Document Sample
122 Powered By Docstoc

                          Running Nonlocal Lagrangians12

                               Vineer Bhansali and Howard Georgi
                        Lyman Laboratory of Physics,Harvard University
                                           Cambridge,ÌÀ 02138

           We investigate the renormalization of "nonlocal" interactions in an effective field
           theory using dimensional regularization with minirnal subtraction. In à scalar field
           theory, we write an integro-differential renormalization group equation for every pos-
           sible classof graph at one loop order.

1.     Introduction

           In its traditional form, an effective field theory calculation goes like this: Start at à very
large scale, that is with the renormalization scale,JJ,very large. In à strongly interacting theory
or à theory with unknown physics at high energy, this starting scalc should Üå sufficiently large
that nonrenormalizabIeinteractions produced at higher scalesare too small to Üå relevant. In à                  l'

renormalizable,weakly interacting theory, one starts at à scaleabovethe masses all the particles,
where the effective theory is given simply Üó the renormalizabletheory, with ïî nonrenormalizabIe
terms. ÒÜåtheory is then evolved down to lower scales.As long as ïî particle màâsås encoun-
tered, this evolution is describedÜó the renormalization group. However,when JJgoes below the
mass, Ë, of one of the particles in the theory, we must changethe effective theory to à new theory
without that particle. In the process,the parametersof the theory change,and new, nonrenormal-
izable interactions màó Üåintroduced. Both the changes existing par"meters, and the coefficients
of the new interactions èå computed Üó "matching" the physicsjust below the boundary in the
two theories. It is this processthat determines the relative sizesof the nonrenormalizableterms
associatedwith the heavy particles.

     'Research supported in ðè! Üó (Üå N&tional ScienceFound&tion under Grøt #ÐÍÓ-8714á54.

     'Research supported in ðè! Üó (Üå Òåõ.. N&tional Research1&bor&1ory Commission, under Grøt   #RGFÓ910á.

        Âåñàèçåmatcl1ing is done for ð ~ Ë, the rule for the size of the coeflicients of the new
operatorsis simple for ð ~ Ë. At this scale,all the new contributions scalewith Ë to the appropriate
power (set Üó dimensiona!analysis) èð to factors of coupling constants, group theory or counting
                                  etc.) 1. Then when the new effective theory is evo!ved down
factors and loop factors (of 1611'2,
to sma!ler ð, the renormalization group introduces additional factors into the coeflicients. ÒÜèç
à heavy particle òàâç àððåìâ in the parametersof an effective field theory in two ways. There
is power dependence the òàâç that arisesfrom matching conditions. There is also logarithmic
         that arisesfrom the renormalization group.
       The matcl1ingcorrection at tree level is simply à differencebetweenà calculation in the full
theory and à calculation in the low energy effectivetheory

                                    J  îr.Î(Ô)   = ~í+r-(Ô) - ~(Ô)
                                         = J {virt~h"V)'tr... }(ô )

where ~í+r.(Ô)     denotes the !ight partic!e effective action in the fu!! theory and ~(Ô)   denotes the
âàòå in the !ow energy theory 2. The matching correction so obtained is non!ocal because it depends
îï ð/ ë throughthe virtual heavypartic!epropagators. is alsoanalytic in ð/ ë in the region
re!evant to the !ow energy theory, i.e. for characteristic momentum«         ë. Thus it ñàï Üå expanded
in powers of ð/ ë with the higher order terms dåñråàçing in importance:       this corresponds to à !îñà!

operator product expansion in the domain of ana!yticity,         equiva!ent to à !îñà! nonrenormalizabIe
Lagrangian which ñàï Üå treated àç an honest-to-goodness !îñà! fie!d theory. However in genera!

an infinite series of terms of inñråàçing!ó higher dimension ìå generated Üó matching at tree !eve!.
These cause ïî    probIem when the sca!es are we!! separated, because their effects quick!y Üåñîòå
neg!igibIe. But if there are two or more scales c!ose together (such àç ò, and Mz òàó Üå), then we
òàó not Üå justified in ignoring terms at higher orders in the expansion. How does one understand
how to interpo!ate   smooth!y between the we!!-understood situation         in which the sca!es are very
different and the we!!-understood situatidn in which the sca!es ìå very c!ose together? How ñàï we
keep track of al! the infinite number of higher derivative operators eflicient!y? Is it possibIe to deal
with the non!ocal effective Lagrangian direct!y, without expanding? In the particu!ar        context of à
scalar fie!d theory, we wi!! attempt to answer questions at the one !îîð        !eve!. The approach wi!!
Üå direct.   We wi!! manipu!ate the non!ocal interactions àç if they ìå       expanded in à momentum
expansion, and then show that the resu!ting {3 functions for the terms in the momentum expansion

                                           Pl                               ðç


                                      Figure 1: Basic nîn!îñà! ô4 vertex.

ñàï Üå co!!ected into integro-differentia!      renormalization        groupequations   for the nîn!îñà!   coup!ings.
           The present paper is organized as fo!!ows: in section 2 we ca!cu!ate the (i function for à
nîn!îñà!     four point coup!ing arising from à înå-!îîð             graph with two internal!ines    in nîn!îñà!   ô4
theory. We give this åõàøð!å before the genera! resu!ts of the fo!!owing sections in order to point
out the important         features of our method.      In section Ç the method is then app!ied to obtain the
contribution     to the renormalization    group equation for à genera! c!ass of graphs in àn arbitrary
massive non!ocal, non-renorma!izable           parity invariant sca!ar fie!d theory.

2.    Basic Example

       ÒÜå matrix e!ements in à genera! non!oca! fie!d theory 3 are ca!cu!ated Üó writing down à

Lagrangian with 'smeared' vertices.            For instance, Cor à non!ocal ô4 interaction          in à sca!ar fie!d
theory with the discrete symmetry Ô -- -ô, the interaction term in the action is

                 84   =   J dÕldÕ2dõýdÕ4F(Õl      -   Õ4, Õ2   -   Õ4,ÕÝ   - Õ4)Ô(Õl)Ô(Õ2)ô(Õý)Ô(õJ                (2)

where Õ; àãå spacetime coordinates and F is à non!oca! 'Corm-Cactor'. Energy-momentum                        conser-
vation at åàñÜ vertex îÑ the corresponding Feynman graph is expressed in terms îÑ the Fourier
transCorm G îÑ F: G = G(Ðl,Ð2,Ðý) Ñîã the basic ô4 interaction shown in figure 1. Bose symmetry

imp!ies that the non!ocal coup!ing G is symmetr\c and satisfies

                                    G(Ðl,Ð2,Ðý)=G(Ðl,Ð2,-Ðl-Ð2-Ðý).                                                (3)
 .                                                                                                        125
                                    Ðl                  k                   ðç

                              -îÑ'" ,"'~(--~~)                   ~""         "',, +ð]+~)

                                                   . . - '-                114
                                    ð2             k+Pl +Ð2                 Ð4

                      Figure 2: Feynman graph contributing    to the renormalization   of G.

