5 Supersymmetric actions: minimal supersymmetry

Transcription

1 5 Supersymmetrc actons: mnmal supersymmetry In the prevous lecture we have ntroduced the basc superfelds one needs to construct N = supersymmetrc theores, f one s not nterested n descrbng gravtatonal nteractons. We are now ready to look for supersymmetrc actons descrbng the dynamcs of these superfelds. We wll frst concentrate on matter actons and construct the most general supersymmetrc acton descrbng the nteracton of a set of chral superfelds. Then we wll ntroduce SuperYang-Mlls theory whch s nothng but the supersymmetrc verson of Yang-Mlls. Fnally, we wll couple the two sectors wth the fnal goal of dervng the most general N =supersymmetrc acton descrbng the nteracton of radaton wth matter. In all these cases, we wll consder both renormalzable as well as non-renormalzable theores, the latter beng relevant to descrbe e ectve low energy theores. Note: n what follows we wll deal wth gauge theores, and hence gauge groups, lke U(N) andalke. Inordertoavodconfuson,ntherestoftheselectureswewll use callgraphc N when referrng to the number of supersymmetry, N =, or4. 5. N = Matter actons Followng the general strategy outlned n 4.3 we want to construct a supersymmetrc nvarant acton descrbng the nteracton of a (set of) chral superfeld(s). Let us frst notce that a product of chral superfelds s stll a chral superfeld and a product of ant-chral superfelds s an ant-chral superfeld. Conversely, the product of a chral superfeld wth ts hermtan conjugate (whch s ant-chral) s a (very specal, n fact) real superfeld. Let us start analyzng the theory of a sngle chral superfeld. Consder the followng ntegral d d. (5.) Ths ntegral satsfes all necessary condtons to be a supersymmetrc Lagrangan. Frst, t s supersymmetrc nvarant (up to total space-tme dervatve) snce t s the ntegral n superspace of a superfeld. Second, t s real and a scalar object. Indeed, the frst component of s whch s real and a scalar. Now, the component of a superfeld, whch s the only term contrbutng to the above ntegral, has the same tensoral structure as ts frst component snce does not have any free space-tme ndces and s real, that s ( ) =. Fnally, the above ntegral has

2 also the rght physcal dmensons for beng a Lagrangan,.e. [M] 4. Indeed, from the expanson of a chral superfeld, one can see that and have both dmenson [M] / (compare the frst two components of a chral superfeld, (x) and (x), and recall that a spnor n four dmensons has physcal dmenson [M] 3/ ). Ths means that the component of a superfeld Y has dmenson [Y ]+ f [Y ] s the dmenson of the superfeld (whch s that of ts frst component). Snce the dmenson of s, t follows that ts component has dmenson 4 (notce that R d d s dmensonless snce d (d )hasoppostedmensonswthrespectto ( ), gven that the d erental s equvalent to a dervatve, for Grassman varables). Summarzng, eq. (5.) s an object of dmenson 4, s real, and transforms as a total space-tme dervatve under SuperPoncaré transformatons. To perform the ntegraton n superspace one can start from the expresson of and nthey (resp. ȳ) coordnatesystem,taketheproductof (ȳ, ) (y, ), expand the result n the (x,, ) space, and fnally pck up the component, only. The computaton s left to the reader. The end result s L kn = d d µ µ µ + FF+totalder. (5.) What we get s precsely the knetc term descrbng the degrees of freedom of a free chral superfeld! In dong so we also see that, as antcpated, the F feld s an auxlary feld, namely a non-propagatng degrees of freedom. Integratng t out (whch s trval n ths case snce ts equaton of moton s smply F =0)onegetsa (supersymmetrc) Lagrangan descrbng physcal degrees of freedom, only. Notce that after ntegratng F out, supersymmetry s realzed on-shell, only, as t can be easly checked. The equatons of moton for, and F followng from the Lagrangan (5.) can be easly derved usng superfeld formalsm readly from the expresson n superspace. Ths mght not look obvous at a frst sght snce varyng the acton (5.) wth respect to we would get = 0, whch does not provde the equatons of moton we would expect, as t can be easly nferred expandng t n components. The pont s that the ntegral n eq. (5.) s a constraned one, snce s a chral superfeld and hence subject to the constrant D =0. Onecanrewrtetheabove ntegral as an unconstraned one notcng that d d = d D. (5.3) 4 In gettng the rght hand sde we have used the fact that R d = D,uptototal space-tme dervatve, and that s a chral superfeld (hence D = 0). Now,

3 varyng wth respect to weget D =0, (5.4) whch, upon expanson n (x,, ), does correspond to the equatons of moton for, and F one would obtan from the Lagrangan (5.). The check s left to the reader. Anaturalquestonarses. Canwedomore? Canwehaveamoregeneral Lagrangan than just the one above? Let us try to consder a more generc functon of and, call t K(, ), and consder the ntegral d d K(, ). (5.5) Agan, n order for the ntegral (5.5) to be a promsng object to descrbe a supersymmetrc Lagrangan, the functon K should satsfy a number of propertes. Frst, t should be a superfeld. Ths ensures supersymmetrc nvarance. Second, t should be a real and scalar functon. As before, ths s needed snce a Lagrangan should have these propertes and the component of K, whchstheonlyone contrbutng to the above ntegral, s a real scalar object, f so s the superfeld K. Thrd, [K] =,sncethents component wll have dmenson 4, as a Lagrangan should have. Fnally, K should be a functon of and butnotofd and D. The reason s that, as t can be easly checked, covarant dervatves would provde -term contrbutons gvng a hgher dervatve theory (thrd order and hgher), whch cannot be accepted for a local feld theory. It s not d cult to get convnced that the most general expresson for K whch s compatble wth all these propertes s K(, ) = X m,n= c mn m n where c mn = c nm. (5.6) where the realty condton on K s ensured by the relaton between c mn and c nm. All coe cents c mn wth ether m or n greater than one have negatve mass dmenson, whle c s dmensonless. Ths means that, n general, a contrbuton as that n eq. (5.5) wll descrbe a supersymmetrc but non-renormalzable theory, typcally defned below some cut-o scale. Indeed, genercally, the coe of the form cents c mn wll be c mn (m+n) (5.7) wth the constant of proportonalty beng a pure number. The functon K s called Kähler potental. Thereasonforsuchfancynamewllbecomeclearlater(see 5..). 3

