m om ent we may pick off the meson masses in term s of the constants B m u, B m d and B m , .
m 2T± = 2 B m as 2 B m 2 m K± = B ( m u + m ,) 2 TTljfo = B ( m d + m , ) m l ~ \ B { m + 2m,)
The mixing m ay be removed by diagcnalizing £2 w ith a redefinition of the neutral scalar fields.
2.1 P arity eve.: sector 8
rjg = — tc° sin 9 + rj8 cos 9
W ith the appropriate choice for the mixing angle 9, the Lagrangian will be free of the term proportional to t°<78.
£a = | (d''7r°dM7r0 - m 2„7r°2) + i ( ~ ™1, Vb2) + • • • = | ( d ,lv 0dltv 0 - (m2„ cos2 9 + m 2t sin2 9)n°2^
+ | (d^'OedvVs ~ (mlo sin2 9 + m 2s cos2 9jr/g2) - cos 6 sin 9 ( m \ „ - m2 )w°t]8 + . . .
M atching term s with the expression of Eq. (2.2) we have
i sin 29 (m 2, - m ;„) = ^ B ( m d - m u )
mlo cos2 0 sin2 0 = 2Brh
m20 sin2 9 + m 2# cos2 9 — |5 ( m + 2m ,) (2-4) which may be solved for 9 by dividing the first equation above by the difference of the second and third.
ta n 2 9 = V S(2 i ^ ) 2 (m, — m)
This mixing angle is clearly very small; less than 2° if we take the isospin breaking scale to be around jmu - m d\ £ 10 MeV and flavour SU(3) breaking at m , — m ~ 150 MeV. Making a small angle expansion in Eq. (2.4), the first order corrections to th e 7r° and rj8 masses are
t( m d - mu )2 2 = 2B m -
iB'-4 (m , - m)
We consider this neutral meson mixing to be negligible in comparison w ith the error introduced in our calculations by computing to a low order in the m om entum expansion. To simplify our calculations we will therefore continue to write the Lagrangian in term s of the mixed fields tt° and tjs rath er than the diagonalized fields f ° and fj8.
By taking the appropriate com bination of the meson masses it is clear th a t we can now reproduce the Gell-M ann-Okubo mass relation for the pseudoscalar mesons.
2(m x± + mjfo) - 3m 2t - m2 ss
2(B(m „ -f m ,) + B ( m d + m ,)) - 3 \ B ( m + 2 m ,) - 2B m = o + o f n ^ - ^ n + o ^ )
2.1 P arity even sector 0
If we use the physical kaon and charged pion masses in this formula ( m K± = 493.7 MeV, m*o — 497.7 MeV and m T = 139.6 MeV) we find th e predicted value for th e octet t}b
mass to be m v, = 566.7 MeV as compared to the physical rj mass of m v = 547.5 MeV, a difference of less th a n 4%. Indeed this agreement seems fortuitously good since the relation does not take into account rj — rf mixing or, within this context of x P T , the truncation of th e chiral expansion a t lowest order.
We now consider the term s of the parity even sector which are of higher order in momentum. Each of the expressions
£
4, £ 6, . .
. may be constructed by writing dov. n all independent term s of the required order in m om entum , according to the chiral counting rules, which m ay be formed of the building blocks 17, D^U, F£"R and x- The full Lagrangian, and therefore each term of a particular order in m om entum , m ust be Lorentz covariant and invariant under p arity and charge conjugation since these are symmetries of the strong interaction. We quote the m ost general such expression for £ 4 from the papers of Gasser and Leutwyler—Refs. [11, 15] considered the case of two flavours of light quark (u and d), while Ref. [17] considered the case w ith which we are concerned involving three light flavours (u, d and s).£ 4
=
L^D pU 'D ^v ) 2 + L i ( D liU1Duu)(D>iU]D,'u)
+ L 3( D ltU i D>i U D l/W D ,,u ) + L i ( D r U ' D ltu ) { x ' U + x & )
+ L i ( p itV ' D » U {x' U + XU]) ) + L 6{ x ]U + x ^ 1) ’
+ I r ( x ftf - XU')* + Lb( x 1UX]U + x t f W )
- i U l F g D p U D v U ' + F ? D J J ' D VU ) + L l0( u ' F»v UFLliV)
+ H 1( F ^ F R, y + F ? F Llll) + i5r,<xtX> (2.5)
The factors L \ — L10 are the low energy constants of the 0(p*) parity even sector. Their values are not given by x P T and m ust be determined within the -context of a particular model of low energy hadronic interactions or by comparison w ith experim ent. These constants are a param eterization of those aspects of the strong interaction which are not determined solely by the sym m etry constraints placed on the chiral Lagrangian. In principle the low energy constants may be obtained from quantum chromodynamics (QCD), if we believe th a t theory to be the full and correct description of the strong interaction, using a non-perturbative m ethod of calculation such as lattice field theory. Unfortunately, lattice calculations are not currently of a sufficient level of sophistication to give predictions for these constants. All of the term s in Eq. (2.5) except those w ith coefficients L 3 and L 7 act as counter-term s during renorm alization of the divergent 0 ( p 4) loop integrals which arise from the £ 3 Lagrangian. The term s w ith coefficients
Hx
and2,2 Parity odd sector 10
H 2 are required as renorm alization counter-terms b u t have no contribution to physical processes involving incoming or outgoing mesons. We will conduct our discussion of renormalization in the following chapter.
