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Running Nonlocal Lagrangians∗†

arXiv:hep-ph/9205242v1 28 May 1992#HUTP-92/A0605/92Running Nonlocal Lagrangians∗†Vineer Bhansali and Howard GeorgiLyman Laboratory of Physics, Harvard UniversityCambridge, MA 02138AbstractWe investigate the renormalization of “nonlocal” interactions in an effective fieldtheory using dimensional regularization with minimal subtraction. In a scalar fieldtheory, we write an integro-differential renormalization group equation for every pos-sible class of graph at one loop order.∗Research supported in part by the National Science Foundation under Grant #PHY-8714654.†Research supported in part by the Texas National Research Laboratory Commission, under Grant #RGFY9106.

1IntroductionIn its traditional form, an effective field theory calculation goes like this: Start at a very large scale,that is with the renormalization scale, µ, very large. In a strongly interacting theory or a theory withunknown physics at high energy, this starting scale should be sufficiently large that nonrenormaliz-able interactions produced at higher scales are too small to be relevant.

In a renormalizable, weaklyinteracting theory, one starts at a scale above the masses of all the particles, where the effectivetheory is given simply by the renormalizable theory, with no nonrenormalizable terms. The theoryis then evolved down to lower scales.

As long as no particle masses are encountered, this evolutionis described by the renormalization group. However, when µ goes below the mass, Λ, of one of theparticles in the theory, we must change the effective theory to a new theory without that particle.In the process, the parameters of the theory change, and new, nonrenormalizable interactions maybe introduced.

Both the changes in existing parameters, and the coefficients of the new interactionsare computed by “matching” the physics just below the boundary in the two theories. It is thisprocess that determines the relative sizes of the nonrenormalizable terms associated with the heavyparticles.Because matching is done for µ ≈Λ, the rule for the size of the coefficients of the new operatorsis simple for µ ≈Λ.

At this scale, all the new contributions scale with Λ to the appropriate power(set by dimensional analysis) up to factors of coupling constants, group theory or counting factorsand loop factors (of 16π2, etc.) [1].

Then when the new effective theory is evolved down to smaller µ,the renormalization group introduces additional factors into the coefficients. Thus a heavy particlemass appears in the parameters of an effective field theory in two ways.

There is power dependenceon the mass that arises from matching conditions. There is also logarithmic dependence that arisesfrom the renormalization group.The matching correction at tree level is simply a difference between a calculation in the fulltheory and a calculation in the low energy effective theoryZδL0(Φ) = S0LH+L(Φ) −S0L(Φ)=Z nvirtual heavyparticle treeso(Φ)(1)where S0LH+L(Φ) denotes the light particle effective action in the full theory and S0L(Φ) denotesthe same in the low energy theory [2].

The matching correction so obtained is nonlocal because2

it depends on p/Λ through the virtual heavy particle propagators.It is also analytic in p/Λin the region relevant to the low energy theory, i.e. for characteristic momentum << Λ. Thusit can be expanded in powers of p/Λ with the higher order terms decreasing in importance: thiscorresponds to a local operator product expansion in the domain of analyticity, equivalent to a localnonrenormalizable Lagrangian which can be treated as an honest-to-goodness local field theory.However in general an infinite series of terms of increasingly higher dimension are generated bymatching at tree level.

These cause no problem when the scales are well separated, because theireffects quickly become negligible. But if there are two or more scales close together (such as mt andMZ may be), then we may not be justified in ignoring terms at higher orders in the expansion.

Howdoes one understand how to interpolate smoothly between the well-understood situation in whichthe scales are very different and the well-understood situation in which the scales are very closetogether? How can we keep track of all the infinite number of higher derivative operators efficiently?Is it possible to deal with the nonlocal effective Lagrangian directly, without expanding?

In theparticular context of a scalar field theory, we will attempt to answer questions at the one looplevel. The approach will be direct.

We will manipulate the nonlocal interactions as if they areexpanded in a momentum expansion, and then show that the resulting β functions for the terms inthe momentum expansion can be collected into integro-differential renormalization group equationsfor the nonlocal couplings.The present paper is organized as follows: in section 2 we calculate the β function for a nonlocalfour point coupling arising from a one-loop graph with two internal lines in nonlocal φ4 theory.We give this example before the general results of the following sections in order to point out theimportant features of our method. In section 3 the method is then applied to obtain the contributionto the renormalization group equation for a general class of graphs in an arbitrary massive nonlocal,non-renormalizable parity invariant scalar field theory.2Basic ExampleThe matrix elements in a general nonlocal field theory [3] are calculated by writing down a La-grangian with ‘smeared’ vertices.

For instance, for a nonlocal φ4 interaction in a scalar field theorywith the discrete symmetry φ →−φ, the interaction term in the action isS4 =Zdx1dx2dx3dx4F(x1 −x4, x2 −x4, x3 −x4)φ(x1)φ(x2)φ(x3)φ(x4)(2)3

✒❅❅❘❅❅❅❅❅❅■❅❅❅❅✠p3p1p2p4✇−iG(p1, p2, p3)Figure 1: Basic nonlocal φ4 vertex.where xi are spacetime coordinates and F is a nonlocal ‘form-factor’. Energy-momentum conser-vation at each vertex of the corresponding Feynman graph is expressed in terms of the Fouriertransform G of F: G = G(p1, p2, p3) for the basic φ4 interaction shown in figure 1.

