Neutral scalar penguin, 4D

Demonstration of the “matching” of the loop cutoff with the EFT cutoff (largest KK mass)

by Flip Tanedo & Yuhsin Tsai; to complement written calculation. 
Highlighting results from 1004.2037.

3 May 2012

ClearAll["Global`*"]

Amplitude

The amplitudes in the written calculation map on to the following integrands.
Amp1 is the diagram with the mass insertion to the left of the photon insertion, Amp2 has the mass insertion after.

Matching notation to the write up:
M1=(Subscript[m, (m)]), M2=(Subscript[m, (n)]), m=(Subscript[m, h])

Amp1[k_]:=(((((k^2)))^2)/(((((k^2)-(((M1))^2)))^3)((k^2)-((( M2))^2))((k^2)-(m^2))))+(((((k^2)))^2)/(((((k^2)-((( M1))^2)))^2)((k^2)-((( M2))^2))((((k^2)-(m^2)))^2)))-((2((k^2)))/(((((k^2)-((( M1))^2)))^2)((k^2)-((( M2))^2))((k^2)-(m^2))))-((((k^2))(((M1))^2))/(((((k^2)-((( M1))^2)))^3)((k^2)-((( M2))^2))((k^2)-(m^2))))
Amp2[k_]:=((((( M2))^2) (k^2))/(((((k^2)-((( M1))^2)))^2)((((k^2)-((( M2))^2)))^2)((k^2)-(m^2))))+(((((M2))^2) (k^2))/(((k^2)-((( M1))^2))((((k^2)-((( M2))^2)))^3)((k^2)-(m^2))))+((((( M2))^2) (k^2))/(((k^2)-((( M1))^2))((((k^2)-((( M2))^2)))^2)((((k^2)-(m^2)))^2)))-((2 ((( M2))^2) )/(((k^2)-((( M1))^2))((((k^2)-((( M2))^2)))^2)((k^2)-(m^2))))+(((k^2) )/(((k^2)-((( M1))^2))((((k^2)-((( M2))^2)))^2)((k^2)-(m^2))))-(((((k^2)))^2)/(((((k^2)-((( M1))^2)))^2)((((k^2)-((( M2))^2)))^2)((k^2)-(m^2))))-(((((k^2)))^2)/(((k^2)-((( M1))^2))((((k^2)-((( M2))^2)))^3)((k^2)-(m^2))))

Now write the KK masses in terms of KK number (n1) times the KK scale (M):

M2=n2 M;
M1=n1 M;

The Integrand

Now let’s plot the Wick-rotated integrand for some typical values.

m=.1; (* Higgs mass *)
M=1; (* KK mass *)

KKmax =20; (* maximum number of KK modes *)
Plotmax = 40; (* maximum loop momentum to plot *)

Plot[Sum[(l^3) (Amp1[I l]+Amp2[I l]),{n1,1,KKmax},{n2,1,KKmax}],{l,0,Plotmax},PlotRange-> All]

Clear[m]; Clear[M];

Note that for low values of loop momentum, the integrand is large and negative but then becomes small and positive. Thus as one integrates infinity (i.e. well past the largest KK mass) this small positive contribution cancels off the large negative contribution at lower scales.

Remarks

Recall that Lisa pointed out that it that we could write
 
 (∫_0^∞)Integrand = (∫_( 0)^( (Subscript[nM, KK])))Integrand + (∫_(  (Subscript[nM, KK]))^( ∞))Integrand
 
The full (0, ∞) integral is small because the cutoff isn’t matched to the maximum KK scale, i.e. one integrates beyond the cutoff of the EFT. The main contribution comes from the integral from the (0, n (Subscript[M, KK])) part. Lisa pointed out that this seems strange since it would imply that the (n (Subscript[M, KK]), ∞) integral must contribute with an opposite sign.

As we see from the plot above, this is precisely the behavior that we observe from the integrand.

Doing the monentum integral incorrectly

Do the loop integral with Λ->∞, but leave the maximum KK number finite

This is the wrong thing to do! Here we demonstrate that the (((1/(Subscript[M, KK])))^2) vanishes when we do the integral this way so that it appears that the integral goes like (((1/(Subscript[M, KK])))^4).