4 If renormalzablty s an ssue, the lowest component of K should not contan operators of dmensonalty bgger than, gven that the component has dmensonalty [K] +. In ths case all c mn but c should vansh and the Kähler potental would just be equal to, the object we already consdered before and whch leads to the renormalzable (free) Lagrangan (5.). In passng, notce that the combnaton + respectsallthephyscalrequrements dscussed above. However, a term lke that would not gve any contrbuton snce ts component s a total dervatve. Ths means that two Kähler potentals K and K 0 related as K(, ) 0 = K(, ) + ( ) + ( ), (5.8) where s a chral superfeld (obtaned out of ) and sthecorrespondngantchral superfeld (obtaned out of ), are d erent, but ther ntegrals n full superspace, whch s all what matters for us, are the same d 4 xd d K(, )0 = d 4 xd d K(, ). (5.9) Ths s the reason why we dd not consder m, n =0ntheexpanson(5.6). Thus far, we have not been able to descrbe any renormalzable nteracton, lke non-dervatve scalar nteractons and Yukawa nteractons, whch should certanly be there n a supersymmetrc theory. How to descrbe them? As we have just seen, the smplest possble ntegral n superspace full-fllng the mnmal and necessary physcal requrements,, already gves two-dervatve contrbutons, see eq. (5.). Not to menton the more general expresson (5.6). What can we do, then? When dealng wth chral superfelds, there s yet another possblty to construct supersymmetrc nvarant superspace ntegrals. superfeld (whch can be obtaned from products of t n full superspace would gve Let us consder a generc chral s, n our case). Integratng d 4 xd d =0, (5.0) snce ts component s a total dervatve. Consder nstead ntegratng n half superspace d 4 xd. (5.) D erently from the prevous one, ths ntegral does not vansh, snce now t s the component whch contrbutes, and ths s not a total dervatve for a chral superfeld. 4

5 Note that whle = (y, ), n computng (5.) one can safely take y µ = x µ snce the terms one s mssng would just provde total space-tme dervatves, whch do not contrbute to R d 4 x. Another way to see t, s to notce that n (x µ,, )coordnate, the chral superfeld reads (x,, ) =exp( ) (x, ). Besdes beng nonvanshng, (5.) s also supersymmetrc nvarant, snce the component of a chral superfeld transforms as a total dervatve under supersymmetry transformatons, as can be seen from eq. (4.59)! An ntegral lke (5.) s more general than an ntegral lke (5.5). The reason s the followng. Any ntegral n full superspace can be re-wrtten as an ntegral n half superspace. Indeed, for any superfeld Y d 4 xd d Y = 4 d 4 xd D Y, (5.) (n passng, let us notce that for any arbtrary Y, D Y s manfestly chral, snce D 3 =0dentcally). Thssbecausewhengongfromd to D the d erence s just a total space-tme dervatve, whch does not contrbute to the above ntegral. On the other hand, the converse s not true n general. Consder a term lke d 4 xd n, (5.3) where s a chral superfeld. Ths ntegral cannot be converted nto an ntegral n full superspace, essentally because there are no covarant dervatves to play wth. Integrals lke (5.3), whch cannot be converted nto ntegral n full superspace, are called F-terms. All others, lke (5.), are called D-terms. Comng back to our problem, t s clear that snce the smplest non-vanshng ntegral n full superspace, eq. (5.), already contans feld dervatves, we must turn to F-terms. Frst notce that any holomorphc functon of W ( ) such =0,sachralsuperfeld,fsos D W ( D, namely a functon. D =0. (5.4) The proposed term for descrbng nteractons n a theory of a chral superfeld s L nt = d W( ) + d W ( ), (5.5) where the hermtan conjugate has been added to make the whole thng real. The functon W s called superpotental. Whch propertes should the (otherwse arbtrary) functon W satsfy? Frst, as already notced, W should be a holomorphc 5

6 functon of. Ths ensures t to be a chral superfeld and the ntegral (5.5) to be supersymmetrc nvarant. Second, W should not contan covarant dervatves snce D snotachralsuperfeld,gventhatd and D do not (ant)commute. Fnally, [W ]=3,tomaketheexpresson(5.5)havedmenson4. Theupshots that the superpotental should have an expresson lke X n W ( ) = n= a n (5.6) If renormalzablty s an ssue, the lowest component of W should not contan operators of dmensonalty bgger than 3, gven that the component has dmensonalty [W ]+. Snce has dmenson one, t follows that to avod non-renormalzable operators the hghest power n the expanson (5.6) should be n =3,sothatthe component wll have operators of dmenson 4, at most. In other words, a renormalzable superpotental should be at most cubc. The superpotental s also constraned by R-symmetry. Gven a chral superfeld, f the R-charge of ts lowest component s r, thenthatof s r andthat of F s r. Ths follows from the commutaton relatons (.7) and the varatons (4.59). Therefore, we have R[ ] =, R[ ] =, R[d ] =, R[d ] = (5.7) (recall that d andsmlarlyfor ). In theores where R-symmetry s a (classcal) symmetry, t follows that the superpotental should have R-charge equal to R[W ]=, (5.8) n order for the Lagrangan (5.5) to have R-charge 0 and hence be R-symmetry nvarant. As far as the Kähler potental s concerned, frst notce that the ntegral measure n full superspace has R-charge 0, because of eqs. (5.7). Ths mples that for theores wth a R-symmetry, the Kähler potental should also have R-charge 0. Ths s trvally the case for a canoncal Kähler potental, snce has R- charge 0. If one allows for non-canoncal Kähler potental, that s non-renormalzable nteractons, then besdes the realty condton, one should also mpose that c nm =0 whenever n 6= m, seeeq.(5.6). The ntegraton n superspace of the Lagrangan (5.5) s easly done recallng the expanson of the superpotental n powers of. Wehave W ( ) = W ( F W 6, (5.9)

7 are evaluated at =. So we have, modulo total space-tme dervatves L nt = d W( ) + W ( ) @ + h.c., (5.0) where, agan, the r.h.s. s already evaluated at x µ. We can now assemble all what we have found. The most generc supersymmetrc matter Lagrangan has the followng form L matter = d d K(, ) + d W( ) + For renormalzable theores the Kähler potental s just at most cubc. In ths case we get L = d d + d W( ) + where d W ( ). (5.) and the superpotental d W ( ) (5.) µ µ + @ W ( ) = + h.c., 3X a n n. (5.3) We can now ntegrate the auxlary felds F and F out by substtutng n the Lagrangan ther equatons of moton whch read Dong so, we get the on-shell Lagrangan, (5.4) L on shell µ µ µ From the on-shell Lagrangan we can now read the scalar potental whch W (5.5) V (, = FF, (5.6) where the last equalty holds on-shell, namely upon use of eqs. (5.4). All what we sad so far can be easly generalzed to a set of chral superfelds where =,,...,n. In ths case the most general Lagrangan reads L = d d K(, )+ d W( )+ d W ( ). (5.7) 7