     ÎÑ course, Lorentz invariance dictates that in the final expression for the matrix element in øî-
     mentum çðàñå, only scalar products îÑ momenta will àððåè àç arguments of G. lmplicit               in the
     definition   î/ G is à mass scale À which sets the limit /or the region î/ analyticity    î/ G, and for
     characteristic momenta ð < À, G i8 analytic. We go to the effective low energy theory byexpanding
     in à Taylor expansion in ð/ À. One ñàï think of this Tay!or expansion àç the formal implementation
     of à !îñà! operator product expansion îÑ G.      At this point, in going to the effective low energy
     theory, we have actually changed the high energy behavior of the theory çî that integrals which
     were convergent in the full theory are divergent in the effective theory. This trades !ogs of À in the
     ful! theory for anomalous dimensions in the !ow energy theory. This trade allow8 us to ca!culate
     the !ogs more 8imp!y and to sum them using the the renormalization       group.
            Let us first di8CUSS an example th~ renorma!ization        of the non!ocal ô4 interaction    from

     the graph shown in figure 2. For màçs!ess fie!ds, this is the only contribution    in one !ðîð.    In t~e
     mas8ive ñàâå, there is à!âî à contribution   Crom à tadpo!e graph. In addition, in either ñàâå, the~e
     are renorma!ization   of ô2k interactions for k > 2. We will systematical!y consider them in the next

            The Feynman integra! for the graph in figure 2 i8

                               - J ~         G(Ðl,Ð2,k)G(Ðý,Ð4,k+Ð1+Ð2)                                     4.
                                   (27Ã)4(k2- ò2 + if)((k + Ðl + Ð2)2 - ò2 + if)                           ( )



Combine denominators

                                      [k2   -   ò2                     -
                                                     + if][(k + ð1+ Ð2)2 ò2 + if]

and shift momenta
                                                      k = [-à(PI +Ð2)

                - r     da   J~       G(pl,P2,i - à(ð! + Ð2»)         +
                                                            G(Ðç,Ð4,i (1- à)(ð! + Ð2»)                                        (7)
                   10            (271")4                            -
                                              [t'l + à(l à)(ð! + Ð2)2 ò2 + iE]~

Now, here is the crucial point.                  We get the low energy effective theory Üó expanding
the àâ in à momentum expansion. Àn equivalent procedure is to treat the àâ àç if they
were analytic everywhere in momentum âðàñå. Thus we ñàï deal with the 1.dependence
G Üó writing à symbolic Taylor åõðànsiînÝÍåãå we àãå effectively doing the momentum
expansion. But the key is that we ñàï resum the final result into àn finite integral îóåã
the original nonlocal couplings.

                             G(Pl,P2,l-         à(Ðl + Ð2))G(Ðç,Ð4,l + (1- à)(Ðl + Ð2))
                                                  .                                                                           (8)
                              = el" )f(Pl,P2, q) G(Ðç,Ð4, + Ðl + P2)]q=-Q(~l+"')

        Now all the dependence the loop momentumis in the denominatorsand in the exponential,
80 we jU8t have to do the Feynma.n integra! of the exponential

                                       d4l                             elJ;
                                  J   (ù   [Ð + a(l-                      -
                                                              à)(Ðl + Ð2)2 ò2 + iE]2

or to Üå more precise, the dimensionally regu!arized integra!

                               J (2w)- ,2,
                                    4 11-
                                                     [fl + à(l -l!)(Pl + Ð2)2 ò2 + if] 2                                     (10)

   3Analyticity î! the nonlocal couplings òàó alternatively Üåexploited Üóformally Laplace transforming in theloop dependent
arguments, which yield. instead î! .rl;   G an exponential .-'ÎÉ       in terms î! 'Üå Laplace transform î! à, along with an extra
Laplace inversion integral. ÒÜå r..t î! 'Üå compntation i. essentially identical.

                 We wi!! do this Üó manipulating the exponential !ike à power såriåç,becausethe basic àâ-
         sumption is analyticity in the momenta. It is at this point that we have irrevocably changedthe
         high energy behavior and gone over the effective !ow energy theory. Becauseof symmetry, on!y
         even terms in l contribute,

    .                                                                                      ([89)2'
                             ~ J
                              ~          1
    !                               (2;:)1         (21Ã)4-'ð-2' [fl + 0(1 - 0)(PI + Ð2)2- ò2 + if]2                 (11)
         We calculate this àç follows

                                                                    dfl(          """.,

                                                                           (2r)!((2~!) 'enns                        (12)
                                                 = À,.   (l2)'          ", . .. 9",.-, +

                                                                        î=! dOto                                   (13)
         Contracting   with 9"""    gives

                                                 A,.-l = (4 + 2(ò- 1» À. = 2(ò+ 1)À,.
                                                                   À,. = 2'"(r + 1)!
                                            J~n ([.!!.- -- 4'r!(r1+ 1)![]Ð'
         and thus                 -2-                              2'                              l    .!!.-2J

                                  (2r)!           aq                                                   (aq) .
        80 the integral is

                         ~           1            J         ~-'l                          [Ð]' [(1;)1'
                        ~    4rr!(r + 1)!                        [Ð
                                                     (2'K)4-'1J-2C +à(l               - à)(ð; + Ð2); - ò2 + if]2   (17)
        \'lhich we can calculate in the èçèàl \'lay. Wick rotate

                                        .    ~      (-1)'           J      ~-'l           [ÐÓ [(1;)2]r
                                     '~          4'r!(r+l)!             {2'K)4-'1J-2'
                                                                                   {Ð+À(à)2]2                      (18)