Bose symmetryimplies that the nonlocal coupling G is symmetric and satisfiesG(p1, p2, p3) = G(p1, p2, −p1 −p2 −p3) . (3)Of course, Lorentz invariance dictates that in the final expression for the matrix element in mo-mentum space, only scalar products of momenta will appear as arguments of G. Implicit in thedefinition of G is a mass scale Λ which sets the limit for the region of analyticity of G, and forcharacteristic momenta p < Λ, G is analytic.

We go to the effective low energy theory by expandingin a Taylor expansion in p/Λ. One can think of this Taylor expansion as the formal implementationof a local operator product expansion of G. At this point, in going to the effective low energytheory, we have actually changed the high energy behavior of the theory so that integrals whichwere convergent in the full theory are divergent in the effective theory.

This trades logs of Λ in thefull theory for anomalous dimensions in the low energy theory. This trade allows us to calculatethe logs more simply and to sum them using the the renormalization group.Let us first discuss as an example the renormalization of the nonlocal φ4 interaction from thegraph shown in figure 2.For massless fields, this is the only contribution in one loop.In themassive case, there is also a contribution from a tadpole graph.

In addition, in either case, thereare renormalization of φ2k interactions for k > 2. We will systematically consider them in the nextsection.4

................................................................................................................................................................................................................✒❅❅❘❅❅❅❅❅❅■❅❅❅❅✠✛✲p3p1p2p4k + p1 + p2k✇✇−iG(p1, p2, k)−iG(p3, p4, k + p1 + p2)Figure 2: Feynman graph contributing to the renormalization of G.The Feynman integral for the graph in figure 2 is−Zd4k(2π)4G(p1, p2, k) G(p3, p4, k + p1 + p2)[k2 −m2 + iǫ][(k + p1 + p2)2 −m2 + iǫ](4)Combine denominators1[k2 −m2 + iǫ][(k + p1 + p2)2 −m2 + iǫ]=Z 10 dα1[(1 −α)k2 + α(k + p1 + p2)2 −m2 + iǫ]2(5)and shift momentak = ℓ−α(p1 + p2)(6)to obtain−Z 10 dαZd4ℓ(2π)4G(p1, p2, ℓ−α(p1 + p2)) G(p3, p4, ℓ+ (1 −α)(p1 + p2))[ℓ2 + α(1 −α)(p1 + p2)2 −m2 + iǫ]2(7)Now, here is the crucial point. We get the low energy effective theory by expandingthe Gs in a momentum expansion.

An equivalent procedure is to treat the Gs as if theywere analytic everywhere in momentum space. Thus we can deal with the ℓdependence ofG by writing a symbolic Taylor expansion1 Here we are effectively doing the momentum1Analyticity of the nonlocal couplings may alternatively be exploited by formally Laplace transforming in theloop dependent arguments, which yields instead of eℓ· ∂∂q G an exponential e−ℓ·s ˜G in terms of the Laplace transformof G, along with an extra Laplace inversion integral.

The rest of the computation is essentially identical.5

expansion. But the key is that we can resum the final result into an finite integral overthe original nonlocal couplings.G(p1, p2, ℓ−α(p1 + p2)) G(p3, p4, ℓ+ (1 −α)(p1 + p2))= eℓ∂∂q [G(p1, p2, q) G(p3, p4, q + p1 + p2)]q=−α(p1+p2)(8)Now all the dependence on the loop momentum is in the denominators and in the exponential,so we just have to do the Feynman integral of the exponentialZd4ℓ(2π)4eℓ∂∂q[ℓ2 + α(1 −α)(p1 + p2)2 −m2 + iǫ]2(9)or to be more precise, the dimensionally regularized integralZd4−ǫℓ(2π)4−ǫµ−2ǫeℓ∂∂q[ℓ2 + α(1 −α)(p1 + p2)2 −m2 + iǫ]2(10)We will do this by manipulating the exponential like a power series, because the basic assumptionis analyticity in the momenta.

It is at this point that we have irrevocably changed the high energybehavior and gone over the effective low energy theory. Because of symmetry, only even terms in ℓcontribute,∞Xr=01(2r)!Zd4−ǫℓ(2π)4−ǫµ−2ǫℓ∂∂q2r[ℓ2 + α(1 −α)(p1 + p2)2 −m2 + iǫ]2(11)We calculate this as followsZdΩℓΩℓµ1ℓµ2 · · · ℓµ2r= Arℓ2r(2r)!/(2rr!) termsz}|{hgµ1µ2 · · · gµ2r−1µ2r + permsi.(12)whereΩ≡ZdΩℓ.

(13)Contracting with gµ1µ2 givesAr−1 = (4 + 2(r −1)) Ar = 2(r + 1) Ar(14)orAr =12r(r + 1)! (15)6

and thus1(2r)!ZdΩℓΩ ℓ∂∂q!2r=14rr! (r + 1)!hℓ2ir ∂∂q!2r.

(16)So the integral is∞Xr=014rr! (r + 1)!Zd4−ǫℓ(2π)4−ǫµ−2ǫ[ℓ2]r∂∂q2r[ℓ2 + α(1 −α)(p1 + p2)2 −m2 + iǫ]2(17)which we can calculate in the usual way.

Wick rotatei∞Xr=0(−1)r4rr! (r + 1)!Zd4−ǫℓ(2π)4−ǫµ−2ǫ[ℓ2]r∂∂q2r[ℓ2 + A(α)2]2(18)whereA(α)2 = −α(1 −α)(p1 + p2)2 + m2 ,(19)and A(α)2 is positive for Euclidean momenta.