8 For renormalzable theores we have K(, ) = and W ( )=a + m j j + 3 g jk j k, (5.8) where summaton over dummy ndces s understood (notce that a quadratc Kähler potental can always be brought to such dagonal form by means of a GL(n, C) transformaton on the most general term Kj j,wherekj s a constant hermtan matrx). In ths case the scalar potental reads V (, ) = F F, (5.9) where 5.. Non-lnear sgma model I = (5.30) The possblty to deal wth non-renormalzable supersymmetrc feld theores we alluded to prevously, s not just academc. In fact, one often has to deal wth e ectve feld theores at low energy. The Standard Model tself, though renormalzable, s best thought of as an e ectve feld theory, vald up to a scale of order the TeV scale. Not to menton other e ectve feld theores whch are relevant beyond the realm of partcle physcs. In ths secton we would lke to say somethng more about the Lagrangan (5.7), allowng for the most general Kähler potental and superpotental, and showng that what one ends-up wth s nothng but a supersymmetrc non-lnear -model. Though a bt heavy notaton-wse, the e ort we are gong to do here wll be very nstructve as t wll show the deep relaton between supersymmetry and geometry. Snce we do not care about renormalzablty here, the superpotental s no more restrcted to be cubc and the Kähler potental s no more restrcted to be quadratc (though t must stll be real and wth no covarant dervatves actng on the chral superfelds K = W = ). For later purposes t s convenent to defne the K(, ), K ), K j K(, W ( ), W = W, W @ j W ( ), Wj = W j, where n the above fromulæ both the Kähler potental and the superpotental are meant as ther restrcton to the scalar component of the chral superfelds, whle 8

9 stands for the full n-dmensonal vector made out of the n scalar felds (smlarly for ). Extractng the F-term contrbuton n terms of the above quanttes s pretty smple. The superpotental can be wrtten as W ( ) = W ( )+W + W j j, (5.3) where we have defned and we get for the F-term d W( )+ d W ( ) = (y) = (y) = p (y) F (y), (5.3) W F W j j + h.c., (5.33) where, see the comment after eq. (5.), all quanttes on the r.h.s. are evaluated n x µ. Extractng the D-term contrbuton s more trcky (but much more nstructve). Let us frst defne whch read (x) = (x), (x) = j (x) = p j (x)+ j (x) F j (x) (x) (5.34) µ j (x) µ j(x) = p j(x) j(x) Fj (x)+ p µ j(x) 4 j (x) 4 j(x). Note that j k = j k = 0. Wth these defntons the Kähler potental can be wrtten as follows K(, ) = K(, )+K + K + K j + Kk j j k + Kj k j k + 4 Kkl j j + Kj j + K j j + j k l. (5.35) We can now compute the D-term contrbuton to the Lagrangan. We get d d K(, ) = 4 K 4 K + K j F Fj µ j 4 K j@ µ j 4 µ j Kk j µ k@ µ j + j µ k@ µ µ j µ µ j j Fk 4 Kj k (h.c.)+ + 4 Kkl j j k l, (5.36) 9

10 up to total dervatves. Notce now that K(, ) =K + K +K µ j@ µ + K µ j + K µ j. (5.37) Usng ths dentty we can elmnate K j and K j,andrewrteeq.(5.36)as d d K(, ) = K j F Fj µ j µ j µ µ j + agan up to total dervatves. + 4 Kk j µ k@ µ j + j µ k@ µ j Fk 4 Kj k (h.c.)+ + 4 Kkl j j k l, (5.38) A few mportant comments are n order. As just emphaszed, ndependently whether the fully holomorphc and fully ant-holomorphc Kähler potental components, K j and K j respectvely, are or are not vanshng, they do not enter the fnal result (5.38). In other words, from a practcal vew pont t s as f they are not there. The only two-dervatve contrbuton enterng the e ectve Lagrangan s hence K j.thsmeansthatthetransformaton K(, )! K(, )+ ( )+ ( ), (5.39) known as Kähler transformaton, s a symmetry of the theory (n fact, such symmetry apples to the full Kähler potental, as we have already observed). Ths s mportant for our second comment. The functon K j whch normalzes the knetc term of all felds n eq. (5.38), s hermtan,.e. K j = K j,sncek(, ) sarealfuncton. Moreover,tspostve defnte and non-sngular, because of the correct sgn for the knetc terms of all non-auxlary felds. That s to say K j has all necessary propertes to be nterpreted as a metrc of a manfold M of complex dmenson n whose coordnates are the scalar felds themselves. Ths s the (supersymmetrc) -model. The metrc K j s n fact the second dervatve of a (real) scalar functon K, snce K j K(, ). (5.40) j In ths case we speak of a Kähler metrc and the manfold M s what mathematcally s known as Kähler manfold. The scalar felds are maps from space-tme to ths Remannan manfold, whch supersymmetry dctates to be a Kähler manfold. 0

11 Actually, n order to prove that the Lagrangan s a -model, wth target space the Kähler manfold M, we should prove that not only the knetc term but any other term n the Lagrangan can be wrtten n terms of geometrc quanttes defned on M: the a ne connecton, the curvature tensor, etc.... Wth a bt of an e ort one can compute the a ne connecton and the curvature tensor R out of the Kähler metrc K j and, usng the auxlary feld equatons of moton F =(K ) kw k (K ) kk k lm l m (5.4) (remark: the above equaton shows that when the Kähler potental s not canoncal, the F-felds can depend also on fermon felds!), get for the Lagrangan L = K µ j + D µ µ j µ D µ j (K ) jw W j (5.4) where W j k jw j k W j j k W k j + 4 Rkl j j k l, V (, ) =(K ) jw W j (5.43) s the scalar potental, and the covarant dervatves for the fermons are defned as D µ µ + jk@ µ k D µ µ + µ k j. Wth our conventons on ndces, jk =(K ) mkjk m whle kj =(K ) m Km kj and Rj kl = Kj kl Kj m (K ) n mkn kl. As antcpated, a complcated component feld Lagrangan s unquely characterzed by the geometry of the target space. Once a Kähler potental K s specfed, anythng n the Lagrangan (masses and couplngs) depends geometrcally on ths potental (and on W ). Ths shows the strong connecton between supersymmetry and geometry. There are of course nfntely many Kähler metrcs and therefore nfntely many N =supersymmetrc normalzable case, K j = j,sjustthesmplestsuchnstances. -models. The The more supersymmetry the more constrants, hence one could magne that there should be more restrctons on the geometrc structure of the theores wth extended supersymmetry. -model for Ths s ndeed the case, as we wll see explctly when dscussng the N =versonofthesupersymmetrc -model. In ths case, the scalar manfold s further restrcted to be a specal class of Kähler manfolds, known as specal-kähler manfolds. For N = 4 constrants are even