While £ 2 has the constant F to be extracted from experim ent, and £ 4 has the ten
Li, it has been found by Fearing and Scherer in Eef. [18] th a t the 0 ( p 6) parity even sector £ 8 contains 111 independent term s, each with its own low energy constant to be fit to experiment or derived from a model. The authors of Ref. [18] in addition found terms which are proportional to the classical equation of m otion derived from £ 2. They argue th at those term s cannot contribute to physical processes since their effects m ay be removed by making unitary transform ations of the field U—such transform ations leave the S-matrix unchanged.
In our 0 ( p 6) calculation of the 7r°7*7’ form factor we will not require term s from the parity even expression £ 0 or from any higher order term s. We will therefore now leave the parity even sector and consider the form of the parity odd term s £4 and £ 0 .
2.2
Parity odd sector
2 .2 .1 T h e a x ia l a n o m a l y in c h ir a l p e r t u r b a t i o n t h e o r y
The chiral Lagrangian as described so far has all of the symmetries of QCD, bu t in addition has an unw anted symmetry not found in QCD. The Lagrangian of §2.1 is invariant under the parity operation x z, t — t. It is also invariant under U U*— this operation preserves the number of mesons in an interaction, modulo two. Thus only even numbers of mesons can interact at a vertex. This is not a property of QCD, which is invariant only under the combined transform ation
x «-» - x , t —>• t, U — t / t
and does allow the interaction of odd numbers of mesons a t a vertex. We therefore wish to introduce the m ost general term s to the chiral Lagrangian which break this unw anted symmetry. At lowest order in the m om entum expansion these ex tra term s arise from the axial anomaly.
The anomaly term s will account for the coupling of odd numbers of mesons to the electromagnetic external field. In addition they will give rise to purely hadronic vertices coupling odd numbers of mesons. In Ref. [19] Wess and Zumino derived a set of consistency relations to be satisfied by an action describing the axial anomaly. W itten [20] later developed an explicit representation of such an action. Interestingly, when this action is w ritten in term s of the exponentiated representation, U, of the meson field it
2.2 P arity odd sector 11
cannot be expressed in the usual fashion as a space-time integral over some Lagrangian density. W itten instead wrote the action as an integral over a five dimensional open surface labelled Q. This surface has as its boundary a four dimensional closed surface which is defined to be norm al four dimensional space-time.
S w z w = [ dsxe'jklmwijklm where i , . . . , m = 0,1,2,3 ,4
J
q
Wijklm S ~ 2{ ( U 1di U ) ( W d j U ) ( W d kU ) ( W d lU ) ( W d mU ) ) (2.6) Here e*’ klm is the totally antisym m etric object in five dimensions, w ith c01234 = +1, and so dsx.€ijklm is a n infinitesimal element of the five-surface Q. One can expand the field U to any desired order in the meson field <f> and m anipulate the resulting expression into a four dimensional integral over a Lagrangian density by w riting the integrand as a to tal derivative and applying a higher dimensional analog of Stoke’s theorem ,
= j x f x e ^ T ^ p (2.7)
with being an arb itrary tensor. Examples of such Stoke’s theorem extensions are given for other dimensionalities in Ref. [21, §6].
As an example we trea t the lowest order in such an expansion of the field U, which contains five occurances of the meson field.
w =
- ^(didj<f>)(dk<l>)(d,(l>){dm<f>)^
- ( W i M d u M d A M d ^ ) )
- ( H d M d k ^ i M K d i d ^ ) } ] + o(<t>7)
Due to the antisym m etry of the four term s with sym m etric derivatives like (didj<f>) vanish. We m ay apply Eq. (2.7) to find the result
S w z w = j f x C f a + O W ')
2.2 Parity odd sector 12
As it stands, this action gives only the coupling of the meson fields to one another. W itten used a trial-and-error m ethod to find the gauge invariant addition to T w z w which would give the photon couplings. His result, though gauge invariant, was flawed in th a t it failed to respect parity invariance, which is of course a sym m etry of the electrom agnetic and strong interactions we wish to model. This flaw was corrected by Kaym akcalan, Rajeev and Schechter [22] whose result we quote below.
S w z w = S w z w + N c f # z ( - e A » J * + ^ t r ^ i d p A J A a T , , ) ( 2 .8 )
where
J>1 s 4 ^ * ia'al,( Q ( W u ' ) ( a au u ' ) ( d pu u ' ) + Q ( W d vu ) ( u ' d a U ) { u ' d fiv ) ) Tp = ( Q 2{ dp UW) + Q 2{Wd pU) + a W Q U Q U \ d pU) + b Q W Q{ d p U )}.