Now doing the ℓintegral gives2i(4π)2ǫµ−2ǫ∞Xr=014rr!2 A(α)2r ∂∂q!2r+ · · ·(20)where we have dropped everything but the 1/ǫ pole and the associated µ dependence. We can writethe result as2i(4π)2ǫµ−2ǫ∂∂x∞Xr=0x4rr!

(r + 1)! [−xA(α)2]r ∂∂q!2rx=1.

(21)This can then be turned back into the (Euclidean) angular average of an exponential, just inverting(16),2i(4π)2ǫµ−2ǫ∂∂x"Z dΩeΩx e√xA(α) e ∂∂q#x=1(22)where e is a Euclidean unit vector. Now that we have done the ℓintegral and extracted the 1/ǫpole, we can use the Taylor series to put the form (22) in terms of the Gs.

The result isi(8π)2ǫµ−2ǫ∂∂x"Z dΩeΩZ 10 dα xhG(p1, p2, √xA(α)e −α(p1 + p2))G(p3, −p1 −p2 −p3, √xA(α)e + (1 −α)(p1 + p2))i#x=1(23)with the derivative evaluated at x = 1. Thus we have reduced the divergent part of the originalFeynman graph to an integral of the Gs and their derivatives over finite ranges (for Euclideanmomenta).These last integrals are well-behaved and could be done numerically.Hence, with7

................................................................................................................................................................................................................❅❅❅❅❅❅❅❅✇✇✇❅❅❅❅❅Figure 3: Feynman graph contributing to the running of φ6 coupling.Ω= 2π2 in for Euclidean four space, we obtain the one-loop β function for the nonlocal interaction,G:βG(p1,p2,p3) =116π4∂∂x"ZdΩeZ 10 dα xhG(p1, p2, √xA(α)e −α(p1 + p2))G(p3, −p1 −p2 −p3, √xA(α)e + (1 −α)(p1 + p2))i#x=1+ crossterms(24)Now in the massless case, where there are no other contributions to the renormalization of theφ4 coupling, the “running” nonlocal coupling satisfies the integrodifferential renormalization groupequationµ ∂∂µ G(p1, p2, p3) = βG(p1,p2,p3). (25)A useful check on this result is obtained by going to the ‘local limit’, i.e.

G →g where g is theusual coupling constant of local φ4 theory. Indeed, we know that the contribution to the β functionin local φ4 theory at one loop isβ(g) = 3g216π2 > 0.

(26)Substituting G = g, inserting an explicit symmetry factor of 1/2, and summing over the crossedgraphs, (24) indeed yieldsβG(p1,p2,p3) →β(g)asG →g. (27)Now the nonlocal quartic coupling G can induce changes in the coupling terms with more fields.For instance, the one-loop diagram relevant to the renormalization of the φ6 coupling is shown infigure 3.8

................................................................................................................................................................................................................✇❅❅❅❅❅❆❆❆❆❆✁✁✁✁✁Figure 4: Feynman graph that vanishes for massless φs.For massless fields, this does not mix back into G in a mass independent renormalization scheme,because the relevant Feynman graph of figure 4 vanishes. However, in the massive case, the graphof figure 4 gives a non-vanishing contribution.

We will now systematically compute the β functionsof the general nonlocal couplings in a massive nonrenormalizable theory.3General ResultsWorking within the formalism of a massive nonlocal effective theory induced by some unknownfull theory, we are forced to consider an effective Lagrangian with operators with an arbitrarynumber of low energy fields. In this section, we will compute the one-loop running of a general non-renormalizable, nonlocal scalar effective theory with φ →−φ symmetry.

Specifically, our purposeis to isolate the dimensional regularization pole of a 2m point function of type n (i.e.with nvertices or propagators) at one loop. The plan is as follows: we first construct the expressionscorresponding to the Feynman integral of a general graph with a special choice of momentumlabelling conventions.

We then describe the application of the method of the last section to tworelevant examples: the tadpole renormalization of the nonlocal ‘φ4’ coupling with a φ6 operatorinsertion, and the renormalization of a nonlocal φ6 coupling with three φ4 operator insertions. Thelast example is useful in highlighting the special features of renormalizing graphs with more than twointernal lines.

Finally, we attack the general case, and give the complete expression for the 1/ǫ poleof the 2m point function of type n at one loop as a surprisingly compact multi-Feynman-parameter,multi-dimensional angular integral.9

....................................................................................................................................................................................................................................................................................................................✇✇✇✇✇✇✇kk + Q1k + Q2k + Q3k + Qn−1k + Q4k + Q5k + Q6PnP2P4P6P1P3P5· · ·✛Figure 5: Type-n Feynman graph. The blobs signify arbitrary φ2r insertions.3.1PreliminariesWe assume that the dimensionally continued d = 4 −ǫ dimensional non-local Lagrangian has aφ →−φ symmetry and hence has an interaction term proportional toLint.

=∞Xr=1µǫ(r−1)G2rφ2r. (28)The mass scale µ is introduced to keep the dimensions of the nonlocal couplings G2r fixed underdimensional continuation.

The interaction is not normal ordered2, and each G2r is some nonlocalfunction which is analytic in the region under consideration, depends as a consequence of momentumconservation on 2r −1 linearly independent momenta, and may have dimensions proportional tosome power of an implicit scale of nonlocality Λ.A diagram with n internal lines will be called type-n. At one loop, a type-n graph has n verticesconnected to external lines. LetN =nXi=1vi ,(29)where vi is the number of external lines emanating from the ith vertex.