12 sharper. In fact, n ths case the Lagrangan turns out to be unque, the only possble scalar manfold beng the trval one, M = R 6n,fns the number of N =4vectormultplets(recallthataN =4vectormultpletcontanssxscalars). So, for N =4supersymmetrytheonlyallowedKählerpotentalsthecanoncal one! As we wll dscuss later, ths has mmedate (and drastc) consequences on the quantum behavor of N =4theores. 5. N = SuperYang-Mlls We would lke now to fnd a supersymmetrc nvarant acton descrbng the dynamcs of vector superfelds. In other words, we want to wrte down the supersymmetrc verson of Yang-Mlls theory. Let us start consderng an abelan theory, wth gauge group G = U(). The basc object we should play wth s the vector superfeld V, whch s the supersymmetrc extenson of a spn one feld. Notce, however, that the vector v µ appears explctly n V so the frst thng to do s to fnd a sutable supersymmetrc generalzaton of the feld strength, whch s the gauge nvarant object whch should enter the acton. Let us defne the followng superfeld and see f ths can do the job. W = 4 D DD V, W = 4 DD D V, (5.44) Frst, W s obvously a superfeld, snce V s a superfeld and both D and D commute wth supersymmetry transformatons. In fact, W s a chral superfeld, snce D 3 =0dentcally. ThechralsuperfeldW s nvarant under the gauge transformaton (4.6). Indeed W! W 4 D DD + = W + 4 D D D = W + 4 D { D,D } =W + D =W. (5.45) Ths also means that, as antcpated, as far as we deal wth W, we can stck to the W-gauge wthout botherng about compensatng gauge transformatons or anythng. In order to fnd the component expresson for W t s useful to use the (y,, ) coordnate system, momentarly. In the W gauge the vector superfeld reads V W = µ vµ (y)+ (y) (y)+ µ v µ (y)). (5.46) It s a smple exercse we leave to the reader to prove that expandng n (x,, ) coordnate system, the above expresson reduces to eq. (4.65). Actng wth D we

13 get D V W = µ v µ + + D +( µ µ v + (5.47) (where we used the dentty µ µ = µ and y-dependence s understood), and fnally W = + D + ( µ ) F µ + µ, (5.48) where F µ µ v µ s the usual gauge feld strength. So t seems ths s the rght superfeld we were searchng for! Indeed, W s the so-called supersymmetrc feld strength. Note that W s an nstance of a chral superfeld whose lowest component s not a scalar feld, as we have been used to, but n fact a Weyl fermon, gaugno. For ths reason, W s also called the gaugno superfeld., the Gven that W s a chral superfeld, a putatve supersymmetrc Lagrangan could be constructed out of the followng ntegral n chral superspace (notce: correctly, t has dmenson four) d W W. (5.49) Pluggng eq. (5.48) nto the expresson above and computng the superspace ntegral one gets after some smple algebra d W W = F µ F µ µ + D + 4 µ F µ F. (5.50) One can get a real object by addng the hermtan conjugate to (5.50), havng fnally L gauge = d W W + d W W = F µ F µ 4 µ +D. (5.5) Ths s the supersymmetrc verson of the abelan gauge Lagrangan (up to an overall normalzaton to be fxed later). As antcpated, D s an (real) auxlary feld. The Lagrangan (5.5) has been wrtten as an ntegral over chral superspace, so one mght be tempted to say t s a F-term. Ths s wrong snce (5.5) s not atrue F-term. Indeed, t can be re-wrtten as an ntegral n full superspace (whle F-terms cannot) d W W = d d D V W, (5.5) and so t s n fact a D-term. As we wll see later, ths fact has mportant consequences at the quantum level, when dscussng renormalzaton propertes of supersymmetrc Lagrangans. 3

14 All what we sad, so far, has to do wth abelan nteractons. What changes f we consder a non-abelan gauge group? Frst we have to promote the vector superfeld to V = V a T a a =,...,dm G, (5.53) where T a are hermtan generators and V a are dm G vector superfelds. Second, we have to defne the fnte verson of the gauge transformaton (4.6) whch can be wrtten as e V! e e V e. (5.54) One can easly check that at leadng order n ths ndeed reduces to (4.6), upon the dentfcaton =. Agan, t s straghtforward to set the W gauge for whch e V =+V + V. (5.55) In what follows ths gauge choce s always understood. The gaugno superfeld s generalzed as follows W = 4 D D e V D e V, W = 4 DD ev D e V (5.56) whch agan reduces to the expresson (5.44) to frst order n V. Let us look at eq. (5.56) more closely. Under the gauge transformaton (5.54) W transforms as W! D D he e V e D e e V e 4 = 4 D D e e V D e V e + e V D e = 4 e D D e V D e V e = e W e, (5.57) where we used the fact that, gven that (and products thereof) s a chral superfeld, D e = 0, D e = 0 and also D DD e = 0. The end result s that W transforms covarantly under a fnte gauge transformaton, as t should. Smlarly, one can prove that W! e W e. (5.58) Let us now expand W n component felds. We would expect the non-abelan 4

15 generalzaton of eq. (5.48). We have W = D D apple V + 4 V D +V + V = = = 4 D DD V 8 D DD V + 4 D DV D V 4 D DD V 8 D DV D V 8 D DD V V + 4 D DV D V 4 D DD V + 8 D D [V,D V ]. The frst term s the same as the one we already computed n the abelan case. As for the second term we get 8 D D [V,D V ]= ( µ ) [v µ,v ] hv µ µ,. (5.59) Addng everythng up smply amounts to turn ordnary dervatves nto covarant ones, fnally obtanng W = (y)+ D(y)+ ( µ ) F µ + µ D µ (y) (5.60) wth F µ µ v µ [v µ,v ], D µ µ [v µ, ], (5.6) whch provdes the correct non-abelan generalzaton for the feld strength and the (covarant) dervatves. In vew of couplng the pure SYM Lagrangan wth matter, t s convenent to ntroduce the couplng constant g explctly, makng the redefnton V! gv, v µ! gv µ,! g, D! gd, (5.6) whch mples the followng changes n the fnal Lagrangan. Frst, we have now F µ µ v µ g [v µ,v ], D µ µ g [v µ, ]. (5.63) Moreover, the gaugno superfeld (5.60) should be multpled by g and the (nonabelan verson of the) Lagrangan (5.5) by /4g.TheendresultfortheSuperYang- Mlls Lagrangan (SYM) reads L SY M = 3 Im h = Tr 4 F µ F µ d TrW W µ D µ + D + YM 3 g TrF µ F µ, (5.64) 5