                                                 À(à)2      = -à(l      - à)(Ðl + P2f + ò2,


and À(à)2 is positive for Euclidean momenta. Now doing the f. integral gives

                                        2i              00       -À   1         à   2r   [( -a)2] r +...                                           20
                                 (41r)2t/,-2'           ?;4rr!2 ()                         aq                                                    ( )

where we have dropped e.verythingbut the 1ft pole and the àçsîñiàtåd/' dependence.We ñàï write
the result àç

                      ~ ~[~                                ~;:~"::j:1)! [-xA(a)2Jr               [(~)2]r]r=t.                                    (21)

This ñàï then Üå turned backinto the (Euclidean) angular a.verage an exponential,just in.verting

                                                   2i      ~
                                                                      []~   Ë
                                                                                    õ e,fiA(Q)./t    ]

\vhere å is à Euclidean unit vector. Now that we ha.vedone the !. integral and extracted the 1ft
pole, we ñàï use the Taylor seriesto put the {îãò (22) in terms of the àâ. The rcsult is
                 (81Ã)2fð-2(     Â; l1 n Jo da Æ[G(Pl,P2,
                                 iJ    an. ['                                              fiA(a)e             - à(Ðl    + Ð2»)
                                                             î                                                                                  (23)

                         G(Ðç,-Pt         -Ð2 -Ðç"fiÀ(à)å+(l-à)(Pt                                   +Ð2»)] ]Æ=l

with the derivative evaluated at æ                  = 1.     Thus we have reduced the divergent part of the original

Feynman graph to an integral of the Gs and their derivatives over finite                                                 ranges      (for Euc]idean
momenta).    These ]ast integrals are wel]-behaved and could Üå done numerically.                                                      Hence, with
n = 211'2 CorEuclidean
        in                     Cour space, we obtain the one-loop fJ Cunction Cor the nonlocal interaction,


             fJG(p"P2'n)    =   ~ t;                [ 1 an.l1                        ,fiA(a)e
                                                                          dàæ[G(Pl,P2,                            -     à(Ðl + Ð2»)
                                                 ,î                                                                                             (24)
                    G(Ðç, -Ðl - ð2 - Ðç, ,fiA(a)e                         + (1 - à)(ð! + Ð2»)]]                       + cross
Now in the massless ñìå,         where there are ïî                       other contributions                  to the renormalization         of the
ô. coupling, the "running"       nonlocal coupling satisfies the integrodifferential                                      renormalization     group

                                                   Ðä; G(Ðl,Ð2,Ðç) = fJG(P',P2d'3).                                                             (25)

À useful check îï    this result is obtained Üó going to the 'locallimit',                                        i.e. G    --+   9 where 9 is the
usual coupling constant of 'îñà] ô. theory. Indeed, we know that the contribution
usua.l coupling
                                                                                                                                  to the fJ function    -.


             in !îñà! ô4 theory at one !îîð   is
                                                          .8(g)= l6;i > Î.
                                                                  3g2                                                  (26)
             Substituting   G = g, inserting an exp!icit symmetry factor of 1/2, and summing over the crossed

             graphs, (24) indeed yie!ds
                                                              -+   .8(g) as G -+ g.                                    (27)
                    Now the non!oca! quartic coup!ing G ñàï induce changes in the coup!ing terms with more
             fie!ds. For instance, the one-!oop diagram re!evant to the renormalization              of the ô6 coup!ing is
             shown in figure 3.
                    For mass!ess fie!ds, this does not mix Üø         into G in à mass independent renormalization

             scheme, because the re!evant Feynman graph of figure 4 vanishes. However, in the massive ñàçå,
             the graph of figure 4 gives à non-vanishin!..contribution.         We wi!! now systematica!!y compute the
             {ç Cunctions îÑ the general non!oca! coup!ings in à massive nonrenorma!izable theory.

             number of !ow energy fie!ds. In this section, we wi!1 compute the one-!oop running of à general non-
             renorma!izabIe, nonloca! sca!ar effective theory with     4> -+   -4>   symmetry. Specifica!ly, our purpose is

             to iso!ate the dimensional regularization pole of à 2ò point function of type n (i.e. with n vertices or
             propagators) at one !îîð.    The plan is àç fo!!ows: we first construct the expressions corresponding to



                        Figure 4: Feynman graph that vanishes for massless ôs.

.the Feynman integra.l of à genera.l graph with à special choice of momentum labelling conventions.
We then describe the application               of the method of the last section to two relevant examples: the
tadpole renorma.lization of the nonloca.l 'ô.' coupling with à ô8 operator insertion, and the renor-
ma.lization of à nonloca.l ô8 coupling with three ô. operator insertions. ÒÜå last example is useful in

higblighting   the specia.l features of renormalizing graphs with more than two interna.llines.                                                       Fina.lly,
we attack the genera.l ñàâå, and give the complete expression for the 1/~ ðîlå of the 2ò point func-
tion of type n at one 'îîð     as à surprisingly                        compact multi-Feynman-parameter,                               multi-dimensiona.l

angular integra.l.
       3.1.    Preliminaries     We assume that the dimensiona.lly continued d                                                        =4-      ~ dimensional

non-loca.l Lagrangian has à Ô -.          -ô     symmetry and hence has an interaction term proportiona.l to
                                                    1'.             -    '"' ILqr-l)   G               Ô   2r

                                                 '--1ò.             -    L..,'"             2r.

                               The interaction
                                     --                       ---       is not normal             --   ordered4,
                                                                                                           ---       and each G2r is some nonlocal
                                                                                                                          ---r---g-      -..
The mass scale Jl is introduced       to keep the dimensions of the nonlocal
dimensional continuation.      The interaction                          is not normal ordered4, and ,
                               I the region               under            consideration,                  depends   as à consequence          of momentum
function which is analytic in the region under consideration, depends as à..
                             ---1.. :--'  '--.             ---' --  ,

ààòå pawer à! animplicit scaleà! nanlacality Ë.
        À diagram with n intemallines will Üå called type-n. At one loop, à type-n graph has n




                       Figure 5: Òóðå-n Feynman graph. ÒÜåbIobs

                                                           N=LV"                                              (29)
             where vi is the number of externallines   ema.nating from the ith vertex. ÒÜå loop integral for the

             type-n one-loop renorma1ization of the N point function (shown in figure 5) is

                                          I=   J~(2"1Ã)4
                                                       (k2)(k + Ql)2.. .(k + Qn-l)2'

             where the upper index îï   the G' s is used to number the vertices àç one goes along the loop.
                    Here à\1 external momenta are taken to Üå incoming (signified Üó ...,      in the numerator,

             different for different Gi) and we Üàóå defined

                                                  Qo   =   Î                                                  (31)
                                                  Ql   =   P1

                                                  Q2   =   P2+Ql

    .-                                            Qç   =   Ðç+Q2

                                                  "; =                  = 1...,Ð;
    r                                                                   = L r;
                                                           Ó; + "i-1
                                                            Ð, + Qi-1
                                                 Qn = Q..-I + Ð..= j=1
                                                      Q"-1 + Ð" = Î,Î,

             and Ð; is the sum î! the external momenta Howing into the ith vertex. Energy momentum conser-

             vation is simply Q.. = Î. Symmetry factors are suppressed.