W itten ’s original work had a = 1, b = 0 in the above expression, and while the authors of Ref. [22] found th at any such expression having a + b = 1 satisfies gauge invariance, they also found it necessary to impose a - b = 0 in order to make the action parity invariant. W ith the definitions L M = U^d^U and R* = d^UU^ we have our fined form for J M and Tp.
= ^ - 2^ a0( Q ( R . R aR^ + L vL aLp))
Tp = ( Q 2{Rp + Lp) + i Q U Q U ' R p + \ QT J' QVL p )
More recently several authors [23, 24, 25] have rederived the action of the axial anomaly coupled to external fields, avoiding the tried-and-error technique used by W it ten. They instead begin w ith the Beudeen form [26] of the anomalous axied current and from it derive the action. W hen the only gauge couplings are to vector fields, such as the photon field, each of their results reduces to the action given above.
In Ref. [27] Adkins, Nappi and W itten note th a t solitonic solutions of the non-linear SU(2)xSU(2) model, the Skyrme model, have precisely the quantum numbers of QCD baryons provided one includes the effects of the W ess-Zum ino-W itten term given in W itten ’s earlier papers [20, 28]. This is contradicted by the later claim of Bijnens et al. [29] th a t the W ZW term must enter the chiral Lagrangian with sign opposite to th a t given by W itten. No detailed argum ent is made in Ref. [29] in support of this claim, and so we take the relative sign of the W ZW term from W itten ’s original work. Though this sign does not enter any observables com puted using th e chiral action, it is im po rtan t th at we establish the convention since it determines how we define the sign of the low energy constants of the 0 ( p e) parity odd sector of the action.
2.2 P arity odd sector 13
In com puting the tree level and one-loop contributions to the tt° 7 7 and t?s7 7 vertices we shall not require the W ess-Zum ino-W itten action Sw z w itself, b ut only the meson- photon couplings given in Eq. (2.8).
2 .2 .2 T h e p a r i t y o d d L a g r a n g i a n o f 0 ( p 6)
We take the £ £ tree level contribution to the ir°7 7 and r?87T vertices from the 0 ( p 6) parity odd Lagrangian of Fearing and Scherer [18]. These authors use a very compact notation, designed to make the construction of the general form for the chiral Lagrangian a m ore straightforw ard task. Our goal in this section is to remove this no tational shell and to see which of the term s in the Lagrangian of Ref. [18] contribute to our processes of interest. At an interm ediate stage we shall also be in a position to relate the low energy constants of the Fearing and Scherer Lagrangian to those of other authors who have previously considered this p art of the 0 ( p 6) parity odd sector.
The additional notation used in Ref. [18] is = f^ U + U F ^ H nv _ f^ u _ UF
[A)± = \ ( A U ' ± U A ' ) (2.9)
where A is any operator. Since in our case we deal with only an electrom agnetic external field, we have
F£v = F£v = - e F ^ Q where = d^ A" - d v A» and hence
G*v = - e F ,lv { Q ,U } RV* = - e F ^ v [Q,U].
We are interested only in those term s of the £g Lagrangian which m ight contribute to the meson-photon-photon vertices. From the classification scheme of Ref. [18] we see th a t we need consider th e term s listed in their Table V.
cs
= ( l* W ] - ( [ |D 'x].,[G-®jt ] - [M t ,[D 'G ’ »]+]) ) + i ^ e , „ , { [ I W ] - { [ i > ,'G‘"]t ,[G ,fl+} )+ i A „ , y a f ( l D ‘ U)- ( { p x G ‘"]+, [G « ]+} - { [ 0 ,G “' ] +, [GJ'J + } ) )
2.2 Parity odd sector 14
The notation used here is extremely terse. The brackets ( ...) imply a trace over flavour indices, and are regular com m utators and anti-com m utators, while the [.. .]± are the operators defined in Eq. (2.9) above. Only th e braces ( . . . ) Me simple grouping symbols. During our calculation we shall ignore term s containing two or more meson fields or three or more photon fields. We Me interested in the £ £ couplings of the 7T° and T}8, and since term s from the pMt of the Lagrangian enter only at tree level in our calculation we may take U to be composed only of neutral mesons. This implies th a t [f7, Q] = 0. In pMticulM the covMiant derivative reduces to an ordinMy derivative in this circumstance; D^U — d^U. In order th a t we can unambiguously compare our notation to th a t of Refs. [30, 31] we do not yet use the relation [{7, Q] = 0 to further simplify the remaining terms. The building blocks of the expressions above Me
[x]±
=
B ( m U ' ± U m )[G«/>]+ = - e F all{ Q , u } u 1 [Hap)+
=
- e F af>[ Q , u ] w[■D'lx]+ = ~ 2 i e BA >ijjm, Q] = 0 since [m, Q] = 0 = 8^UU]
[D,iC a0]+ = - \ e F a0 [ [ Q , d ^ u } U ' + u { Q , d >,Ut }) . We now consider each te rm ’s contribution to the m eson-photon-photon vertex in turn. • A x :
A x t ^ i D ' U ] - ( [ [ i r x ] +,[ G ^ ] +] - [[X]+ ,[ D " G ^ ] +]) )
This term has, with the loss of the covMiant derivatives due to the neutral meson condition, only a single photon field. It does not, therefore, contribute to the vertices 7r° 7 7 or 77877.