16 where we have ntroduced the complexfed gauge couplng = YM + 4 g (5.65) and the dual feld strength F µ = µ F, (5.66) whle gauge group generators are normalzed as Tr T a T b = ab. 5.3 N = Gauge-matter actons We want to couple radaton wth matter n a supersymmetrc consstent way. To ths end, let us consder a chral superfeld transformng n some representaton R of the gauge group G, T a! (TR a) j where, j =,,...,dmr. Under a gauge transformaton we expect! 0 = e, = a TR a, (5.67) where must be a chral superfeld, otherwse the transformed would not be a chral superfeld anymore. Notce, however, that n ths way the chral superfeld knetc acton we have derved prevously would not be gauge nvarant snce! e e 6=. (5.68) So we have to change the knetc acton. The correct expresson s n fact ev, (5.69) whch s ndeed gauge nvarant and also supersymmetrc nvarant (modulo total space-tme dervatves), when ntegrated n superspace. Therefore, the complete Lagrangan for charged matter reads L matter = d d ev + d W( ) + d W ( ). (5.70) Obvously the superpotental should be compatble wth the gauge symmetry,.e. t should be gauge nvarant tself. Ths means that a term lke a... n... n (5.7) s allowed only f a... n s an nvarant tensor of the gauge group and f R R R n tmes contans the snglet representaton of the gauge group G. 6

17 As an explct example, take the gauge group of strong nteractons, G = SU(3), andconsder quarksas matterfeld. In thscase R s the fundamental representaton, R =3. Snce =+... and jk s an nvarant tensor of SU(3), whle t follows that a supersymmetrc and gauge nvarant cubc term s allowed, but a mass term s not. In order to have mass terms for quarks, one needs R =3+ 3 correspondng to a chral superfeld n the 3 (quark) and a chral superfeld n the 3 (ant-quark). In ths case s gauge nvarant and does correspond to a mass term. Notce ths s consstent wth the fact that a chral superfeld contans aweylfermononlyandquarksaredescrbedbydracfermons. Thelessons that to construct supersymmetrc actons wth colour charged matter, one needs to ntroduce two sets of chral superfelds whch transform n conjugate representatons of the gauge group. Ths s just the supersymmetrc verson of what happens n ordnary QCD or n any non-abelan gauge theory wth fermons transformng n complex representatons (G = SU() s an excepton because ' ). Let us now compute the D-term of the Lagrangan (5.70). We have (as usual we work n the W gauge) ev = + V + V. (5.7) The frst term s the one we have already calculated, so let us focus on the D-term contrbuton of the other two. After some algebra we get V = v µ v µ µv µ + p p + D V = v µ v µ. Puttng everythng together we fnally get (up to total dervatves) ev = D µ D µ µd µ + FF + p p + D, (5.73) where D µ µ va µtr a. Performng the rescalng V! gv and rewrtng µd µ = µ D µ (recall the spnor dentty µ = µ )wegetfnally egv = D µ D µ µ D µ + FF+ p g p g +g D, (5.74) where now D µ µ gvµt a R a. So we see that the D-term n the Lagrangan (5.70) not only provdes matter knetc terms but also nteracton terms between matter 7

18 felds, and gaugnos,wheretsunderstoodthat and smlarly for the other couplngs. = (T a R) j a j, (5.75) To get the most general acton there s one term stll mssng: the so called Fayet- Ilopulos term. Suppose that the gauge group s not sem-smple,.e. t contans U() factors. Let V A be the vector superfelds correspondng to the abelan factors, A =,,...,n, where n s the number of abelan factors. transforms as a total dervatve under super-gauge transformatons, snce The D-term of V A V A! V A + : D A! D A µ (...). (5.76) Therefore a Lagrangan of ths type L FI = X A A d d V A = X A D A (5.77) s supersymmetrc nvarant (snce V A are superfelds) and gauge nvarant, modulo total space-tme dervatves. We can now assemble all ngredents and wrte the most general N =supersymmetrc Lagrangan (wth canoncal Kähler potental, hence renormalzable, f the superpotental s at most cubc) L = L SY M + L matter + L FI = = 3 Im d TrW W +g X A d d V A + A + d d egv + d W( ) + d W ( ) (5.78) h = Tr + g X A p g 4 F µ F µ µ D µ + D + YM 3 g TrF µ F µ A D A + D µ D µ µ D µ + FF + p g + @ F W W j j j. Notce that both D a and F are auxlary felds and can be ntegrated out. The equatons of moton of the auxlary felds read D a = g T a g a ( a =0 f a 6= A). (5.79) 8

19 These can plugged back nto (5.78) leadng to the followng on-shell Lagrangan h L = Tr 4 F µ F µ µ D µ + YM 3 g TrF µ F µ + D µ D µ µ D µ + + p g p W W j j ), (5.80) j where the scalar potental V (, ) s + g X (T a ) j a j + a = (5.8) = FF + D on the soluton 0. We see that the potental s a sem-postve defnte quantty n supersymmetrc theores. Supersymmetrc vacua, f any, are descrbed by ts zero s whch are descrbed by sets of scalar feld VEVs whch solve smultaneously the so-called D-term and F-term equatons F ( )=0, D a (, ) =0. (5.8) That supersymmetrc vacua correspond to the zeros of the potental can be easly seen as follows. Frst recall that a vacuum s a Lorentz nvarant state confguraton. Ths means that all feld dervatves and all felds but scalar ones should vansh n a vacuum state. Hence, the only non trval thng of the Hamltonan whch can be d erent from 0 s the non-dervatve scalar part, whch, by defnton, s ndeed the scalar potental. Therefore, the vacua of a theory, whch are the mnmal energy states, are n one-to-one correspondence wth the (global or local) mnma of the scalar potental. Now, as we have already seen, n a supersymmetrc theory the energy of any state s sem-postve defnte. Ths holds also for vacua (whch are the mnmal energy states). Hence for a vacuum we have h P 0 X Q + Q 0. (5.83) Ths means that the vacuum energy s 0 f and only f t s a supersymmetrc state, that s Q =0, Q =0 8. Conversely, supersymmetry s broken (n the perturbatve theory based on ths vacuum) f and only f the vacuum energy s postve. Hence, supersymmetrc vacua are ndeed n one-to-one correspondence wth the zero s of the scalar potental. To fnd such zero s, one frst looks for the space of scalar feld VEVs such that D a (, ) =0, (5.84) 9