The loop integral for thetype-n one-loop renormalization of the N point function (shown in figure 5) isI =Zd4k(2π)4Qni=1 Givi+2(. .

. , k + Qi−1)(k2)(k + Q1)2 .

. .

(k + Qn−1)2. (30)where the upper index on the G′s is used to number the vertices as one goes along the loop.2Hence fields at the same point can be contracted, and tadpoles will occur explicitly.10

Here all external momenta are taken to be incoming (signified by . .

., in the numerator, differentfor different Gi) and we have definedQ0=0(31)Q1=P1Q2=P2 + Q1Q3=P3 + Q2.........Qi=Pi + Qi−1 =j=iXj=1PjQn=Qn−1 + Pn = 0,and Pi is the sum of the external momenta flowing into the ith vertex. Energy momentum conser-vation is simply Qn = 0.

Symmetry factors are suppressed.Using this notation, the loop integral for the for the φ4 coupling that we computed in the lastsection is obtained by setting n = 2, v1 = 2, v2 = 2:I =Zd4k(2π)4G14(p1, p2, k)G24(p3, p4, k + Q1)(k2 + iǫ) [(k + Q1)2 + iǫ]. (32)Using the generalized Feynman identity1Aρ11 Aρ22 .

. .

Aρnn = Γ(ρ1 + ρ2 + . .

. + ρn)Γ(ρ1)Γ(ρ2) .

. .

Γ(ρn)(33)×Z 10 dα1 . .

. dαnαρ1−11αρ2−12.

. .

αρn−1nδ(1 −α1 −α2 −. .

. −αn)(α1A1 + α2A2 + .

. .

+ αnAn)ρ1+ρ2+...+ρn,combine the denominators1(k2)(k + Q1)2 . .

. (k + Qn−1)2(34)=(n −1)!Z Yidαi1[k2(1 −Pn−1i=1 αi) + Pn−1r=1(k + Qr)2αr]n,and shift the loop momentumk = ℓ−n−1Xs=1Qsαs(35)so thatk2 = ℓ2 + (n−1Xs=1Qsαs)2 −2ℓ·n−1Xs=1Qsαs,(36)11

which, when put into the denominator givesD=[ℓ2 + (n−1Xs=1Qsαs)2 −2ℓ·n−1Xs=1Qsαs −ℓ2n−1Xt=1αt(37)−(n−1Xs=1Qsαs)2n−1Xt=1αt + (2ℓ·n−1Xs=1Qsαs)(n−1Xt=1αt) +n−1Xr=1ℓ2αr+n−1Xr=1(n−1Xs=1Qsαs)2αr −2n−1Xr=1ℓ·n−1Xs=1Qsαsαr +n−1Xr=1Q2rαr+2n−1Xr=1ℓ· Qrαr −2n−1Xr=1Qr(n−1Xs=1Qsαs)αr]n.After cancellation (37) yields the denominatorD = [ℓ2 −A2]n(38)whereA2 ≡(n−1Xs=1Qsαs)2 −n−1Xr=1Q2rαr. (39)For the massive case, we simply substitute in (34)k2→k2 −m2(40)(k + Qr)2→(k + Qr)2 −m2(41)so that we just get an additional term in (37) when combining denominators after the shift (35):−m2(1 −n−1Xs=1αs) +n−1Xs=1(−m2)αr = −m2.

(42)So the general form for the denominator of the right hand side of (34) with massive fields isD = [k2 −(n−1Xs=1Qsαs)2 + (n−1Xr=1Q2rαr) −m2]n.(43)Now the shift (35) changes the argument of the numerator factors in (30) also, to yield the finalexpression for the Feynman integral3(n −1)!Z n−1Yj=1dαjZddℓ(2π)dQni=1 Givi+2(. .

. , ℓ−Pn−1s=1 Qsαs + Qi−1)[ℓ2 −(Pn−1s=1 Qsαs)2 + (Pn−1r=1 Q2rαr) −m2 + iǫ]n(45)3The reader may check that with n = 2, v1 = v2 = 2, the above relations give the correct integral for the exampleof the first section:ZdαZddℓ(2π)dG4(p1, p2, ℓ−α(p1 + p2))G4(p3, p4, ℓ+ (1 −α)(p1 + p2))[ℓ2 + α(1 −α)Q21 −m2]2(44)with Q1 = p1 + p2.12

Important remarks on notation : (1) In intermediate steps of the computations, we denote thedependence of each nonlocal function on the (distinct!) external momenta by ellipsis.

This is usefulsince the external momenta play a trivial part in the loop integration, and may be reinstated byexamination in the final expressions. (2) Also, since at one loop any vertex shares two and only twolines with the loop, by energy momentum conservation the loop momentum appears only once asan argument of any Gi and will be put in its last slot.

(3) The first vertex will have only the loopmomentum as its last entry.Crossing : The external momenta can be exchanged amongst themselves.The final result,however, depends only on the number of distinct Lorentz invariants of the four momenta pi (i = 2mfor renormalization of the 2m point function) under the condition P2mi=1 pi = 0 and p2i = m2. Thisequals the total number of graphs related by ‘crossing’ where the crossed graphs can be obtainedby exchanging external momenta.

We will give explicit expressions only for one member of eachcrossed set.Counting i′s : Ignoring for the moment the i′s appearing due to Wick rotation and integration(see below), the integrand itself gives no powers of i. This is seen as follows: each propagator isip2−m2+iǫ, and each nonlocal vertex has a Feynman rule −iG2r.