• A 2:
; ^ w ( [ ^ ] - { P ^ " ] +,[g/ ] +}}
= ie2A i e,va0&'FXaFxl3( d * U U ' { { Q , u } u \ { Q , t/}* /1} )
=
2ie2A2eMt/a0d v F Xa F f (d^U [ U'Q2 + QU'Q)-
P V ' (QUQ+
UQ2))
We choose to write^ va0d vF XaF ^
= €„„a/J
{ - d xF ovF / - d aF vXF>? ) = ^ apdxF vaFxp - eliVa0d vF XaF ^ = W ^ xF vaF ^2.2 P arity odd sector 15
where in the first line we used the Bianchi identity for d xF lu' + d * F ' x + d ' F * 11 = 0
and in the second line we used the antisym m etry under index interchanges of F ^ and of e^ap. The A 2 term is finally
i e2A 2efl^ d xF vaFx^ d > ‘U ( U ' Q 2 + Q U' Q) - d ^ U ' (QUQ + U Q 2) ).
({[£>aG ^]+,[ G ^ ] + } - {[2?*G“'5]+,[G**']+}) )
= ie2A3^ vap (dxF XvF ap - dxF a0F Xv) ( d * U U ' { { Q , u } u \ { Q . l f } ^ } ) = 2ie2A 3€ltvap (dxF XvF a^ - dxF a? F Xv) x ( ^ C 7 ( U 'Q 2 + QU'Q) - 8 HJ' (QUQ + U Q2) ) • j4 4 : ^ 4w ([x] - [ G ^ ] + [ G ^ ] +) = ie2A i e^vaPF ^ F a^ B ( m W - U m ) {<?, v } W {q, u) W ) = ie2A±<itlvapF*>' F a^ ( B m (ZQ2(U' - U ) + U ' Q U Q U ' - U Q U ' Q U ) ) where we have used [m, Q] — 0 in the last step.
• A s:
^ w ( i x l + [[ff',1+ .[G 8' ] +] )
= ie2A ^ vapF>"'Faf,( B (mU' + Um) [|^Q,«7]Z7% { Q ,u}u' ] )
This is proportional to [Q , U] and so gives no contribution when applied to neutral mesons.
• A e :
^ e w ( [ x ] - ) ( [ G H +[G“*]+)
- ie2A 6‘M„a0F<"Fa(,( B m ( U ' - U ) ) ( {q,u}u' {q,u}u' )
2 4 P arity odd sector 16
There are also term s in the C f Lagrangian which are proportioned to the classical equa tion of m otion. It has been proposed in Ref. [18] th a t such term s can always be removed by a suitable unitary transform ation of the field U. The S-m atrix is invariant under such uniiary transform ations and so it seems th a t these equation of m otion term s have no physical content. For completeness, however, we shall compute the contribution of these term s to the ir° 7 7 and ^ 7 7 vertices. By exam ination of Table V m in Ref. [18] we find th a t there is only one equation of m o tfm term which we must consider. Defining
O 'iL = 2[ D , D UU}_ - 2[x]_ + § ([* ]_ )
this term is
«,^ ( &
r w
G
a']+)
= iei E „ e lu,aPF' u' F af>( ( [ d i UU' - Ud2U'){Q + U Q U ' ) 2) - 2B ( ( m W - Um)(Q + U Q U ' f ) + l B { m ( W ~ U ) ) (q{q, U ' } U ) ) .
For later reference we collect the term s of which contribute to the meson-photon- photon
vertex-C\$ = ie2A 2e ^ a/}9 xF ,' aF ^ { d >iU {U'Q2 + QU' Q) - d»U' (QUQ + U Q2) ) +2ie2A 3efil,a/} {dxF x,/F aP - d . F ^ F ^ )
( I FQ 2 + QU' Q) - W U 1 (QUQ + UQ2) ) +ie?A 4eliual}F li' 'Fal3( B m (3Q 2(U' - U) + U ' Q U Q U * - UQU 'Q U) ) + 2 ie2A 6e ^ afiF liVF a^ B m { U ' - U ) ) (q{q,u}u' )
+ie2E 29ellvaf}F f" ' F af)( ( ( d 2UU' - U d 2U'){Q + U QU f)2) - 2 B ( ( m l U - Um){Q + U Q U ' ) 2) + i B { m ( V ' - U ) ) (q{q, U ' }u) )
We -ire now in a position to compare our four term s with the parity odd, 0 { p 6) Lagrangians of other authors: Bijnens, Bram on and Cornet in Ref. [30], and Donoghue and Wyler in Ref. [31]. Neither of these groups considered term s proportional to the classical equation of m otion and so we set E29 = 0 for the comparisons. The Lagrangian used in Ref. [31] is
2.2 Parity odd sector 17
3C. Bm{U - U ' ) ) ( U Q U ' Q )
+
c < ( 2 B m{ U Q WQ U-
U ' Q U Q U ' j ) ) t*-*? {c6d xF ^ F aX{ Q 7U DpW - Q 7U ' D fiu )+ced xFllvFaX( Q U Q D !3U' - Q U ' Q D p l f ) +c7d xF ^ F a0( Q 2W D „ U - Q 7U D VU ' ) +ced xFXflFa0( Q U ' Q D vU - Q U Q D , . U ' ) ) .