20 whch s called the space of D-flat drectons. If a superpotental s present, one should then consder the F-term equatons, whch wll put further constrants on the subset of scalar feld VEVs already satsfyng the D-term equatons (5.84). The subspace of the space of D-flat drectons whch s also F-flat,.e. whch also satsfes the equatons F ( )=0, (5.85) s called (classcal) modul space and represents the space of (classcal) supersymmetrc vacua. Clearly, n solvng for the D-term equatons, one should mod out by gauge transformatons, snce solutons whch are related by gauge transformatons are physcally equvalent and descrbe the same vacuum state. That the modul space parametrzes the so-called flat drectons s because t represents the space of felds the potental does not depend on, and each such flat drecton has a massless partcle assocated to t, a modulus. Themodulrepresent the lghtest degrees of freedom of the low energy e ectve theory (thnk about the supersymmetrc -model we dscussed n 5..). As one moves along the modul space one spans physcally nequvalent (supersymmetrc) vacua, snce the mass spectrum of the theory changes from pont to pont, as genercally partcles masses wll depend on scalar feld VEVs. In passng, let us notce that whle n a non-supersymmetrc theory (or n a supersymmetry breakng vacuum of a supersymmetrc theory) the space of classcal flat drectons, f any, s genercally lfted by radatve correctons (whch can be computed at leadng order by e.g. the Coleman-Wenberg potental), n supersymmetrc theores ths does not happen. If the ground state energy s zero at tree level, t remans so at all orders n perturbaton theory. Ths s because perturbatons around a supersymmetrc vacuum are themselves descrbed by a supersymmetrc Lagrangan and quantum correctons are protected by cancellatons between fermonc and bosonc loops. Ths means that the only way to lft a classcal supersymmetrc vacuum, namely to break supersymmetry, f not at tree level, are non-perturbatve correctons. We wll have much more to say about ths ssue n later lectures Classcal modul space: examples To make concrete the prevous dscusson on modul space, n what follows we would lke to consder two examples explctly. Before we do that, however, we would lke to rephrase our defnton of modul space, presentng an alternatve (but equvalent) way to descrbe t. 0

21 Suppose we are consderng a theory wthout superpotental. For such a theory the space of D-flat drectons concdes wth the modul space. The space of D-flat drectons s defned as the set of scalar feld VEVs satsfyng the D-flat condtons M cl = {h /D a =0 8a}/gauge transformatons. (5.86) Genercally t s not at all easy to solve the above constrants and fnd a smple parametrzaton of the M cl. An equvalent, though less transparent defnton of the space of D-flat drectons can help n ths respect. It turns out that the same space can be defned as the space spanned by all (sngle trace) gauge nvarant operator VEVs made out of scalar felds, modulo classcal relatons between them M cl = {hgauge nvarant operators X r ( )}/classcal relatons. (5.87) The latter parametrzaton s very convenent snce, up to classcal relatons, the constructon of the modul space s unconstraned. In other words, the gauge nvarant operators provde a drect parametrzaton of the space of scalar feld VEVs satsfyng the D-flat condtons (5.84). Notce that f a superpotental s present, ths s not the end of the story: F- equatons wll put extra constrants on the X r ( ) s and may lft part of (or even all) the modul space of supersymmetrc vacua. In later lectures we wll dscuss some such nstances n detal. Below, n order clarfy the equvalence between defntons (5.86) and (5.87), we wll consder two concrete models wth no superpotental term, nstead. SQED. ThefrstexamplewewanttoconsdersSQED,thesupersymmetrcverson of quantum electrodynamcs. Ths s a SYM theory wth gauge group U() and F (couples of) chral superfelds (Q, Q )havngoppostechargewthrespecttothe gauge group (we wll set for defnteness the charges to be ±) and no superpotental, W =0. Thevanshngofthesuperpotentalmplesthatforthssystemthespace of D-flat drectons concdes wth the modul space of supersymmetrc vacua. The Lagrangan s an nstance of the general one we derved before and reads L SQED = 3 Im d W W + d d Q ev Q + Q e V Q (5.88) (n order to ease the notaton, we have come back to the most common notaton for ndcatng the hermtan conjugate of a feld).

22 The (only one) D-equaton reads D = Q Q Q Q =0 (5.89) where here and n the followng a hs understood whenever Q or Q appear. What s the modul space? Let us frst use the defnton (5.86). The number of putatve complex scalar felds parametrzng the modul space s F. We have one D-term equaton only, whch provdes one real condton, plus gauge nvarance Q! e Q, Q! e Q, (5.90) whch provdes another real condton. Therefore, the dmenson of the modul space s dm C M cl =F =F. (5.9) At a generc pont of the modul space the gauge group U() s broken. Indeed, the abovecorrespondstothecomplexfeldwhch,togetherwthtsfermonc superpartner, gets eaten by the vector superfeld to gve a massve vector superfeld (recall the content of a massve vector multplet). One component of the complex scalar feld provdes the thrd polarzaton to the otherwse massless photon; the other real component provdes the real physcal scalar feld a massve vector superfeld has; fnally, the Weyl fermon provdes the extra degrees of freedom to let the photno become massve. Ths s nothng but the supersymmetrc verson of the Hggs mechansm. As antcpated, the vacua are physcally nequvalent, genercally, snce e.g. the mass of the photon depends on the VEV of the scalar felds. Let us now repeat the above analyss usng the defnton (5.87). The only gauge nvarants we can construct are M j = Q Qj, (5.9) the mesons. As for the classcal relaton, ths can be found as follows. The meson matrx M s a F F matrx wth rank one snce so s the rank of Q and Q (Q and Q are vectors of length F,sncethegaugegroupsabelan). Thsmplesthatthe meson matrx has only one non-vanshng egenvalue whch means Recallng that for a matrx A det (M ) = F ( 0)( ) F. (5.93)... F A j A j...a F jf =deta j j...j F (5.94)