Integrate[(l^3)(Amp1[I l]+Amp2[I l]),{l,0,∞},Assumptions-> {{m,M,nf,n1,n2,ϵ}∈Reals,m>0,M>0,nf>0,n1>0,n2>0,ϵ>0}]

Write this as a function of the KK scale:

Result[M_]:=((m^2) (4 (m^2) ((n1^2)-(n2^2)) ((m^6)-3 (m^2) (M^4) (n1^2) (n2^2)+(M^6) (n1^2) (n2^2) ((n1^2)+(n2^2))) Log[m]-2 (m^2) (n1^2) ((((m^2)-(M^2) (n2^2)))^3) Log[(M^2) (n1^2)]-((m^2)-(M^2) (n1^2)) (((n1^2)-(n2^2)) (-(m^2)+(M^2) (n2^2)) (-3 (m^4)+(M^4) (n1^2) (n2^2)+(m^2) (M^2) ((n1^2)+(n2^2)))-2 (m^2) ((((m^2)-(M^2) (n1^2)))^2) (n2^2) Log[(M^2) (n2^2)])))/(2 ((((m^2)-(M^2) (n1^2)))^3) ((n1^2)-(n2^2)) ((((m^2)-(M^2) (n2^2)))^3))

Leading (((1/(Subscript[M, KK])))^2) term vanishes

M2pole=Limit[Result[M]*(M^2),M-> ∞]

Next-to-leading (((1/(Subscript[M, KK])))^4) term is non-zero

M2pole=Limit[Result[M]*(M^4),M-> ∞]

Doing the momentum integral with cutoff matching

We now show how the leading term does not vanish when we integrate up to a finite cutoff and make sure that this cutoff is roughly matched with the mass of the heaviest KK modes. This is consistent with doing the integral over a 5D Lorentz-invariant region in 5D momentum space. 

We introduce a parameter that measures the difference between the loop cutoff Λ and the heaviest KK mode (Subscript[n, f]) (Subscript[M, KK]):  λ = (Λ/ (Subscript[n, f]) (Subscript[M, KK])) where (Subscript[n, f]) is the largest KK number.

Integrate[(l^3)(Amp1[I l]+Amp2[I l]),{l,0,λ nf M},Assumptions-> {{m,M,nf,n1,n2,λ}∈Reals,m>0,M>0,nf>0,n1>0,n2>0,λ>0}]

Now put this into a function of the KK scale M:

Place the result into a function, MatchedResult[M] (hide this block, it’s just a mess)