132                                                                                                                                                                                                "::!t.-

           Using this notation, the loop integral for the for the ô4
last section is obta.inedÜó setting n = 2, t1}= 2, v, = 2:

                                                                                                                                                                                       (32)            j

                                     1                   Ã(Ð1 ð2+ . . . + ð,.)
                                                             +                                                                                                                                     ;~
                         À:'Àð;'        ...À::"        = r(P1)r(P2)...r(PR)                                                                                                            (33)

                             1d  1
                                                            Ðl-1_Ð2-1 ""'-1"
                                                           à1 à2- ... ö;, î (1 - à1 - à2                                                                                      )
                                                                                                                                                                           - QR
                                                                                                                                                                              '                    "

                                 (k2)(k QJ2... (k + Q"-l)2
                                      +                                                                                                                                               (34)

                             (n -1)!            JÏ . dai~i                          : ¨:-:! ai)+ 1I:;';;:(k                                             + Qr)2ar]" ,


                                                                   k=l-):=Qoao                                                                                                        (35)
50 tha.t
                                                                          ..-1                                       ..-1
                                               k3 = 1.3+        <L Q,a,)2 - 2/..                                     L Q,a"                                                           (36)
                                                                          ,=1                                        0=1

                                                ..-1                                    ..-1                                     ..-1
                     D            [[2   + (}:::Q,a,)2-è. }:::Q,a, -[2}:::à,                                                                                                           (37)
                                                ,=1                                     ,=1                                      '=1                                                          ~,
                                        ..-1                ..-1                                   ..-1                          ..-1                         ..-1
                                  -(}::: Q,a,)2}::: à, + (2(. }::: Q,a,)(}::: à,) + }::: (2à.                                                                                                  l'

                                     n-1        n-1                                     n-1                    n-1                                      n-1                                    ..,
                                  + 2:::(2:::Q,a,fa.                             - 22:::[.2:::                       Q,a,a. + 2:::Q~a.
                                     .=1 ,=1                                            .=1                    ,=1                                      .=1                                   ..
                                         n-l                                  n-l                  n-l
                                  +2     L ,-. Qrar - 2 L                               Qr(L Q.a.)ar]n.
                                         -=1                                  -=1                  0=1
After cancellation (37) yields the denQminator

                                                                          D   = [Ð- À2]n                                                                                              (38)

                                                               n-1                 n-1
                                                 À2= (}::: Q,a,)2- L Q~ar'                                                           (39)
                                            ,=1                                    "=1
    For the òassive ñìå, we simply substitute in (34)

                                                               k2    -+    k2 - ò2                                                   (40)
                                                 (k + Qr)2           -..   (k + Qr)2 - ò2                                            (41)

    50 that we jU5t get an additional           term in (37) when combining denominators after the shift (35):

                                                         n-l               n-l
                                             - ò2(l -    L à.)+ L(               -ò2)à. = -ò2.                                       (42)
                                                         .=1               .=1
    50 the general form for the denominator of the right hand side of (34) with massive fields ~s
                                                         n-1                     n-l
                                         D   = [k2 -    (}:::: Q,a,)2       + (}::::   Q~ar)    - ò2]n.                              (43)
                                                         ,=1                     ,=1
    Now the shift (35) changes the argument of the numerator factors in (30) also, to yie!d the fina!
    expression for the Feynman integra15

                                                                 Ï ,=1àî",+2(... ,l-
                                                                     N                         "n-l
                                                                                        '--,:1 Q,a, + Qi-l)
                      (ï - 1)1
                               !BdQj! ~                                       Q,a,)2+ ():::;;;;.:              -
                                                         [fl    -    ():::;~;;.:               Q~a.) ò2 + if)n                       (45)

              Iòportant      reòarks îï      notation : (1) 1n intermediate                steps of the computations,         we denote
    the dependence of åàñÜ nonlocal function îï                      the (distinct!)       external momenta Üó ellipsis. This is
    useful since the external momenta play à trivial part in the loop integration,                                 and òàó Üå reinstated
~   Üó examination in the final expressions. (2) Also, since at one loop any vertex shares two and only
J   two lines with the loop, Üó energy momentum conservation the 'îîð                                 momentum àððåàãç qnly once

    àç an argument of any à; and will Üå put in its làçt slot. (3) ÒÜå first vertex will have only the
l   loop momentum àç its làçt entry.

               Crossing : ÒÜå external momenta can Üå exchanged amongst themselves. ÒÜå final result,
    however,dependsonly îï the number of distinct Lorentz invariants of the four momentaÐ; (i = 2ò

    first section:
                                J da J     d"l G,(JI"P2,l-a ,+Ð2»G,(JIý,ð"l+                     l-a(JI,+P2»
                                          (2rF              [l'+a(1-a)~-m2]
    with   Q, = 1'1 + 1'0.

forrenormalization of the 2ò point function) under the condition LP; = Î and ó. = ò2. This
equals the total number of graphs related Üó 'crossing' where the crossedgraphs can Üå obtained
Üó exchanging                              We wi1l give explicit expressionsonly for one member of åàñÜ
crossed set.

       Coènting i's: 19noring for the moment the i's appearing due to Wick rotation and integration
(see below), the integrand itself gives ïî          powers of i. This is seen as follows: åàñÜ propagator is                     ~
~,          and eachnonlocal vertex has à Feynmanrule -iG2r' Sincethe number of vertices equals
the number of internallines       at one loop, for à type-n graph we obtain (-i)"i"                           = (-1)"( i)2" =   "\
(-1 )2" = 1. 80 the only source of i's and minus signs is the integration formula.