Since the number of vertices equalsthe number of internal lines at one loop, for a type-n graph we obtain (−i)nin = (−1)n(i)2n =(−1)2n = 1. So the only source of i’s and minus signs is the integration formula.Computational Tools :In order to do the dimensionally regularized loop integrals we will need the well-known integra-tion formula (for Euclidean momenta):Zddq(q2)r(q2 −A2)n = iπd/2(−1)n+r(A2)r−n+ d2 Γ(r + d2)Γ(n −r −d2)Γ( d2)Γ(n),(46)and the expansionΓ(−n + ǫ) = (−1)nn!

[1ǫ + (1 + 12 + . .

. + 1n −γ) + O(ǫ)],(47)where γ = 0.5772157 is Euler’s constant.Also note that in 2κ −2 dimensions (κ a positive integer)ZdΩ2κ−2ℓΩ2κ−2 ℓµ1ℓµ2 · · · ℓµ2r= A2κ−2rℓ2r(2r)!/(2rr!) terms,z}|{hgµ1µ2 · · ·gµ2r−1µ2r + permsi(48)13

where we have definedΩd ≡ZdΩd = 2πd/2Γ(d/2) . (49)Contracting with gµ1µ2 givesA2κ−2r−1 = [2κ −2 + 2(r −1)] A2κ−2r= (2κ + 2r −4) A2κ−2r(50)SoA2κ−2r=12r(κ + r −2)!

(51)and thus in 2κ −2 dimensions1(2r)!dΩ2κ−2ℓΩ2κ−2 ℓ∂∂q!2r=14rr! (r + κ −2)!hℓ2ir ∂∂q!2r.

(52)The importance of the result of (52) will be seen when we isolate the pole pieces appearing fromgraphs with more than two internal lines.The point is that the leading terms in the Taylorexpansion then give Feynman integrals which are manifestly convergent by power counting, anddivergences appear only at some higher order in the Taylor expansion. This implies that the sum ofdimensional regularization poles does not begin at zero, and we cannot immediately write the resultas an angular integral over a 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 bysubtracting offa finite sum (which vanishes under the action of differential operators in the dummyparameters).

Alternatively, using (52), we can re-sum the poles for a graph with n > 2 internallines, without reference to any dummy variables , in terms of a 2n −2 dimensional angularintegral! The example of the 2 −2 −2 graph below will give the explicit details of how this is done.The φ6 operator can be renormalized at one loop by a type-1 (tadpole) with a G8 couplinginsertion, a type-2 graph with G6, G4 insertions, or the most convergent graph, of type-3, withthree G4 insertions.

For the last one, the relevant graph with incoming external momenta is givenin figure 6.As mentioned in the last section, for massless fields this will not mix back into G4 in a massindependent renormalization scheme, because the Feynman graph in figure 7 vanishes. However,for massive light fields the tadpole graph does not vanish, and we will therefore evaluate this type-1‘tadpole’ graph first.14

.......................................................................................................................................................................................................................................................❅❅❅❅❅❅❅❅✇✇✇❅❅❅❅❅p1p2p3p4p5p6−iG4(p1, p2, k)−iG4(p5, p6, k + Q2)−iG4(p3, p4, k + Q1)k✛Figure 6: Type-3 Feynman graph contributing to βG6................................................................................................................................................................................................................✇❅❅❅❅❅❆❆❆❆❆✁✁✁✁✁p1p2p3p4−iG6(p1, p2, p3, p4, k)k✛Figure 7: Tadpole graph that vanishes for massless φs.15

3.2Tadpole diagram contribution to βG4We need to do the integral (see figure 7), where r = 1 in (28), and n = 1, v1 = 4, p4 = −(p1+p2+p3),µ2ǫZddk(2π)dG6(p1, p2, p3, p4, k)(k2 −m2 + iǫ). (53)No momentum shift is needed since the denominator is purely quadratic in the loop momentum.Now because G6 is analytic, we can deal with the k dependence by writing a symbolic TaylorexpansionG6(p1, p2, p3, p4, k) = ek ∂∂q [G6(p1, p2, p3, p4, q)]q=0(54)Then we just have to do the dimensionally regularized integralZd4−ǫk(2π)4−ǫµ−2ǫek ∂∂q[k2 −m2 + iǫ].

(55)We again do this by manipulating the exponential like a power series, because the basic assump-tion is analyticity in the momenta. Because of symmetry, only even terms in k contribute, so theintegral to be done equals∞Xr=01(2r)!Zd4−ǫk(2π)4−ǫµ−2ǫk ∂∂q2r[k2 −m2 + iǫ](56)which finally yields the integral (just as in example of last section)∞Xr=014rr!

(r + 1)!Zd4−ǫk(2π)4−ǫµ−2ǫ[k2]r∂∂q2r[k2 −m2 + iǫ]. (57)After performing a Wick rotation, and doing the integral using (46) and (47) we obtain for the 1/ǫpole and the associated µ dependence:im2(4π)2ǫµ−2ǫ∞Xr=014rr!

(r + 1)! [m2]r ∂∂q!2r+ · · ·(58)This can then be turned back into the (Euclidean) angular average of an exponential, by justinverting (16),im2(4π)2ǫµ−2ǫZdΩ4eΩ4 em e ∂∂q(59)where e is a Euclidean unit vector, and Ω4 = 2π2.