MatchedResult[M_]:=(-4 (m^10) (M^2) (n1^4) (n2^2) (nf^2) (λ^2)+6 (m^8) (M^4) (n1^6) (n2^2) (nf^2) (λ^2)-2 (m^6) (M^6) (n1^8) (n2^2) (nf^2) (λ^2)+4 (m^10) (M^2) (n1^2) (n2^4) (nf^2) (λ^2)-6 (m^6) (M^6) (n1^6) (n2^4) (nf^2) (λ^2)+2 (m^4) (M^8) (n1^8) (n2^4) (nf^2) (λ^2)-6 (m^8) (M^4) (n1^2) (n2^6) (nf^2) (λ^2)+6 (m^6) (M^6) (n1^4) (n2^6) (nf^2) (λ^2)+2 (m^6) (M^6) (n1^2) (n2^8) (nf^2) (λ^2)-2 (m^4) (M^8) (n1^4) (n2^8) (nf^2) (λ^2)-2 (m^12) (n1^2) (nf^4) (λ^4)+2 (m^10) (M^2) (n1^4) (nf^4) (λ^4)+2 (m^12) (n2^2) (nf^4) (λ^4)-5 (m^8) (M^4) (n1^4) (n2^2) (nf^4) (λ^4)+6 (m^6) (M^6) (n1^6) (n2^2) (nf^4) (λ^4)-3 (m^4) (M^8) (n1^8) (n2^2) (nf^4) (λ^4)-2 (m^10) (M^2) (n2^4) (nf^4) (λ^4)+5 (m^8) (M^4) (n1^2) (n2^4) (nf^4) (λ^4)-7 (m^4) (M^8) (n1^6) (n2^4) (nf^4) (λ^4)+4 (m^2) (M^10) (n1^8) (n2^4) (nf^4) (λ^4)-6 (m^6) (M^6) (n1^2) (n2^6) (nf^4) (λ^4)+7 (m^4) (M^8) (n1^4) (n2^6) (nf^4) (λ^4)-(M^12) (n1^8) (n2^6) (nf^4) (λ^4)+3 (m^4) (M^8) (n1^2) (n2^8) (nf^4) (λ^4)-4 (m^2) (M^10) (n1^4) (n2^8) (nf^4) (λ^4)+(M^12) (n1^6) (n2^8) (nf^4) (λ^4)-3 (m^10) (M^2) (n1^2) (nf^6) (λ^6)+4 (m^8) (M^4) (n1^4) (nf^6) (λ^6)-(m^6) (M^6) (n1^6) (nf^6) (λ^6)+3 (m^10) (M^2) (n2^2) (nf^6) (λ^6)-3 (m^6) (M^6) (n1^4) (n2^2) (nf^6) (λ^6)-4 (m^8) (M^4) (n2^4) (nf^6) (λ^6)+3 (m^6) (M^6) (n1^2) (n2^4) (nf^6) (λ^6)+(m^2) (M^10) (n1^6) (n2^4) (nf^6) (λ^6)+(m^6) (M^6) (n2^6) (nf^6) (λ^6)-(m^2) (M^10) (n1^4) (n2^6) (nf^6) (λ^6)+4 (m^4) ((n1^2)-(n2^2)) ((m^6)-3 (m^2) (M^4) (n1^2) (n2^2)+(M^6) (n1^2) (n2^2) ((n1^2)+(n2^2))) ((n1^2)+(nf^2) (λ^2)) ((n2^2)+(nf^2) (λ^2)) ((m^2)+(M^2) (nf^2) (λ^2)) Log[m]-4 (m^4) (n1^2) ((((m^2)-(M^2) (n2^2)))^3) ((n1^2)+(nf^2) (λ^2)) ((n2^2)+(nf^2) (λ^2)) ((m^2)+(M^2) (nf^2) (λ^2)) Log[n1]+4 (m^12) (n1^2) (n2^4) Log[n2]-12 (m^10) (M^2) (n1^4) (n2^4) Log[n2]+12 (m^8) (M^4) (n1^6) (n2^4) Log[n2]-4 (m^6) (M^6) (n1^8) (n2^4) Log[n2]+4 (m^12) (n1^2) (n2^2) (nf^2) (λ^2) Log[n2]-12 (m^10) (M^2) (n1^4) (n2^2) (nf^2) (λ^2) Log[n2]+12 (m^8) (M^4) (n1^6) (n2^2) (nf^2) (λ^2) Log[n2]-4 (m^6) (M^6) (n1^8) (n2^2) (nf^2) (λ^2) Log[n2]+4 (m^12) (n2^4) (nf^2) (λ^2) Log[n2]-8 (m^10) (M^2) (n1^2) (n2^4) (nf^2) (λ^2) Log[n2]+8 (m^6) (M^6) (n1^6) (n2^4) (nf^2) (λ^2) Log[n2]-4 (m^4) (M^8) (n1^8) (n2^4) (nf^2) (λ^2) Log[n2]+4 (m^12) (n2^2) (nf^4) (λ^4) Log[n2]-8 (m^10) (M^2) (n1^2) (n2^2) (nf^4) (λ^4) Log[n2]+8 (m^6) (M^6) (n1^6) (n2^2) (nf^4) (λ^4) Log[n2]-4 (m^4) (M^8) (n1^8) (n2^2) (nf^4) (λ^4) Log[n2]+4 (m^10) (M^2) (n2^4) (nf^4) (λ^4) Log[n2]-12 (m^8) (M^4) (n1^2) (n2^4) (nf^4) (λ^4) Log[n2]+12 (m^6) (M^6) (n1^4) (n2^4) (nf^4) (λ^4) Log[n2]-4 (m^4) (M^8) (n1^6) (n2^4) (nf^4) (λ^4) Log[n2]+4 (m^10) (M^2) (n2^2) (nf^6) (λ^6) Log[n2]-12 (m^8) (M^4) (n1^2) (n2^2) (nf^6) (λ^6) Log[n2]+12 (m^6) (M^6) (n1^4) (n2^2) (nf^6) (λ^6) Log[n2]-4 (m^4) (M^8) (n1^6) (n2^2) (nf^6) (λ^6) Log[n2]+2 (m^12) (n1^4) (n2^2) Log[(n1^2)+(nf^2) (λ^2)]-6 (m^10) (M^2) (n1^4) (n2^4) Log[(n1^2)+(nf^2) (λ^2)]+6 (m^8) (M^4) (n1^4) (n2^6) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^6) (M^6) (n1^4) (n2^8) Log[(n1^2)+(nf^2) (λ^2)]+2 (m^12) (n1^4) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]+2 (m^12) (n1^2) (n2^2) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]-4 (m^10) (M^2) (n1^4) (n2^2) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]-6 (m^10) (M^2) (n1^2) (n2^4) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]+6 (m^8) (M^4) (n1^2) (n2^6) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]+4 (m^6) (M^6) (n1^4) (n2^6) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^6) (M^6) (n1^2) (n2^8) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^4) (M^8) (n1^4) (n2^8) (nf^2) (λ^2) Log[(n1^2)+(nf^2) (λ^2)]+2 (m^12) (n1^2) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]+2 (m^10) (M^2) (n1^4) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]-4 (m^10) (M^2) (n1^2) (n2^2) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]-6 (m^8) (M^4) (n1^4) (n2^2) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]+6 (m^6) (M^6) (n1^4) (n2^4) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]+4 (m^6) (M^6) (n1^2) (n2^6) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^4) (M^8) (n1^4) (n2^6) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^4) (M^8) (n1^2) (n2^8) (nf^4) (λ^4) Log[(n1^2)+(nf^2) (λ^2)]+2 (m^10) (M^2) (n1^2) (nf^6) (λ^6) Log[(n1^2)+(nf^2) (λ^2)]-6 (m^8) (M^4) (n1^2) (n2^2) (nf^6) (λ^6) Log[(n1^2)+(nf^2) (λ^2)]+6 (m^6) (M^6) (n1^2) (n2^4) (nf^6) (λ^6) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^4) (M^8) (n1^2) (n2^6) (nf^6) (λ^6) Log[(n1^2)+(nf^2) (λ^2)]-2 (m^12) (n1^2) (n2^4) Log[(n2^2)+(nf^2) (λ^2)]+6 (m^10) (M^2) (n1^4) (n2^4) Log[(n2^2)+(nf^2) (λ^2)]-6 (m^8) (M^4) (n1^6) (n2^4) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^6) (M^6) (n1^8) (n2^4) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^12) (n1^2) (n2^2) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]+6 (m^10) (M^2) (n1^4) (n2^2) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]-6 (m^8) (M^4) (n1^6) (n2^2) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^6) (M^6) (n1^8) (n2^2) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^12) (n2^4) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]+4 (m^10) (M^2) (n1^2) (n2^4) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]-4 (m^6) (M^6) (n1^6) (n2^4) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^4) (M^8) (n1^8) (n2^4) (nf^2) (λ^2) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^12) (n2^2) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]+4 (m^10) (M^2) (n1^2) (n2^2) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]-4 (m^6) (M^6) (n1^6) (n2^2) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^4) (M^8) (n1^8) (n2^2) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^10) (M^2) (n2^4) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]+6 (m^8) (M^4) (n1^2) (n2^4) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]-6 (m^6) (M^6) (n1^4) (n2^4) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^4) (M^8) (n1^6) (n2^4) (nf^4) (λ^4) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^10) (M^2) (n2^2) (nf^6) (λ^6) Log[(n2^2)+(nf^2) (λ^2)]+6 (m^8) (M^4) (n1^2) (n2^2) (nf^6) (λ^6) Log[(n2^2)+(nf^2) (λ^2)]-6 (m^6) (M^6) (n1^4) (n2^2) (nf^6) (λ^6) Log[(n2^2)+(nf^2) (λ^2)]+2 (m^4) (M^8) (n1^6) (n2^2) (nf^6) (λ^6) Log[(n2^2)+(nf^2) (λ^2)]-2 (m^12) (n1^4) (n2^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^12) (n1^2) (n2^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+6 (m^8) (M^4) (n1^6) (n2^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^6) (M^6) (n1^8) (n2^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-6 (m^8) (M^4) (n1^4) (n2^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^6) (M^6) (n1^4) (n2^8) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^12) (n1^4) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^10) (M^2) (n1^4) (n2^2) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+6 (m^8) (M^4) (n1^6) (n2^2) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^6) (M^6) (n1^8) (n2^2) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^12) (n2^4) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^10) (M^2) (n1^2) (n2^4) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+4 (m^6) (M^6) (n1^6) (n2^4) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^4) (M^8) (n1^8) (n2^4) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-6 (m^8) (M^4) (n1^2) (n2^6) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-4 (m^6) (M^6) (n1^4) (n2^6) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^6) (M^6) (n1^2) (n2^8) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^4) (M^8) (n1^4) (n2^8) (nf^2) (λ^2) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^12) (n1^2) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^10) (M^2) (n1^4) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^12) (n2^2) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+6 (m^8) (M^4) (n1^4) (n2^2) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+4 (m^6) (M^6) (n1^6) (n2^2) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^4) (M^8) (n1^8) (n2^2) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^10) (M^2) (n2^4) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-6 (m^8) (M^4) (n1^2) (n2^4) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^4) (M^8) (n1^6) (n2^4) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-4 (m^6) (M^6) (n1^2) (n2^6) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^4) (M^8) (n1^4) (n2^6) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^4) (M^8) (n1^2) (n2^8) (nf^4) (λ^4) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^10) (M^2) (n1^2) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^10) (M^2) (n2^2) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]+6 (m^6) (M^6) (n1^4) (n2^2) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]-2 (m^4) (M^8) (n1^6) (n2^2) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]-6 (m^6) (M^6) (n1^2) (n2^4) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)]+2 (m^4) (M^8) (n1^2) (n2^6) (nf^6) (λ^6) Log[(m^2)+(M^2) (nf^2) (λ^2)])/(2 ((((m^2)-(M^2) (n1^2)))^3) ((n1^2)-(n2^2)) ((((m^2)-(M^2) (n2^2)))^3) ((n1^2)+(nf^2) (λ^2)) ((n2^2)+(nf^2) (λ^2)) ((m^2)+(M^2) (nf^2) (λ^2)))

Identify leading ((((Subscript[M, KK])))^(-2)) term

Now we take the limit of a large KK scale in order to identify the leading term in powers of  ((((Subscript[M, KK])))^(-2)).

HeavyKKLimit=Limit[MatchedResult[M]*(M^2),M-> ∞]

(This replaces the expression mentioned in 1004.2037 which is not quite correct, but has the same limiting behavior.) 

Now write this term as a function:

LeadingTerm[λ_,nf_,n1_,n2_]:=-(((nf^2) (λ^2))/(2 ((n1^2)+(nf^2) (λ^2)) ((n2^2)+(nf^2) (λ^2))))

Integral result for leading term when loop momentum and KK cutoff are matched

ListPlot[Table[N[Sum[LeadingTerm[λ,nf,n1,n2]/.{λ->1},{n1,1,nf},{n2,1,nf}]],{nf,10,300,10}]]

The final result goes to -0.307 as the KK sum goes to ∞.

Behavior when varying the ratio of the max KK scale and the loop cutoff (fix (Subscript[n, KK])=100)

Now we consider what happens when we sum only to the 100th KK mode, but vary the loop momentum cutoff relative to this. We vary Λ from between 1/10th to 10 times the heaviest KK scale. Note that we do this by varying λ = (Λ/ (Subscript[n, f]) (Subscript[M, KK])) for fixed (Subscript[n, f])and (Subscript[M, KK]).

ListPlot[Table[N[Sum[LeadingTerm[λ,100,n1,n2],{n1,1,100},{n2,1,100}]],{λ,0.1,10,.2}]]

Let’s go even further and see what happens as we take the loop cutoff even further beyond the heaviest KK scale:

ListPlot[Table[N[Sum[LeadingTerm[λ,100,n1,n2],{n1,1,100},{n2,1,100}]],{λ,10,1000,20}]]

Observe that the leading term becomes negligibly small as the loop cutoff is taken arbitrarily far away from the heaviest KK scale. In this sense it is clear that even though the loop integral is manifestly finite, the value of the integral is sensitive to the value of the loop cutoff, i.e. one cannot integrate to ∞ since this is going beyond the regime of the EFT.

This is because the loop integral in this regime is breaking 5D Lorentz invariance.

Note that this doesn’t mean that one necessarily has to take many KK modes into account---it only means that one should match the highest KK number (even if this is only a few) with the [finite] loop cutoff. Below we show the value of the leading term for a matched loop cutoff and highest KK numbers from 1 to 100.

Table[N[Sum[LeadingTerm[λ,highestKK,n1,n2]/.{λ->1},{n1,1,highestKK},{n2,1,highestKK}]],{highestKK,1,100,5}]