       Compètational    Tools :
       ln order to do the dimensionally             regularized loop integrals we will need the weIl-known
integration formula (for Euclidean momenta):

                  J trlq(q2(q2)r
                            - À2)"
                                          = iJr"/2(-1    ) ,,+r   ( A2 ) r-,,+~ r(r      + i)r(n - r - i)
                                                                                           r(i)r(n)       ,              (46)

and the expansion

                     Ã( -n + Å) =    ~          [~ + (1 + ~ + . . . + ~ - -Ó)+ Î( Å)] ,                                  (47)

where 'ó   = 0.5772157is Euler's constant.
       Also note that in 2" - 2 dimensions (" à positive integer)

                                             J  -nf;=2l",l",
                                                dn2K-2                      .. .l"'r
                                                                  (2r)!/(2'r!)    I=-.                                   (48)
                                                    ,~                                        ,
                                = A~K-2(å)'         [9"",             9"'r-"'.r+ perms]

where we have defined
                                             nd = J dnd= "f(di'i)'
                                                                         2'Kd/2                                                  ~
                                                                                                                         (49)   -,
Contracting with 9""'2 gives
                       A~~12   = [2,. -    2 + 2(r -1») A~~-2= (2,. + 2r - 4) A~"-2                                      (50)

                                             À2ê-2=                     1
                                               ,.         2'"(,,+r-2)!                                                   (51)

                                            ð2                                                ðâ

                                       -iG4(Pl                                                s,Ðâ,k + QJ

                                            Pl                                            ps

                                                   ðç        I       Ð4
                                                    -iG4(Ðç,Ð4,k + Ql)

                                Figure á: Òóðå-ÇFeynmangraph contributing to Ðà..

     and thus in 2,.    - 2 dimensions
                                                     1      df!2"-2    ([-ä )   2'

                                                   (2r)! f!2"-2            Bq
                                                                            [          2] r                                (52)
                                              4rr!(r+K-2)!            []        (aq)          .

     The importance of the result of (52) wil1 Üå seen when we isolate the pole pieces appearing from
     graphs with more than two internal            lines.     The point is that the leading terms in the Taylor
     expansion then give Feynman integrals which àãå manifestly convergent Üó power counting, and
     divergences appear only at some higher order in the Taylor expansion. This implies that the sum of
     dimensional regularization        poles does not begin at zero, and we cannot immediately write the result
     as an angular intcgral over à Euclidean unit vector in four dimensions, as we did in the last section.
     One choice is to redefine the infinite sum to start at zero and compensate for the redefinition Üó

     subtracting off à finite sum (which vanishes under the action of differential operators in the dummy

     parameters).     Alternatively,     using (52), we ñàï re-sum the poles for à /?raph with n > 2 internal
     lines, without    reference       to ànó dummy         variabIes      , in terms of à 2n - 2 dimensional angular
     integral! The example of the 2        - 2 - 2 graph    below will give the explicit details of how this is done.

            The ôá operator ñàï Üå renormalized at one 'îîð                 Üó à type-l           (tadpole) with à Ñâ coupling

     insertion, à type-2 graph with Gá, Ñ. insertions, or .he most convergent graph, of type-3, with
     three Ñ. insertion..    For the last one, the relevant graph with incoming external momenta is given
     in figure á.
                                                        fields this will not rnix back into Ñ. in à mass
            As mentioned in t,helast section,for massless


                                         ð,                                                Ð4

                         Figure 7: Tadpole graph that vanishesfor masslessôs.

independent renorma!ization scheme,becausethe Feynman graph in figure 7 vanishes. However,
for massive!ight fie!ds the tadpo!e graph doesnot vanish, and we wi!! therefore eva!uatethis type-l
'tadpo!e' graph first.
       3.2. Tadpo!e diagram contribution                         to        Ðà.We need to           do the integra! (see figure 7),
wherer = 1 in (28),andn = 1,v, = 4,ð. = -(ð, + ð2+ Ðç),

                                     IJ J ~
                                       2'      Gâ(ÐI,Ð2.Ðç,Ð4,k)
                                          (271")" (k2 - ò2 + if) .                                                           (53)

No momentum shift is neededsince the denominator is purely quadratic in the loop momentum.
Now Üåñàèçå is analytic, we ñàï deal with the k dependenceÜó writing à symbolic Taylor
       .                                   =
                           Gâ(ÐI,Ð2,Ðç,Ð4,k) ek/q [G6(ÐI,Ð2,Ðç,Ð4,q)]q=î                                                     (54)

       Then we just have to do the dimensionally regularized integral

                                     J              d4-'k                       ek/q

                                             (27Ã)4-'ð-2'            [k2    -     ò2   +    it].

       We again do this Üó manipulating the exponential like à power series, Üåñàèçå basic
assumptionis analyticity in the momenta. Âåñàèçå symmetry, only even terms in k contribute,
çî the integral to Üå done equals
                                00       1        J           ~-'k                     (k/q) 2..

                               ~~                           {21t")4-'Jl-2' - ò2 + it]
                                                                        [k2                                                  (56)

            which fina.lly yields the integra.l(just as in example îÑ last section)

                                               1              J    ~-'k      [k2],"[(/q)1'"
                                ~ 4rr!(r + 1)' J (21Ã)4-'ð-2'[k2-- Lò2 + iE].
                                 00       4

                                ~ 4'"r!(r + 1)! (21Ã)4-'ð-2'[k2 ò2 + it:]'
                                '""            1   à-,.              ,-., J

     After performing à. Wick rotation, a.nd doing the integra.l using (46) and (47) we for the 1fE

tI   ðî!å and the associated ð dependence:
                                . 2            ~   00
                                                               1             [(    ä
                                                                                  "ij";j"   ) ) " +...
                                                                                             2                   (58)

            This ñàï then Üå turned into the (Euclidea.n) a.ngu!ar average of an exponentia!, Üó just

     inverting (16),                                      .
                                                               2   J dn4 òe~

     where å is à Euc!idea.n unit vector, a.nd {!4 = 21Ã2.Now that we have done the !îîð                 integra! and
     extra.cted the lfE ðî!å, we ca.n use the Òàó!î! series to put the form (59) in terms of Gá. The result

     for the ðî!å pa.rt then

                               im2            J       -Ð!        òå))] .
                             321Ã4Åð-2'dÏ: [Gá(Ðt,Ð2,Ðç, - ð2- Ðç,                                               (60)