Now that we have done the loop integral andextracted the 1/ǫ pole, we can use the Taylor series to put the form (59) in terms of G6. The resultfor the pole part then equalsim232π4ǫµ−2ǫZdΩ4e [G6(p1, p2, p3, −p1 −p2 −p3, me))] .

(60)16

Thus we have reduced the divergent part of the original Feynman graph to an integral of G6 overa finite region (for Euclidean momenta). Hence we have obtained the tadpole contribution to theone-loop β function for the nonlocal interaction, G4:βG4[4](p1,p2,p3) ≡µ ddµG4[4](p1, p2, p3)(61)= m232π4ZdΩ4eG6(p1, p2, p3, −p1 −p2 −p3, me),where the subscript on G on the LHS denotes the coupling that is renormalized, with the numberof external lines at each vertex for the graph under consideration in square brackets (i.e.

type ofgraph considered).43.32-2-2 diagram contribution to βG6The Feynman integral from figure 6, with Q1 = p1 + p2, Q2 = p1 + p2 + p3 + p4,Pi=6i=1 pi = 0; andv1 = v2 = v3 = 2, equals= µ3ǫZd4k(2π)4G4(p1, p2, k) G4(p3, p4, k + Q1) G4(p5, p6, k + Q2)(k2 −m2 + iǫ)[(k + Q1)2 −m2 + iǫ][(k + Q2)2 −m2 + iǫ]. (62)Combining denominators and making the shift k = ℓ−(Q1α1 + Q2α2), we get the denominatorD =hℓ2 −Q21α1(α1 −1) −Q22α2(α2 −1) −2Q1 · Q2α1α2 −m2i3(63)so that the integral becomes (with a factor of 2 from the Feynman trick)= 2Z YidαiZddℓ(2π)d1DhG4(p1, p2, ℓ−(Q1α1 + Q2α2))(64)G4(p3, p4, ℓ−(Q1(α1 −1) + Q2α2))G4(p5, p6, ℓ−Q1α1 −Q2(α2 −1))i.Define the numeratorN(ℓ, p, α)=[G4(p1, p2, ℓ−(Q1α1 + Q2α2))(65)G4(p3, p4, ℓ−(Q1(α1 −1) + Q2α2))G4(p5, p6, ℓ−Q1α1 −Q2(α2 −1))].4Also note than in the local limit, replacing G6 by a ‘constant’ g6, we get βg4[4] = m2g616π2 which may be directlyobtained in the local limit as a check.17

Now, because the Gs are analytic, we can deal with the ℓdependence of G by writing a symbolicTaylor expansionN(ℓ, p, α)=eℓ∂∂qhG4(p1, p2, q)(66)G4(p3, p4, q + Q1)G4(p5, p6, q + Q2)iq=−(Q1α1+Q2α2)and we just have to do the dimensionally regularized integralZd4−ǫℓ(2π)4−ǫµ−3ǫeℓ∂∂q[ℓ2 −A2 + iǫ]3(67)where A2 = Q21α1(α1 −1) + Q22α2(α2 −1) + 2Q1 · Q2α1α2 + m2.Again, we do this by manipulating the exponential like a power series, because the basic as-sumption is analyticity in the momenta. Because of symmetry, only even terms in ℓcontribute, anddoing the ℓintegral yields a sum for the pole pieces5:i8π2ǫµ−3ǫ∞Xr=114rr!

(r −1)! (A2)r−1 ∂∂q!2r(68)where we have retained only the 1/ǫ pole and the associated µ dependence.

We cannot yet convertthis sum to an angular integral, because it starts at r = 1, which just reflects the fact that theleading term in the Taylor expansion gives a convergent Feynman integral, with no 1/ǫ pole piece.To put the contribution of the sum back into the G’s, we attempt to massage it further:∞Xr=114rr! (r −1)!

(A2)r−1 ∂∂q!2r=∂2∂x2∞Xr=114rr! (r + 1)!

(A2)r−1xr+1 ∂∂q!2rx=1=∂2∂x2∞Xr=014rr! (r + 1)!

( xA2)(A2)rxr ∂∂q!2r−xA2x=1=∂2∂x2∞Xr=014rr! (r + 1)!

( xA2)(A2)rxr ∂∂q!2rx=1(69)Note that the finite sum subtracted offto redefine the sum to start at zero gets annihilated by thedifferential operator.5Note also that the factor of 2/ǫ gets compensated by a factor of 1/2 from a Γ function18

The terms in the last sum (69) are exactly of the form seen before in (16), and it can be writtenin terms of a four dimensional Euclidean angular integral over a finite range:∂2∂x2" xA2Z dΩ4eΩ4 e√xAe· ∂∂q#x=1(70)which can be put back into the G’s by inverting the Taylor expansion. The contribution to the βfunction for the nonlocal φ6 vertex arising from a graph with three G4 vertices is thusβG6[2,2,2](p1,...p5) =(71)18π2∂2∂x2"Z dΩ4eΩ4Zdα1 dα2xA2G4(p1, p2, √xAe −(Q1α1 + Q2α2))G4(p3, p4, √xAe −(Q1α1 + Q2α2) + Q1)G4(p5, p6, √xAe −Q1(α1 −1) −Q2(α2 −1))#x=1+ crosstermswhere e is a Euclidean unit vector and the square root of A2 is real for Euclidean momenta, so wehave written it as A.Alternatively, we can eliminate reference to the extraneous parameter x by the following trick.Consider the sum of (68) again:∞Xr=114rr!

(r −1)! (A2)r−1 ∂∂q!2r.

(72)Withp ≡r −1(73)we obtain∞Xp=014p+1p! (p + 1)!

(A2)p ∂∂q!2p+1(74)=14 ∂∂q!2 ∞Xp=014pp! (p + 1)!

(A2)p ∂∂q!2p(75)=14 ∂∂q!2 "Z dΩ4eΩ4 eAe· ∂∂q#,(76)19

and using (49) for four dimensions,R dΩ4 = 2π2, we get a more compact form for the nonlocal βfunction:βG6[2,2,2](p1,...p5) =(77)164π4 ∂∂q!2 ZdΩ4eZdα1 dα2hG4(p1, p2, Ae + q)G4(p3, p4, Ae + q + Q1)G4(p5, p6, Ae + q + Q2)iq=−(Q1α1+Q2α2)+ crossterms,The last form is free of extraneous parameters.We reiterate that the evaluation of the 2−2−2 diagram above highlights two important featureswhich will be crucial for the application of the method to the general case below. First, for graphswith more than two internal lines, the integral of the leading (zeroth) term in the formal Taylorexpansion does not give a divergence (it is power-counting convergent), so the infinite sum for thepoles starts only at some higher order: for a type n graph (n > 1) the first divergent contributioncomes from the (n −2)th term in the Taylor expansion, so the sum of poles starts at n −2.

Torewrite this sum in terms of the angular integral over a Euclidean unit vector, we can subtract offafinite sum which vanishes under the action of the differential operator of order n −1 in the dummyparameter x. In fact, we can do much better - the parameter x can be eliminated by writing thepole sum a power of∂∂q2acting on a 2n −2 dimensional Euclidean unit angular integral ofG’s.

Let us explain why this makes sense: For one-loop graphs with one or two internal lines, thefour dimensional Feynman integral has a divergence which begins with the zeroth order term in theTaylor expansion (and higher terms in the Taylor expansion are even more divergent), so the sumof poles starts at zero. However, for graphs with more than two internal lines, enough powers ofmomentum in the numerator are required to cancel the denominator powers - thus the pole sumstarts at a higher order.

This sum can be made to start at the zeroth order by defining theloop integral in the appropriate number of higher dimensions. Hence the resummation ofthe pole terms for a sum starting at zero gives a higher dimensional angular integral.20

3.42m-point function of type n at one loopFor the non-trivial general graph with n ≥2, after combining denominators and shifting the loopmomenta, we get6 the integral(n −1)!Z 10n−1Yj=1dαjZℓddℓ(2π)d1DnYi=1µ∆(i)Givi+2(. .

. , ℓ−n−1Xs=1Qsαs + Qi−1)(78)where vi is the number of external lines emanating from the ith vertex, αj are the Feynman param-eters,D = [ℓ2 −(n−1Xs=1Qsαs)2 + (n−1Xr=1Q2rαr) −m2]n(79)is the combined denominator, and∆(i) = ǫ2vi(80)carries the renormalization scale dependence.

The ellipsis denote the dependence of each Gi on theexternal momenta and the momenta Qi are defined as in (31).7For the renormalization of a 2m point function with n internal linesi=nXi=1vi = 2m(82)so summing up the µ dependence from all the vertices gives the overall power of µ appearing asµ∆m where∆m ≡i=nXi=1vi( ǫ2) = mǫ. (83)Now, using again the fact that all the G’s are analytic, we can writenYi=1Givi+2(.

. .

, l −n−1Xs=1Qsαs + Qi−1)(84)=eℓ∂∂q" nYi=1Givi+2(. .

. , q + Qi−1)#q=−Pn−1s=1 Qsαs.6The overall factor of (n −1)!

comes from the Feynman trick.7By momentum conservation at the last vertex, we may make the structure of the last vertex simpler by makingthe replacementGnvn+2(. .

. , ℓ−n−1Xs=1Qsαs + Qn−1) →Gnvn+2(.

. .

, −ℓ+n−1Xs=1Qsαs). (81)21

Now we just have to do the dimensionally regularized integralZd4−ǫℓ(2π)4−ǫµ−∆mǫeℓ∂∂q[ℓ2 −A2 + iǫ]n(85)whereA2 = (n−1Xs=1Qsαs)2 −(n−1Xr=1Q2rαr) + m2. (86)Remembering the analyticity in momenta, we do this as usual by manipulating the exponentiallike a power series, with only even terms in ℓcontributing.

Doing the integral as before using thekey formula (46), and expanding the Γ function using (47), withΓ(n −r −2 + ǫ2) = 2ǫ(−1)2+r−n(2 + r −n)! (87)for n ≤r + 2, we get the pole piece(n −1)!i8π2µ−∆mǫ∞Xr=n−2(−1)2+r−n(−1)n+r(A2)2+r−n(n −1)!4rr!

(2 + r −n)! ∂∂q!2r. (88)Quite nicely, the (n −1)!

coming from the integration formula cancels with the (n −1)! appearingfrom the Feynman trick.

The sum (88) can then be written asi8π2µ−∆mǫ∞Xr=n−2(A2)2+r−n4rr! (2 + r −n)! ∂∂q!2r=i8π2µ−∆mǫ∂n−1∂xn−1∞Xr=n−2(A2)2+r−nxr+14rr!

(r + 1)! ∂∂q!2rx=1=i8π2µ−∆mǫ∂n−1∂xn−1x(A2)n−2∞Xr=0(A2)rxr4rr! (r + 1)! ∂∂q!2r−Tx=1(89)where we have defined the finite sumT =x(A2)n−2n−3Xr=0(A2)rxr4rr!

(r + 1)! ∂∂q!2r. (90)taken to be vanishing for n < 3.

As in the last example, this finite sum is subtracted offto makethe first sum start at r = 0. Now the differential operator is of order n −1, and the finite sum is oforder n −2, so the finite sum gets annihilated by the differential operator, giving finally the polepiece:i8π2µ−∆mǫ∂n−1∂xn−1x(A2)n−2∞Xr=0(A2)rxr4rr!

(r + 1)! ∂∂q!2rx=1. (91)22

Now, inverting as usual using (16) for a Euclidean unit vector, we may cast this pole contributionas an integral over a finite four-dimensional Euclidean angular region:Pole =i8π2ǫµ−mǫ∂n−1∂xn−1(92)"Z dΩ4eΩ4Z 10j=n−1Yj=1dαjx(A2)n−2i=nYi=1Givi+2 . .

. , √xAe −(n−1Xs=1Qsαs) + Qi−1!#x=1which gives the β function in the form8βG2m[v1,...,vn](p1,...,p2m−1) =(94)116π4∂n−1∂xn−1ZdΩ4eZ 10j=n−1Yj=1dαjx(A2)n−2i=nYi=1Givi+2 .

. .

, √xAe −(n−1Xs=1Qsαs) + Qi−1!#x=1+ crossterms,However, we can again do much better, and obtain a more compact expression in terms of a higherdimensional angular integral. Referring back to (88), we can write∞Xr=n−2(A2)2+r−n4rr!

(2 + r −n)! ∂∂q!2r(95)in terms ofp ≡r −n + 2,(96)forn ≥3,(97)8With the property that at the last vertex in the product we may make the replacement, by momentum conser-vation,Gnvn+2(. .

. , √xAe −(n−1Xs=1Qsαs) + Qn−1)≡Gnvn+2(.

. .

, √xAe + (n−1Xs=1Qsαs)). (93)23

as∞Xp=0(A2)p4p+n−2p! (p + n −2)! ∂∂q!2p+n−2(98)=14n−2 ∂∂q!2n−2∞Xp=0(A2)p∂∂q2p4pp!

(p + n −2)!=14n−2 ∂∂q!2n−2 "Z dΩ2n−2eΩ2n−2 eAe· ∂∂q#(99)so that we finally get the polePole =i22n−1π2ǫµ−mǫ ∂∂q!2n−2"Z dΩ2n−2eΩ2n−2Z 10j=n−1Yj=1dαj(100)i=nYi=1Givi+2 (. .

. , Ae + q + Qi−1)#q=−(Pn−1s=1 Qsαs)which, using (49) yields the contribution to the β function in terms of a 2n−2 dimensional angularintegral over a finite Euclidean region:βG2m[v1,...,vn](p1,...,p2m−1) =(101)(n −2)!4nπn+1 ∂∂q!2n−2 "ZdΩ2n−2eZ 10j=n−1Yj=1dαji=nYi=1Givi+2 (.

. .

, Ae + q + Qi−1)#q=−(Pn−1s=1 Qsαs)+ crossterms.Expressions (94), (93) and (101) for the β function of an arbitrary nonlocal coupling are themost general results of this paper.99It is verified easily that with n = 2, m = 2 the β function calculation for the nonlocal φ4 coupling, as in section1, is reproduced. For n = 3, m = 3, and n = 1, m = 2, the previous results for the φ6 maximally convergent graph(2 −2 −2) and φ4 tadpole are also verified from the general expression.24

4Concluding RemarksWe have illustrated explicitly how to obtain the renormalization group coefficients in a nonlocalscalar field theory. Note that our general results were computed with massive fields, and for thisreason techniques such as the Gegenbauer polynomial method of [5], valid only for massless propa-gators, are not useful here.We regard our results as suggestive, but not final.

We have obtained integro-differential renor-malization group equations by the brute-force technique of exanding, renormalizing and then re-summing. We would like to find the rule that allows us to do this in one step, rather than three.It would be interesting if, by modifying the analytic structure of the nonlocal theory, we couldactually give a one-step prescription to isolate the divergent part of the Feynman integral, withoutever expanding in terms of a formal Taylor expansion.

Work along these lines is in progress [6].We would like to thank Peter Cho, Mike Dugan and Ben Grinstein for numerous discussions.References[1] S. Weinberg, Physica 96A (1979) 327. A. Manohar and H. Georgi, Nucl.

Phys. B234, 189(1984).

[2] H. Georgi, HUTP-91/A014. [3] C. Bloch, Dan.

Mat. Fys.

Medd. 27, 8 (1952); P. Kristensen and C. Moller, Dan.

Mat. Fys.Medd.

27, 7 (1952); D.A. Kirzhnits, Sov.

Phys. Usp.

9, 692 (1967) and references therein; M.Chretien and R.E. Pierls, Proc.

Roy. Soc.

A223, 468 (1954). [4] P. Cho, HUTP-91/A046 to appear in Nucl.

Phys. B.

[5] K.G. Chetyrkin and F.V.

Tkachov, Nucl. Phys.

B192(1981) 159; K.G. Chetyrkin, A.L.

Kataevand F.V. Tkachov, Nucl.

Phys. B174(1980) 345.

[6] V. Bhansali, H.Georgi, under preparation.25

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