23 wth... F the fully antsymmetrc tensor wth F ndces, we have... F (M j j )...(M F jf FjF )= F ( 0)( ) F j j...j F (5.95) whch means that from the left hand sde only the coe cents of the terms F and F survve. The next contrbuton, proportonal to F,shouldvansh,thats These are the classcal relatons: (F... F M j M j j j...j F =0. (5.96) ) complex condtons the meson matrx M should satsfy. Snce the meson matrx s a complex F F matrx, we fnally get that dm C M cl = F (F ) =F. (5.97) whch concdes wth what we have found before! The parametrzaton n terms of (sngle trace) gauge nvarant operators s very useful f one wants to fnd the low energy e ectve theory around the supersymmetrc vacua. Indeed, up to classcal relatons, these gauge nvarant operators (n fact, ther fluctuatons) drectly parametrze the massless degrees of freedom of the perturbaton theory constructed upon these same vacua. Usng equaton (5.89) and the fact that the meson matrx has rank one, one can easly show that on the modul space Tr Q Q =Tr Q Q =Tr p M M. (5.98) Therefore, the Kähler potental, whch s canoncal n terms of the mcroscopc UV degrees of freedom Q and Q, onceprojectedonthemodulspacereads h K =Tr Q Q + Q Q =Tr p M M. (5.99) The Kähler metrc of the non-lnear -model hence reads ds = K MM dmdm = p M M dmdm (5.00) whch s manfestly non-canoncal. Notce that the (scalar) knetc term d 4 x p M µm@ µ M (5.0) s sngular at the orgn, snce the Kähler metrc dverges there. Ths has a clear physcal nterpretaton: at the orgn the theory s un-hggsed, the photon becomes massless and the correct low energy e ectve theory should nclude t n the descrpton. Ths s a generc feature n all ths busness: sngulartes showng-up at 3

24 specfc ponts of the modul space are a sgnal of extra-massless degrees of freedom that, for a reason or another, show up at those specfc ponts Sngulartes! New massless d.o.f.. (5.0) The correct low energy, sngularty-free, e ectve descrpton of the theory should nclude them. The sngular behavor of K MM at the orgn s smply tellng us that. SQCD.Letusnowconsderthenon-abelanversonoftheprevoustheory. Wehave now a gauge group SU(N) andf flavors. The quarks superfelds Q and Q are F N matrces, and agan there s no superpotental, W =0. LookngattheLagrangan, whch s the obvous generalzaton of (5.88), we see there are two ndependent flavor symmetres, one assocated to Q and one to Q, SU(F ) L and SU(F ) R respectvely SU(N) SU(F ) L SU(F ) R Q a N F Q b j N F (5.03) where, j =,,...,F and a, b =,,...,N. The conventon for gauge ndces s that lower ndces are for an object transformng n the fundamental representaton and upper ndces for an object transformng n the ant-fundamental. The conventon for flavor ndces s chosen to be the opposte one. Gven these conventons, the D-term equatons read D A = Q b (T A N ) c b Q c + Q b (T Ā N )b c Q c = Q b (T A ) c b Q c Q b (T A ) b c Q c = Q b (T A ) c b Q c Q b (T A ) c b Q c =0, (5.04) where A =,,...,N, and we used the fact that (TN A)a b = (T Ā N )a b (T A ) a b. Let us frst focus on the case F<N. Usng the two SU(F ) flavor symmetres and the (global part of the) gauge symmetry SU(N), one can show that on the modul space (5.04) the matrces Q and Q can be put, at most, n the followng form (recall that the maxmal rank of Q and Q s F n ths case, snce F<N) 0 v v Q = B A = Q T (5.05) v F... 4

25 Ths means that at a generc pont of the modul space the gauge group s broken to SU(N F ). So, the complex dmenson of the classcal modul space s dm C M cl =FN N (N F ) = F. (5.06) Let us now use the parametrzaton n terms of gauge nvarant sngle trace operators (5.87). In ths case we have M j = Q a Q a j (5.07) (notce the contracton on the N gauge ndces). The meson matrx has now maxmal rank, snce F < N,sotherearenoclasscalconstrantsthastosatsfy: tsf entres are all ndependent. In terms of the meson matrx the classcal modul space dmenson s then (trvally) F, n agreement wth eq. (5.06). Agan, playng wth global symmetres, M can be dagonalzed n terms of F complex egenvalues V, whch, obvously, are nothng but the square of the ones n (5.05), V = v. AsmlarreasonngastheoneworkngforSQEDwouldholdaboutpossble sngulartes n the modul space. On the modul space we have Q a Q b = Q a Q b. Usng ths dentty we get M M j = Q aq a k Q k b Q b j = Q a Q a k Q k b Q b j (5.08) whch mples Q Q = p M M as a matrx equaton. So the Kähler potental s K =Tr p M M. (5.09) The Kähler metrc s sngular whenever the meson matrx M s not nvertble. Ths does not only happen at the orgn of feld space as for SQED, but actually at subspaces where some of the N (N F ) =(N F )F massve gauge bosons parametrzng the coset SU(N)/SU(N to be ncluded n the low energy e ectve descrpton. Let us now consder the case F F ) become massless, and they need N. Followng a smlar procedure as the one before, the matrces Q and Q can be brought to the followng form on the modul space 0 v v Q = v N C A ṽ ṽ , QT = ṽ N C A (5.0)

26 where v ṽ = a, wtha a -ndependent number. Snce F N, atagenerc pont n the modul space the gauge group s now completely hggsed. Therefore, the dmenson of the classcal modul space s now dm C M cl =NF N. (5.) The parametrzaton n terms of gauge nvarant operators s slghtly more nvolved, n ths case. The mesons are stll there, and defned as n eq. (5.07). However, there are non-trval classcal constrants one should take nto account, snce the rank of the meson matrx, whch s at most N, snowsmallerthantsdmenson, F. Moreover, besdes the mesons, there are now new gauge nvarant sngle trace operators one can buld, the baryons, whch are operators made out of N felds Q respectvely N felds Q, wthfullyant-symmetrzedndces. As an explct example of ths rcher structure, let us apply the above ratonale to the case N = F. Accordng to eq. (5.), n ths case dm C M cl = F +. The gauge nvarant operators are the meson matrx plus two baryons, B and B, defned as B = a a...a N Q a Q a...q N a N B = a a...a N Qa Qa... Q a N N. Snce F = N the ant-symmetrzaton on the flavor ndces s automatcally taken care of, once ant-symmetrzaton on the gauge ndces s mposed. All n all, we have apparently F +complexmodulspacedrectons. Thereshoweveroneclasscal constrants between them whch reads det M B B =0, (5.) as can be easly checked from the defnton of the meson matrx (5.07) and that of the baryons above. Hence, the actual dmenson of the modul space s F + = F +, as expected. As for the case F<N,theresasubspacenthemodulspace, whch ncludes the orgn, n whch some felds become massless and the low energy e ectve analyss should be modfed to nclude them. In fact, all we sad, so far, s true classcally. As we wll later see, (nonperturbatve) quantum correctons sensbly change ths pcture and the fnal structure of SQCD modul space d ers n many respects from the one above. For nstance, for F = N SQCD, t turns out that the classcal constrant (5.) s modfed at the quantum level; ths has the e ect of excsng the orgn of feld space, 6

27 Q a = Q jb = 0 from the actual quantum modul space, removng, n turn, all sngulartes and correspondng new massless degrees of freedom, whch are then just an artfact of the classcal analyss, n ths case. We wll have much more to say about SQCD and ts classcal and quantum propertes at some later stage The SuperHggs mechansm We have alluded several tmes to a supersymmetrc verson of the Hggs mechansm. In the followng, by consderng a concrete example, we would lke to show how the superfeld degrees of freedom rearrangement explctly works upon hggsng. As we already notced, t s expected that upon supersymmetrc hggsng a full vector superfeld becomes massve, eatng up a chral superfeld. Let us consder once agan SQCD wth gauge group SU(N) andf flavors and focus on a pont of the modul space where all scalar feld VEVs v are 0 but v. At such pont of the modul space the gauge group s broken to SU(N ) and flavor symmetres are broken to SU(F ) L SU(F ) R.Thenumberofbroken generators s N (N ) =(N ) + (5.3) whch just corresponds to the statement that f we decompose the adjont representaton of SU(N) ntosu(n ) representatons we get Adj N = N N Adj N! G A = X 0,X,X,T a (5.4) where G A are the generators of SU(N), T a those of SU(N ) and the X s the generators of the coset SU(N)/SU(N ) (A =,,...,N, a =,,...,(N ) and =,,...,N ). Upon ths decomposton, the matter felds matrces can be re-wrtten schematcally as! Q =!0! Q 0!!, Q 0 =! Q0 (5.5) where, wth respect to the survvng gauge and flavor symmetres,! 0 and! 0 are snglets,! and! are flavor snglets but carres the fundamental (resp ant-fundamental) representaton of SU(N ), (resp ) aregaugesngletsandtransformnthe fundamental representaton of SU(F ) L (resp SU(F ) R ), and fnally Q 0 (resp 7

28 Q 0 )arenthefundamental(respant-fundamental)representatonofsu(n ) and n the fundamental representaton of SU(F ) L )(respsu(f ) R ). By expandng the scalar felds around ther VEVs (whch are all vanshng but v )andpluggngthembackntothesqcdlagrangan,aftersometedousalgebra one fnds all v -dependent fermon and scalar masses, together wth massve gauge bosons and correspondng massve gaugnos. On top of ths, there remans a set of massless felds, belongng to the massless vector superfelds spannng SU(N ) and the massless chral superfelds Q 0 and Q 0.Werefrantoperformthscalculaton explctly and just want to show that the number of bosonc and fermonc degrees of freedom, though arranged d erently n the supersymmetry algebra representatons, are the same before and after Hggsng. We just focus on bosonc degrees of freedom, snce, due to supersymmetry, the same result holds for the fermonc ones. For v = 0 we have a fully massless spectrum. As far as bosonc degrees of freedom are concerned we have (N ) of them comng from the gauge bosons and 4NF comng from the complex scalars. All n all there are bosonc degrees of freedom. (N ) + 4NF (5.6) For v 6= 0 thngs are more complcated. As for the vector superfeld degrees of freedom, we have (N ) massless ones, whch correspond to [(N ) ] bosonc degrees of freedom, and + (N ) massve ones whch correspond to 4 + 8(N ) bosonc degrees of freedom. As for the matter felds (5.5), the massve ones are not there snce they have been eaten by the (by now massve) vectors. These are! and!, whch are eaten by the vector multplets assocated to the generators X and X,andthecombnaton! 0! 0 whch s eaten by the vector multplet assocated to the generator X 0.Wehavealreadytakenthemnto account, then. The massless chral superfelds Q 0 and Q 0 provde (N )(F ) bosonc degrees of freedom each, the symmetrc combnaton S =! 0 +! 0 another bosoncdegreesoffreedom,andfnallythemasslesschralsuperfelds and (F ) each. All n all we get (N ) +4+8(N )+4(N )(F )++4(F ) = (N )+4NF, whch are exactly the same as those of the un-hggsed phase (5.6). (5.7) 8

29 5.3.3 Non-lnear sgma model II In secton 5.. we dscussed the supersymmetrc non-lnear -model for matter felds, whch s relevant to descrbe supersymmetrc low energy e ectve theores. Though t s not often the case, t may happen to face e ectve theores wth some left over propagatng gauge degrees of freedom at low energy. Therefore, n ths secton we wll generalze the -model of secton 5.. to such a stuaton: a supersymmetrc but non-renormalzable e ectve theory coupled to gauge felds. Note that the choce of the gauge group cannot be arbtrary here. In order to preserve the structure of the non-lnear -model one can gauge only a subgroup G of the sometry group of the scalar manfold. Followng the prevous strategy, one gets easly convnced that the pure SYM part changes smply by promotng the (complexfed) gauge couplng to a holomorphc functon of the chral superfelds, gettng d TrW W! d F ab ( ) W a W b, (5.8) where the chral superfeld F ab ( ) should transform n the Adj Adj of the gauge group G n order for the whole acton to be G-nvarant. Notce that for F ab = TrT a T b one gets back the usual result (recall that we have normalzed the gauge group generators as Tr T a T b = ab ). For ths reason the functon F ab (actually ts restrcton to the scalar felds) s dubbed generalzed complex gauge couplng. As for the matter Lagrangan, gven what we have already seen, namely that whenever one has to deal wth charged matter felds the gauge nvarant combnaton s ( e gv ), one should smply observe that the same holds for any real G-nvarant functon of and. In other words, the -model Lagrangan for charged chral superfelds s obtaned from the one we derved n secton 5.. upon the substtuton K(, )! K(, ( e gv ) ). (5.9) The end result s then L = apple 3 Im d F ab ( ) W a W b + + d d K(, ( e gv ) )+ d W( )+ d W ( ). (5.0) By expandng and ntegratng n superspace one gets the fnal result. The dervaton s a bt lengthy and we omt t here. Let us just menton some mportant d erences 9