     Thus w.e have reduced the divergent part of the origina! Feynman graph to an integra! of Gá îóå!
     à finite region (for Euc!idea.n momenta).             Hence we Üàóå obtained the tadpole contribution     to the

     one-loop Ðfunction for the nonloca.linteraction, à4:

                              fJa'r'] (ÐI.Ð2J'Ç) IId;G.[.)(ð"Ð2,ÐÇ)                                              (61)
                                      =:12;i   f        dÏ:G6(Ðl,Ð2,Ðç,-Ðl    -Ð2 -Ðç,òå),

        3.3. 2-2-2 dingram colltributioll                   to ;JG,TII~ \'cYllmall illtcgra'\ from figure 6, with QI =
Ð1 + ]l-J, Q2   = ÐI +~J + Ðý+ Ð4,2:::~~ Ðî = î;             1\11..1V.   = =
                                                                          V2    VÝ   = 2, equ1\ls

                 =        ý(!f~            G4{Ð.,1'2,~)G4(Ðç,Ð4,k+Q1)G.(ÐS,Ð6,k+Q2)                                   62
                     /l      } (2Ir)4(J.,2-m2+if)[(k+Qtf-ò2+if][(k+Q2f-m2+if]'                                    (        )

        Combining denorninatorsand rnaking the shift k                    =l-(QtQt       +Q2Q2),weget the denorninator

                          D   = [å -   Q~Qt(Q1     - 1) - Q~Q2(Q2        - 1) - 2Qt ,Q2atQ2 - m2]Ý                (63)

so that the integral becomes (with à factor îÑ 2 from the Feynrnan trick)

                     = 2J Ï              J ~])Jl     1
                                                                         - (QtQt + Q2Q2»)
                          . - (Qt(at
                                   dQ;                   [G.(Pt,1'2,.t

                                                   - 1) + Q2Q2)G4(Ðs,Ðá,l-             QtQt - Q2(a2 - 1»].

Define the numerator

                                 N(l,p,a)     = [G4(Pl,P2,l- (Ql0l + Q20J)                                        (65)
                                                G4(Ðç,Ð4, (Ql(Ol - 1) + Q202))
                                                    G~(ð5,Ðâ,l- Ql0l - Q2(O2-1))].

       Now, because the Gs are ana!ytic, we ñàï dea! with the (                                      î! G Üó writing à
symbolic Tay\or expansion

                                  N(i,p, à)           elJ. [G4(pl,V'l,Q)                                          (66)
                                                      G4(Ðç,Ð41q + QJ

                                                      G4 (Ð51 Ðá, q + Q2)]     q=-(Q,al+Q2a2)
       and we just have to do the dimensionally regularizedintegral
                                             J        J4-'l          el/o
                                                      ).-'ð-Çñ [i2 - À2+ if]3                                    (67)
where À2   = Q~Ql(Ql -1)                  -                 +
                                  + Q~Q2(Q2 1) + 2Ql . Q2Q1Q2 ò2.
       Again, we do this Üó manipulating the exponentiallike à power series, becausethe basic
assumption is analyticity in the momenta. Because symmetry, only åóån terms in f. contribute,
                                                                                                                                '~ ...

                                                                                                                 (68)               ~



             \vherewe have retained only thc Iff pole and the associaLed dependence.We cannot yet convert
             this sum to an angular integral, hecauseiL starts at r = 1, \vhich just rcnects tlle fact that the
             leading term in the Taylor expansiongives à convergentFeynman integral, with ïî 1ft pole piece.
             Òî put the contrihution of the sum hack into the G's, we attempt to massage further:
                                    ~           1      (A2)r-1
                                                                   ä   l(        ).
                                    f::: 4rr!(r     -
                                                   1)!            ä; J
                                     ä2 [ ~          1       À2 ,-I ,+I  ä 2] r ]     [(          )
                                      =     ~   ~(             ) æ       aq       ~=I

                                        ~ [~~-;:!(~(~)(À2)rær [(~)2]r-                                            ~]~=I

                                      = ~  [~~÷~(~)(À2)rær                                [(~)2]r]~=1                         (69)

             Note that the finite sum subtracted off to redefine the sum to start at zero gets annihilated Üó the

             differential operator.
                    The terms in the last sum (69) àãå exactly îÑ the Corm seen beCore in (16), and it ñàï Üå
             written in terms îÑ à Cour dimensional Euclidean angular integral over à finite range:

     ~                                              ~
                                                          [ ..:..
                                                                    J ~e.;%A'.I.
                                                                       Ï4                 ] ~=l                               (70)

             which ñan Üå put back into the G's Üó inverting the Taylor expansion. The contribution                       to the (Ç
      ~      function for the nonloca! ô6 vertex arising from à graph with three à4 vertices is thus

                                                    [f dÏ4 f dal da2:4";;
                                           1 82        W                         Õ                                            (71)

                                          G4(Pl,P2,v'XAe        - (Qlal      + Q2a2))
                                          G4(Ðç,Ð~, v'XAe       -    (Qlal   + Q2aJ + Ql)
                                          G..(ps,pG, v'XAe      -    Ql(al   -   1)   -   Q2(a2       -   1))
                                                                                                                ] ~al

         I                                  cross
                                          ,                                                  ,
             where å i5 à Euclidean unit vector and the 5quare root îÑ À2 i5 [åà! Cor Euclidean momenta, 50 we

    .-       have written it àç À.