The relationship between their low energy constants c< and our A 2, A 3, A+ and A e may then be extracted by inspection. Though the condition [17, Q) = 0 has been used to exclude term s from o ur Lagrangian it has not been used to m anipulate the form of the remaining term s. The following relations between these remaining term s are therefore quite general. -32tTr2F 2' / - 3 2 t i r 2F 2\ J ( C l + C 3 ) - ^ — J Ae / - 3 2 ,iw2F 7\ A
(o + *>
= ( — j
7
c— )-«<
f - 32ii r2F 2\ n i . (cs + c6) - ^ — j ( A 2 - 2A3j ( —64iir2F 2 \ , (cr + c3) -(
— N— j As N c ( 2 . 1 0 ) This would imply th a t the coefficients of the Donoghue and W yler Lagrangian can only appear in a m eson-photon-photon process in the linear combinations (ct + c3), (c2
- f c 4 ) ,(cs + c3) and (c7 + c8). The authors of Ref. [31] do indeed find this to be the case when they com pute the 0 ( p 6) contribution to 7r° — 7 7 and r/8 —*• 7 7. The orthogonal linear combinations of c{ coefficients contribute only to vertices w ith three or more mesons. In the Fearing and Scherer formalism employed by us the term s In' £g coupling three or more mesons to two photons have the coefficients A 7 — A 26. We find th a t, for instance, (ci - c3) ~ A 26 and (c2 - c4) ~ A 2S. The 0 ( p 6) parity odd Lagrangian presented in Ref. [31] for the ir° ■-> 7 7 and q8 —1• 7 7 vertices is therefore missing no term s in comparison w ith our work bu t does have four excess degrees of freedom. Also, it appears th a t the coefficients c, have been defined rath er oddly in th a t their numeric values would be purely imaginary.
We may repeat this comparison for the case of Ref. [30]. The relevant term s from their Lagrangian are
2.2 P arity odd sector 18
+a2( m { U Q U QU - U ' Q U Q W ) ) ) + i e ^ a0d xFXuFaP ( 3 l ( Q 2W d ^ U - Q ' U d J l ' )
+ B i ( Q U ' Q d ltU - Q U Q d ' U ' ) ) .
Here we have om itted several term s which, as the authors of Ref. [30] stated in a later publication [29], provide no contribution to the 7r°7 7 or r\977 vertices. We also om it term s which are specific to the T/177 vertex since we have not included the singlet eta in our treatm ent. The Lagrangian above also seems to be missing some term s proportioned to e.,ivaPd xF)iVFxa—these are the term s with coefficients c5 and c6 in Donoghue and W yler’s work and w ith coefficient ( A2 - 2A3) in our analysis—and in addition are missing a term of th e form e ^ aP F ^ F ap { Q U Q W ) { m { U —{/*)). For neutral mesons only composing U we have [U, Q] = 0 and so this last term reduces to the form of th a t in the Bijnens, Bram on and Cornet Lagrangian having coefficient a[. In making comparisons w ith this term of coefficient a[ we therefore assume th a t [U, Q] = 0. Below we relate as many coefficients as possible between our work and th a t of Ref. [30], given the apparent omissions in the form of their Lagrangian.
a4 = (-127raB )A 4 a\ = ( -167raB )A6 a2 = ( -47raJ9)A4
B x = ( 8 t t q ) A 3
B 2 ~ (8ira)A3
In the above B is the constant appearing in our definition x = 2 Bm . This is of less utility than the previous comparison w ith the work of Donoghue and Wyler [31]. A calculation m ade by the authors of Rex. [30] w ith particular values for their constants Bx and B 2 would have a different result th an the same calculation m ade w ith our Fearing-Scherer Lagrangian where we had set A3 = jBli2/8rra using the expressions above. Agreement would only be achieved if we were to ignore the term s of the form ~ A 3efil,a^ d >'Ffil,Fxa in our Lagrangian, since these term s have been om itted during the construction of the Lagrangian of Ref. [30].
Before proceeding w ith the calculation of particular processes we shall in the next chapter consider the one-loop renorm alization of the chiral Lagrangian.
19
C hapter 3
R en orm alization to O ne Loop
In chiral p erturbation theory the low energy constants a t each order in the m om entum expansion can pick up renorm alization contributions from a lim ited num ber of sources. For example the constants L \ . . . Li0, Hi, H 2 of £ 4 are required to absorb divergences only from the one-loop structure of £ 2. Similarly the thirty-tw o low energy constants of Cq absorb divergences from one-loop diagrams composed of a vertex from £4 and vertices from C2. The param eters of this effective theory can therefore be renormalized once and for all, with a finite am ount of effort. This is to be contrasted w ith theories such as quantum electrodynamics, referred to as renormalizable theories, which have a finite num ber of constants to be renormalized b ut which m ust absorb new divergent contributions in these constants a t each level of the perturbative expansion.
As has been shown previously [15, 17, 29, 31] it ic possible to perform the one- loop p art of this renorm alization by expanding the appropriate p art of the chiral action about a held configuration U which satisfies the classical equation of m otion. Taking the p ath integral over field configurations of this expanded action one may w rite down a form al expression for the action’s one-loop structure. The divergent term s in this one- loop effective action may then be extracted using the heat kernel m ethod, which will be described shortly. Using this technique we shall first com pute the one-loop structure of the lowest order p art of the chiral action, S 2 — f d*z £2. The divergent p a rts of this provide the renorm alization coefficients for the param eters L i . . . L 10, H lf H 2 of th e next order in p 2 in the parity even sector, S4 = f d4z £ A. In §3.2 we briefly describe how this procedure could be extended in a future project to calculate the renorm alization of the 0 ( p 6) parity odd sector.
3.1 R enorm alization o f £ 4 20
3.1
R enorm alization o f £ 4
The action we m ust consider is
5 2 = j f x C 7 = J * z * l ( D ' U D > U i + X W + U ' x )
-We expand the field U about a configuration U which we shall set to be a solution of the classical equation of m otion for S2. This expansion may be param eterized in m any equivalent ways. We choose to use U = Ue'* w ith A = A°A° H erm itian so th a t U and U m ay be kept unitary simultaneously. The A° are therefore a set of real fields.
The action S 2(U) is now expanded to second order in the fields A° about U. This will allow us to extract the first and second variations of 52( t/) —the first, variation will be set to zero, giving us the classical equation of m otion to this order in p 2, while the second variation after p ath integration over the field variable A a will give us the one-loop effective action for S2.
S 2(U) = J d * z C 2 =
j
d4z ^ { ^ ( & e ’A)D ''(e -‘At / 0 + Xt ( ^ ) + ( € - 'At / t )x ) Expanding to second order in A we haveD ^ U e * )
= d ^ U e ' * ) - i r j j e * +
= ( d f i + i U d ^ A - i r j j + * W % c - ’A) e iA
= ( d j j + iU d^ A - i r j j + iUt„ + # [/„ , A] - i f f [[/„, A], A ] ) e i& + 0 ( A S) and the adjoint of this operator is
D ^ e - ^ W )
= - id*A t/ 1 + i f f V - i l * # ' - [tM, A } V ' + £[[/„, A], A ]fff) + 0 ( A 3) while for the m ass term s we have
( x ' ^ + c - ' ^ ’ x ) = ( x ' U + U ^ + i ^ x ' U - U ' x )
— i A ^ x ^ + ^ x ) + 0 ( A 3). The action is then
s 2(U) = / ^ ( ( D f j r V + x W + V x )
+ ( - 2if f t £ , if f lP A + i A { y } U - U ' X))
3.1 R enorm alization o f 2 1
where we have defined a derivative operator for A Z?mA = <9mA - i[fM, A] and have used the relation
D ^ U ' U = D »( fPU ) - U ' D ^ U = - U ' D » U since U W = 1.
We m atch this expression against variations of the action 52(17) w ith respect to the fields A° about the so-called background field U.
S , m = S, (U ) + ^ j ^ ( V ) + i A - j ^ i (U)A'’ + 0 ( A 3)
i A ‘ < ^ » ( e ) A * = / <r‘iC^ “ {jD'‘AZ>‘‘A + i ^ [ l < " . A ) .A ] - i A ’ ( x 't / + P Tx)} Here we have defined = W D ^ U . Setting the first variation to zero will give us th< equation of m otion satisfied by U ,
* aj £ - a(U) =
J
* * = £ - ( - D ^ A - \ A ( XW - W X)) = 0after integration by p arts of the first term . We have used the fact th a t the covariant derivatives have suitable definitions th a t they may be m anipulated like ordinary deriva tives, and also th a t the volume integral over all space of a to tal ordinary derivative is zero. In particular we used
A) = A + L p D ^ A
which may be checked explicitly using the definitions of the covariant derivatives of U and A. Setting the first variation of the action to zero gives us
( \ a ( D ^ L li + \ { X' U - t / tx ) ) ) = 0. In m atrix form this is
D H U ' D f l ) + Ux'U - U'x) - \ ( X'U - W X) = 0
where the trace term arises due to the m inim ization of the action under the constraint det(t7) = 1, since U is an SU(3) m atrix. We take the form of this ex tra term directly from Ref. [17].
3.1 R enorm alization o f £4 22
We now define an effective action Z which is equal, up to 0 { A 2), to the action Si (U ) after it has been p ath integrated over all field configurations A “.
j [ d ^ ] e iS’W = j [rfA.j exp (,•$ ,(# ) + i A aj ~ ^ ( U ) A b + 0 ( A 3))
We take the result of this integration over a Gaussian function directly from Ref. [32, pp. 75]. The second variation is w ritten in the form
A ° - A^ - 6A i> = C / d * x Aa{d!ld„ + m2 + <r)akA b (3.1) where C, m2 are constants, the operator d^ is defined as d^ = + T“b, and a ab, r “b contain no derivatives. The result of the p ath integral is then
/ [ I A o]e«s3(tf) _ ^ e x p ^ iS^iU) ~ 2
J
d*x ( ( x \ log(d^d^ + m2 + < r)|z )^ .Here a, b are SU(7V) indices and is a divergent constant resulting from the p ath integration. The |z) are some complete set of coordinate-space states. We shall find it convenient to work in N flavours throughout our calculation since the various powers of N which arise provide useful tags with which to group the m any term s th a t appear and therefore help to keep the algebra under control—a t the end we shall reduce our result to N = 3.
The effective action is defined by
Z = S i i U ) + z loop
Zloop = \ j di x((x\\o%(d>ldll + m2 + o')!1 )) (3.2) where Zloop contains the full one-loop structure of our original action S 2(U). This one- loop effective action is of 0(p*) and contains divergences which m ust be absorbed w ithin the low energy constants of the 0 { p A) p art of the chiral action
S< =
J
d*x£4(U-,L1. . . L 10, H 1, H 2). To 0 ( p A) the full effective action is thereforeS
2(U) +
+ z};;p) + s 4(t/; l x .. . j 10, h u k 2)
which we wish to write as
3.1 R enorm alization o f £ 4 23
Clearly we have to define renormalized constants L
\ ...
L \ 0, HI, such th a t the follow ing is satisfied,S<(U
;
L\ . .
L \ 0, H[ , HI)= S4(U;
L, ...
L 10, H u H 2)+ Z £ p.
The divergent structure Z A0^p may be extracted from .&DOp of Eq. (3.2) using a technique called the heat kernel m ethod. The renorm alization of the 0 ( p 4) parity-even sector’s low energy constants m ay then be performed by comparing the functional forms in ZtdJ'p w ith those in S4(U). We shall quote the necessary results of the heat kernel approach directly from Appendix B of Ref. [33] and from Ref. [34].
Using the dimensional m ethod to regulate divergences, the one-loop effective action may be w ritten
- i / m 2 \ ~ * °°
(x\\ogV\x) =
£ a n(m2)2- ’T ( n - 2 + e)Here the space-time dimension is defined to be 2(2 - e). The divergences which occur in this expression in the lim it f - * 0 are contained wi bin the T-function, and T(z) is singular for z = 0 , - 1 , - 2 , — It can be seen, therefore chat divergences arise only in the terms n =0, 1 and 2. We quote the heat kernel coefficients a0, a : and a2 from Ref. [34],
do — 1 0*1 — —CT
W H y + s v i
where we have defined
• p a 6 _ _ q -p afc Q *po& i - p o c T x : ' p o c - p c l * fiu — 1/ C/j/I ^ ~r L p L v L p 1 ^ • The divergent p art of the 0(p*) one-loop effective action is then
z?::P
= i /<?*{(*
ii° g ^ i* ))Given this equation our p i.x ed u re is straightforward—we must w rite the second variation of our expanded action in the form of Eq. (3.1), pick off the structures T*b and <rab, tn d then insert them into the equation above for Z d”p. The renorm alization of S*(U; Li. . . L l0, Hi, H 2) m ay then be done by inspection.
3.1 R enorm alization o f £ 4 24
The second variation of the action is
*A° « i ^ A' = f
- LA’(x'V + U<x))
=
J
d x — ^D^AD^A + L ^ A D ^ A - D * A A ) - §A 2{X' U + U ' x ) ) where we have used the relationA D ^ A - D ^ A A = A d * A - iA [f, A] - d * A A + A]A
=
* [ ^ 4
We complete the square on D* A by defining
D * A = D^A + \[L,A] = &* - * [ f A ] where f ** = I* + \ L ^ .
= f
' ^ ( x ' v + u 'x ))
W ith a small am ount of further m anipulation we have the form of the differential operator appearing in the p ath integral of Eq. (3.1). We must compare term s in the expression
j d*x((d» - i[ f ", A])(3m - i[ f „ A]) - i[ j " , A }[L,,A] - IA 2(X 'U + U ' X )) =
J
d x C A a(($“ 0M + r “e)(«tbd" + r cb") + crob) A b.The overall constant C is determined by comparison of th. ordinary derivatives,
J
d x ^ A d ^ A ) =J
d x i d ^ i A d ^ A ) -A d ^ A )
= - ( A aAb)
J
d*xAad >idllA b=
( - 2 ) 6 abJ
d4x A ad,id>iA band so C — - 2 . We may now identify by inspection th a t suitable definitions for r ab and <rah are
r;* s }{[A”,A‘]f„ )
» “ s | ( [ i “, . \ “][i„ A * ] + {A“,A *}(x, £' + P ^ ) ) -The omy term where it is not obvious th a t these are the correct choices is
3.1 R enorm alization o f £ 4 25
We use the SU( N ) completeness relation for N flavours
E < * • ) # < * > = (3.4)
0=1 ' IV J
to w rite this as
pocpctM _ _ i ^ A«Ae fM _ AcA°fe ^A*A*f„ - A*AT„ )
= — j ^ A aAfcf Mf M - A“f <4A6F ‘ - Aaf /'A6f M + A-f ' ‘f„A‘)
which is exactly the term we require.
