Phys 731: String Theory - Assignment 2

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1 Phys 73: String Theory - Assignment Andrzej Pokraka October 8, 07 Problem Part a) We are given the action S S P + S J, S P 4πα d σδ ab a Xσ) b Xσ) S J d σjσ)xσ). Here, σ a, ) like in a regular QFT. First, we write the action in terms of Fourier space S d σ 4πα δab a Xσ) b Xσ)+Jσ)Xσ) d σ 4πα Xσ) Xσ)+Jσ)Xσ) d d κ d κ κ σ π) π ) 4πα Xκ) Xκ )+ Jκ) Xκ )+ Jκ ) Xκ) e iσ κ+κ ) d κ κ π) 4πα Xκ) X κ)+ Jκ) X κ)+. The generating functional is ZJ DXe S DX exp DX exp d κ π) κ d κ κ π) 4πα 4πα Xκ) X κ)+ Jκ) X κ)+ Xκ) πα πα X κ)+ Jκ) X κ)+ κ κ We can change integration variables to put this is the form of agaussianintegral.ifwelet. Xκ) χκ) πα κ Jκ),

2 then the measure of the path integral DXx) x dxx) does not change under such a constant shift DXx) Dχx) and the path integral becomes Gaussian Z J d κ κ πα πα DX exp Xκ) X κ)+ Jκ) X κ)+ π) 4πα κ κ d κ κ ) πα Dχ exp π) 4πα χκ) χ κ)+ J κ) Jκ) κ exp d κ πα π) κ Z 0 exp πα Z 0 exp πα d κ π) d κ π) J κ) Jκ) Z 0 Z J J0 is a possibly infinite) constant and Jκ) κ J κ) Dχ exp Jκ) κ) J κ) Dχ exp κ) κ d κ π) κ 4πα χκ) χ κ) d κ κ π) 4πα χκ) χ κ) is the massless propagator. This can be recast as a functional ofj ZJ Z 0 exp πα d σd σ Jσ) σ σ ) Jσ ) is the position space massless propagator. Part b) d σ σ κ e iκ σ σ ) π) κ ) We are asked to find the momentum space propagator by expanding Z J/Z 0 to second order in J. Expanding, Z J exp πα d κ π) Jκ) κ J κ) πα d κ π) Jκ) κ J κ)+...

3 we see that 0 T Xκ) Xκ ) 0 δ κ κ ) κ. Part c) In this problem we are asked to explicitly evaluate the position space propagator d σ σ κ e iκ σ σ ) π) κ dκ dθ π) κ dκ π) the integral representation for BesselJ is J n N x) π π ) π) eiκ σ σ cos θ J 0 κ σ σ ) κ dθ π π einθ x sin θ) e i dθ nπ 0 π einθ x cos θ). As expected, we see that this integral diverges. The divergence comes from the IR part of the integral. In the IR region, J0κ σ σ ) κ κ κ σ σ which diverges as k 0. Thisintegralis UV finite. For κ, J 0 sin cos and the integrand drops off like /κ 0. Part d) Adding an extra mass to the propagator we obtain d m σ σ κ e iκ σ σ ) ) π) κ + m dκ dθ π) κ π) eiκ σ σ cos θ dκ κ π) κ + m J 0 κ σ σ ) K 0 m σ σ ) K 0 is a modified Bessel function of the second kind. This last integral was performed in Mathematica; I have found no such integral tabulated in references such as Abramowitz and Stegun. Now in the limit of small separation, σ σ, thepropagatorislogarithmick 0 m σ σ ) log m σ σ ). 3

4 3 K 0 x) -logx) Part e) Differentiating the above result we obtain 0 T a Xσ) bxσ ) 0 a b 0 TXσ)Xσ ) 0 a bk 0 m σ σ ) 0 T Xσ) Xσ ) 0 m Σ ) Σ Σ ) Σ +Σ ) Σ K 0 m Σ ) K m Σ ) Σ 3/ 0 T Xσ) Xσ ) 0 m Σ Σ Σ K m Σ ) 0 T Xσ) Xσ ) 0 m Σ Σ Σ K m Σ ) 0 T Xσ) Xσ ) 0 m Σ ) Σ Σ ) Σ +Σ ) Σ K 0 m Σ )+ K m Σ ) Σ 3/ Σ a σ a σ a. Part f) The complex coordinates are defined to be z σ + iσ z σ iσ. The scalar product in complex coordinates is σ σ ) σ σ )z z ) z z )z z ) z z ) z z. With the above, we can rewrite the results of part d) and e): 0 TXσ)Xσ ) 0 K 0 m z z ), 4

5 Part g) z z 0 T a Xσ) b Xσ ) 0 m z z K m z z ) K 0 m z z ) 4 K 0 m z z z z ) z z K m z z ) Expanding the generating functional to 4 th order in J yields ZJ exp πα d σd σ Jσ) σ σ )Jσ ) Z 0 πα d σd σ Jσ) σ σ )Jσ ) + πα ) d σ d σ d σ 3 d σ 4 Jσ ) σ σ )Jσ )Jσ 3 ) σ 3 σ 4 )Jσ 4 ) The four point correlation function is related to the third term above which implies πα ) d σ d σ d σ 3 d σ 4 Jσ )Jσ )Jσ 3 )Jσ 4 ) σ σ ) σ 3 σ 4 ) πα ) d σ d σ d σ 3 d σ 4 Jσ )Jσ )Jσ 3 )Jσ 4 ) 3 σ σ ) σ 3 σ 4 )+ σ σ 3 ) σ σ 4 )+ σ σ 4 ) σ σ 3 )) 0 Xσ )Xσ )Xσ 3 )Xσ 4 ) 0 σ σ ) σ 3 σ 4 )+ σ σ 3 ) σ σ 4 )+ σ σ 4 ) σ σ 3 ). ). 5