         Alternatively, we ñàï eliminate referenceto the extraneous parameter õ Üó the following
trick. Consider the sum of (68)

                                       ~~;Ö~-=-i)T(À2)r-l                                [(~)2]'
With                                                                                                                                                 ~
                                                                        p=r-l                                                            (73)
we obtain                                                                                                                                            '}

                                        1           )
                             ~ 4P+lp!(p l)!(A-}" l\aq)]                              [( ä;]
                                                                                        ä     2

                           = (ä ) [ ~~+1)!(A2)P
                                           aq          [( aq)
                                                       2     ÑÞ              1                        'ä    2] Ð]                        (75)
                           =                                                                                                                          ~
                               (~)2 [~4PP~"i)1(A2)" [(~)2]"]
                           = ~
                              (~ )2[] ~eA../q
                               (~)                     2
                            ~ aq [] ~eA'.I; '
                            4  4

                                          !1. ]
                                           aq                     {14            ]   '                                                   (76)

          (49) {îò {îèò dimensions,Jd{14 = 2?Ã2, get à òîòå
                                               we                                                      compact [îòò {îò the nonlocal fJ
and using
function:                                                                                                                                            ;~

                                                                                                                                         (77)        j.
                                                           J dn:J dal da2[G4(Pl,P2, + q)
                                                           +q+Ql)                                                                                     '"
                            G4(PS,PG,Ae +                     q + Q2)
                                                                             ] 9=-(0,0,+0202)                                                         "
                            + terms'

The last Cormis Cree extraneousparaIneters.
        We reiterate that the evaluation îÑ the 2                            - 2-        2 diagram ,,:Üîóå highlights two important
       which will Üå crucial for the application îÑ the method to the'gencral casebelo\v. First,
Corgraphs with more than two internallines, the integral îÑ the leading (zeroth) term in the Cormal
Taylor expansiondoes not give à divergence(it is power-collDtingconvergcnt), so the infinite sum
Ñîñ the poles starts only at some higher order: Corà type n graph (n > 1) the first divergent
contribution   com~   Crom the (n      -        2)th       term         in theTaylor       cxpansion,       so the sum îÑ poles starts          at

n - 2. Òî rewrite this sum in terms îÑ the angular integral îóñã à Euclidean unit vector, we ñàï
subtract off à finite sum which vanishesunder the action îÑ the differential operator îÑ order n - 1

       in the dummy parameter õ. In fact, we ñàï do mèñÜbetter - the parameter õ ñàï Üå eliminated
/-"    Üó writing the pole sum à power of [                   (/;)2] acting           îï à 2ï   - 2 dimensionalEuclidean unit angular
       integral of G's. Let us explain why this makessense:For one-loopgraphs with înå or two internal
       !ines, the four dimensiona! Feynman integra! has à divergencewhich begins with the zeroth order
       term in the Tay!or expansion(and higher terms in the Tay!or expansionare even more divergent),
       so the sum of po!es starts at zero. However,for graphs with more than two interna!!ines, enough
       powers of momentum in the numerator are required to ñàïñå! the denominator powers - thus the
       pole sum starts at à higher order. This sum ñàï Üå made to start at the zeroth order
       Üó defining the loop integral in the appropriate                                      number of higher dimensions. Íånñå
       the resummation of the pole terms for à sum starting at zero gives à higher dimensional angular

              3.4. 2m-point function of type ï at one loop For the non-trivia! general graph with
       n ?: 2, after combining denominatorsand shifting the !îîð momenta, we get8 the integra!
                                1"-1 da   ~~ " 4(i).
                                            "' l         1            "-1
                         (n -1)!       î   D ' 1(21r)dDÏIJG:,i+2(...,l-LQoaO+Qi-l)                                              (78)
                                           ,-1                           0=1                           0=1
       whereVi is the number îÑ externallines emanatingCromthe ith vertex, à, are the Feynmanparam-

                                          n-l           n-l
                                D = [Ð - (}:::; Qoao)2 (}:::; Q~ar) - ò2]"                                                      (79)
                                          0=1           ,=1
       is the combined denominator, and
                                                t..(i) = ':Vi                                    (80)
       carries the renormalization scaledependence.The ellipsis denote the dependence each Gi îï the
       external momenta and the momenta Qi ûå defined as in (31).9
                For the renormalization îÑ à 2ò point Cunctionwith n internallines
                                                                        }:::; Vi   = 2ò                                         (82)

         'The overall factor of (ï   - 1)! ñîòåî      ([îò   Ihe Feynm_n     t[ick.

         .Âó momentum conserv_lion -1 Ihe làçt vertex, we ò_ó make the struclure î{ Ihe làçl vertex simpler Üó makjng the

                                                             ï-1                                      ï-1
                                       G~.+,(...,l-          }:::;Q",,+Qï-1)-G~.+,(...,-l+            }:::;Q",,).                (81)
                                                             ,.1                                      ,.1


    ðÀò   where
                                                         ~ò = L Vi (~) = òÅ.                                                                  (83)
    Now, using again the fact that ìl the G's are analytic, we ñàï write
                                              n                             n-1
                                             ÏG'..+2(...             ,1-    L:Q.a. Qi-J
                                                                                 +                                                            (84)
                                             i=1                            .=1

                                      = e~1.                                                                                                         ~
                                                       i=1                                 ] q-- ~"-l Q,à,
                                                                                              - ,,--,.,

            Now we just have to do the dimensionally regularized integral

                                                   J       ~-'!             el/o
                                                        (27Ã)4-'Jl-~m'[(1 - À2 + it]n                                                        (85)
    where                                                .. -                      .. -
                                             À2    =    <2::: QoQo)2        -     <2::: Q:Qr)    + ò2.                                       (86)
                                                         0=1         ,=1                                                                     (86)
            Remembering the analyticity                in momenta, we do this as u~ual Üó manipulating                              the exponen-
    tia!!ike à power series,with only even terms in l contributing. Doing the integra! àç before using
    the key formu!a (46), and expanding the ã.funñtiîn using (47), with
                                                                 f          2
                                      Ã(ï -r           -2 + 2) = ~(-1)2+r-"(2+r -ï)!

    for n ~ r + 2, we get the pole piece
                                         i              ~       (-1)2+r-"(-I)"+r(A2)2+r-"                      [( ä )2] '                             .
                      (ï -l)!~                         '~2           (ï -1)!4rr!(2 + r - ï)!                     ä";]       .

    Quite nicely, the (ï   -   1)! coming from the integration fornmla cancels with the (ï                                      -   1)! appcaring

    from the Feynman trick. The sum (88) ñàï then Üåwritten as
                       87r2p-A"'f 'õ' 4'r!(2 + r - n)! aq 2
                            i            (À2)2+'-n                                   l( a)       j'

                                  i           än-l
                                                            [   00    (À2)2+r-,."ær+l       [(        a)2] r
                                             a;;;::t "~2 4rr!( r + 1)!