W ith the identification of <rab and T“b we are in a position to com pute th e necessary heat kernel coefficients an and hence the one-loop divergent structure Z £ l . At this point we note th a t we have defined T®6 and <r“6 such th a t the m ass-type term m2 in our derivative operator V ab
=
+ m2 + <r)abis zero. The expression we require therefore simplifies to* £ . =
■
3
^
/ ^
>
“ ( ~ 7 - r '<1) + 0 ( t ) ) / ' ' * * ( “ ■)■ We in fact choose to redefine Z ^ p as
Z t cp = - D { e )
J
where D{e) = j - 1 + 7* - log4ir^ and Euler’s constant is 7# = - r ' ( l ) « 0.577. We have introduced some finite constants into the definition of Z*”p in order th a t we be consistent with the MS-scheme dimensional regularization procedure to be used in our later p erturbation theory calculations. The way we choose to split Zloop into its finite and divergent p a rts Z f * £ and Z f ”f is not unique, and we are free to move finite constants in this fashion between the two parts.It now rem ains to com pute in term s of r®k and r ab. We recall th a t ( a , ) = ( f a ’ + i r v r " )
3.1 R enorm alization o f £ 4 26
where
•p a 6 Q 'p a fc Q - n a 6 1 r p a c T 'cfcl X fit/ C/^X v Ui/L p I i“ j* J ^ y J
r ? = i([A*,A4]f„)
<r“4 = |([X^,A°][XM,A4] + {Aa,A 4}M )
M
= ( X ^ + ^ X )-The first term in is
( ^ 2} = l E ^ b“ a,fc
= ifc E **]&•> ^b] + ^ °> A6}M)([X% A “][£„, A4] + {A°, a4}m )
a,6
= jig 2 (2i-A°X,A4 - {Aa, A4} ( i ^ -
M )) (2 L ''\ai „ \ b
- {A°, A4}(X‘/XV - M )).a,fc
Using the SU(7V) completeness relation of Eq. (3.4) to sum over the indices a and b we find, after some algebra,
8 \ / \ ^ / 16 \ " / 16
The remaining term in (^a3\ is of the form
( n r »»r " ') -Recalling th at
r ; ‘ = *([A-,A4] f „ ) we can write the field strength T“4 as
K = i([A “,A4] ^ f , ) - i ( [ A tt,A4] ^ f , )
- l ( ( [ A a,Ac] f p )([A%A4] f v) - ^[A°, Ac] f v)([A c, A4] f M) ) = f([A a,A4] f MV)
where in the final line we have used Eq. (3.4) to sum on index c, and where we have defined a field strength for f M,
3.1 R enorm alization o f £ 4 27
The te rm in is of the form
a,S
= &£([Aa.A
a,b*]M([A
°’A
‘lf'"'>
= + i ( f
where in the last step we have once more used the SU(JV) completeness relation to sum on a and b.
We m ust now com pute T ^ in term s of the objects D j J F r et cetera which appear in the 0(p*) parity even action. Recall th a t F£v and F g ' are defined in Eq. (2.1) as field strengths for the left-handed and right-handed external fields I ** and t . We expand the function T*1 to give
f t * _ Qvfv _ Q*fM _ ipf1'*, r*']
= i ( 2d ^ & ' V - i U ' r vU - i t ) - 2dv{Uxd ,1V - i U ' t U - i t )
+ [ W D ^ U - 2i t , U ' D VU - 2i t ] ) , (3.5)
employ the result
d ^ U ' F U ) ~ d ' i U ' d ' U )
= - ff ' f j ' & ' U + I Vd » dvU - U i d vd ,1U - - U ^ U W f f ' U + U ' f f ' U U ' d llU
= - \ W d » U , U ' d vu] (3.6)
and the relation between the com mutators of p artial and covariant derivatives ] p ' D * V i & D vV\ = + [ - i U ' t U + i t , U1d uu] + [ W d ^ U , - i U 1r vU + i t ] + [ - i U ' t U + i t , - i U ' r vU + i t ] (3.7) to w rite t tiV as f '" ' = U ' D vu] - 2i d * r + 2i d pt - 2 [ t , t ] - U U ' d ^ U + 2 i u ' d vt u - 2U 1 [r* \ r 1'] u ) .
There are a great many term s which arise upon the substitution of Eqs. (3.5,3.6) into Eq. (3.7), and subsequently cam el amongst themselves. Finally then we have