                                              än-l      õ         (À2Óõ"
                                             a;;;::t[ (À2ð 00 4rr!(r + 1)! [( oq 2] " - ò ] ~=l
                                                                                 ] ~=l



        taken to Üå vanishing for n < 3. Àç in the làçt example, this finite çnò is snbtracted off to make
        the first sum start at r    = Î.     Now the differential operator is î! order n -1, and the finite çnò is î!
        order n   - 2, âî the linite çnò         gets annihilated Üó the differential op~rator, giving finally the pole

        piece:                81Ã211-ÀòÅ
                                                a;;=t [ ~ Z
                                                                            ~   4rr!( r + 1)!
                                                                                  (A2)rZr                   [( aq
                                                                                                               ä    ) ] ']
                                                                                                                         2           .
                                                                                                                                   %=1   (91)

        Now, inverting àç nçnà! nsing (16) for à Enclidean nnit vector, we òàó ñàçt this pole contribution
        àç àn integral over à finite fonr-dimensional                  Enclidean angular region:

                                   Pole =                 i         -0"-1                                                                (92)
                                                 8?Ã2Åð-ò, äæ..-1
                                             [1 dÏ: ;=..-1   æ
                                                W 10 JJ da;(A2F2
f                                        Ï ~o+2
                                                              ""     .j"iAe -    (I: Q.a.) Q;-l
                                                                                              )] =1
        which gives the   .8function     in the forml0
                                   'âG2m!."...ï](Ð"...,Ð2m-,)          =
 1                                       --1 8"-1
                                         -- 1 8"-1                    l
                                                                [1 dW 1 1
                                                                   dn4 li=n-1 da.-=-
                                         167Ã~ äõn-1                     å î
                                                                         å î     Ö              J (A2)n-2

.                                        Ï d.;+2
                                                              '."    .j'iAe - (I: Q.a.)+ Qi-I
                                                                                  .=1                               )]       ~=I
\                                        +


                                                     0':.+2(...' .;"iAe - (L          QO"O)     +   QO-l)

                                                          =     a=.+2(      ;"iAe + (LQO"O»).                                            (93)

      dimensional angular integral. Referring back to (88), we ñàï write
                                                    00                         (A2)2+r-n                       [(    ä   )2] '

                                                   '~2 4rr!(2+ r - n)!                                              aq                                                       (95)
      in terms of
                                                                               ð = r - n + 2,                                                                                (96)
                                                                                         n ?; 3,                                                                             (97)
                                                                               (À2)"                           [(        a)  2


                                     =                                                                                                                                                .
                                          ~                       [(~)2]"_2                           [ f:~~~l~~)j~ -
                                                                                                        ð=Î 4Ðð.(ð + n                    2)!   ]                                     I
                                              1                   [       ä       2] ..-2                    dW..-2                                                                  '-'
                                     =   ~                            (à÷)                        [j~eA..I.]                                                                 (99)
" ~                              ,
      50 that we fina11y get the pole
                                 Pole    =                            i                   [       -ä         2] ..-2

                                         dË~"-2 11;=-1 da;
                                      [1 ~          Ï                                                                                                                               ~
                                                                              î     ;=1

                                      Ï~;+2(...,Àå+q+Qi_J                                                                     .-1
                                                                                                                    ] 9=-(}:::.., Q
                                      0=1                                                                                                           )

      which, using (49) yields the contribution                               to the {J function in terms of à 2n                                       - 2 dimensional   angular

1:    integral over afinite   Euclidean region:                                                                                                                                     1,



                                               (JG2M(., I'2M-l) =

                                                                        [( )
                                                                 .!!.- 2] "-2           [J dfl2n-2 11;=      ï..-1dà"

      f                                                "4"'/f..+1          aq                    å                        J
      .                                                                                                 Î    ;=1

     I                                                 Ï~;+2(...,Àå+q+Q;-1)
                                                       ;=1                                       ] q=-(E;~: Q.a.)
                                                       +            .

                       Expressions (94), (93) and (101) for the fJ function of an arbitrary                               nonlocal the
         j    most genera.l results of this paper}l
              4.     Concluding        Remarks

      .                We have illustrated        explicitly            how to obtain the renormalization               group coeflicients in à nîn-
     Ã,       local scalac field theory. Note that our general results were computed with massive fields, and Cor

     -        this reason techniques such as the Gegenbauer polynomial                               method îÑ 5, valid only for massless

      ..      propagators, àñå not useful here.

       j               We regard our results as suggestive, but not final.                           We have obtained integro-differential
      ;       renormalization       group equations Üó the brute-force technique of exanding, renormalizing and then
              resumming. We would like to find the rule that allows us to do this in înå step, rather than three.
              1t would Üå interesting          if, Üó modiCying the analytic structure îÑ the nonlocal theory, we could
              actually give à one-step prescription to isolate the divergent part of the Feynman integral, without
              ever expanding in terms îÑ à Cormal TayJor expansion. Work along these lines is in progress á.
                       We would like to thank Petcr ÑÜî, Mike Dugan and Âån Grinstein for numerous discussions.


                1. s. Weinberg, Physica 9áÀ (1979) 327. À. Manohar and Í. Georgi, Nucl. Phys. Â2Ç4, 1!!9

               "It   is verified easily that \vith n   = 2,m = 2 the         IJ function calculation for the nonlocal ô' coupling, ..-in section 1, is
              reproduced. n = 3,m = 3, andn = 1,m = 2, the previous      for                      graph(2 - 2
                                                                  results the ô' maximallyconvergent                                         - 2)and

2. Í. Gcorgi, HUTP-91/ ÀÎ14.

3. Ñ. Bloch, Dan. Mat. Fys. Medd. 27, 8 (1952); Ð. Kristensen and Ñ. Moller, Dan. Mat. Fys.
  Medd. 27, 7 (1952); D.A. Kirzhnits, Sov. Phys. Usp. 9, 692 (1967) and referencestherein; Ì.
  Chretien and R.E. Pierls, Proc. Roy. Soc. À22Ç, 468 (1954).
4. Ð. Cho, HUTP-91/ ÀÎ46 to appear in Nuc1.Phys. Â.

5. K.G. Chetyrkin and F.V. Tkachov, Nuc1.Phys. Â192(1981)159; K.G. Chetyrkin, A.L. Kataev
  and F.V. Tkachov, Nuc1.Phys. ÂI74(1980) 345.

6. Ó. Bhansali, H.Georgi, under preparation.


Shared By: